DOCUMENT RESUME ED 300 252 AUTHOR Steen, …ED 300 252 AUTHOR TITLE INSTITUTION REPORT NO PUB DATE...

ED 300 252

AUTHORTITLE

INSTITUTION

REPORT NOPUB DATENOTEAVAILABLE FROM

PUB TYPE

EDRS PRICEDESCRIPTORS

ABSTRACT

DOCUMENT RESUME

SE 050 088

Steen, Lynn Arthur, Ed.Calculus for a New Century: A Pump, Not a Filter.Papers Presented at a Colloquium _(Washington, D.C.,October 28-29, 1987). MAA Notes Number 8.Mathematical Association of America, Washington,D.C.

ISBN-0-88385-058-388267p.

Mathematical Association of America, 1529 18thStreet, NW, Washington, DC 20007 ($12.50).Viewpoints (120) -- Collected Works - ConferenceProceedings (021)

MF01 Plus Postage. PC Not Available from EDRS.*Calculus; Change Strategies; College Curriculum;*College Mathematics; Curriculum Development;*Educational Change; Educational Trends; HigherEducation; Mathematics Curriculum; *MathematicsInstruction; Mathematics Tests

This document, intended as a resource for calculusreform, contains 75 separate contributions, comprising a very diverseset of opinions about the shap, of calculus for a new century. Theauthors agree on the forces that are reshapinc calculus, but disagreeon how to respond to these forces. They agree that the current courseis not satisfactory, yet disagree about new content emphases. Theyagree that the neglect of teaching must be repaired, but do not agreeon the most promising avenues for improvement. The document contains:(1) a record of presentations prepared fcr a colloquium; (2) acollage of reactions to the colloquium by a variety of individualsrepresenting diverse calculus constituencies; (3) summaries of 16discussion groups that elaborate on particular themes of importanceto reform efforts; (4) a series of background papers providingcontext for the calculus colloquium; (5) a selection of finalexaminations from Calculus I, II, and III from universities,colleges, and two-year colleges around the country; (6) a collectionof reprints of documents related to calculus; and (7) a list ofcolloquium participants. '.PK)

***********************************************************************

Reproductions supplied by EDRS are the best that can be madefrom the original document.

******************************************************k****************

CALCULUSfor a

NEW CENTURYA PUMP, NOT A FILTER

A National Colloquium

October 28-29, 1987

Edited by Lynn Arthur Steen forThe Board on Mathematical Sciences and

The Mathematical Sciences Education Board ofThe National Research Council

National Academy of SciencesNational Academy of Engineering

Institute of MedicineMathematical Association of America

The image used on the cover of this book and on the openingpages of each of its seven parts shows a periodic minimalsurface discovered in 1987 by Michael Callahan, DavidHoffman, and Bill Meeks III at the University ofMassachusetts. This computer generated image was createdby James T. Hoffman, ''',1987.

The questions from the AP calculus test in the Examinationsection are reproduced with the permission of The CollegeEntrance Examination Board and of Educational TestingService, the copyright holder.

r 1988 by the Mathematical Association of America.Permission for lirmied reproduction of these materials foreducational purposes but not for sale is granted.

Library of Congress Card Number 87. 063378ISBN 0. 88385-058-3

Printed in the United States of America

Current printing (last digit) 6 5 4 3 2 1

4

Calculus for a New Century

A Pump, Not a Filter

Preface .;z Bernard L. MadisonIntroduction xi Lynn Arthur Steen

Plenary Presentations


Castles in the Sand

Calculus of Reality

Calculus Today

Calculus Tomorrow

Views from Client Disciplines

Colloquium

3 Frank Press4 Ronald G. Douglas6 Robert M. White

10 Lynn Arthur Steen14 Thomas W. Tucker

Views from Client Disciplines 18

Calculus for Engineering Practice 18

Calculus in the Biological Sciences 20

Calculus for Management: A Case Study 21

Calculus and Computer Science 23

Calculus for the Physical Sciences 25

Views from Across Campus

A Chancellor's Challenge

National Needs

Now is Your Chance

Involvement in Calculus Learning

Calculus in the Core of Liberal Education

Mathematics as a Client Discipline

Calculus and the Computer in the 1990's

The Role of the Calculator Industry

Calculus; Changes for the Better

Cathleen S. MorawetzW. Dale ComptonHenry S. HornHerbert MoskowitzAnthony RalstonJames R. Stevenson

27 Daniel E. Ferritor29 Homer A. Neal30 Michael C. Reed32 Linda B. Salamon35 S. Frederick Starr

Responses

41 George E. Alberts

42 William E. Boyce43 Michael Chrobak44 Ronald M. Davis

iv

Imperatives for High School Mathematics

An Effective Class Size for Calculus

Don't!Evolution in the Teaching of Calculus

Collective Dreaming or Collaborative Planning

CalculusA Call to ArmsSurprises

Calculus for a Purpose

Yes, Virginia ...

Calculus for Physical Sciences

Calculus for Engineering Students

Calculus for the Life Sciences

Calculus for Business and Social Science Students

Calculus for Computing Science Students

Encouraging Success by Minority Students

Role of Teaching Assistants

Computer Algebra Systems

Objectives, Teaching, and Assessment

Innovation

45 Walter R. Dodge46 John D. Fulton48 Richard W. Hamming49 Bernard R. Hodgson51 Genevie,,e M. Knight52 Timothy O'Meara53 William M. Priestley54 Gilbert Strang55 Sallie A. Watkins

Reports

59 Jack M. Wilson and Donald J. Albers60 Ronald D. Archer and James S. Armstrong62 Donald E. Carlson and Denny Gul lick63 Charles W. Haines and Phyllis 0. Boutilier65 Carl A. Erdman and J. J. Malone66 William Bossert and William G. Chinn68 Dagobert L. Brito and Donald Y. Goldberg70 Gerald Egerer and Raymond J. Cannon, Jr.71 Paul Young and Marjory Blumenthal74 Rogers J. Newman and Eileen L. Poiani76 Bettye Anne Case and Allan C. Cochran78 John W. Kenelly and Robert C. Es linger79 Paul Zorn and Steven S. Viktora81 Alphonse Buccino and George Rosenstein83 Philip C. Curtis, Jr. and Robert A. Northcutt84 Lida K. Barrett and Elizabeth J. Teles

Issues

Calculus Reform: Is it Needed? Is it Possible? 89 Gina Bari KolataRecent Innovations in Calculus Instruction 95 Barry A. Cipra

Science, Engineering, and Business

Calculus for Engineering

The Coming Revolution in Physics Instruction

Calculus in the Undergraduate Business Curriculum

Calculus for the Biological Sciences

104 Kaye D. Lathrop106 Edward F. Redish112 Gordon D. Prichett116 Simon A. Levin

n0

TABLE OF CONTENTS

Teaching and Learning

The Matter of Assessment

Calculus Reform and Women Undergraduates

Calculus Success for All Students

The Role of Placement Testing

Institutional Concerns

Calculus from an Administrative Perspective

Innovation in Calculus Textbooks

Perspective from High Schools

A Two-Year College Perspective

Calculus in a Large University Environment

Mathematical Sciences

A Calculus Curriculum for the Nineties

Computers and Calculus: The Second Stage

Present Problems and Future Prospects

Final Examinations for Calculus I

Final Examinations for Calculus II

Final Examinations for Calculus III

A Special Calculus Survey: Preliminary Report

Two Proposals for Calculus

Calculus in Secondary Schools

Calculators in Standardized Testing of Mathematics

NSF Workshop on Undergraduate Mathematics

Transition from High School to College Calculus

Calculators with a College Education?

Who Still Does Math with Paper and Pencil?

Computing in Undergraduate Mathematics

122 Donald W. Bushaw125 Rhonda J. Hughes129 Shirley M. Malcom and135 John G. Harvey

141 Richard S. Millman145 Jeremiah J. Lyons149 Katherine P. Layton154 John Bradburn157 Richard D. Anderson

Uri Treisman

162 David Lovelock and Alan C. Newell168 R. Creighton Buck172 Gail S. Young

179

194

207

215

216

218

219

221

224

229

231

233

Calculus for a New Century: Participant List 243

Examinations

Readings

Richard D. Anderson and Don LoftsgaardenLeonard GillmanNCTM-MAA Joint LetterCollege Board-MA A Recommendations

Donald B. SmallThomas TuckerLynn Arthur SteenPaul Zorn

Participants

National Research Council

Task Force on Calculus

RONALD G. DOUGLAS (CHAIRMAN), Dean, College ofPhysical Sciences and Mathematics, State Univer-sity of New York, Stony Brook.

LIDA K. BARRETT, Dean, College of Arts and Sci-ences, Mississippi State University.

JOHN A. DOSSEY, Professor of Mathematics, IllinoisState University.

ANDREW M. GLEASON, Hollis Professor of Mathe-maticks and Natural Philosophy, Harvard Univer-sity.

JEROME A. GOLDSTEIN, Professor of Mathematics,Tulane University.

PETER D. LAX, Professor of Mathematics, CourantInstitute of Mathematical Sciences, New York Uni-versity.

ARTHUR P. MATTUCK, Professor of Mathematics,Massachusetts Institute of Technology.

SEYMOUR V. PARTER, Professor of Mathematics,University of Wisconsin.

HENRY 0. POLLAK, Bell Communications Research,(Retired).

STEPHEN B. ROD!, Chair, Division of Mathematicsand Physical Sciences, Austin Community College.

Supporting Staff

LAWRENCE H. Cox, Staff Director, Board on Mathe-matical Sciences, National Research Council.

FRANK L. GILFEATHER, Consultant to the Board onMathematical Sciences, University of Nebraska.

THERESE A. HART, Research Associate, Mathemat-ical Sciences in the Year 2000, National ResearchCouncil.

BERNARD L. MADISON, Director, Mathematical Sci-ences in the Year 2000, National Research Council.

PETER L. RENZ, Associate Director, MathematicalAssociation of America.

LYNN ARTHUR STEEN, Consultant to the Task Forceon Calculus, St. Olaf College.

MARCIA P. SWARD, Executive Erector, Mathemat-ical Sciences Education Board, National ReseaichCouncil.

ALFRED B. WILLCOX, Executive Director, Mathemat-ical Association of America.

Board on Mathematical Sciences

PHILLIP GRIFFITHS (CHAIRMAN), Provost, Duke Uni-versity.

PETER BICKEL, Professor of Statistics, University ofCalifornia, Berkeley.

HERMAN CHERNOFF, Professor of Statistics, HarvardUniversity.

RONALD DOUGLAS, Dean, College of Physical Sci-ences and Mathematics, State University of NewYork, Stony Brook.

E.F. INFANTE, Dean, Institut, of Technology, Univer-sity of Minnesota.

WILLIAM JACO, Professor of Mathematics, OklahomaState University, Stillwater.

JOSEPH J. KOHN, Professor of Mathematics, Prince-ton University.

CATHLEEN S. MORAWETZ, Director, Courant Insti-tute of Mathematical Sciences, New York Univer-sity.

ALAN NEWELL, Chairman, Department of Mathemat-ics, University of Arizona.

RONALD PYKE, Professor of Statistics, University ofWashington.

GUIDO WEISS, Professor of Mathematics, WashingtonUniversity.

SHMUEL WINOGRAD, Mathematical Sciences Depart-ment, IBM T.J. Watson Research Center.

viii

Mathematical Sciences Education Board

SHIRLEY A. HILL (CHAIRMAN), Curators Profes-sor of Mathematics and Education, University ofMissouriKansas City.

J. MYRON ATKIN, Professor, School of Education,Stanford University.

C. DIANE BISHOP, State Superintendent of PublicInstruction, Arizona Department of Education.

DAVID BLACKWELL, Professor of Statistics, Universityof CaliforniaBerkeley.

Irtis M. CARL, Elementary Mathematics InstructionSupervisor, Houston Independent School District.

RICHARD DE AGUERO, Director of Student Activities,Miami Senior High School, Florida.

WILLIAM J. DENNIS, JR., Senior Research Fellow,National Federation of Independent Business, Inc.

JOHN A. DOSSEY, Professor of Mathematics, IllinoisState University.

JAMES T. FEY, Professor of Curriculum & Instructionand Mathematics, University of Maryland.

SHIRLEY M. FRYE, Director of Curriculum & Instruc-tion, Scottsdale School District, Arizona.

ANDREW GLEASON, Hollis Professor of Mathematicksand Natural Philosophy, Harvard University.

NEAL GOLDEN, Chairman, Computer Science Depart-ment, Brother Martin High School, Louisiana.

KENNETH M. HOFFMAN, Professor of Mathematics,Massachusetts Institute of Technology.

DAVID R. JOHNSON, Chairman, Department of Math-ematics, Nicolet High School, Wisconsin.

ANN P. KAHN, Past-President, The National PTA.DONALD L. KREIDER, Professor of Mathematics and

Computer Science, Dartmouth College.

MARTIN D. KRUSKAL, Professor of Mathematics andAstrophysical Sciences, Princeton University.

KATHERINE P. LAYTON, Beverly Hills High School,

California.

STEVEN J. LEINWAND, Mathematics Consultant,Connecticut State Department of Education.

GAIL LowE, Principal, Acacia Elementary School,California.

PEGGY C. NEAL, Burney Harris Lyons MiddleSchool, Georgia, (Retired).

HENRY 0. POLLAK, Bell Communications Research,New Jersey, (Retired).

JACK PRICE, Superintendent of Schools, Palos VerdesPeninsula Unified School District, California.

ANTHONY RALSTON, Professor of Computer Scienceand Mathematics, State University of New York,Buffalo.

LAUREN 13. RESNICK, Learning, Research and Devel-opment Center, University of Pittsburgh.

STEPHEN B. RODI, Division Chairperson, Mathemat-ics and Physical Sciences, Austin Community Col-lege.

YOLANDA RODRIGUEZ, Mathematics Teacher, MartinLuther King Open School, Massachusetts.

FREDERICK A. ROESCII, Senior Vice President,Global Electronic Marketing, Citibank, N.A.

THOMAS ROMRERG, Professor, Department of Cur-riculum & Instruction, University of Wisconsin,Madison.

TED SANDERS, Superintendent of Education, Illinois.LYNN ARTHUR STEEN, Professor of Mathematics, St.

Olaf College.

DOROTHY S. STRONG, Director of Mathematics,Chicago Public Schools.

NELLIE C. WEIL, Past - ?resident, National SchoolBoards Associntion.

CALVIN J. WOLFBERG, Past-President, PennsylvaniaSchool Boards Association.

Preface

On October 28-29, 1987, over six hundred mathematicians, scientists, and educators gatheredia Washington to participate in a Colloquium, Calculus for a New Century, sponsored by theNational Academy of Sciences and the National Academy of Engineering. Centering on thatColloquium and containing 75 separate background papers, presentations, responses, and otherselected readings, Calculus for a New Century: A Pump, Not a Filter conveys to all who areinterested the immense complexity of issues in calculus reform..

Conducted by the National Research Council in collaboration with the Mathematical As-sociation of America, the Colloquium is a part of Mathematical Sciences in the Year 2000 (MS2000), a joint project of the Board on Mathematical Sciences and the Mathematical Sciences Ed-ucation Board. The National Research Council is the principal operating agency of the NationalAcademy of Sciences and the National Academy of Engineering and is jointly administered byboth academies and the Institute of Medicine.

The National Academy of Sciences is a private, non-profit, self-perpetuating society of dis-tinguished scholars engaged in scientific and engineering research, dedicated to the furtherance ofscience and technology and to their use for the general welfare. Upon the 'Lori ty of the chartergranted to it by the Congress in 1863, the Academy has a. mandate that requires it to advise thefederal government on scientific and technical matters.

The National Academy of Engineering was established in 1964, under the charter of the Na-tional Academy of Sciences, as a parallel organization of outstanding engineers. It is autonomousin its administration and in the selection of its members, sharing wit.. the National Academy ofSciences the responsibil:ty for advising the federal government. The National Academy of Engi-neering also sponsors engineering programs aimed at meeting national needs, encourages educationand research, and recognizes the superior achievement of engineers.

The Institute of Medicine was established in 1970 by the National Academy of Sciences tosecure the services of eminent members of appropriate professions and the examination of policymatters pertaining to the health of the public. The Institute acts under the responsibility givento the National Academy of Sciences by its congressional charter to be an advisor to the federalgovernment and, upon its own initiative, to identify issues of medical care, research, and education.

The National Research Council was organized by the National Academy of Sciences in 1916to associate the broad community of science and technology with the Academy's purposes offurthering knowledge and advising the federal government. Functioning in accordance with generalpolicies determined by the Academy, the Council has become the principal operating arm of boththe National Academy of Sciences and the National Academy of Engineering in providing servicesto the government, the public, and the scientific and engineering communities. Frank Press andRobert M. White are chairman and vice chairman, respectively, of the National Research Council.

The Mathematical Association of America is an organization of about 26,000 members ded-icated to the improvement of mathematics, principally at the collegiate level. It has played along-term role in improving mathematics education through the work of its committees and its

z

publications, of which this volume is one example.The Board on Mathematical Sciences of the Commission on Physical Sciences, Mathematics,

and Resources was established in 1984 to maintain awareness and active concern for the healthof the mathematical sciences and to serve as the focal point in the National Research Council forissues connected with the mathematical sciences. The Mathematical Sciences Education Boardwas established in 1985 to provide a continuing national overview and assessment capability formathematics education. These boards, joint sponsors of MS 2000, selected a special Task Forceon Calculus, chaired by Ronald G. Douglas, to direct the planning of the colloquium Calculus fora New Century.

Calculus for a New Century was made possible by a grant from the Alfred P. Sloan Foundationto the National Research Council. Support for the umbrella pxoject, MS 2000, was provided by theNational Science Foundation. This volume was prepared at St. Olaf College under the directionof editor Lynn Arthur Steen, and the publishing was directed for the Mathematical Associationof America by Peter L. Renz.

Bernard L. MadisonMS 2000 Project Director

U1

Introduction

Nearly one million students study calculus each year in the United States, yet fewer than 25%of these students survive to enter the science and engineering pipeline. Calculus is the critical filterin this pipeline, blocking access to professional careers for the vast majority of those who enroll.The elite who survive are oo poorly motivated to fill our graduate schools; too few ill number tosnstain the needs of American business, academe, and industry; too uniformly white, male, andmiddle class; and too ill-suited to meet the mathematical challenges of the next century.

These facts led Robert White, President of the National Academy of Engineering, to suggestthat calculus must become a pump rather than filter in the nation's scientific pipeline. Othersused a different metaphorto become a door, not a barrier. To make calculus a pump is a chal-lenge to educators and scientists; to walk through the door that calculus opens is a challenge tostudents. Regardless of the metaphor, calculus must change so that students will succeed.

All One SystemIn a narrow sense, calculus can be viewed simply as a sequence of courses in the mathematics

curriculum, of concern primarily to those who teach high school and college mathematics. Butin fact, calculus is a dominating presence in a number of vitally important educational and socialsystems. Calculus is:

A capstone for school mathematics, the culmination of study in the only subject (apart fromreading) taught systematically all through K-12 education.A pre-requisite to the majority of programs of study in colleges and graduate schools.The dominant college-level teaching responsibility of university departments of mathematics,intimately linked to the financial support of graduate education in mathematics.A course whose techniques are rapidly being subsumed by common computer packages andpocket calculator,An important comp,dnent of liberal education, part of the core learning that is the hallmarkof an educated person.

These interlocking systems make calculus extraordinarily resistant to change. Calculus has im-mense inertia ''at is rooted in tradition, reinforced by client disciplines, and magnified by massesof students. Yet calculus is failing our students: no one is well served by the present course. Thosewho apply calculus want students to have more mathematical power rather than mere mimicryskills; those concerned for education fear that for far too many able students, calculus is the endof ambitions rather than thte key to cncroRc; atilt others f9resee that comp tors will make much ofwhat we now teach irrelevant:

Computers and advanced calculators can now do most of the manipulations that studentslearn in a typical calculus course.College administrators report that calculus is a 'ightning rod for students' complaints.Cimputer scientists, one of the largest of mathematics' many clients, advocate discretelatherrrit',s rather than calculus as a student's first college mathematics course.

laity large universities, fewer than half of the students who begin calculus finish the termpassing grade.

xii

Little in the typical calculus course contributes much to the aims of general education,although for most students, calculus is the last mathematics course they ever take.

For all these reasons, a broad coalition of organizations has now undertaken an effort to revi-talize calculus. Calculus for a New Century is a first visible public step in thi3 long but cruciallyimportant process.

Issues and ControversiesCalculus for a New Century: A Pump, Not a Filter is intended to be a resource for calculus

reform, rooted in the October 28-29 Colloquium, but incly.ding much additional material. Over 80authors from mathematics, science and engineering convey in 75 separate contributions (totallingover 165,000 words) a very diverse set of opinions about the shape of calculus for a new century.The authors agree on the forces that are reshaping calculus, but disagree on how to respond tothese forces; they agree that the current course is not satisfactory, yet disagree about new contentemphases; they agree that neglect of teaching must be repaired, but do not agree on the mostpromising avenues for improvement. Readers must judge for themselves how they will respond tothis diverse yet realistic sample of informed views on calculus:

Colloquium: A record of presentations prepared for the first of the two-day Colloquium.These include four plenary ,cresentations and prepared remarks from two panels.

Responses: A collage of reactions to the Colloquium by a variety of individuals represent-ing diverse calculus constituencies. These commentaries, each a mini-editorial, were written illresponse to what was said and unsaid during the formal proceedings.

Reports: Summaries of sixteen discussion groups that elaborate on particular themes ofimportance to reform efforts. Each report was prepared jointly by the discussion leader andreporter.

Issues: A series of background papers providing context for the calculus Colloquium. Thefirst two, on Innovation, are journalistic analyses based on extensive interviews with many mathe-maticians, educators, and scientists. The remaining sixteen, in four sections, provide backgroundon issues in Science, Engineering, and Business; Teaching and Learning; Institutional Perspectives;and Mathematical Sciences.

Examinations: A selection of final examinations from Calculus I, II, and III from universities,colleges, and two-year colleges around the country. These exams are intended to illustrate calculusas it is today, to document exactly what students are currently expected to achieve.

Readings: A collection of reprints of documents related to calculus. Many of these papersare referred to either in the Colloquium talks or in the background papers; they are reproducedhere as a convenience to the reader.

Participants: A complete list of names, addresses, institutions, and telephone numbers fromthe registration list of the October 28-29 Colloquium, provided here to facilitate future correspon-dence among those with similar interests.

Process of RenewalCalculus for a New Century is itself a middle chapter in a long process of calculus reform.

It builds on a much smaller workshop organized by Ronald G. Douglas at Tulane University inJanuary, 1986 which led to the much-cited publication Toward a Lean and Lively Calculus (MAANotes No. 6, 1986). Shortly afterwards, the Mathematical Association of America appointed aCommittee on Calculus Reform chaired by Douglas to make plans for an appropriate follow-up

INTRODUCTION

to the Tulane meeting. Then in January 1987, the National Science Foundation proposed toCongress a major curriculum initiative in reform of calculus. To help frame a national agenda forcalculus reform, and to insure broad participation of the scientific and engineering communities,the Douglas Committee recommended that the National Academies of Science and Engineeringsponsor a national colloquium cn calculus reform.

At the same time, the two mathematics boards of the National Research Councilthe Boardon Mathematical Sciences and the Mathematical Sciences Education Boardjointly launchedMathematical Sciences in the Year 2000 (MS 2000), a project led by Bernard L. Madison toprovide a comprehensive assessment of collegiate and graduate education in the mathematicalsciences, analyzing curricular issues, resources, personnel needs, and links to science, engineer-ing, and industry. Since calculus is a central ingredient in the agenda of MS 2000, the CalculusColloquium became the first undertaking of that project.

Calculus for a New CenturyThree hundred years ago, precisely, the first edition of Newton's Principia Mathematica was

published. Two hundred years ago, more or less, calculus was first offered as a regular subjectin the university curriculum. One hundred years ago the mathematical revolution launched byNewton gave birth in the New World to what is now the American Mathematical Society.

"Calculus for a New Century" celebrates these three centenaries, not by looking back but bylooking forward to the 21st Century, when today's students will be our scientific and mathemat-ical leaders. Calculus determines the flow of personnel in the nation's scientific pipeline. To fillthis pipeline, we must educate our youth for a mathematics of the future that will function insymbiosis with symbolic, graphical, and scientific computation. We must interest our students inthe fascination and power of mathematicsin its beauty and in its applications, in its history andin its future. Calculus for a New Century: A Pump, Not a Filter offers a vision of the future ofcalculus, a future in which students and faculty are together involved. in learning, in which calculusis once again a subject at the cutting edgechallenging, stimulating, and immensely attractiveto inquisitive minds.

ACKNOWLEDGEMENTS: Details of Calculus for a New Century were handled efficiently byBernard Madison and Therese Hart, staff members of Project MS 2000. Calculus for a New Cen-tury: A Pump, Not a Filter was prepared in an extraordinarily short timethree weeks for prepa-ration of the camera-ready copyusing a TEX system at St. Olaf College. Mary Kay Petersondeserves special thanks for typing and correcting the entire volumeall 1,083,000 charactersonsuch a very tight timetable.

Lynn Arthur SteenSt. Olaf CollegeNorthfield, MinnesotaNovember 20, 1987

COLLOQUIUM

t

.1\)21).ps--L._ C41&EJrJT FR- 0? -1) Afv-t-s

e-ve.... ck- ve-r. a 1 e_. si--/A AR--vx.-- vu ovyl-k---t,

5 e_k_. -tLe e c i-e., 14,,e/in.--1-- 1 tI A

:L_I,, -n,s. ct c) e_ o Ve.A. ) i fi i -t-c

---)

\ e_ Ss11. Lertn -1-D 1-ievvtuk.s-1-- ov- -i- it_

rex, A -two -1, con 0\_e_a

ef cLoL.i-;014s7, .._.1

c---

c---cc 0 ukck eim.-Rc..ave.. . .71 .. ... . _

T i.., a 1-\ P_AAJ c a l c_ ulAA. .s IAA 0 1-e.,. --;

.

..

,A t'W c. LA- Lt--

in Ls ,C-o-u_c_s e. , " Re 14.

Aire,.._

uL EN .e_c s-t-L. 0(.Q.,t ovy-e_ 10.e._-1-1-e_r_i_

_ .. .

17- ,...

- "

.:-pc-e_ea,v-esi _ .,

cv\-- 0 v\ 1,-,e_ctLA i. rc,_ovC. \---c-0,--4- ' v -t1- \rs ark_-1-f---v1c

LA) ° J d\ k- CLAI I k -A +0 in. c:Ii

__/--Cra_ c_Lr 0- t2-, e

. 016


Frank Press

NATIONAL ACADEMY OF SCIENCES

Good morning. I would like to welcome all of youto this National Colloquium on Calculus For A NewCentury.

Last night I welcomed to this auditorium another au-dience for a preview of a film called "The Infinite Voy-age," which will be shown all over the country in prime-time television, both commercial and public broadcast-ing. It's a family-oriented program that will continuefor three years. Last night's program featured manythings, but computational mathematics was an ess_n-tial feature of all of the science components in the film.

I would like to say a few words about the stronginterest of our own organization here in mathematicseducation. I guess that everyone recognizes that math-ematics education is centrally important as a foundationfor science, for engineering, and also for the social sci-ences. Here at the National Research Council where wework in all of these fields, we certainly recognize that,as you-must too in your own schools and organizations.

Mathematics is increasing in importance, not only inthe fields that I just mentioned, but also in the servicesector and in banking and finance. (In fact, some math--matical geniuses may be responsible for programmingthose computers that caused the automatic selling thatcontributed to the stock market crash. They shouldhave had academic careers ....)

Why are we losing so many bright peoplelosingthem from mathematics and science in favor of WallStreet and business schools and law schools? Mathe-matics is a critical filter in that it can knock you out ofthe pipeline, permanently, particularly for women andminorities who might otherwise have careers in the pro-fessions until their mathematics courses destroy theirhopes and aspirations.

1, myself, had a traumatic experience. I was knockedout in the sense that I took the most advanced under-graduate course in mathematics and received the onlyA in the course. Yet the professor, after giving memy grade, said, "I don't want you to major in math-ematics." So I was knocked out of mathematics intophysicsand it had a traumatic effect on me.

We are very proud and pleased that our Mathemat-ical Sciences Education Board is providing a new type

of national leadership in mathematical education. Itis appropriate for us to have this emphasis in our ownorganization. It is appropriate for the country to takeinitiatives as represented by this Colloquium becauseof the importance of mathematics in so many differentways. We are pleased also that the Mathematical Sci-ences Education Board is interested in the whole rangeof mathematics, from the youngest children through col-lege.

This particular conference is under the initiative ofour Board on Mathematical Sciences which works inpartnership with our Mathematical Sciences EducationBoard. Together these two Boards launched the ProjectMS2000, a comprehensive review of mathematics ed-ucation through college and university systems. ThisColloquium on the calculus is the lead activity of thisthree-year project. We are putting together our com-mittee that will be responsible for MS2000 and havejust received the acquiescence of a major industrialistto serve as chairman of the comm:ttee that will super-vise MS2000.

As I have said, calculus is an important fou Itionfor the scientific, engineering, and research communi-ties. It is also an important foundation for anyone whowants a good solid education in the modern era. Laterthis morning you will hear from Robert White, the Pres-ident of our sister organization, the National Academyof Engineering, who will certainly emphasize the impor-tance of calculus to our technological society.

I know that you have a lot of work to do in these twodays, with a program and workshops that deal withmany different groups covering essentially everybodywho will need calculus for their training. I look forwardto hearing the results of your Colloquium.

__FRANK PRESS is President of the National Academy of

Sciences. A former Science Advisor to President Carter, Dr.Press served for many years as head of the Department ofEarth and Planetary Sciences at the Massachusetts Insti-tute of Technology. He received a Ph.D. in geophysics fromColumbia University.

4 COLLOQUIUM: PLENARY PRESENTATIONS

Castles in the SandRonald G. Douglas

STATE UNIVERSITY OF NEW YORK AT STONY BROOK

I want to welcome you to this Colloquium on be-half of the Calculus Task Force. I am delighted thatyou have come and that so many people are interestedin improving the teaching of calculus. There are oversix hundred people here today, about five hundred ofwhom are mathematicians. About one hundred arefrom research universities, four hundred from four-yearcolleges, fifty from two-year colleges, and fifty from highschools. There are over a hundred people from other("client") disciplines; some are publishers and othersjournalists.

I am delighted that ...so many peopleare interested in improving theteaching of calculus.

Why are we all here? Why is this Colloquium beingheld? Let me try to explain this with a little personalhistory.

Five years ago, I became Chair of the MathematicsDepartment at my university for the second time. I

found a large difference between this time and the last,which had been about ten years earlier. Teaching wasdifferent! Although everyone still did their teaching,morale was low and there was an overwhelming senseof futility, felt by all. Since calculus involved the largestnumber of students, much of this feeling was centeredin the calculus.

About the same time I was confronted with the issueof the continuing relevance of the subject of calculus.Debate on the rising importance of discrete mathemat-ics was sweeping across university campuses, and I oftenwas called on to defend the role of calculus in these dis-cussions.

My interest in calculus was treated asa curiosity. No one ever talked aboutteaching. Teaching was something wehad to do and get over with.

For both of these reasons I was forced to think aboutcalculus and calculus teaching. Moreover I began to

ask questions about calculus when I travelled to otheruniversities.

My interest in calculus was treated as a curiosity. Noone ever talked about teaching. Teaching was some-thing we had to do and get over with. No one said thatmathematics teaching was good; everyone believed im-plicitly that since nothing could be done about it, thenwhy talk about it.

Most curricular and teaching activities had stoppedaronnd 1970 in the "post post-Sputnick" era. But theneed for change had not. Now there were more anddifferent students; faculty cutbacks had led to largerclasses, and to more teaching assistants and adjunctinstructors. And finally there were more powerful hand-held calculators; computers had gone from a buildingsomewhere across campus to a room down the hall toyour desk top.

A few "oddballs" persevered. ...Butthey were not able to get others to jointhe effort and ultimately the seaoverwhelmed the castles in the sandwhich they had so painstakingly built.

I found in my travels, however, that not all facultyhad given up on curriculum and teaching reform. A few"oddballs" had persevered. Through nearly superhu-man efforts they had mounted experimental programswith different methods of teaching and new curricula.Perhaps surprisingly, they often achieved consideraolesuccess.

But they were not able to get others to join the effortand ultimately the sea overwhelmed the castles in thesand which they had so painstakingly built. Therefore,I concluded that isolated innovations are not the answerto the problems in mathematics teaching. This posed areal dilemma since change certainly cannot be dictatedfrom the top down.

What was needed was a strategy to place the im-provement of mathematics teaching and curriculum onthe national agenda. Support had to be provided forlocal efforts, and this didn't mean just money. Further,a mechanism needed to be provided to coordinate and

16

DOUGLAS: CASTLES IN THE SAND 5

network the results of innovation, both the good andthe bad.

Change certainly cannot be dicta .,edfrom the top down.

To start to accomplish all of this I asked the SloanFoundation to fund a Calculus Workshop, which washeld at Tulane University two years ago. There a di-verse group agreed that both the teaching and contentof calculus should change, and that it could change.We reached substantial agreement on some changes tomake. All this is reported in Toward a Lean and LivelyCalculus [1].

However, changing the calculus is an enormous andcomplex undertaking. Calculus is taught to over threequarters of a million students a term; about a half bil-lion dollars a year is spent on tuition for teaching cal-culus; and calculus is a prerequisite for more than halfof the majors at colleges and universities. Almost ev-eryone has a stake in calculus.

This Colloquium has been organized to discuss howto proceed, not to z atify some preordained plan. Wehope to plant many seeds and to help provide the sunand the rain necessary for these seeds to grow.

Changing the calculus is an enormousand complex undertaking. ...Almosteveryone has a stake in calculus.

A number of excellent position papers have alreadybeen prepared to provide background and to set thestage. Robert White will provide an engineer's perspec-tive on the role and importance of calculus. Lynn Steenwill discuss calculus as it is today and Thomas Tuckerwill discuss some of the possibilities for the calculus ofthe future. The morning session concludes with a de..scription of the new NSF calculus initiative by JudithSunley and Robert Watson.

After lunch there will be two panel discussions. Thefirst, moderated by Cathleen Morawetz, will allow rep-resentatives from the "client disciplines" to discuss their

disciplines. The second, moderated by Andrew Glea-son, will allow campus administrators to provide a viewof calculus from the campus. Finally, all participantswill get a chance to contribute their ideas in the discus-sion groups tomorrow.

Isolated innovations are not the answerto the problems in mathematicsteaching.

Any change in the way calculus is taught, or in theway mathematics is taught, will depend on all of us,on those here today and on the thousands of otherswho teach calculus and all the rest of mathematics inthe colleges (both two-year and four-year), in the highschools, and in flu- universities.

My title, based on my early experiences, refers tocastles in the sand. I had intended to close by stat-ing that we wanted to avoid building more castles inthe sand. However, another metaphor, also based onthe sea, might be better. The number of people at thisColloquium, and the interest that has been manifestedin this topic suggest that a bigger phenomenon is in-volved. 3o I conclude by urging all of us to "catch thewave" and try to direct it to the Calculus for a NewCentury that we will choose.

Reference

[1] Douglas, Ronald G. (ed.) Toward a Lean and Lively Cal-culus. MAA Notes Number 6. Washington, D.C.: Math-ematical Association of America, 1986.

RONALDG. DOUGLAS is Dean of the College of Physi-cal Sciences and Mathematics at the State University of NewYork at Stony Brook and Chairman of the National ResearchCouncil's Task Force on the Reformation of the Teaching ofCalculus. A specialist in operator theory, Douglas is a pastholder of both Sloan and Guggenheim fellowships. He isa member of the Board on Mathematical Sciences of theNational Research Council and Chairman of the AmericanMathematical Society's Science Policy Committee. He re-ceived a Ph.D. degree in mathematics from Louisiana StateUniversity.


Calculus of Reality

Robert M. White

NATIONAL ACADEMY OF ENGINEERING

Let me extend to all of you a welcome on behalf ofthe National Academy of Engineering. I have rarelyseen this auditorium as filled as it is today. It is agood sign that the topic we will be discussing today hasgenerated great concern throughout the mathematicsand the non-mathematics community also.

I am probably one of the few non-mathematicians inthis auditorium today. I am one of the "clients," as youcall them. I would like to talk to you from the viewpointof a client. I have views that stem from my personalexperiences in science and engineering, and from myexperience in trying to make scientific and technologicalinstitutions function.

We were asked to have faith in whatwe were being taught ...and faith thatsometime in the future what we werelearning would have application.

My introduction to the calculus was an introductionto the mysteries. It was assumed, when I was goingthrough school, that we would learn even without un-derstanding. We were asked to have faith in what wewere being taught, faith in our instructors, teachers,and professors, and faith that sometime in the futurewhat we were learning would have application.

I graduated many years ago from the Boston PublicLatin School. It's the oldest public school in the coun-try, founded in 1635. Harvard University was foundedin 1636. Those of us who went to the Boston PublicLatin School always claimed that Harvard was foundedas a place where Latin School graduates could go aftergraduation.

Today, however, students will not takeeducation in any field on faith.

My introduction to the calculus was as a freshmanat Harvard. They didn't teach calculus at Boston Pub-lic Latin School in my days. My instructor, althoughhe may not have been famous then, is well recognized

20

these days as a famous mathematician. He was OscarZariski. He might have been a great mathematician,but he was not the greatest instructor of freshman cal-culus. I survived.

Later, at MIT, when I became involved in the atmo-spheric and oceanographic sciences, it became clear inhindsight why calculus was important and why it wasso necessary for me as a person interested in geophysicalfluid dynamics.

If ever a field needed to be brought outof mystery to reality, it is the calculus.

Today, however, students will not take educatic inany field on faith. They want to know why what they'restudying is important and how it's going to help them.If ever a field needed to be brought out of mystery toreality, it is the calculus.

Calculus now is more important than ever. Calcu-lus, as the mathematics of change, is the skeleton onwhich the flesh of our modern industrial society grows.The public does not understand this fundamental roleof the calculus. But as people who are responsible forimparting the calculus, you must understand it, and youmust understand the consequences of failure to impartan understanding of this field of mathematics.

Restoring Economic GrowthThere are many realities that we need to face in this

country, not the least being the recent events in thestock market. But the most serious reality we face to-day is the need to harness science and technology foreconomic growth. And harnessing science and technol-ogy for economic growth means harnessing the calculus.

We are "a nation at risk," as that famous report in-dicated. However, we are at risk in many waysnotonly in our educational system for which the phrasewas first introduced. Science and technology are thetouchstones of economic growth which is fundamentalto the standard of living, job creation, health care, pro-vision of a good environment, and much more. During,the past century this country has done very well in har-nessing science and technology. Investment in science

WHITE: CALCULUS OF REALITY 7

and techaology has created new industries of all kinds,e.g., in semiconductors and biotechnology. Many of themost progressive and successful companies depend onnew products that are based cn science and technol-ogy for their economic growth. For example, I recentlyvisited the 3M Company and learned that 25% of itsrevenues are based on products that did not exist justfive years ago and were the result of their research anddevelopment activities.

We have done well in this country because over thepast century we have had far-sighted policies that led toan educational system that has produced the talent weneed to run our industrial, academic, and governmen-tal enterprises. We have made major investments inresearch and development enterprises, both those sup-ported by the government and by private industry, toproduce scientific and engineering knowledge. We havean economic system that has provided the incentivesand rewards for innovation and application.

Harnessing science and technology foreconomic growth means harnessing thecalculus.

Nevertheless, in recent years that picture has beenchanging, as anyone who reads the headlines or watchesnightly television news will know. Industry has movedmany production systems abroad. Many industries havebeen severely hurt in terms of jobs. At the end of WorldWar II, the United States accounted for 40% of theworld's GNP; today we account for 20%. We now runa $200 billion trade deficit.

We are in a continuing competitive battle with in-dustries in other countries:

In aviation, the European Airbus now has over 30%of the world market share in large transport aircraft.On U.S. highways, 30% of the automobiles are foreignimports.

In consumer electronics, it is difficult to find productsproduced in the United States.Even in heavy machinery, the last manufacturer oflarge steam turbines has just given up.We are in a competitive battle, across the board. In

industry it is a battle for market share, but it is a newkind of a battle. It is a battle that is fought with tariffs,wage rates, and economic policies. But above all else,it is a battle that is fought with trained people. Lack ofan adequate pool of trained people will, in the long run,lose that battle for us. And losing that battle means a

loss of jobs and a lowering of standards of living. It be-comes a central responsibility for all of us in education,especially those in science and engineering education,to make sure that this country has an adequate pool oftrained talent.

We are in a continuing competitivebattle with industries in othercountries. ...Above all else, it is abattle that is fought with trainedpeople.

There are many disturbing signs. Some have beenhighlighted by the work of the Mathematical SciencesEducation Board. There appears to be a lessening ofinterest among citizens of the United States in careersin science- and technology. We know, on the basis ofsurveys, that the mathematical attrainment of our stu-dents are inferior, at least in K-12 grades, to those ofmost of our industrial allies.

Schoolhouse to the WorldAt the college level, the United States has become

schoolhouse to the world. In 1986 we trained 340,000foreign students in our universities. A very large num-ber of these, about 130,000, go into engineering.

That's most welcome. I think we should train foreignstudents. The problem is that at the very highest levels,at the Ph.D. levels in schools of engineering, 50% of thestudents are not United States citizens. Now that's notnecessarily bad, because about 60% of the foreign-bornstudents eventually remain in the United States. Theybecome productive participants in the industrial andgovernmental apparatus of this country.

We are not getting adequate responseout of our own pool of talent in thiscountry.

This country has been built on immigrants. But ahigh percentage of foreign students in science and tech-nology is an indication of the fact that we are not get-ting adequate response out of our own pool of talent inthis country. We are coming to depend more and moreon the in-flow of talent from other countries.

COLLOQUIUM: PLENARY PRESENTATIONS

Intellectual CapitalThe availability of intellectual capital is now world-

wide. Just as the economy has become intemational,with the economy of one nation dependent on the econ-lily of others, so the scientific and engineering enter-prise has also become internationalized. Since the endof World War H, we have seen the growth of center::of excellence in many countries of the world with verysubstantial capabilities in science and technology. Theissue needs to be framed in terms more familiar in indus-try where investment in capital equipment and physicalplant is accepted. We now need to think in terms ofinvestment in intellectual capital.

Development of intellectual capital requires training.so training must therefore be looked upon as an invest-ment. We must begin to think of our investmentseducation as being just thatinvestments in buildingthe intellectual capital of this country on which our in-dustry, our government, and our universities can thaw.

Intellet.tual capital needs nurturing, it nee.ls protec-tion, and it needs renewal. Yet our ability to form in-tellectual capital is decreasing in many ways. The tech-nological workforce in America is graying. We are allfamiliar with the effects of decreasing birth rates, withthe changing demographics and evolving ethnic compo-sition of the work force. We need to be deeply concernedabout how we are going to build this intellectual cap-ital in the face of a changing and diminishing pool ofindividuals on whom this intellectual capital is based.

Intellectual capital needs nurturing, itneeds protection, and it needs renewal....As teachers of mathematics, you arein the front lines of building thisintellectual capital.

As teachers of mathematics, you are in the front linesof building this intellectual capital. The question is:How can we repair our failings, how can we buttress ourstrengths? As you know better than I, one of our failingsis in teaching mathematics. Buttressing our strengthsis one of the purposes of MSEB.

Making Calculus ExcitingAmong the difficult parts of teaching mathematics

is teaching calculus. As it is now taught, and as youappreciate better than I, it tends to be a barrier to stu-dents. Many of them drop out; many of them fail; many

lose interest. What we need to do is to find ways to en-courage and not discourage students, to keep them inthe pipeline. I don't think it is necessary for us to re-duce standards to maintain students in the pipeline. Ithink instead there need to be new approaches to teach-ing. You have heard about some of these already, andyou will hear about many more ideas in the days ahead.

Calculus is really exciting stuff, yet weare not presenting it as an excitingsubject.

Calculus is really exciting stuff, yet we are not pre-senting it as an exciting subject. At MIT, when I wastaking my graduate degree, I went through the calculuscourse for engineers. By that time I understood whyI was taking calculus and what it would be used for.It is clear to all of you, but not entirely clear to allthe people who need to make decisions, that engineer-ing and scientific applications just cannot exist withoutthe calculus. Whether it is the design of a bridge, anelectronic circuit, aircraft, or calculations of chemicalprocesses, the calculus is at the core. Without calculuswe would revert back to the engineering empiricism ofa century ago. had that we don't want.

W-.ather ForecastingMy own experience is, I think, illustrative. I am a

aometime weatherman. I used to be chief of the UnitedStates Weather Bureau. The field of weather forecastingis a good illustration of the fundamental iniportance ofthe calculus. It dominates work in this field.

Back in 1904 the idea that one could forecast theweather on the basis of physical law was first broachedby Vilhelm Bjerknes in Norway. But of course at thattime there was no way to apply the laws of motion. Itwasn't until 1922 that an Englishman named Lewis FryRichardson set those partial difflrential equations up infinite difference form, and sought to calculate by handthe time changes that would occur given a knowledge ofthe initial state of the atmosphere at a grid of points. Itwas the first application in my field of finite differencemethods. He did it by hand, if you can imagine that,and the results were just all wrong.

In the 1940's and 1950's, when I earned my spurs as aweather forecaster, the best we could do was to use per-turbation theory in trying to understand the growth ofdisturbances in a fluid system. But that only gave youinitial tendencies of the growth of these disturbances.

WHITE: CALCULUS OF REALITY 9

It wasn't until 1955 that the use of digital machinesto calculate the weather was attempted. First applica-tions were run on ENIAC, one of the first digital com-puters developed by the Signal Corps. Under the over-all supervision of John von Neumann of the Institutefor Advanced Study at Princeton, the first experimentswere conducted. That was the beginning of the trans-formation of weather forecasting from an art to a sci-ence.

The ability to apply ... equationstransformed an entire science. Thevalue to the world of modern weathercalculations is enormous. The centralrole of calculus ... to produce importantpractical results is evident.

The ability to apply those partial differential equa-tions in finite difference form transformed an entire sci-ence. The value to the world of modern weather cal-culations is enormous. The central role of calculus indealing with these problems to produce important prac-tical results is evident.

Teaching for FlexibilityThe issues that this Conference must address are well

laid out in your background documents. As I look atit, what we need to do now is to teach calculus in away that provides a body of understanding which con-tributes to the flexibility and adaptability required of

-scientists and engineers, of social scientists and man-agers.

What we need to build into ourstudents (and eventually into thepeople in our work force) is an abilityto move from field to field. ...To dothat you need ... an appreciation of thecalculus.

The reason why flexibility is important is that inan era of very rapid technological change, with newlyemerging fields of all kinds, what we need to build intoour students (and eventually into the people in our workforce) is an ability to move from field to field. To do

that you need the kind of understanding that comesfrom an appreciation of the calculus.

We need to teach the calculus in a way that facilitatescomplex and sophisticated numerical computation in anage of computers. Somehow or other you have to makecalculus exciting to students. The question as to therole of calculus in an age of computational mathematicsis one that clearly this Colloquium needs to address.

In the National SpotlightWe confront a real challenge. It is clear that there

is growing appreciation of the role of mathematics. En-rollments in mathematics departments have increased.We do have some problems with teaching assistantsover half come from outside the United States. But wecan't meet these challenges just by arm waving or bygeneralized statements about what needs to be donethat high schools should prepare students better, orthat students should work harder. A Colloquium likethis ha:: r, real opportunity to come up with suggestionsfor how to attack this problem.

Calculus is a critical way-station forthe technical manpower that thiscountry needs. It must become apump instead of a filter in the pipeline.

The national spotlight is turning on mathematics aswe appreciate its central role in the economic growthof this country. The linkage between mathematics andeconomic growth needs to be made, and needs to bemade stronger than it has been to date. Calculus isa critical way-station for the technical manpower thatthis country needs. It must become a pump instead ofa filter in the pipeline. It is up to you to decide how todo that.

---..- _ ,. -.-ROBERT M. WHITE in President of the National

Academy of Engineering. He has served under five U. S.Presidents in leadership positions concerning science andtechnology policy. He continues to be active in an advisorycapacity to the United States Government, and has also re-tained an active role in academic affairs. He received hisSc.D. degree in meteorology from the Massachusetts Insti-tute of Technology.


Calculus Today

Lynn Arthur Steen

ST. OLAF COLLEGE

It is clear from the size of this colloquium that thecalculus enterprise we are embarking on is of immenseinterest. One of the things I would like to do today is todocument with some figures that it is also of immenseproportions.

Calculus is our most important course.... The future of our subject dependson improving it.

I begin with a quotation from Gail Young, from thebackground paper he prepared for this colloquium. Itis a summary assessment by a long-time leader of themathematics community, speaking to mathematicians:

Calculus is our most important course .... The futureof our subject depends on improving it.

In support, I offer this evidence:

Three-quarters of collegiate-level mathematics is cal-culus. (By "collegiate level" I refer to those partsof higher mathematics that are calculus-level andabove, since all other courses taught in colleges anduniversities are really school-level mathematics.)

Calculus is the last mathematics coursetaken by our national leaders.

Calculus is the last mathematics course tahen by ournational leaders. This is a very important issue. Ifyou think about the career patterns of our nationalleaders, and of what students study in universities,the best students who enter universities by and largedo take calculus, either in high school or in college,whether or not they are going to be scientists or engi-neers. Future lawyers, doctors, clergy, public schoolleadersall professional leaders seem to take a littlebit of calculus. But for virtually everybody, it is thelast mathematics course they take. The entire pub-lic image of leaders of the United States concerningthe nature of mathematics, and of the mathematicalenterprise, is set by the last course that they take,which is calculus.

Calculus is among the top five collegiate courses inannual enrollment. Calculus not only dominates themathematics curriculum, but it dominates the entireuniversity curriculum. Also in that top five is pre-calculus. When you put the two of them together,those two enrollments make up a substantial fractionof enrollment in higher education.Most of what students learn in calculus is irrelevantto the workplace. This observation came out of manyof the background papers: an awful lot of what stu-dents actually learn in the current calculus course isno longer relevant to the way mathematics is used inscience or industry.

Calculus EnrollmentsHere is a crude portrait of calculus enrollment,

rounded to the nearest hundred thousand. At the highschool level, there are about 300,000 students enrolledin calculus courses of some lzind. Only about 15-20%of these students are in AP calculus, so there is alarge number of studentswell over 200,000who gothrough calculus in a once-over in high school.

There are 100,000 calculus enrollments in two-yearcolleges, and another 600,000 in the four-year collegesand universities. According to recent data from aspecial calculus survey conducted by Richard Ander-son and Donald Loftsgaarden (21, about half of these600,000 are in mainstream "engineering" calculus, withthe remainder in non-mainstream ("soft") calculus orvarious summer and extension courses.

Only one-fourth of calculus-levelenrollments (or one-eighth ofmathematics enrollments in highereducation) are in courses at or abovecalculus.

Looking at mathematics vertically, a little more thanhalf of total college mathematics enrollments are belowthe calculus level. Of the remaining half, 75% are in cal-culus. So only one-fourth of calculus-level enrollments(or one-eighth of mathematics enrollments in higher ed-ucation) are in courses at or above calculus.

STEEN: CALCULUS TODAY 11

Characteristics of CalculusThe 1985-86 CBMS survey [1] shows that the average

section size in calculus is about 34. That's no surprise.What is surprising is that only about 7% of calculuscourses use computers. The more recent survey [2] re-veals that 3% of the calculus courses require the use ofcomputers.

In a sample of final examinations that appears in theproceedings of this colloquium, we asked about whethercalculators are permitted on final examinations. Itseems to split about 50-50, in a way that is not cor-related with the type of institution. So there is a realdivision in the community on that issue.

Only about 7% of calculus courses usecomputers.

Since these sample examinations also contain infor-mation on grade distribution, I looked at the data todetermine the percentage of students who withdrew orfailedthose who were enrolled at the two or three weekmark, but who did not finish the course with a pass-ing grade. Looking at it institution by institutionwith an unscientific but very diverse sampleit lookslike a uniform distribution for withdrawal and failure of5% to 60%. For comparison, in the recent Anderson-Loftsgaarden study, of the 300,000 students who tookmainstream calculus, only 140,000 finished the yearwith a grade of D or higher.

Ron Douglas and others have conjectured that suc-cess in calculus is correlated with feedback on home-work. So we looked at what percentage of courses cor-rect homework regularly. The recent survey shows that55% rarely or never pick up homework and grade it.

ExamsWhen I looked at the questions on the sample of

final examinations representing colleges of every type,from community colleges to ivy-league institutions, eas-ily 90% of the questions were asking students to

Solve EvaluateSketch DetermineFind CalculateGraph What is?

Most questions asked for straightforward calculationsor posed template problems that are taught over andover again in the course and that are in the textbook innice boxed examples. Anybody who is wide awake and

pays attention ought to lot able to figure out how to dothese kinds of problems.

About 10% of the questions posed higher-order chal-lenges; most of those were template word problems.Those of you who teach calculus know what thatmeansproblems that fit a standard pattern. Someinstitutions and some courses have dramaticall ,f differ-ent patterns, but the mainstream examinations are likethis: 90% calculation, 10% thought.

You find very rarelyonly one problem in 1 out of i;examinationsthe kind of question that used to be verycommon 20 and 30 years ago: "State-and-prove ...."Problems dealing with the theory of calculus or withrigorous calculus have simply vanished from Americancalculus examinations.

Problems dealing with the theory ofcalculus ...have simply vanished fromAmerican calculus examinations.

Look at what students are asked to do for 90% oftheir examination problems; look at the verbs solve,skebh, find, evaluate, determine, calculate, graph, inte-grate, differentiate. What these commands correspondto, more or less, are the buttons on an !I? 28C. Whatwe are actually examining students on, what we reallyexpect them to learn, and what they know we really ex-pect despite whatever the general goals of calculus areclaimed to be, is the ability to do precisely the kinds ofthings that calculators and computers are now doing.

Pushing buttons, whether mechanical or mental, isone of the things we have to look at very carefully, tofigure out how we are going to adapt calculus to bettermeet the needs of students.

What we are actually examiningstudents on, what we really expectthem to learn, and what they know wereally expect ... is the ability to doprecisely the kinds of things thatcalculators and computers are nowdoing.

Ron Douglas talked about riding the wave as anothercalculus metaphor from the sea. If you pay attentionto the general concerns that are coming out of discus-sions of higher education, there is a great deal ofcon-cern about making sure that freshmen courses and other


courses that are taken as part of a student's general ed-ucation make a significant contribution to the broadaims of undergraduate educationthat they help stu-dents learn to think clearly, to communicate, to wrestlewith complex problems ..., etc.

There is very, very little in the calculus Course oftoday that does any of these things. Word problemsare a small step in that direction, but they arc veryrare and most of us who teach calculus know that ifyou put too many of those problems on your test, youare at great risk in the student evaluations.

Innovation: TechnologyOne other item that will appear in the proceedings

of this colloquium is a paper by Barry Cipra based oninterviews with many people who are now doing ex-perimental things with calculus. These are innovationsalready under way, out of the mainstream, at least a lit-tle bit. The areas in which people are currently workingfall into two broad categories: technology and teaching.

Disks. A lot of textbooks, as well as individual au-thors, offer supplementary PC or Macintosh disks tohelp illustrate what is going on in calculus. Becauseof the power of this equipment, these packages tendto be exclusively numerical and graphing, since theydo not have enough power to handle symbolic ma-nipulation.

Symbolic Algebra. These systems perform the ma-nipulative routines of algebra and calculus in purelysymbolic form-as we teach them in class. Packageslike MACSYMA, SMP, and Maple run on large sys-tems or workstations (e.g., VAX or Sun). Some ofthese arc being compressed to fit in to the small desk-top machines.Programming. In some places, instructors embed theteaching of programming into calculus, usually in Ba-sic, in many cases now in TrueBasic; rarely is it inPascal. It still is pretty uncommon anywhere to ex-pect calculus students to also do programming.SuperCalculators. Current top-line calculators fromHP, Sharp, and Cask are in the $80-200 price range,but we all know that soon thcy will be one-fourththat price. All can do graphs; some can do symbolicmanipulation; most can do a great deal of what stu-dents normally accomplish in their freshman course.Electronic Blackboards. Some experiments use tech-nology to make lecturing and presentation more dy-namic. With a good classroom setup with a com-puter and a screen, you can do more examples, morerealistic examples, and more dynamic examples. Cal-

cult. ;s the study of change; wit}- an electronic black-boar you can actually demonstrate that change inreal time, so students can sec what these conceptsarc all about.Tiectronic Tutors. At the further out research level,

there are people in artificial intelligence who are try-ing very hard to adapt techniques from symbolic al-gebra and the electronic blackboard and put it alltogether into sophisticated program that wouldamount to an electronic tutor for calculus.Now a lot of people have been working on this for

school mathematicsfor algebra and geometry. Pmfrankly skeptical that this work will ever come to much,since it seems to me that the subtleties that are involvedin learning calculus are probably a few generations be-yond the ability of the artificial intelligence cl -nmunityto catch up with it. But 1-know there are people in ar-tificial intelligence who believe that I am wrong, peoplewho believe 1,;tat in five years they will have these tu-tors really working. Some of them arc probably in theaudience right now ....

Innovation: TeachingIf you move away from the technology arena, there

is not too much else going on. Technology is certaintywhat has captured the most interest. Here are threeimportant areas related to teaching whet c serious workis taking place.

Teaching Assistants. Many universities are trying todevise means of incorporating TA's into the teaching ofthe calculus in a way that makes the experience for thestudents more satisfactory. As you all know., the budgetstructure of the major universities essentially requiresheavy dependence on teaching assistants in calculus insome form or other, so there is a lot of experimentinggoing on to figure out what forms are better than others.

Even in a calculus course that is verywell done ...students can go through...getting a grade of .13, maybe even agrade of A, acrd never write a completesentence.

Writing. Some people have taken up the task of inte-grating into calculus the objectives of teaching studentsto write and communicate by making writing an impor-tmt component of calculus. It certainly is the case nowthat even in a calculus course that is very well donewith good lecturing and small classes, where students

STEEN: CALCULUS TODAY 13

ask questions and the instructor answers them, whereinstructors or assistants help students, where studentsturn in homework and take regular examinationsit isprobably the case that students can go through sucha course getting a grade of B, maybe even a grade ofA, and never write a complet, sentence in the entiresemester, and probably never even talk at length aboutcalculus with anyone.

There are people who are trying to correct that. Ifyou look in the Tulane Proceedings, Toward a Lean andLively Calculus, there are a set of objectives for calculusthat emphasize expanding its goals beyond core math-ematics goals to include a lot more general educationobjectives. Some far-sighted instructors are working onbroad issues like that.

Constructivism. There rs a movement in educationalpsychology that is actually led by physicists, that pointsout (with good supporting data) that students do notapproach the study of subjects like mathematics andphysicsor probably anything elsewith a blank slateon which we teachers can fill in details by writing themon the blackboard and expecting students to xerox theminto their brain.

Students do trot approach the study ofmathematics with a blank slate....When we instruct them, it is likepushing on a gyroscope. The studentmoves in a different direction than wepush.

Students come with their own preconceptions aboutwhat mathematics should be, with their own repertoireof means of coping with mathematics as they encounterit. In many cases it may be evasive behavior, but it ispart of a variety of prior experiences that students have.

When we instruct them, it is like pushing on a gyro-scope. The student moves in a different direction thanwe push. So in order to teach students what we wantthem to learn, we have to understand the interactionthat goes on when students construct their own imagesof mathematics which are quite likely different than theones we have in our minds or that we are trying to con-vey to them.

Options for the FutureLet me close with an outline of issues for Tom Tucker,

who will be telling you his view of what calculus maybe like in the future.

When I talked with Tom about these presentations, Isuggested that he could view this in a manner that fits

his own expertise as Chairman of the AP ExaminingCommittee which sets the most widely-used multiplechoice examination for calculus. So I gave Tom a mul-tiple choice question which he will answer shortly. Iwould like to conclude by telling you what the questionis, so you can think about how you would answer it.

Where is calculus headed? Here are five choices forcalculus tomorrow:

A.

B.

C.

D.

It will disappear completely as client disciplines dis-cover that they can teach students to run computersbetter than the mathematics department can.It will become the first modern classic--a scholarlyrefuge, like Latin, in which arcane insights of a pastage are rehashed for those who wish to understandthe history of our present culture.it will remain totally unchanged due to the inabilityof forces acting from different directions to move anobject with such large mass.It will grow to double its present gargantuan mass,under pressure from tle many client disciplines whowant students who enter college knowing nothing tolearn everything before they are sophomores.

E. It will explode into a supernova, with every disciplineteaching calculus its own way.

We will shape the answer to this questionnot today,but in the next few yearsand in so doing respond toGail Young's challenge. Reforming calculus is our mostimportant task.

[1.]

References

Albers. Donald J. et al. Undergraduate Programs in theMathematical and Computer Sciences: The 1985-1986Survey. MAA Notes No. 7. Mathematical Associationo' America, 1987.

[2.] Anderson, Richard D. and Loftsgaarden, Donald 0. "ASpecial Calculus Survey: Preliminary Result." Calculusfor a New Century, Mathematical Association of Amer-ica, 1987.

LYNN ARTHUR STEEN is Professor of Mathematics atSt. Olaf College. He is Past President of the MathematicalAssociation of America, a member of the Executive Com-mittee of the Mathematical Sciences Education Board, andChairman-Elect of the Conference Board on MathematicalSciences. He also holds offices in the Council of ScientificSociety Presidents and the American Assoc- on for theAdvancement of Science. He received a F 1.). degree inmathematics from the Massachusetts Institute of Technol-ogy.

0 "rX.


Calculus Tomorrow

Thomas W. Tucker

COLGATE UNIVERSITY

Is there a "crisis" in calculus? If I were to ask thatof students in one of my calculus classes, they wouldall say "yes," but their interpretation of "crisis" wouldbe at a more personal level, especially with that hourexam cgming up next week.

Leonard Gillman, in the most recent issue of Focus[1], writes that the phrase "crisis in calculus" remindshim of a 1946 murder trial in Indiana. The case shouldhave been open and shut, but by continually referring tothe murder as "an unfortunate accident" ("Where wereyou the afternoon of that unfortunate accident?"), theseasoned defense attorney managed to get his client offwith only 2 1/2 years for manslaughter.

Maybe Gillman is right. Maybe there is no crisis incalculus. Maybe it's just ...an unfortunate accident.My students like that phrase too. They use it a lotthemselves: "Professor Tucker, about that unfortunateaccident of mine on problem 3."

No, there is a crisis today in mathematics, and inscience. The crisis lies in the infrastructure of science,which is fed by the undergraduate mathematics curricu-lum. Like it or not, calculus is the entry for the entireundergraduate program in mathematics as well as thefoundation for the sciences.

We have in our calculus classes acaptive audience. ...If we cannotproduce from this audience themathematicians and scientists thecountry needs, we must ask "Whynot?"

We spend enormous amounts of time and effortteaching calculus to masses of students. We have inour calculus classes a captive audience, at least for thetime being. If we cannot produce from this aud'.tncethe mathematicians and scientists the country needs,we must ask "Why not?" Are we doing the right thingsin this course? Can we change what we teach and howwe teach it? What will calculus be like tomorrow?

Business as UsualI propose three pictures of the future for calculus.

The first is the obvious one: business as usual. Text-

books will continue to get bigger; using the logisticequation with data 640 grams in 1934 (Granville, Smithand Long ley), 1587 grams in 1960 (Thomas, Third Edi-tion), and 2617 grams in 1986 (Grossman, Third Edi-tion), I get a limiting mass of 3421 grams. The content,however, will be unchanged.

Nearly half of all calculus students will be enrolledin classes of size 80 or more. Many smaller classesand recitation sections will be taught Li graduate stu-dents whose native language is not English. Calculatorsand computers will be banned from most examinations.Pencil-and-paper algebraic manipulation will be the or-der of the day. Students will fail or withdraw in largenumbers. And no one will complain because, after all,calculus is calculus. It's too familiar, too respected, toocomfortable, and too big to change.

Calculus in High SchoolThe second picture is that the mainstream college

calculus course, like a river in the Great Basin, willgradually disappear into a number of sinkholes. It willseep away into the secondary school curriculum. Al-ready 60,000 students a year are taking the AdvancedPlacement calculus exams and that number has beengrowing steadily at 10% a year ever since 1960. (Infact, it jumped 20% last year.)

It would not be surprising to see 200,000 AP Calculusexams in year 2000. And that is only the tip of theiceberg. From surveys, it appears that fewer than halfof the students enrolled in an AP course actually takethe exam, and even more students are taking non-APcalculus. Within a decade, there could be more studentstaking calculus in secondary school than presently takeit in college.

High school calculus can be very, verygood. ... The hundreds of AP teachersI have met ... embarrass me with theirdedication, enthusiasm, and expertise.

Before we wring our hands over this state of affairs,let me say that high school calculus can be very, verygood. It should be. They have 150 meetings to cover

TUCKER: CALCULUS TOMORROW 15

what we in college do in 50. Teaching calculus in sec-ondary school is viewed as a pleasure and a reward; incollege, it is a burden. The hundreds of AP teachers Ihave met through my involvement in the AP programembarrass me with their dedication, enthusiasm, andexpertise. And of course they teach to the select few insecondary school, while we teach to the masses in col-lege. It is no wonder that in my fall Calculus II class, theincoming freshman always outperform the sophomoreswho took our own Calculus I the previous semester.

But can secondary schools also teach calculus to themasses? In Russia, all 16 and 17-year olds are supposedto learn calculus. In Europe, general education takesplace in secondary school, not in the universities. Many,students, perhaps a majority, who are enrolled in ourmainstream calculus courses are there to get a goodgeneral education: a calculus course on the transcriptis the sign of an educated person.

If the calculus of the future takes placein secondary school, what will collegemathematics departments have toteach?

There is but one question: if the calculus of the fu-ture takes place in secondary school, what will collegemathematics departments have to teach? Already I amseeing more and more students arrive on campus withAP Calculus credit and never set foot in a mathematicsclassroom again. This could be an omen.

Calculus Across the CurriculumMainstream calculus may also disappear into the

client disciplines. Mathematicians are not the only peo-ple smart enough to teach calculus, and itshouldn't besurprising if other users wish to tailor a calculus courseto their own needs. Some of the most innovative text-books to appear in recent years have been for alternativetracksfor business, for life sciences, even for computerscience. When we were making syllabi recommenda-tions at the Tulane Calculus Conference in 1986, wefound that many of our suggestions had already beenadopted in a nonstandard text published independentlyby the Institute for Electrical and Electronic Engineer-ing (IEEE).

The mathematics community could learn somethingfrom alternative courses. If it doesn't, mainstream cal-culus may find its flow of students diverted more andmore. Already more than one in three college calcu-lus students is enrolled in an alternative course, and, aswith secondary school calculus, this number will grow.

The truth of the matter is that our clients have beenremarkably tolerant of mainstream calculus. At my in-stitution, we don't even teach exponential growth infirst semester calculus, and yet the economics depart-ment, which urges students to take at least one semesterof calculus, doesn't seem to notice. Out of sight, out ofmind, perhaps; but we shouldn't count on laissez-faireforever.

The truth of the matter is that ourclients have been remarkably tolerantof mainstream calculus.

Finally, mainstream calculus may disappear intocomputers and calculators. Long division, root ex-traction, use of log and trig tables are all fading fromthe precollege curriculum (not as fast, however, as onewould expect: even though every student has a $10 sci-entific calculator close at hand, most textbooks still in-clude tables of values of sines, cosines, natural logs, andexponentials).

Many traditional calculus topics such as curve sketch-ing, relative maxima and minima, even formal differen-tiation and integration may become just as obsolete inthe face of symbolic manipulation and curve plotting oncomputers and calculators. How long will mathemat-ics faculty be able to maintain discipline in the ranksof students "digging and filling intellectual ditches," asLynn Steen so aptly put it in a recent article [2] in theChronicle of Higher Education?

What happens when our calculus clients find we arestill teaching the moral equivalent of long division whilethey simply want their students to know how to pushbuttons intelligently? Of course we would find it bar-baric if students could only recognize f: 1/(1 + x2) dxas .785398, but what do we do when their calculatorsactually answer "r/4"? That will happen, you know,and as usual, before we're ready for it.

What happens when our calculusclients find we are still teaching themoral equivalent of long division whilethey simply want their students toknow how to push buttonsintelligently?

In the future, we may only need a few people whoknow the inner workings of the ..0.1culus, a cadre of the

0


same size as the numerical analysts who keep our cal-culators going today, and I guarantee you that numberis a lot less than 600,000 students a year.

Cleaning HouseIn contrast to this second picture of calculus tomor-

row, which we might call the twilight of mainstreamcalculus, let us consider a third picture, a vista of amore conceptual, intuitive, numerical, pictorial calcu-lus. The first step in that direction is honesty, withour students and with ourselves. If we teach techniquesof integration because it builds character, let's admitit to ourselves and to our students. If we like 71-/4 asan answer rather than .785398 because it is beautifulrather than useful, let's tell our students that. If com-plex numbers are both too useful and too beautiful toignore, then let's include them.

Do we need PHOpital's rule to know that ex growsfaster than xl°°? Do we ever need to mention thecotangent and cosecant? Do we really want to knowthe volume of that solid of revolution? Are all func-tions encountered in real life given by c'osed algebraicformulas? Are any?

Are all functions encountered in reallife given by closed algebraic formulas?Are any?

We should be asking these questions. In fact, weshould have asked these questions long ago. And whenwe have figured out really why we teach what we do,let's tell it to our students.

After we clean house, what will the new calculus belike? I hope that it uses calculators and computers,not for demonstrations but as tools, tools that raiseas many questions as they answer. The first time oneplots the graph of a random polynomial on a computer,one learns that polynomials lead very dull lives most ofthe time, just going straight up or straight down, andthat singularities are just that, singular. That is animportant lesson, and one which is lost on most calculusstudents.

I remember when a. colleague arranged a classroomcomputer demonstration to show the limit of quotientsAy/Ax is the derivative dy /dx and was shocked andembarrassed to find after a few iterations that the quo-tients diverged. (I warned him about roundoff error buthe didn't believe me.) Finding good numerical answerscan be just as difficult, just as instructive, and just asrewarding as slick algebraic manipulation.

Computers and calculators can also be used as a toolto infuse new mathematics into a staid course. We com-plain that the lay view of mathematics is that there isnothing to studyit was all finished off a long time ago.That shouldn't be surprising if the calculus we teach to-day we could have taught one hundred years ago, maybeeven two hundred.

What have mathematicians been doingfor the last century? Our calculusclasses say the answer is "Nothing."

For example, leafing through those boxed-in biogra-phies of mathematicians in a well-known text, I couldnot find a single mathematician active after Riemann'sdeath in 1866. What have mathematicians been doingfor the last century? Our calculus classes say the answeris "Nothing."

Computers could change that. We could play withcontemporary mathematics: the dynamics of functionaliteration, fractals, stability, three-dimensional graphics,optimization, maybe even minimal surfaces. I know, Iknow: just becalm. it's new, doesn't mean it's good.But a little "live" mathematics in a lean and lively cal-culus wouldn't hurt, even if it's only a commercial.

I hope the use of computers and calculators will alsoteach students to think about the reasonableness oftheir answers. As it is, they work so hard to get solu-tions that they never even give their solutions a secondthought. I remember a problem we graded on an APexam, which asked for the largest possible volume for awater tank meeting certain restrictions. We kept trackof the largest and smallest answers. The largest wassomewhat bigger than the universe, and the smallestwas much less than an atom (not counting, of course,all the negative answers).

We may even end up in the future notonly with "machines who think" butalso with "students who think."

Students who just push buttons and write down theanswer will find out quickly that that is not enough. Wemay even end up in the future not only with "machineswho think" but also with "students who think."

The Day After TomorrowSuppose this third picture of calculus tomorrow

comes true. It's a nice picture, an exciting picture,but there is still the day after tomorrow. Reforms have

30

TUCKER: CALCULUS TOMORROW 17

ways of becoming undone. In the late 1960's the NSF-supported Committee on the Undergraduate Programin Mathematics (CUPM) recommended that multivari-able calculus be taught in the full generality of n di-mensions with a whole semester of linear algebra as aprerequisite. By the early 1970's, textbooks for such acourse had been published and many, if not the major-ity, of colleges and universities taught their multivari-able calculus that way.

But something happened in the next decade with-out any urgings or direction from on high. A recentsurvey for the AP program revealed that 90% of the re-spondents now teach multivariable calculus in two- andthree-dimensions only, out of a standard thick calculustextbook, with no linear algebra prerequisite. Mathe-maticians tried to twist the calculus sequence and cameback later to find it, just like those metals with memory,back in the same old shape again.

There are strong forces that molded today's calculus,and those forces will still exist tomorrow. Students willalways follow the principle of least action. Textbookpublishers will still be guided by the laws of the mar-ketplace. Faculty will still have limited time to devoteto calculus teaching in a system which rewards moreglamorous professional activity. The reform movementmust take these forces into account.

Students will always follow theprinciple of least action. Textbookpublishers will still be guided by thelaws of the marketplace.

Unfortunately, some forces are societal and beyondour control. The conference held here last January oninternational comparisons of mathematics achievementwas particularly depressing in this regard. As muchas one might like to blame American shortcomings onspiral curricula which circle instead, or on middle schoolmathematics which is mostly remedial, it seems clearthat the real problems are much more deeply rooted inour society.

Japanese students think that mathematics is hard,but that anyone can learn it by working enough. Amer-ican students think mathematics is a knack only few areborn with, and if you don't have it, extra work won'thelp. Japanese parents are intensely involved with theirchildren's education and are generally critical of the aca-demic program in their local schools. American parents

think their children are doing fine, even when they arenot, and are generally happy with the academic pro-gram in their local schools, even when they should notbe.

If American children spend their after-school hoursworking for spending money at fast-food franchises, itcannot be surprising that their mathematics achieve-ment might suffer. On my campus, every student wantsnow to become an investment bankerat least they diduntil last week; the old favorite, pre-med, is dwindlingbecause it's too much science, too much "academics,"too much hard work.

The most visible rewards in our society go to enter-tainers, athletes, and corporate raiders, but I have yetto hear of our nation being at risk because of a short-age of, say, TV personalities. If our calculus students donot learn, if their attention wanders, or if they do noteven show up in the first place, we should not burdenourselves with all the blame.

I am not arguing that we should not try to maketomorrow's calculus different. I am sure we can do bet-ter and doing better could have a dramatic effect onthe infrastructure of mathematics and science. We can-not do it alone, however. Like any other educationalenterprise toda:,-, calculus reform needs broad support,from gove*.tment, from private industry, from collegesand schools, from professional societies, from the media,from teachers, from students, from parents. Changingcalculus may be more of a battle than we would everimagine, but it is a battle worth fighting.

References

[1] Gillman, Leonard. "Two Proposals for Calculus." Focus,7:4 (September 1987), p. 3, 6.

[2] Steen, Lynn Arthur. "Who Still Does Math with Paperand Pencil?" The Chronicle of Higher Education, Octo-ber 14, 1987, p. A48.

THOMAc W. TUCKER is Professor of Mathematics atColgate University and author of the Content Workshop Re-port at the Tulane Conference on Calculus. He chaired theCollege Board's Advanced Placement Calculus Committeefrom 1983-1987 and currently serves on a number of com-mittees for the Mathematical Association of America, theCommittee on the Undergraduate Program in Mathemat-ics, the College Board, and the Educational Testing Service.He received a Ph.D. degree in mathematics from DartmouthCollege.

Views from Client DisciplinesCathleen S. Morawetz

COURANT INSTITUTE OF MATHEMATICAL SCIENCES

Welcome. We are here to hear a panel representingthe client disciplines. They are the users of the stu-dents we mathematicians turn out. I am here to rep-resent applied mathematics in the channel connectingmathematics to other disciplines.

As a professor at the Courant Institute, every oncein a while I have taught the calculus. But I have toadmit, not very often. I am a user, and as a user I feelthat I belong with the client disciplines.

Someone mentioned this morning that Leonard Gill-man wrote in Focus that things were all right with cal-culus. Why fix what isn't broke?

I would like to read to you a little bit of what PeterLax wrote to Gillman in answer to that. Peter is sorrythat he isn't here to give the message himself.

Dear Lenny:

At one time the historian-essayist-moralist Carlylewas irritated by a friend who didn't believe in theexistence of the devil. So Carlyle took him to thegallery of the House of Commons, and after listeningto the goings-on, turned to his friend and remarked,"D'ye believe in the devil noo?"

When I encounter a skeptical colleague, I feel likeshowing him today's most widely used calculus textand ask him "D'ye believe in the crisis in the calculusnoo?"

I think this story has another moral too, which is thatwe should do a lot of listening as well as talking aboutthe calculus. Many of you have a first loveperhapstopology, or algebra, or some other very abstract field.You do not use the calculus today, not as I do. So

Calculus for Engineering Practice

W. Dale Compton

NATIONAL ACADEMY OF ENGINEERING

In considering the Calculus for a New Century, withits obvious emphasis on the next century, it is impor-tant that we be sensitive to th... context in which thestudent will be studying this important subject. The

I am a client and I represent the link that binds coremathematics to the client disciplines..

Someone once said that I am a card-carrying appliedmathematician. If someone else wants to dispute that,and some people will, then I'll see them afterwards.

As a card carrying member, I form a link to physicsand to engineering. I am interested in the links to biol-ogy, although I have to confess that it was only ratherlate in life as a mother of an economics student that Ilearned of the importance of calculus to economics. I

also come from an institution which in its research spe-cializes in this linkage. So I am here today to help posequestions for my panelists who come from the clientdisciplines.

Everyone whO can learn calculus shouldlearn calculus.

I have a profound interest in what gets taught in thecalculus. I think I understand the needs of industry andthe other sciences in continuing the scientific educationof students beyond the calculus. I would like to suggesta slogan for the future: Everyone who can learn calculusshould learn calculus.

CATHLEEN S. MORAWETZ is Director of the CourantInstitute of Mathematical Sciences, New York University.She is a member of the Board on Mathematical Sciences ofthe National Academy of Sciences.

future environment will be determined, in a significantway, by the competitive position of the U.S. in the worldmarketplace. We must consider, therefore, the role thatcalculus can have in helping this nation achieve an im-

COMPTON: CALCULUS FOR ENGINEERING PRACTICE 19

proved competitiveness.While many measures could be given of our current

competitiveness, it is sufficient to remind ourselves ofthe impact of the larr trade deficit and of the changesin employment levels tnat have resulted from the move-ment of manufacturing off-shore. With roughly seventypercent of our current manufacturing output facing di-rect foreign competition, we can expect the competi-tive pressures on the manufacturing sector to continue.Even the service sector is not immune to these pres-sures. One example of this is the increasing fraction ofengineering services being contracted to off -shore com-panies.

Calculus should encourage students toproceed to an engineering careernotby being easy, but by being exciting.

Many factors determine our competitiveness andmany actions will be required to improve it. Most peo-ple agree, however, that one of the principal tools forrecovering our competitive position will be a more effec-tive use of technology. This requires that the practiceof engineering be more effective. To affect the practiceof engineering one must focus on industry, the employerof about 75% of all engineers. Therefore, in consideringthe future capability of industry to effectively use tech-nology, an important issue becomes the availability andqualifications of future engineers.

If an adequate supply of qualified engineers is criticalfor industrial competitiveness, it is important to exam-ine the trends in engineering enrollment. First, studentdemographics predict a sharp drop in the college-agepopulation. Second, a decreasing percentage of new col-lege students are indicating an interest in science andengineering. Finally, a larger proportion of entering col-lege students will be minorities and females, a groupthat have not been strongly attracted to engineering asa profession. An important conclusion is that we mustfind a way to reduce the fraction of students who with-draw from the study of engineering, even though theymay have expressed an early interest in the field andare qualified to study it.

We must find a way to reduce thefraction of students who withdrawfrom the study of engineering.

It is here that calculus becomes so important. Math-ematical skills are a prerequisite to the successful prac-tice of engineering, and calculus is the first major step

in acquiring the skills needed for an engineering career.Calculus should encourage students to proceed to anengineering careernot by being easy, but by being ex-citing. It must net be an artificial barrier that is usedto discourage students from proceeding. It should en-courage students to explore the possibilities further. Itshould help convey to the student the sense of excite-ment that the practitioner of engineering experiences.

Engineering, in the words of the Accreditation Boardfor Engineering and Technology (ABET), is "the profes-sion in which knowledge of the mathematical and natu-ral sciences gained by study, experience, and practice isapplied with judgment to develop ways to utilize, eco-nomically, the materials and forces of nature for thebenefit of mankind."

The operative words in this statement are knowledgeand judgment. Whereas mathematics has most oftenbeen considered as a requirement for knowledge, it istime for us to begin to consider its role in judgment.The calculus course can be a place to start creatingthis sense of judgment and a sense of the excitement ofthe field through the examples that are used.

Whereas mathematics has most oftenbeen considered as a requirement forknowledge, it is time for us to begin toconsider its role in judgment.

Consider the following possibilities. Engineeringdeals with systems. Many systems are large and thuscomplex. Most systems are non-linear. Hence, approx-imate solutions are required to many system problems.It follows that the practitioner must have a good senseof the reasonableness of a solution.

It is my guess that students would react positivelyto a calculus that includes examples that require theexercise of good judgment. What better time to intro-duce the student to a sense of engineering than throughexamples of this type. What better way to introducesome excitement into calculus.

W. DALE COMPTON is currently a Senior Fellow at theNational Academy of Engineering. Previously, he served atotal of 16 years with Ford Motor Company, first as Directorof the Chemical and Physical Sciences Lai.Noratory and from1973-1986 as Vice President of Research. From 1961-1970Dr. Compton was a Professor of Physics at the Universityof Illinois at Urbana, serving as Director of the CoordinatedSciences Laboratory from 1965-1970. He received a Ph.D.degree in physics from the University of Illinois.

20 COLLOQUIUM: VIEWS FROM CLIENT DISCIPLINES

Calculus in the Biological SciencesHenry S. Horn

PRINCETON UNIVERSITY

After describing my own deplorable formal mathe-matics background, I would like to make an idiosyn-cratic list of the techniques that my own research andteaching now require. Then I shall condense that listinto a recommendation for the kind of calculus coursethat I wish I had had as an undergraduate.

My. perspective comes from doing empirical researchin ecology, studying the behavior of birds, butterflies,and trees, with a strong conceptual bias, and with in-terests in population genetics, evolution, development,and biomechanics.

My formal courses started with developing geometricintuition and the notion of rigorous proofs in high schoolgeometry, and it ended in college with baby-calculusthrough an introduction to linear differential equations.My fanciest mathematics, namely complex phase planeanalysis, was learned in an engineering course called"Electrical Engineering for Engineers who Aren't Elec-trical."

The help that I currently give my children on mid-dle school and high school homework should also countas formal coursework in higher math. This is partlybecause of their appalling textbooks and the last ves-tiges of the New Math, but it is also partly becauseof the depth and breadth of substance in the currentseconeqry school mathematics curriculum. My researchand that of close colleagues has required far more math-ematics than I have learned formally.

Meeting even the elementarymathematical needs of my area ofbiology requires more than the usualattention to the disciplines neighboringcalculus: geometry, differenceequations, nonlinear qualitativeanalysis, linear algebra, and statistics.

consequences of dispersal behavior, and, in the privacyof my own bedroom, 1 calculate the dispersion of eigen-values of a transition matrix to explore the speed andrepeatability of field-to-forest succession. Multivariatecalculus and linear algebra are useful in statistics.

Reality, breadth, and substance arecrucial if examples are to hold themotivation for their own solutions.

Colleagues studying populations, genetics, and neu-ral networks cite crucial differences between differentialand difference equations, with respect to stability andchaotic behavior. They also use combinatorics in theconstruction and analysis of genetic sequences, evolu-tionary trees, and the like. Propagation of noise spectrathrough differential and difference equations is widelypracticed in biology.

A topic that is far more different than is usually rec-ognized is the study of heterogeneous nonlinear systemsin which it is necessary to carry the full complexitythrough the analysis and plot the distribution of the re-sult. An appropriate choice among these techniques isneeded to study effects of varying environments on pop-ulation dynamics, of variation among individuals on pa-rameters of population or behavior, of sensory filteringin physiology, and of error in estimation of parametersin general.

Meeting even the elementary mathematical needs ofmy area of biology requires more than the usual atten-tion to the disciplines neighboring calculus: geometry,difference equations, nonlinear qualitative analysis, lin-ear algebra, and statistics. Jn addition it requires a per-spective more like applied mathematics or engineeringthan pure mathematics.

An ideal calculus course from my perspective wouldhave the following properties:

In my own research I use infinite series to calculate 1.

the photosynthesis of layers of leaves in a forest. I lookup standard derivatives and integrals in the Handbook ofChemistry and Physics to model butterfly movementsas a diffusion process. I use phase-plane analysis ofsystems of differential equations to discover population

34

Start with its historical origin as solutions to prob-lems in physical dynamics, but move on at least tothe qualitative behavior of solutions of difference andnonlinear differential equations, propagation of vari-ance, and propagation of qualitative heterogeneitythrough complex systems. Here I echo the enthusi-

CLIENT DISCIPLINES: MANAGEMENT SCIENCES 21

asm of many symposium speakers for discrete math-ematics.

Even qualitative contrasts with the traditional cal-culus are of use to me, to say nothing of approxi-mate solutions. These are traditionally considered tobe advanced topics, but they are the necessary rudi-ments to treat the real world beyond idealized New-tonian physics. (Incidentally, I have recently enjoyedjust the kind of playing with advanced concepts thatThomas Tucker extolled this morning, using semi-tutorial graphical microcomputer programs that arebeing developed by folks in my field.)

2. Develop and emphasize proofs, not from first prin-ciples, but from agreed-on intuitive principles. Un-fortunately this depends on common sense, which asVoltaire observed is not so common.

3. Use plug-in problems and template word problems to

drill on substantive questions from a variety of aca-demic disciplines. A partial list of disciplines whoseexamples would have direct relevance to my own is:ecology, molecular biology, physics, structural engi-neering, fluid dynamics, fractal geometry, economics,and politics. Reality, breadth, and substance are cru-cial if examples are to hold the motivation for theirown solutions.

HENRY S. HORN, Professor of Biology at PrincetonUniversity, specializes in ecology and the behavior of birds,butterflies, and trees. The adaptive significance in biology ofgeometrical components is of particular interest to him. Heis a member of several professional organizations includingthe Ecological Society of America and the Animal BehaviorSociety. Dr. Horn received a Ph.D. degree in ecology fromthe University of Washington.

Calculus for Management: A Case StudyHerbert Moskowitz

PURDUE UNIVERSITY

My views on calculus for management will be basedon my experience and observations of our undergradu-ate and graduate professional programs in managementat the Krannert School of Management at Purdue Uni-versity. As background, I will overview relevant aspectsof the curricula in each of these programs and relatethese to the need for calculus. Then I will state severalissues regarding the nature of the calculus courses takenby our students. From this, inferences and conclusionswill be drawn regarding whether and how the instruc-tion in calculus should change to meet the immediateand future needs of students in schools of management,or perhaps, whether it is really needed at all!

Undergraduate Management ProgramsThere are three undergraduate programs in man-

agement at the Krannert School: Industrial Manage-ment (IM), Management (M), and Accounting (A). Themathematics requirement in each of these programs arehigh and demanding compared to other comparable uni-versities, far exceeding AACSB guidelines.

Pre-Management students are required to take 2-3semesters of calculus; the 3 semesters applying specif-ically to our IM students, who must minor in a phys-

ical or engineering science discipline. In addition, allpre-Management students must take a mathematicalstatistics course. Once in the management program,students additionally take a managerial statistics course(calculus-based) as well as a management science coursewhose primary emphasis is on optimization.

The Calculus Requirement. As the Krannert School'sundergraduate programs are currently structured, cal-culus is essential:1. It is a pre-requisite for satisfying courses constituting

the minor in the IM program. It is also a prerequisiteor co-requisite for such required courses as Introduc-tion to Probability, Quantitative Methods (statisticsand optimization), Micro and Macro Economics, andsuch functional area courses as Operations Manage-ment and Marketing Management.

2. It serves as a "mild" filter, in the sense that gradesin the course, along with other courses, are used todetermine whether a student can transit successfullyfrom the Pre-Management Division into the Manage-ment Division.

3. It is a virtual necessary condition for acceptance(viz, "license") into a quality graduate professionalor Ph.D. program in Management.

22 COLLOQUIUM: VIEWS FROM CLIENT DISCIPLINE I

The Calculus Sequence. There is an option for one ofthe following two sequences, which is contingent upona student's prior mathematical preparation and ability,determined through testing and review of a student'srecords:

1. The standard 3-course sequence oriented towards thephysical sciences consisting of differential, integral,and multivariate calculus, respectively;

2. A nonstandard sequence for the less prepared stu-dent composed of a fundamental course in algebraand trigonometry, a less rigorous differential calculuscourse, and a less rigorous integral calculus course.More than 80% of all management students begin

with the algebra and trigonometry course. Seventy-five percent then take the nonstandard calculus se-quence (all pre-IM's, however, must take the calculussequence).

Fifty percent of our students obtain a grade of A orB as compared to 40% in the university in the stan-dard calculus sequence. Moreover, almost 60% of ourstudents obtain an A or B in the nonstandard calculussequence as compared to about 40% in the university.This high success rate is attributed in part to an ini-tial evaluation of the student's background and ability,remedial training, and placement in the appropriate cal-culus sequence (standard vs. nonstandard) for our stu-dents.

Nature of Calculus Courses. All calculus courses aretaught by the mathematics department. The coursesare highly structured and automated, in part to copewith delivering mass-produced training efficiently. Theinstructional approach is algorithmically oriented, i.e.,"how to differentiate and integrate." Little or no em-phasis is focused on conceptualization, modeling, or rel-evant and meaningful mrnagement applications. Classsizes are very large (large lecture halls) for calculuscourses in the standard sequence and approximately 40for courses in the nonstandard sequence. Typically, in-structors are either not the best mathematics facultyor are foreign Ph.D. students majoring in mathematics.Neither, presumably, are familiar with calculus applica-tions in management.

Observations and Implications. Relatively speaking,although students are exposed rather extensively to thecalculus and must use it in their economics, engineeringand management courses, they have considerable diffi-culty, particularly in problem and model conceptualiza-tion and formulation. Moreover, even the well-drilledprocedures of differentiating and integrating are forgot-ten much too quickly and must be reviewed. Hence, theeducational impact should and must be improved.

Graduate Professional ProgramsThere are two predominant masters in management

programs at the Krannert S ,hool: the Master of Sciencein Industrial Administration (MSIA), and the Master ofScience in Management (MS) programs. The MSIA isan 11-month general management program designed forstudents with technical degrees. The MS is a two (or 1-1/2) year general management program, also requiringa specialization in a functional area of management.

Historically, virtually 100% of the students in theMSIA prograitt had technical backgrounds, while about80% of such students entered the MS program. To-day, the composition of the students in both programshas changed considerably, the trend being towards motestudents with degrees in business, economics, and theliberal arts. Concommitantly, so have their mathemat-ical backgrounds changed.

The average GMAT of entering students is in the90th percentile. Both programs place strong emphasison the use of quantitative (and computing) skills forproblem solving and decision making, hence studentsare screened carefully for the quantitative aptitudes inthe admissions process (background in calculus appearsto be independent of the GMAT quantitative score).

Calculus is rarely used in any course...due to the personal computer, whichhas been delegated the task ofperforming computation and analysis.

36

Due, in part, to derrographics, only about 70% ofcurrently enrolled student, :rave at least one semesterof college calculus on entering the program. In the nottoo distant past, all students were well trained in cal-culus, and this was reflected in the Masters programcoursework.

For example, our statistics course used to be acalculus-based course in mathematical statistics. How-ever, today, calculus is rarely used in any course in ourMasters in Management Programs, including our statis-tics course. This is, in part, due to the lack of mathe-matical background of students in our program, whichis more quantitatively rigorous than most MBA pro-grams in the nation. But it is due even more so to thepersonal computer, which has been delegated the taskof performing computation and analysis.

Now, with computers, increasing effort can and is fo-cused on solving large-scale, real-world problems withemphasis on a problem's "front end" (problem defini-tion, formulation, modeling) and "rear end" (perform-

RALSTON: COMPUTER SCIENCE 23

ing "what if" and "what's best" analyses, and inter- 4.preting the results from a managerial viewpoint). Toillustrate the trend, the simplex method of linear pro-gramming is no longer taught in our management sci-ence course, but is solved by an appropriate softwarepackage integrated to a spreadsheet. Calculus in theMBA program, for all intents and purposes, has essen-tially vanishedmuch to the delight of students.

Calculus in the MBA program, for allintents and purposes, has essentiallyvanishedmuch to the delight ofstudents.

Is Calculus Needed? The concepts of calculus areclearly relevant to management in obviously pervasiveand important ways. Hence, they should be taught tomanagement students; probably to all college students.However, the nature and focus of what is being taughtin a calculus course must he improved. This can be ac-complished in a variety of ways including the following:

1. Incentive systems must be established to reward in-novative and solid calculus instruction.

2. Calculus classes perhaps ought to be partitionedbroadly into the physical, engineering, and social sci-ences to at least allow the possibility for focusingmore on applications that are relevant to students inthese respective disciplines.

3. Increased coordination and collaboration are neces-sary between mathematics departments and, for ex-ample, schools of management, to maximize topicalrelevance and develop meaningful management ap-plications of concepts.

Calculus and Computer ScienceAnthony Ralston

STATE UNIVERSITY OF NEW YORK AT BUFFALO

I stand second to no one in a belief that the teachingof calculus needs to be changed considerably if Ameri-can university students arc going to be well-served bydepartments of mathematics. But the title of this con-ference epitomizes one of the things that is wrong withcalculus teaching today. It implies that the place of cal-culus in the mathematics firmament is still just what it

5.

Following the innovations in management and engi-neering schools in particular, the computer can as-sume a significant instructional role, via developmentand implementation of appropriate interactive soft-ware and graphics. This will allow instructors to fo-cus more heavily on calculus concepts, modeling, andinterpretation, relegating computational work to thecomputer.Under conditions of mass-produced training, in par-ticular, intelligent tutoring systems could be devel-oped to "teach" novices to become experts in a giventopical domain. Dissertations could and should beencouraged to develop such software systems, per-haps in joint collaboration with faculty in computerscience, engineering, and management. Computerlaboratories for experimentation in interactive in-struction should also be established to try out noveland imaginative instructional technologies. Such ef-forts would simultaneously make both research aswell as teaching contributions.

What are the alternatives? Perhaps business as usualwith its well-known, predictable result; perhaps, schoolsof management teaching calculus to their own students;perhaps, no calculus at all!

HERBERT MOSKOWITZ is the James Brooke HendersonProfessor of Management and is a past director of Gradu-ate Professional Programs in Management at the KrannertGraduate School of Management at Purdue University. Hisarea of specialization is management science and quantita-tive methods with interests in judgment, decision-making,and quality control. He received a Ph.D. in managementfrom the University of California at Los Angeles.

has been for the last century or so, namely the rootof the tree from which all advanced mathematicsatleast all advanced applied mathematicsmust be ap-proached.

You don't have to agree with me that calculus anddiscrete mathematics should be coequal in the first twoyears of college mathematics to recognize that some-

24 COLLOQUIUM: VIEWS FROM CLIENT DISCIPLINES

thing is happening out there which implies a need forthe teaching of calculus to adapt to changes in the waymathematics is applied and in the clientele for collegemathematics. This need includes, at least, a require-ment that new subject matter be considered for thefirst two years of college mathematics and that there besome integration of the discrete and continuous pointsof view in those first two years.

How ironic that there has been asteady, perhaps accelerating trend inrecent years for college calculus to bedominated by the teaching of justthose symbol manipulations whichhumans do poorly and computers dowell.

Let me make a few remarks about the impact of com-puter and calculator technology on calculus and how itshould be taught. Hand-held calculators can lira per-form all but a very few of the manipulations )f K-14mathematics. Within a very few years the "all" willnot have to be qualified and, moreover, the devices willcost no more than about the price of a book. Howironic, therefore, that there has been a steady, perhapsaccelerating trend in recent years for college calculus tobe dominated by the teaching of just those symbol ma-nipulations which humans do poorly and computers dowell and whose mastery, I believe, does not aid the abil-ity to apply calculus or to proceed to advanced subjectmatter.

Too many mathematicians act likemechanical engineers when teachingcalculusthey focus on crank-turning.

Too many mathematicians act like mechanical en-gineers when teaching calculusthey focus on crank-turning. This must be changed. Calculus must becomea mathematics course again, one which focuses on math-ematical understanding and on intellectual mastery ofthe subject matter and not on producing symbol ma-nipulators. The technology available must be integrated(pun intended) with the teaching of calculus in order toget rote mastery of out-dated skills out of the syllabusand to get mathematics back in.

Except perhaps for engineering students, computerscience students are now the largest single potentialclient population for departments of mathematics. Nev-ertheless, 1-) often today departments of mathematicshave, in effect, encouraged computer science depart-ments to each their own mathematics because theyhave not been willing to teach discrete mathematicsthemselves. While I would argue that all undergrad-uates should be introduced to the calculus since it isone of the great artifacts created by humankind, thereis little disciplinary reason for computer science under-graduates to study calculus.

There is little disciplinary reason forcomputer science undergraduates tostudy calculus.

With the almost sole exception of the course in anal-ysis of algorithms, there is no standard course in thecomputer science undergraduate curriculum which leansmore than trivially on calculus. (Yes, I know aboutnumerical analysisI was, after all, once a numericalanalystbut fewer and fewer computer science under-graduates take numerical analysis any longer, and al-most none are required to take it.) Even in undergrad-uate courses in the analysis of algorithms, the use ofcalculus-based material is often non-existent and; evenwhen it is not, only very elementary aspects of calculusare used. For students headed toward graduate work incomputer science, one cannot be quite so unequivocalbut, even at the graduate level, few compUter sciencestudents have anywhere near as much use for continuousanalysis as they do for discrete analysis.

ANTHONY RALSTON is Professor of Computer Scienceand Mathematics at the State University of New York atBuffalo. Dr. Ralston currently chairs the MathematicalSciences Education Board Task Force on the K-12 mathe-matics curriculum and is a past president of the Associa-tion for Computing Machinery. In recent years he has beeninterested in the interface between mathematics and com-puter science education, particularly in the first two yearsof the college mathematics curriculum. He received a Ph.D.degree in mathematics from the Massachusetts Institute ofTechnology.

n

STEVENSON: PHYSICAL SCIENCES 25

Calculus for the Physical SciencesJames R. StevensonIONIC ATLANTA, INC.

My original entry into this discussion came severalyears ago when I was asked by Ronald Douglas to repre-sent the physical sciences at a workshop in New Orleans(also supported by the Sloan Foundation). At that timeI was Professor of Physics and Executive Assistant tothe President at Georgia Institute of Technology. TodayI am emeritus in both those positions, and chief execu-tive officer of a small start-up high_ technology company.Whether I have gained or lost credibility in the interimis an open question. In the few minutes today, I willemphasize a few of the points addressed at New Or-leans.

Physical science is empirical. The description andthe formulation of "La-.vs of Nature" depend on obser-vations rather than on logical development from axioms.Mathematics has always provided a convenient frame-work for the description of physical phenomena. Can itdo the same thing in the future? In the past the phys-ical science community has interacted at' withmathematicians and mathematical description .4ew-tonian mechanics have made a smooth transition fromnon-relativistic to relativistic and from non-quantum toquantum descriptions. The next challenge is one of thereasons for today's discussion.

Observations over the years have demonstrated thatthe important features of nature are imbedded in thedescription of non-linear phenomena. Both physical sci-entists and mathematicians have been very clever at ar-riving at excellent approximate descriptions. Recentlymany scientists and mathematicians have questionedthis approach, believing instead that many secrets ofnature may be hidden because of our insistence to forcephysical observations to be described by available math-ematics.

Many secrets of nature may be hiddenbecause of our insistence to forcephysical observations to be describedby available mathematics.

The computational approach has uncovered theworlds of "chaos" and "fractal geometry." Will we dis-cover sufficient cause to reformulate our "Laws of Na-ture" in a new descriptive format? Is the invasion of

mathematics by the empiricists of the computer goingto result in a completely new approach to mathematicsat the undergraduate level? Quo vadis calculus?

The development of intuitive thinking is a most valu-able asset to the physical scientist. Calculus irstructioncan play an important role. A quotation from Maxwellprovides insight:

For the sake of persons of these different types, scien-tifi.. truth should be presented in different forrns, andshould be regarded as equally scientific, whether it ap-pears in the robust form and the vivid coloring of aphysical illustration, or in the tenuity and paleness of

symbolic expression.

The physical scientist would argue that a similarstatement can be made for presenting introductory cal-culus. The meaning of slope and curvature and their re-lation to the first and second derivative are important.The location of maxima, minima, and inflection pointsare also important, and the physical scientist must havesufficient drill to be able to loot: at a graph and tell im-mediately the sign of the first and second derivatives aswell as to estimate their magnitudes without resortingto calculators or computers.

The content of introductory calculus isprobably not as important to thephysical scientist as the insight to thisform of matheznntical reasoning.

Isimilar vein the area under a curve must have anintuitive relation to integration. Infinite series are usedto approximate analytical fuie-:ions. Some knowledgeof convergence as well as truncation errors are neededon an intuitive basis prior to releasing the power of thecomputer to grind away and produce nonsense. Intu-ition and "back-of-the-envelope" calculations are stillimportant in guiding the physical scientist to under-stand the significance of observations. Mathematicalintuition and physical intuition are frequently interre-lated.

The content of introductory calculus is probably notas important to the physical scientist as the insight tothis form of mathematical reasoning. Many times a

26 CoLLognium: VIEWS FROM CLIENT DISCIPLINES

student in physical science uses techniques of mathe-matical analysis the student has never encountered in amathematics course.

Principles of mathematics taught inthe context of a specific applicationhave the danger of not beingrecognized for the breadth of theirapplicability.

The important parameter with regard to content iseffective communication between the mathematics fac-ulty and the faculty teaching courses for which calculusis prerequisite. Most faculty in the physical scienceshave sufficient background in mathematics so that theycan direct students to appropriate sources or use theirlectures to provide background coverage. The studentcan become a victim if communication between facultyof different disciplines is missing.

Principles of mathematics taught in the context of aspecific application have the danger of not being recog-nized for the breadth of their applicability. Thus the

40

mathematics facult; does have an obligation to lookat content and the order in which it is presented tominimize the amount delegated to other faculty. In de-ciding content, order of presentation, and pedagogicalapproach, an effective dialogue is most important.

In looking at Calculus for a New Century, the math-ematks faculty nas an obligation to look at the edu-cational content of its courses. The educational con-tent must contain r. balance between the teaching ofnew skills and the development of mathematical andintuitive reasoning. In addition, scientists and mathe-maticians must continue to examine the question of theapplicability of mathematics to the description of the"Laws of Nature."

JAMES R. STEVENSON, Chief Executive Officer atIonic Atlar.f.a, Inc., also serves as consultant to the presi-dent at Georgia Institute of Technology. He is a memberof several professional organizations, including the Ameri-can Physical Society, the American Association of PhysicsTeachers, and the American Society for Engineering Educa-tion. He received a Ph.D. degree in physics from the Uni-versity of Missouri.

A Chancellor's ChallengeDaniel E. Ferritor

UNIVERSITY OF ARKANSAS, FAYETTEVILLE

Since it is nearly always more fun to reform someoneelse's discipline than your own, I have looked forwardwith anticipation to this colloquium on calculus insteadof one bent on reforming sociology.

The reforms contemplated here, however, are differ-ent from most. They start, it seems to me, from es-tablished strength, not from disarray. Calculus coursesalready command a respect (even among students whohope never to have to take one) which all college coursesshould command, but which few others do.

Calculus courses already command arespect ... which all college coursesshould command, but which few othersdo.

Along with respect, to be sure, calculus courses evokefear, awe, resignation, delight, and even resentment.Some students find what they learn in our calculusclasses an indispensable tool; others view it as a mingless hurdle. To some it is the pinnacle of mathe-matical achievement; to others it is only a foundationcourse for years of further study. I sympathize withyour challenge to make the experience better for stu-dents regardless of their different views and needs.

While improving the quality of mathematics edu-cation for all Americans is becoming an agenda foraction throughout the country, we in Arkansas maybe slightly behind the curve in recognizing its impor-tance. Before a recent raising of educational standardsin Arkansas, there was no requirement in mathematicsfor high school graduation. In fact, until recently, ourown university, the strongest in the state, had no math-ematics requirement in many of its degree programs.Only last year did the arts and sciences faculty includesignificant college-level mathematics requirements in itsB.A. degree programs.

Students bring to the University of Arkansas au ex-treme range of mathematics skills and experiences, in-cluding, in some cases, the apparent lack of either. Ithas been our mission for 116 years as a land-grant schoolto attempt to meet the needs of students whose diverseskills and needs are ensured by an admissions policywhich opens our doors to most would-be students with

a high school diploma and minimal GPA or nationaltest scores.

Because mathematics education in many of our pub-lic schools has been limited, even able students oftencome to us unprepared for college mathematics. At thesame time, we are the only institution in Arkansas withestablished programs of research and doctoral study,many of our undergraduate programs are unique in thestate, and many of our students come well equippedfor challenges and expect us to provide all educationalexperience which ranks with the best available.

To meet the needs of both groups of students is noeasy task. Our 2,500 entering students each year havemath subscores on the ACT test ranging from 1 to 36,although the average composite score is well above thenational average at nearly 21 (comparable to an SATof 870-900). This range of skills makes initial place-ment quite difficult but enormously important, and wehave six levels at which students may begin the study ofmathematics. These range from a one-semester reme-dial course in algebra to beginning calculus. Placementbeyond the first course in calculus is possible, but highlyunusual. About 20 percent of our entering students areplaced in the remedial course, and about 20 percentare placed in the first semester of calculus, with othersentering at intermediate levels.

Because mathematics education inmany of our public schools has beenlimited, even able students often cometo us unprepared for collegemathematics.

Our calculus courses have many of the same problemsas those at other schools across the country. However,the percentage of students who fail or withdraw is notquite as high as the 50 percent often reported nationally.We feel that our placement scheme has helped increasethe success rate for students in calculus as well as inother mathematics courses.

Better placement, though, is not enough. Nowhereare the differing views of calculus more obvious thanin the classrooms, where the perceptions of instructorand student can be worlds apart. For most students,

COLLOQUIUM: VIEWS FROM ACROSS CAMPUS

calculus is the highest level of mathematical knowledgeto which they aspire, while to many instructors, it isthe lowest legitimate level of mathematical inquiry.

For most students, calculus is thehighest level of mathematicalknowledge to which they aspire, whileto many instructors, it is the lowestlegitimate level of mathematicalinquiry.

The difficulty is further complicated by the fact thatsome students need calculus as a working tool in sub-sequent mathematics courses, while others concludetheir study of mathematics with calculus. Both shouldgain from calculus a larger vision of the mathematicallandscape. One teaching approach and one syllabus,though, may not be right for both kinds of students,and therein lies a major dilemma which must be re-solved as we develop a calculus for a new century.

Colleges and universities must do a better job of rais-ing the success rate of entering students by staffing be-ginning mathematics courses with instructors who areespecially well qualified for the demands of such courses.This would not be an easy task even with a shared phi-losophy and a well-defined policy, and few institutions,I saspect, have either.

Colleges and universities must do abetter job of raising the success rate ofentering students by staffing beginningmathematics crurses with instructorswho are especially well qualified for thedemands of such courses.

Within our own institution we do not even agree onwhich instructorsthose with a broad vision of the fieldor those committed to providing tools for specific usesare likely to be the best teachers. Nor have we begunto consider radical approaches such as more stringentcontrols, supervision, and uniformity in class organiza-tion and conduct in such courses. The current debateshould lead to better definition of the teaching problemand, eventually, to the improvement we all desire.

Exacerbating the problem at an institution such asours is the interaction of our graduate and undergrad-uate programs, which is generally healthy but whichmakes instructor choice and placement a practical as

well as philosophically challenging matter. In lower-division mathematics, we depend heavily on graduateassistants as assigned teachers and as teaching assis-tants.

As is the case nationally, our graduate students in-clude international students for whom English is a sec-ond language. Students who experience difficulty with acourse like calculusand many dolook widely for anexplanation of their difficulty. If the instructor sp. aksEnglish with an accent, he or she is a likely target.While the accent may well be a factor in the student'sfailure, even if it is not, too often students, parents, andlegislators believe it is the primary cause.

"Why can't we have American mathand science teachers?" is a toughquestion, but it is one of the questionsmost frequently asked of me byArkansas legislators.

"Why can't we have American math and scienceteachers?" is a tough question to handle, but it isone of the questions most frequently asked of me byArkansas legislators. We must avoid overreacting tosuch criticism and remind critics of the American tradi-.tion, from our earliest beginnings, to welcome and relyon imported talent. However, by the same token, wecannot rely solely on international students and faculty.The decline in mathematics and science majors amongnative-born students has reached alarming proportionsand should stimulate us to devise ways to attract moreof our own students into such careers.

Finally, in addition to accurate placement and fo-cused and enlightened instruction in calculus courses,I see a need on the institutional level for an increasedawareness among our students and faculty of the impor-tance of mathematics. I hope that efforts like this oneat the national level will help us there. The umbrellaproject "Mathematical Sciences in the Year 2000" is,I understand, designed to broaden this discussion be-yond calculus to the other courses, to the flow of math-ematical talent, and to the issues of resources. Sincethe University of Arkansas programs span a broad areafrom remedial algebra through graduate work and re-search programs in mathematics and statistics, we willbe looking forward to that broader discussion and as-sessment.

I am sure I speak for many university chancellors andpresidents, as well as for my own university, when I ex-press support for the goals of the projects here at the

42

NEAL: NATIONAL NEEDS 29

National Academy. While I can't evaluate the substan-tive changes in calculus which are suggested, I certainlyrecognize the need for a revitalization of the teachingand of the learning of college mathematics.

Returning a sense of discovery and excitement toclassrooms where calculus is taught will be a vital steptoward university preparation of the kinds and num-bers of mathematics graduates needed by U.S. science,industry, and society. It should also ensure for count-less students the pleasure of accomplishment and in-sight which mastery of calculus is meant to bring to theliberally-educated individual. Even if Edna St. Vin-cent Millay was right that "Euclid alone has looked on

National NeedsHomer A. Neal

UNIVERSITY OF MICHIGAN

I am honored to have been asked to appear on thispanel today to discuss the prospects for major revisionsin the undergraduate calculus curriculum.

In 1985, I chaired a National Science Board TaskCommittee to study the state of undergraduate science,engineering, and mathematics education. Our commit-tee completed its work in the spring of 1986, after draw-ing upon published reports and information from inter-views with faculty members, university presidents, vicepresidents and other university officers, representativesof foundations are industry, as well as representativesof various profes .al societies. Included in this groupwas the Presiwnt of the Mathematical Association ofAmerica, a former President of the Ameilcan Math-ematical Society, and an executive officer of the SloanFoundationindividuals who have played a particularlysignificant role in advancing the cause of the conferencehere today.

We found in the work of our committee that therewere numerous reasons to be concerned. There waswidespread evidence of serious problems in the curricu-lum and laboratory instrumentation used in the instruc-tion of both majors and general students. Moreover,related motivational problems existed for students andfaculty. Students found many of their key courses to bedull and uninspired. Faculty were often frustrated bythe lack of student interest, and the faculty themselvesoften found it difficult to keep abreast of the rapid de-velopments in their fields in the absence of special pro-

Beauty bare," every student of mathematics should beable to catch more than a glimpse of a similar vision.

DANIEL E. FERRITOR is Chancellor of the University ofArkansas, Fayetteville, where he has also served as Provost,Vice Chancellor for Academic Affairs, and Chairman of theDepartment of Sociology. The author or co-author of over40 publications in the field of sociology, Dr. Ferritor hasworked for several years in national educational programsfunded by grants from the federal government. He receiveda Ph.D. degree in sociology from Washington University inSt. Louis.

visions for them to have the time and resources to dSO.

Our recommendations for action were extensive, andmany have already been implemented, either directly bythe National Science Foundation, or indirectly throughnew initiatives at the university, regional, or state level.Examples include the immediate launching of the Re-search Experience for Undergraduates program at theNSF, legislative hearings on the health of undergrad-uate science, engineering, and mathematics by statessuch as New York, and the President's request for in-creased support for the College Instrumentation Pro-gram and other related initiatives.

No other discipline is so fundamentalto ensuring a talented pool of futurescientists and engineers, and atechnologically literate generation.

Regarding the focus of today's symposium, our taskcommittee was frequently reminded of the critical roleplayed by mathematics in the training of students inall disciplines. In particular, mathematics is often thedetermining filter for all science and engiLzering disci-plines, not to mention for the advanced mathematicsprograms themselves. No other discipline is so funda-mental to ensuring a talented pool of future scientistsand engineers, and a technologically literate generation.

43

30 COLLOQUIUM: VIEWS FROM ACROSS CAMPUS

Within mathematics, there is strong evidence thatone of the most urgent challenges is to reform the cal-culus curriculum. Experts here today have made thecase for why this is so, and the impact that such re-forms could have on a wide span of disciplines. I onlywish to add my encouragement for this initiative andto congratulate those who have had the insight and thepersistence to proceed with the formulation of strategiesfor addressing head-on such a massive problem.

It would be very easy to view presentcalculus instruction as being aninvariant of nature. The way it wastaught-to-us-could be thought to bethe way it must be taught fo ever.

There are, to be sure, numerous obstacles that lieahead. Faculties in mathematics departments must em-brace the concept that significant changes are called forin the calculus curriculum for the reforms to be success-ful. Colleagues in cognate departments must cooperatein providing feedback on the success of the reforms, asviewed from the perspectives of their disciplines. Deansand provosts must get on board, to provide both moralencouragement and some significant fraction of the ac-tual financial support required to implement and mon-

Now Is Your ChanceMichael C. Reed

DUKE UNIVERSITY

I would like to talk about three things. First, letme say what I think about current calculus courses andtexts.

I think they are awfulbut they're awful for a lot ofunderstandable reasons. They are awful because theyare too technical; they try to teach too much material;they teach very little conceptual understanding; andthey have a tremendous lack of word problems. As weall know, it is the word problems that students hate themost, and yet it is the ability to do word problems thatmakes mathematics applicable for a physics major, achemistry major, or an engineering major.

If you look at the section on differentiation of poly-nomials in the text you are using, you will undoubtedly

itor the revised programs. Funding agencies must com-mit to providing the required external support over asufficiently long period to insure that the revisions canbe implemented, studied and refined. I am particularlypleased to see the Sloan Foundation and NSF take alead in achieving this goal.

It would be very easy to view present calculus in-struction as being an invariant of nature. The way itwas taught to us could be thought to be the way it mustbe taught forever, regardless of the fact that'the tech-nological context has changed by leaps and bounds. Ittakes unusual insights and courage to challenge such atradition. What you are doing is extraordinarily impor-tant and I wish you every success.

HOMER A. NEAL is Chair of the Department of Physicsat the University of Michigan. Dr. Neal was Provost at theState University of New York at Stony Brook before as-suming his current position as Chairman of the Departmentof Physics at the University of Michigan. The recipient ofGuggenheim and Sloan fellowships and a former member ofthe National Science Board, Dr. Neal headed the NationalScience Board Task Committee on Undergraduate Science,Mathematics, and Engineering Education. His research is inthe area of experimental high energy physics. He received aPh.D. degree in physics from the University of Michigan.

discover at the end of that section an extremely longlist of problems. I guess that none of the problems isa word problem and that not a single one of the exam-ples at the beginning of the sectionexamples whichare supposed to motivate why we want to know how todifferentiate polynomialsis a word problem with anykind of interesting application attached to it.

It is the ability to do word problemsthat makes mathematics applicable.

To try to press this point home, I have to tell youa story. I was standing around the common room last

44

REED: Now IS YOUR CHANCE 31

year and there I saw a calculus exam which, as it turnedout, had been prepared by one of our graduate studentswho, unfortunately, was standing near me. The firstproblem on the exam was to differentiate x''''' and Isaid, "Thai is what is wrong with calculus." I was thenvery embarrassed because the poor student was thereand a large discussion ensued. My colleagues challengedme to explain what was wrong with making studentsdifferentiate en'.

Why should I spend all my timeworrying about how to differentiatestupid looking functions like that? Nofunction like that has every occurred inthe history of physics.

Here is my answer. We live in the same buildingas the Physics Department. I guess that not a singlemember of the Physics Department could differentiatethat particular function and say what the answer is.Furthermore they wouldn't be embarrassed by it. Theywould say, "Are you crazy? Why should I spend all mytime worrying about how to differentiate stupid lookingfunctions like that? No function like that has everyoccurred in the history of physics, so why should I beconcerned that I can't differentiate it?"

Well, it sounds like a joke, but it's not a joke. Itmeans that the teaching of calculus has developed intoa seriez of technical hurdles for students to go past,one after the other, bearing very little relation to whatthey're suppose to get out of the course.

Now we come to the reasons why this has happened.First of all, students arrive at the university with verylittle motivation to think about mathematiCs. That'sbecause their training in K-12 is mostly plug-and-chugand we give them what they expect because we get trou-ble when we don't give them what they expect. It'smuch easier to teach plug-and-chug than it is to teacha conceptual course.

It's much easier to teach plug-and-chugthan it is to teach a conceptual course.

Secondly, a great deal of calculus teaching in thiscountry is done by non-tenure-track faculty, by grad-uate students, and by part-time instructors who havebeen hired to fill large gaps on the teaching staff. Ata large university, it is the best you can hope for thatall instructors teach more or less the same thing so that

when students go on to the next course they will havethe same background. In a situation where you have avery large number of instructors who perhaps are nei-ther very well trained nor motivated to teach well bycontinuing attachment to the institution for which theyteach, the best that you can ask for is an adequate,standard job. Finally, the lack of original textbooksthat try to strike out in new directions is really a greathindrance.

Two years ago my colleagues told me that I shouldeither shut up about this or go teach calculus myself.So, I taught it myselfsince I didn't want to shut upabout it, and still have not. I taught out of one of thestandard texts; I tried as much as possible to put wordproblems and applications in the course. I found it verydifficult.

Without excellent standard textual material, innova-tions will surely die out. That means that at the end ofthe projects that many of.you are considering, books orother materials that every student can buy for twenty,thirty, or forty dollars at the bookstore has to comeout. If you are thinking about a project, you have tofigu. out how at the end of the experiment you aregoing to produce something that can be used at otherinstitutions.

Without excellent standard textualmaterial, innovations will surely dieout.

These are some of the reasons why I think theprojects that many of you ale considering will en-counter real obstacles to success. Even if the curriculumchanges, much of the teaching is going to be done bynon-tenure track people. Many of the students are go-ing to arrive not wanting to take your new interestingcourse. Finally, there's the question -)f when these won-derful text books are really going to arrive so that youcan use them. When are the publishers going to agreeto cooperate with youas individuals or as groups--toproduce such textbooks.

There's a third aspect of this issue which I wouldlike to address, not to my fellow administrators here onthe panel, but to my colleagues, the mathematicians.That's what I like to call the G.H. Hardy syndrome.

I trace a lot of the evils in calculus instruction to G.H.Hardy. There is a commc i attitude very well expressedby his posture in the picture on the front of that book (AMathematician's Apology), that matherni ics has littleto do with the rest of the world, and, in fact, shouldproperly be contemptuous of the world.


This attitude has served us rather badly, I think, inthe last thirty or forty years. It has bred an attitudeamong mathematicians not to talk to their colleagues;it has made narrow mathematicians who have very fewinterests outside of eir own discipline; and this hasproduced mathematicians who are not very capable ofenlarging their courses with appropriate and interestingapplications.

So that is a problem within the mathematics commu-nity, not a problem for administrators. However, thereis another issue which is related to administrators, andthat's this: Mathematicians as a group are terrible en-trepreneurs. We are really bad. Mathematicians areembarrassed to stand -up for their subject, to say thatit's important, and to fight for it in their home institu-tions.

Mathematicians are embarrassed tostand up for their subject.

Many mathematicians do not realize that their col-leagues in physics, chemistry, and biology could not ex-ist unless they were excellent entrepreneurs, becausetheir laboratories depend on their skill in administra-tion, as well as on their scientific skills. Those guysfrom those other departments are harassing their deansand their provosts all the time for money on behalf oftheir research, on behalf of their teaching programs, andfor everything else.

Mathematics departments are often afraid to do that.They think they gill not be well thought of; they're too

Involvement in Calculus LearningLinda Bradley Salamon

WASHINGTON UNIVERSITY

It is, I hope, appropriate for an English professorwho's addressing a group of mathematicians to beginwith an allusion to that eccentric figure who bridgedboth fields, Charles Dodgson (a.k.a. Lewis Carroll), tothe effect that deans of arts and sciences like to believethree impossible things before breakfast .... We wantto believe

that our colleges can present a steadily improving,timely curriculum which will provide both studentsand faculty with continuous challenge, and

timid to do it; they feel it's like P.R.; it's self-serving.In many cases, they secretly believe that mathematicsisn't so important anyway.

Here is my message: This is your opportunity. Hereyou are at a national colloquium that says there's sup-posed to be a new agenda for calculus. Go to yourchancellor, go to your president, go to your dean andsay, "All these years you have been complaining like hellabout the calculus instruction in our institution. Howmany calls have I received from you saying that you gota call from so-and-so's mother who said, 'Why does myboy have to be taught by a graduate student?' "

"Now's your chance to invest some money to makeit better. I want so-and-so much released time for thesetwo faculty members in the mathematics department sothey can work on restructuring calculus for three years.I want so-and-so many funds to support a secretary whowill be typing the new manuscripts that the studentsare going to read; and so forth." Administrators hearthose kinds of requests from all other departments allthe time. They hardly ever hear them from mathemat-ics departments because mathematicians are too timidto ask. Now is your chance.

MICHAEL C. REED, co-author of Methods of ModernMathematical Physics, is Chairman of the Mathematics De-partment at Duke University. His research is in the area ofnonlinear harmonic analysis and in the application of math-ematics to biology. He received a Ph.D. degree in mathe-matics from Stanford University.

that political "peace in the valley" can prevail amongthe various disciplines with which we work, with theirvery various intellectual styles and varying currentsuccesses, and

that both those goals can be accomplished within ourbudgetary means, or with clearly foreseeable new re-sources.

The probability that the teaching of calculus can find apoint of intersection in that three-way matrix is smallenough to make Alice blink in Wonderland.

46

SALAMON: INVOLVEMENT IN CALCULUS LEARNING 33

The good news from the dean's office is that one re-peated assertion in the preparatory materials for thisconference is false: on my campus, calculus is not thecourse about which most student complaints are reg-istered; that dubious distinction belongs to chemistry.Nor do my colleagues in the sciences use performancein calculus, even implicitly, as t "weed-out" device,nor does our Medical School use it as a shorthandadmissions indicator. Again, chemistryparticularlyorganic chemistryadmirably and accurately fulfillsthose roles.

Paranoia ill becomes you, though you have real ene-mies. The principal source of complaints about calcu-lus, one where my Lone Ranger's peace-keeping skillsmost frequently intervene, is the "Engine School." I

must admit, as an'outside observer of the mathematicalcommunity but as an experienced student of pedagogy,that engineering educators have a point.

The objective of instruction in calculus...is to bring each student to the mostthorough and functional understandingof this sophisticated subject that sheor he can achieve ...like it or not.

Because I cannot begin to consider whether calcu-lus needs to cover partial fractions in order to preparefor a later encounter with Laplace transformations andother such arcana, I want to make four points from theperspective of a teaching colleague and educational ad-ministrator. They relate to classroom management andclass size, to the use of the computer, to the role ofmathematics in general education, and, of course, tomoney.

All my remarks assume that the objective of instruc-tion in calculusI'll qualify that by saying "at a re-search university," but I really mean anywhereis tobring each student to the most thorough and functionalunderstanding of this sophisticated subject that she orhe can achieve ... like it or not. (Remember, please,that I hail from the only other discipline in our col-lege:: where 80% or more of the beginning students areunwil:ing draftees; I know the consequences.)

The first implication to be drawn from a coal ofbringing each student to his or her best achievementis obvious. We must take them as we find them both inability and in preparation, from effectively near-zero toAdvanced Placement.

Because mathematics is so linear and progressive adiscipline, students' differing readiness at entrance dic-tates that at all but the most select institutions, there

be several different calculus courses. I think they shouldvary not by the student's immediate use for calculus (asa biolor or business major, say, rather than a proto-physicist or engineer), but by the pace at which theymove, the degree to which they pause over relevant pre-calculus topics before introducing new material, andconversely, the depth and sophistication of conceptsthey have time to include. The same textbook, afterallif one's not enslaved to its teaching manualcanbe creatively utilized in quite different ways.

On a large campus, three or four different calculuscoursesincluding an honors effortmight be under-way. One of those can certainly be lean and lively; Idoubt that all can. Selection should be thorough andinformed, and prerequisites should be vigorously en-forced, to the point of requiring preparatory "collegealgebra" or other euphemistic courses, if necessary.

What simply will not do is the model of a commonsyllabus for 1000 students, so that interchangeable Pro-fessor X can, on 15 minutes' notice, give a lecture ona particular topic to an anonymous mass, then whipback to his office and his Fourier transforms untouchedby human minds. Count on your dean's complainingabout that.

The second implication of seeking each student'sachievement is what the NIE has taught us all tocall "involvement in learning." In mathematics, thisterm that describes attempts at personal, internalizedmastery surely means homeworkhomework that's re-quired, evaluated, and included in the final grade.

In mathematics ... mastery surelymeans homeworkhomework that'srequired, evaluated, and included inthe final grade.

Here's where my compulsive friends the engineershave a point, and you know it's true. Even in a mini-mal calculus course that's only teaching calculationsgiven the limited concentration and persistence of to-day's studentschecked homework is needed; if youchoose to concentrate on concepts and what the kidscall "word problems," it's essential.

Now, does this imperative dictate the end of largelectures in favor of 30-student classes? Maybe, but Idoubt it. "Help" sessions and drop-in math labs staffedby grad students go a long way toward giving studentscontrol over their own learning, as do 20 minutes forquestions before or after each lecture.

All across our undergraduate curricula, moreover,the one thing that computers can reliably do better

COLLOQUIUM: VIEWS FROM ACROSS CAMPUS

than we can is rote drilling in mechanics and routines.If the machine can do it for German adjective endings,with repetitions and branching that match the student'space, surely they can do it for power series or what-have-you. (And that's leaving aside the intrinsic rea-sons for introducing students to computers and theiralgorithms.) As a dean, I'd rather pay for new termi-nals or PCs or networked micros than for new assistantprofessorseven supposing the department chair canfind us talented candidates who can articulate clearlyin English and relate comfortably to puzzled and inse-cure students.

As a dean, I'd rather pay for newterminals or PCs or networked microsthan for new assistant professors.

About the issue of surrendering the teaching of thelogic behind integrals and derivatives to some HP cal-culator that can score a B on tests without studying, Ihave no right to comment, but I hope that, as a pro-fession, you won't succumb completely. If mathemat-ics has a role to play in the general education of ourstudentsin shaping their minds as informed and dis-ciplined instrumentsit is in this ..:omain of conceptu-alization.

To deal with numbers in the real world, they needto know not the right formula to plug in to do the jobof the moment but skills like estimation and approx-imation, concepts like scatter and risk, the meaningof graphic representation, the sheer likelihood of trialsolutionsintuitive probability, if you will. Those ideasdon't require a calculus course, trs be sure, and I'd wishall mathematics departments would offer a clever coursein finite mathematics for those students who won't takecalculus.

Teach in the best way you can find topresent the real power (and maybeeven the beauty) of your discipline toan anxious or indifferent kid.

But for those whose only mathematics course will becalculusa very great manyusing computerized cal-culations that avoid fundamental ideas about numbersseriously deprives them, know it or not. What couldreplace the pencil sketching, the concrete visualizationthat I see our best math students doing? Honor theintegrity of your discipline, please; teach what it's allabout.

By the same token, with all due respect to the"user groups" represented on the previous panel, I

hope you won't take the service function of calculus asparamount. Teach the topics they need covered, sure;but teach in the best way you can find to present thereal power (and maybe even the beauty) of your disci-pline to an anxious or indifferent kid. We all know thatthe best means for meeting the objective with whichI beganto help the student achieve the best under-standing she canis a willing, imaginative teacher wholikes the material and will work. hard at expounding it.

The tacit purpose for my invitation to be here to-day is not these opinions of mine, of course, but thequestion, "What will deans pay for?" or, more pre-cisely, "Will deans pay for more mathematics faculty?"I think I know our tribe and its bronzed responses wellenough to answer.

First, we do have those engineering and businessdeans riding like sheepherders into our peaceful valley,and for financial reasons we have to satisfy them. We'lldefend our mathematicians to them if you give us theammo and if what you do is defensible. If what theydemand is more, differentiated sections and more eval-uated homework (and it sometimes is), you should bemaking peace (and common cause) with them, not com-plaining about their students, or their demands.

Calculus is the second largest courseafter English composition on manycampuses, as on mine, and we deanssimply must attend to that brute fact.

Next, when mathematics department chairs ask usfor more slots, of course they have to get in queue withthe other two dozen or so department chairs, and avail-able slots will be allocated on the basis of institutionalpriorities. The best rationale for attaining a high prior-ity is unlikely to be the putative need to teach smallerclasses per se, but if you devise genuine and compellingpedagogical reform that demonstrably requires addi-tional staff used in imaginative and effective ways, wewill certainly listen. Double your teaching staff? No.But relief for a worthy experiment, to continue if it suc-ceeds? Highly probable. Calculus is the second largestcourse after English composition on many campuses, ason mine, and we deans simply must attend to that brutefact.

Demanding folks that are, though, we'll also ex-pect your candidates to be talented differential geome-ters or harmonic analysts, and that requirement raisesdifferent questions. I'm not a graduate dean, but my

4

STARR: CALCULUS IN LIBERAL EDUCATION 35

colleague who is both that and a mathematician reportsmixed news about the pipeline. Are university mathe-matics departments producing enough new Ph.D.s wellprepared to teach undergraduates to satisfy the ambi-tious goals of your proposed curriculum project, on anational scale? Guessing the answer to that, as an un-dergraduate dean I must look to our mathematics ma-jor program and its ability to retain students beyondcalculus, indeed beyond graduation. In this partiallyclosed system, perhaps we do indeed need to invest the

resources that will bring us mathematicians for a newcentury.

LINDA BRADLEY SALAMON, Dean of the College ofArts and Sciences at Washington University in St. Louis, isco-author of the Association of American Colleges' Integrityin the College Curriculum. She is Past Chair of the AACand a trustee of the College Board. A scholar of Elizabethanculture currently teaching Shakespeare, she received a Ph.D.degree from Bryn Mawr College.

Calculus in the Core of Liberal EducationS. Frederick Starr

OBERLIN COLLEGE

Despite being a college president, I still am a histo-rian. I was asked the other day by a student perplexedover the mathematics competency requirement at Ober-lin, "Why do it?" The specific question had to do withcalculus. I suggested that if Newton could invent it inhis sophomore year, he can study it in his.

Speaking as a historian, I would like to note a verycurious exchange that took place between Newton andhis friend John Locke which has very much to do, Ithink, with the subject at hand. When Newton had fin-ished the Principia, realizing full well the importance ofwhat he had done, he sent it over to Locke to get Locke'sestimation of it. Locke locked it over, and couldn't de-cipher the math. So he sent it to a friend in Holland andasked him to check it out. The friend read it and saidthat the math was okay. Locke then read everything ex-cept the mathematics in the Principia and wrote veryintelligently on it.

I wcs asked ... by a student [about]calculus. I suggested that if Newtoncould invent it in his sophomore year,he can study it in his.

Now, the issue here is whether Locke, who obviouslyhad absolutely no contact with the process of math-ematics that underlay the Principia, was able, as aneducated person, to deal with that work or net. The as-sumption on which this conference rests, I would ,ather,

is that Newton should have encouraged his friend Locketo study the mathematics necessary to deal with it be-cause otherwise he would simply be dealing with theproducts of other people's thoughts and never be ableto engage in the process. That really raises the first offive points I would like to lay before you here.

I don't get the sense, reading through the variouspapers that have been prepared, and hearing the dis-cussion, that there is much agreement as to the basicpurpose of the enterprise- -the reform of calculus. Is

calculus a service course? Is it a r urse that it provid-ing techniques, methods, manipulation, and so forth?Or is it truly part of some core learning that an edu-cated person should have dealt with? Is it really deal-ing with concepts, or can someoneas was asked ear-lier this afternoondeal with the manipulations with-out genuine understanding?

As a pedagogue, that is an absolutely preposterousproposition to me. But obviously, if calculus isn't in thecore, if it is instead a service for others, then one canget by with all kinds of mischief. It seems to me thatgreat clarity on that point is required before anything*Ise can proceed coherently.

My colleagues in mathematics at Oberlin have takenthe rather uncompromising view that calcuhis has todo with thinking, with concepts, with the core of a lib-eral education. From that they proceed to deal withother questions. Discrete mathematics, for example, isbeing offered as a separate parallel course in the sopho-more year. There is concern for the verbal dimensionof thought, great concern in fact, among them. There


is also a good deal of experimentation going on withcomputer algebra systems, both within the classroomand without.

But all this flows from some clarity on the basic pur-pose of the enterprise. You might he concerned whetherthis might lead to someone who has thought some finethoughts but can't do anything. The answer, I think,is that the purpose of education isn't to teach you howto do something, but how to be anything.

The purpose of education isn't to teachyou how to do something, but how tobe anything.

Sia,ce the mathematicians at Oberlin are apparentlyenabling their students to go on into careers in scienceat a remarkable rateto get their Nobel Prizes and toget their whole science program selected as first in thecountry among collegesthey're obviously doing some-thing.right. I would suggest that it has a lot to do withtheir very clear beginning point, namely, that calculus ispart of the core of learning. It is not simply a "how-to"course.

The second proposition that I we -r) .Ite to lay beforeyou has to do with the process of study. There was avery interesting body of research carried out by severalinvestigators at Oberlin College in the last four yearson the unclagradiate preparation of scientists in theL'.4ited States. They focused on a group of fifty liberalarts colleges ".11:tt were the most productive of scientists7' a similar :lumber of universitic3.

DaVie Atta and Sam Carrier, who di,; thisresearch, documer.t, 3, among other things, that therehas been a precipitous decline in the the percerta, ifpeople entering careers in science and also r.r. incrk...ein the attri' ion rate of intended scientists as th-.y entercollege rh.c1 1:oceed towards graduation.

Calculus is part of the core of learning.It is not simply a "140 -to" course.

They showed, moreover, that the only institutionsthat are successfully bucking this national trend in thesciences are those that base their pedagogy on a kindof apprenticeship systemthose in which the student isbrought into the laboratory, in which there is a direct,hand-to-hand contact, not with graduate students, butwith real professors.

If that is true, and there is so clear a correlation thatI'm left with no doubt about it, then I think you'vegot to ask whether your discussions are taking this intoaccount. Is mathematics open to that kind of directapprenticeship-based engagement? Does the advent ofthe modern computer in fact provide a wonderful op-portunity to go in that direction and to think of muchmore active forms of pedagogy than have been used inthe past?

My third point is very different in character, I'm con-cerned that you might be replicating in a perverse waysome of the negative featurc3 of a national discussion inwhich I've participated over some years regarding theteaching of foreign languages in the United States. Infact, one of the background papers drew the paralleland I was pleased to see at least that the relevance ofthis parallel was acknowledged.

The attempt to improve foreign language teachingculminated in a Presidential Commission a few yearsback. Frankly, the mountain didn't quite give birth to:a. mouse but something on the scale of a rat. It neverrtally brought about the great transformation.

Because it is in the schools where yourproblem is being formulated ...if youdon't bridge the geological faultseparating you from the schoolstherewon't be any progress.

I think the great flaw in the Commission's approach,and in the thinking of the late seventies and early eight-ies, is the assumption that you can build a house fromthe roof down. I wonder if this flaw also might not bepresent here. Can you really hold conferences and talkseriously about calculus at the university level and notspend an equal amount of time on the secondary schoolmathematics curriculum? It seems inconceivable to me.

I would be most interested l )w what has gone onsince the MAA and the Ncall "n years ago cautionedhigh schools not to teach calculus unless they do it ata university level with university standards. It seemsto me that this question has to be opened wide, notjust for precalculus courses, but for the entire secondaryschool preparation, or you will not progress an inch.Because it is an the schools where your problem is beingformulated, if it's not addressed at that levelif youdon't bridge the geological fault separating you fromthe schoolsthere won't be any progress.

This suggests, by the way, that once you do cross thatfault and deal with the whole process, then you can also

STARR: CALCULUS IN LIBERAL EDUCATION 37

consider calculus in several dimensions, progressively, inthe course of teaching over time.

My fourth point is thisthat calculus was not in-vented here. There are people elsewhere in the worldwho know how to teach it reasonably well. In fact,many demonstrably are doing a better job at it than weare. I think we're simply handicapping ourselves if wedon't examine very carefully the pedagogical systems,pre-calculur and calculus, that obtain in those countriesthat are doing a good job.

You may say that such societies are different. Theysend a lower percentage on to college. Fine. Thendemonstrate that that decisively negates the experienceof country'X. It seems to me the burden of proof lies onsomeone who would argue that international experiencein this area is irrelevant.

Do a survey.' You will find aphenomenal ignorance of just' whatcalculus is all about.

And now, finally, a rather more practical matter,that takes me back also to the Locke-Newton link. Idon't believe that calculusor for. that matter, math-ematics in generalhas a very, strong .constituencyamong faculties of our universities and colleges. 'All ofus, all those outside the mathematically-based fields,.are unfortunately too much in, the position of JohnLocke. And we're not embarrassed about it, as Lockeat least was.

The problem is very serious because if you are say-ing that mathematics, or specifically calculus, belongsin some- core curriculum of what an educated personshould know, you are saying that to a group of col-leagues who themselves haven't taken calculus and whodon't know what it is. Mk them. Do a survey. Youwill find a phenomenal ignorance of just what calculusis all about.

It seems to me that there is .something very seri-ous about this matter. What I would suggest is rather

naively grand, but do-able. If you do proceed from thefirst principle that mathematics in general and calculusin particular should be part of the equipment of an ed-ucated person, then take time to offer an accessible cal-culus course for your colleagues in other departments.

Crack that problem. It's do-able. It's not the po-litical dimension I'm concerned with here, but the in-tellectual dimension. Until this g..:p is bridged in atleast one institution to prove that it is possible, we aretalking about such remote worlds that, although peo-ple might as a matter of political bargaining give youyour required hours in the classroom, or your piece ofthe budget, they won't really understand why they aredoing so.

There are outposts in economics and various areasof.the. social'sciences. where your task will be.easy. Bat:most of .the social scientists and nearly all those in thehumanities are illiterates in mathematics and in calcu-lus. Hence there is a need for teaching that is directedtoward the professorial community itself simply as ameans of making up, even at this late date, for thefatal neglect of mathematically-based learning in ourprimary, Achools, in our secondary schools, and in mostof our colleges. Until the professors are mathematicallyliterate, don't expect them to understand why studentsshould be.

S. FREDERICK STARR, President of Oberlin College, isa specialist on Soviet Affairs, founding secretary of the Ken-nan Institute for Advanced Russian Studies at the Smithso-nian Institution, the author of numerous books in his field,and a member of the Trilateral Commission. As a resultof his initiative on the so-called Oberlin Reports on Un-dergraduate Science, the National Science Foundation andmany private foundations have strengthened their involve-ment with mathematics and science at the undergraduatelevel. Dr. Starr received a Ph.D. degree in history fromPrinceton University.

II

1

r.-.r,.rr

4,...... -1

( rt 1

ELIRTICIPAAir Re S-fonisE.S r)-1

. ri:

LA.est-i cm WI Act+ ck-re- tts.e_i)ri 1,-% ci pc. t L.) eft kL.e.tisex r-1 vi ced ct-s_ILLs i (4.Sti 01'1? (.4

.

77i:1, ylc weAc 0

No se.vtse 4 ad v en ture_ 04-A,c1 ° II' .S ye.) ri..j 0.

.C.0(-:

-p_DD cl__Ir WI Oi- V A. -Fe_cX. c:..A.,,, eX, (cf...

wn.2_9xD. vv-. vtA e A v-1.0 t- -t-t,, it, kT1-. e. ii,N5..tr.c.tLs-- .fi-efin.

(--.;\---- c. i-

____LQ, vv. w 1 4-14,e-4,_d-I-D te-c___,- orl c--,--......,, It P I'l^ Cihe. el^ ,k-(.-* j C.. , 00

.__ai___Re, eckitA.

ct,,A 'el" CLI Vl. 1 tAlc, cl.,

*...k WIL_C,Ina_ln i Ca t".--te-tet.t.te,vi-; tie tA4 t i-t ic Iiv Cci:.-1

1-1-01A., cctan okleeak stt&c,L.Q.A, ft < o L.). e.:___g

tiAL.AY- e...A.\ -,f t2 S-_,..________Zt:,Ao Cz.r-Ct".\ e...-t-iC.-7

_ 1,..,zi- att,os÷ yob° 6t4i is ii nori.: .

c 1 c_ lA_As e_ry e...S. f-oo vvx a-Ait (1 1414-V--irL.Ci.r-E- is vi-o + a .s er-v ce_ co u-r.S e. .

.

---c..

. - .-kJ

Mathematics as a Client DisciplineGeorge E. Alberts

NATIONAL SECURITY AGENCY

As a representative of (arguably) the largest em-ployer of mathematicians in this country, I have foundthe Calculus for a New Century colloquium remark-able. The huge turaeut is a clear sign of growing con-cern aid interest. The candor and quality of the dis-cussion Inzve been impressive. Although the NationalSecurity Agency also employs large numbers of engi-neers and computer scientists, and despite the temp-tation to comment on tht full range of fascinating is-sues and ideas, these remarks will focus on the morenarrow issue of the future health of the mathematicsprofession itself. In that regard, it seems worthwhile toview mathematics itself as its own "client discipline"in discussing reform and revitalization of the calculus.

Tom 'Dicker helped set the tone for candor by sug-gesting it might indeed be appropriate for the sec-ondary schools to assume the responsibility for teach-ing calculus. In his "desert metaphor" the secondarylevel might be the appropriate "sinkhole" for the calcu-lus. "Mainstream" calculus, which controls the pipelinefrom which the mathematics teaching faculty is replen-isned, couid be taught at the secondary level. We mayindeed "only need a few people who know the innerworkings of calculus." I think not.

All of what I have heard described by clients as de-sirable seems essential for professional mathematiciansthemselves. As Gail Young wrote in his backgroundpaper, "the future of our subject depends on improv-ing [calculus]." Calculus has frequently been describedas the first real mathematics course (with the possibleexception of a good Euclidean geometry course in highschool). Although Anthony Ralston eloquently arguesthat calculus is dead, and discrete mathematics musttake its rightful place as the core mathematical sub-ject, I endorse Cathleen Morawetz' remark that trends'in parallel computing suggest a convergence of the twosubjects.

Complaints about lack of understanding, the needfor more conceptual, intuitive, and at the same timemore rigorous calculus echo a growing concern of our

Agency's mathematics communityat a time when weneed to hire more outstanding mathematicians, to docreative mathematics, the supply seems to be declin-ing.

Mathematics' "clients" are said to be leading inno-vators, and might well be approaching the capabilityof teaching their own calculus. Professional mathemat-ics must begirt to learn from their clients. Ron Dou-glas suggested that attempts to reform calculus havebeen well-intentioned but short-lived "castles in thesand." Oberlin President Prederick Starr remarked onthe short attention span of reformers. The present cal-culus curriculum, which Tom 'Ricker aptly describedas unchanged in 100 years, is more a castle made ofstone.

If the mathematics profession is to avoid ossification,its practitioners must take charge of the reform move-ment, and, while meeting the legitimate concerns of theother "clients," focus on reform of the calculus to reju-venate American mathematics as a discipline first anda service second. The .1,4:illative may we:. I.: the cal-culus "super-nova" suggested as a possibility by LynnSteen.

We do indeed need to !solve the basic purpose ofall this activity, to reassert the significance of mathe-matics as core training for all educated people, to ad-dress the demanding full range of problemsnot, pri-marily, because of our other constituencies, but becauseof ourselves. A lean and lively calculus, while servingthe needs of those other constituents (which it can ifthey share in its revitalization) will at the same timedo something more fundamentally important: attractand inspire successively better generations of Americanmathematicians.

GEORGE E. ALBERTS has ser ed for twenty two yearsas a professional mathematician at the Nati:: J Securli.Agency, Fort George G. Meade, Maryland. He is present*an Agency executive and Chairman of its Mathematical Sci-ences Panel.

42 RESPONSES

Calculus and the Computer in the 1990'sWilliam E. Boyce

RENSSELAER POLYTECHNIC INSTITUTE

Whether or not a "crisis" exists, there is widespreadagreement that improvements are possible in the cal-culus courses offered in many coll-ges and universities.Powerful inexpensive calculators .4.,d microcomputers,now widely available, provide numerous opportunitiesfor courses adapted to the contemporary environment.

First, calculus should be taught so that numericalcomputation is seen as a natural consequence of us-ing calculus. Most students who take calculus coursesdo so to improve their capacity to solve problems inother fields. Generally speaking, a numerical answer issought,-so eventually something must be computed inorder to obtain it.

On the other hand, computation can also lead nat-urally to analysis. For example, computation carrieswith it the obligation to consider the accuracy of theresults. Therefore it provides an opening to discusssuch questions as estimation of errors or remainders,or speed of convergence of an infinite series or iterativeprocess. Ideally, analysis and computation should ap-pear as complementary and mutually reinforcing modesof problem solving, each used when appropriate, andeach enhancing the power of the other.

Moreover, at every opportunity students should beencouraged to try to assess the reasonableness of theiranswers. A result that appears reasonable may not becorrect, but one that appears unreasonable is almostcertainly wrong (unless one's idea of what is reasonableis seriously deficient).

In addition to computing numerical approximations,other aspects of computing may also be important incalculus. One is the rapid and accurate generation ofthe graph of a function or an equation. To do this effec-tively it is essential to scale the problem appropriately.Beginners rarely, if ever, give adequate thought to thisquestion, although it is crucial in obtaining a usefulgraph. Consequently, they often have difficulty in in-terpreting the graphs their computer screens depict.

In choosing a proper scale, it may help to do some ofthe things usually taught in the context of curve sketch-ing, such as finding maxima and minima, checking forsymmetry, and locating asymptote.. Thus, comput-ing may make more clear the need to learn somethingabout calculus and the behavior of functions, ratherthen merely resorting to trial-and-error computation.

The use of symbolic computational packages to re-duce the need to perform tedious and repetitive alge-braic procedures is highly attractive. However, the meof such packages also inc Irs a cost. One does not learnto use a sophisticated symbolic computation package(MACSYMA or Maple, for example) instantaneously.At least some of the time saved by using the packagemust be invested up fro_it in learning how to use it.

There is also the "black box" question: should wepermit students to use a symbolic computation pack-age without some understanding of what the packageis actually doing? My view is that in order to thinkconstructively about the behavior of models of physi-cal phenomena, one must have some specific informa-tion about some particular functions. It is probably notnecessary to know how to evaluate f secs x dx, but oneshould certainly know f cos x dx.

The boundary between what is essential knowledgeand what is not may be unclear, but surely we shouldinsist that students must learn to execute some proce-dures themselves, even while relying on a computer tohandle the more complicated cases.

The use of a computer may not save much time in a,calculus course, although it will give the course a some-what different orientation. Assuming that it does saveat least a few class days, what should we do with them?I suggest that a good use would be to attempt to fosterbetter problem solving capability among our students.

The problems might be mathematical ones. Virtu-ally every meeting of a calculus class offers the opportu-nity to expand on the day's assignment, to go a little offthe prescribed path, and to explore interesting relatedmaterial. If there is a little extra time in a course, thiswould be a good way to spend it. It might even helpto attract to our discipline some of the many very goodstudents who now see no attractive future in the seriousstudy of mathematics.

Another possibility is to do more in the area of appli-cations and mathematical modeling. I am not particu-larly enthusiastic about "realistic" applications. Theymust often be couched in terminology unfamiliar tomany students and require too much time to describethe underlying problem.

It is better to use simple problems and models (evenif "unrealistic") so that everyone can understand them.

CHROBAK: THE ROLE OF THE CALCULATOR INDUSTRY 43

Even simple problems can be embellished and modifiedso as to illustrate the ideas and principles of mathemat-ical modeling.. Given students' paucity of experiencein mathematical modeling, even the simplest problems(other than the standard template problems) will provechallenging enough for almost all students.

Finally, what is needed as much as anything in teach-ing calculus is the proper attitude: a recognition thatthe computer is here to stay; that it offers insight intothe phenomena of change; and that it can provide aspringboard for the discussion of interesting mathemat-

ical questions, including many of those that are nowpact of standard calculus courses.

WILLIAM E. BOYCE is Professor of MathematicalSciences at Rensselaer Polytechnic Institute. He is for-merly Managing Editor of SIAM Review and Vice-Presidentfor Publications of SIAM. He is author (with Richard C.Di Prima) of Elementary Differential Equations and Bound-ary Value Problems.

The Role of the Calculator IndustryMichael Chrobak

TEXAS INSTRUMENTS

To make calculus instruction more applicable toreal-world problems, educators must focus more onfundamental ideas, off-loading the mechanics of ex-ecuting formulas and equations to modern comput-ing aids. Embracing such toolsspecifically, hand-held calculatorswill bring about instructional changesneeded to better prepare students for tasks in advanctsieducation and the workplace.

Several issues must be addressed by edu, tors andcalculator manufacturers if this tool is to be incorpo-rated effectively into calculus and lower-level mathe-matics programs.

First, calculator manufacturers must tailor theirproducts, based on guidance from educators, to meetmaintained education requirements. Close communica-tion with the educational community must be, with newdesigns being based on the special requirements of theclassroom. Sometimes this can lie as simple as unclut-tering a keyboard, thereby preserving and accentuatingrequired functions.

The design of the TI-30 SLR+ calculator is one ex-ample where this collaboration between educators andindustry has been effective. Specifically tailored to re-quirements at the high school level, this calculator wasconstructed with a hard shell case for increased dura-bility. Solar power was chosen to eliminate the need forbatteries. Large, brightly-colored keyc were incorpo-rated to identify matlicmatks function groups, to pro-vide for ease-of-use, and to stimulate learning.

Second, proper use of calculators in the classroom re-quires textbooks developed with the calculator in mind.

Indeed, some states already are making this a require-ment. For instance, in California, publishers must nowdetail the use of calculators in their text, not as a sup-plement, but as part of each lesson. To achieve this,manufacturers, educators, and publishers must work to-gether, incorporating effective use of calculators into in-structional materials.

An example of planned technology emerging in thetextbook industry can be found in a classroom calcu-lator kit developed by TI and Addison-Wesley. Thetwo companies have worked together to produce instruc-tional material that incorporates the use of a hand-heldarithmetic calculator for an elementary mathematicscurriculum.

Third, the use of calculators in testing is another keyissue. If students are taught with calculators, it followsthat they should be tested with them as well. Today,testing materials incorporating the use of calculators dvnot exist. Manufacturers can assist in the creation ofnew testing materials, providing development supportas well as inputs based on the design and operation oftheir products.

Currently, the Mathematical Association of Amer-ica (MAA) has begun to reform mathematics educationwith a calculator-based placement test program. Thiseffort is under the direction of the MAA's Committeeon Placement Exams (COPE), managed by John Har-vey of the University of Wisconsin at Madison, with TIproviding funding since 1986.

Testing with calculators at the high school level willhelp to ensure that students have proficiency of calcu-

44 RESPONSES

lator use by the time they reach college. Today, this isusually not the case.

While the calculator itself should never be the mainfocus in mathematics instruction, it should be viewedas an important tool to assist in the educational pro-cess. Calculator manufacturers should be included infuture mathematics reform, ensuring that calculatorsfor the classroom are designed appropriately, and thatthey integrate well into textbook formats and testing

Calculus: Changes for the BetterRonald M. Davis

NORTHERN VIRGINIA COMMUNITY COLLEGE

Calculus is the mathematical study of change. Yet,for the large part, calculus has changed little in the pastthirty years. It has remained static in a continuouslychanging environment.

I strongly agree that our present calculus content andthe way in which we teach it needs revision. Calculusneeds to provide students with the abilities of reasoning,logic, and judgment. Calculus courses need to empha-size clear thinking and not merely symbol manipulation.There is an underlying fundamental importance of cal-culus for the sciences and technologies. We as teachersmust comprehend this and must share this knowledgewith our students.

As teachers of calculus we cannot ignore the vastarray of available technologies that can enhance our ef-forts. We must incorporate calculators and computersas aids for our teaching and as tools for our students.These tools will not only simplify calculation and sym-bol manipulation, but will also require students andteachers to heighten their understanding of and insightinto the concepts of calculus.

I am convinced that change will only be accomplishedthrough a concerted and coordinated effort from indus-try, business, government, and education. For two-yearcollege faculty, the effort will be exceptionally taxing.Teaching loads of fifteen hours or more often provide lit-tle time for curriculum development. Minimal comput-ing support at two-year colleges restricts student andteacher access to computers.

Sine" the transfer of courses requires two-year col-lege calculus courses to be accountable to four-year col-leges, two-year college faculty will move slowly in mak-

materials.

. -

MICHAEL CHROBAK is Scientific Calculator Managerof Texas Instruments, Dallas, Texas. Previously, he servedTexas Instruments as a Project Engineer and a ProgramManager for financial and scientific calculators. He holds aB.S.E. from the New Jersey Institute of Technology, and anM.B.A. from Texas Technological University.

ing changes. Development-of-a dynamic calculus coursewill require a joint effort on the part of the mathematicsfaculties at two-year and four-year colleges.

A sense of isolation exists for the calculus teacher atmany two-year colleges as their departments often havelittle or no travel funds. They have limited opportuni-ties to meet with other calculus teachers from two-yearand four-year colleges. Since calculus constitutes only10% of the course load at two-year colleges, it cannotoften be given the added attention that a revision wouldrequire.

I strongly believe that a mathematics course thatinvolves a dynamic approach to calculus and portionsof discrete mathematics needs to be developed and im-plemented in place of the present-day calculus. Theteaching of such a course will Yeqx- 're much preparationand retraining. I am, therefore, concerned that two-year colleges may be unable to commit the resources toprepare their faculty adequately to teach this dynamiccourse.

As a teacher I become invigorated with each oppor-tunity to rethink course content and instructional ap-proach. I am excited at the prospect of a new, dynamiccalculus course.

.

RONALD M. DAVIS is Professor of Mathematics atNorthern Community College, Alexandria Campus. He hasserved as Second Vice President of the Mathematical As-sociation of America, and as Chair of the AMATYC-MAASubcommittee on Curriculum at Two-Year Colleges. He re-ceived a Ph.D. in mathematics education from the Univer-sity of Maryland.

DODGE: HIGH SCHOOL MATHEMATICS 45

Imperatives for High School MathematicsWalter R. Dodge

NEW TRIER HIGH SCHOOL

The high schools' perspective of calculus is two-pronged. For some of our students calculus will Le acourse they take in high school. This course will con-sist of our ablest students taught in small classes byour most capable teachers. For others, calculus willbe taken following high school. For these studentsthe high school's objective is to prepare them in thebest possible manner with the mathematical conceptsand skills required for success in a college calculus pro-gram.

The universities' view of calculus is quite differentfrom that of the secondary school. For them very of-ten calculus is the lowest course offered, populated bystudents of varying abilities, and with teachers who areoften not the ablest. In addition, quite often the classsize is very large. Although this is a problem on a na-tional scale, I do not believe it can be solved on a na-tional basis. It must be solved within each local insti-tution.

If we are truly to bring about calculus reform wemust not only change the curriculum, but must alsochange drastically our delivery systems. If calculus isto become more conceptual and more intuitive, edu-cational structures must allow this to occur: student-faculty interaction, group discussions, and laboratoryexperiences must be given top priority. A change inthe content of the course without parallel change in themethod and quality of instruction is doomed to fail-ure.

One has to ask if there really is need for change incalculus? One very clear outcome of the Conference isthat there is a compelling need to change the contentas well as the conceptual nature of the course. Thegraphical, algebraic, and numerical capability of calcu-lators and computers has definitely made many of theskills of a traditional calculus course very suspect, if notarchaic.

Numerical methods of solution, implemented via thecalculator or computer, open a whole new arena of cal-culus to the beginning calculus student. The graphi-cal capabilities of both calc..lators and computers canenhance a student's understanding of the fundamentalideas of calculus. Therefore change should occur bothfor the improvement of the curriculum and the under-standing of the student.

Exactly what parts of the traditional course can beeliminated is not clear. Just how much skill work isnecessary and how much can be eliminated is a cru-cial questionoften mentioned at the Conferencethatmust be answered.

The entire cur.iculum of the high school will beaffected by these changes. If students are to besuccessful in college, the school curriculum will alsohave to become more conceptual and will need toincorporate technology. Skills will have to be de-veloped in the use of both the calculator and com-puter. Approximation, estimation, and reasonable-ness of answers must be emphasized. Mathematicswill become more "decimalized." Algorithmic rea-soning will also become a high priority educationalneed.

The Conference has certainly stimulated thinkingand planted the seeds for a change in the calculus.If this change is to cone about, cooperative effortsbetween secondary school teachers and college facultymust be organized. Changes at one level have effectson the other, and the success of one will depend uponchange in the other.

WALTER R. DODGE is a mathematics teacher at NewTrier High School in Illinois. He has been a member ofthe AP Calculus Exam Committee, an AP Calculus Reader,Table Leader and Exam Leader. He is currently President-Elect of the Metropolitan Mathematics Club of Chicago.

46 RESPONSES

An Effective Class Size For CalculusJohn D. Fulton

CLEMSON UNIVERSITY

To create a Calculus for a New Century, more hasto change in the calculus of today than course content.Calculus is not suited for mass education, yet calculusin many if not most colleges and universities. is for theacademic treasury the most productive "cash cow" inthe university.

To be sure, English writing courses have more stu-dent enrollments than does calculus. But English writ-ing classes are taught in sections of sizes twenty totwenty-five students. While writing courses generatemore revenue for the university, they have been suc-cessfully portrayed to unive:sity administrators as be-ing more labor intensive with respect to commitment ofteaching staff. As a result, they cost more per sectionto teach.

In the September 1987 issue of College English, theNational Council of Teachers of English published itsrecently adopted standards "No more than 20 studentsshould be permitted in any writing class. Ideally, classesshould be limited to 15."

The justification given by this Council for the classsize is that

...the work load of English faroity [should} be rea-sonable enough to guarantee that ever) student re-ceive the time and attention for genuine improvement.Faculty members must be given adequate time to ful-fill their responsibility to their students, their depart-ments, their institutions, their profession, the largercommunity, and to themselves. Without that time,they cannot teach effectively. 'Unless English teachersare giv n reasonable loads, students cannot make theprogress the public demands.

Interchange "Mathematics" with "English" and a rea-sonable case for calculus is made as well.

Calculus for a New Century will be labor intensivewith respect to teaching staff. In the early stages ofdevelopment of a new calculus, considerable experimen-tation will be necessary. Guidelines for a new calculuslikely wil' àllow from such a body as CUPM. Text ma-terial incorporating the guidelines can be expected tofollow. In the development period, experimental classestaught by faculty would seem of necessity to be smallclasses, as would the control classes with which theywould be compared. Faculty teaching the classes wouldrequire released time from other duties to lead this de-velopment.

If calculus ultimately incorporates many of the ideasand recommendations discussed at the Colloquium,then it will be essential that the new calculus be con-siderably more labor intensive than the old. Also, itseems clear that a new calculus must rely more heavilyupon experienced and committed faculty and less uponpart-time faculty and graduate teaching assistants.

A new calculus must be taught with more attentionto concepts than was the old. The assignin3 and grad-ing of nonstandard problems as homework, more class-room discussion, board work, and class presentationsby students, more word problems, and more questionsrequiring essay responses address the effective teachingof concepts, but do not lend themselves to a calculusclass with thirty-five or more students per instructor.

If the new calculus uses technology effectivelyas itmustexpect it to be more labor intensive, not less. Inaddition to the concepts of change, limit, and summa-tion of the old calculus, expect hand-held, micro- andmainframe computers to be used creatively in the newcalculus to instill concepts such as approximation, esti-mation, error analysis, asymptotic behavior, and good-ness of fit.

More frequent and longer discourse with students willbe required to instill these numerical concepts. It hasbeen alternately predicted over the years that radio,television, and now computers, videotapes, and elec-tronic blackboards, would make classroom teaching ob-solete. All of those predictions have been wrong. Like-

ise, we expect that the principal value of any cleverlydevised expert system for the teaching of calculus willonly be to enhance the effects of good classroom teach-ing.

The practitioners in our client disciplinesthe en-gineers, the biological and physical scientists, the so-cial scientists, the business and economic scientistsexpect a lean and lively calculus to contain examplesand problems for students which reflect applications intheir disciplines. Faculty will have to lead in determin-ing these applications and translating them into theteaching mode for the new calculus. This effort willnot only be labor intensive, but quite likely it cannotbP done by graduate teaching assistants. We expectapplied problems to be nonroutine, troublesome to stu-dents of calculus, and requiring considerable discourse

FULTON: EFFECTIVE CLASS SIZE 47

with knowledgeable teaching staff.One goal of a Calculus for a New Century must be to

assist more students, especially students from minoritygroups, through the filter which calculus represents andinto the life stream of science, mathematics, and tech-nology. The Professional Development Program for cal-culus students at the University of California, Berkeley(described in the background paper "Success for All" byShirley Malcolm and Uri Treisman) seems to have hadspectacular success for minority students with mathe-matics SAT scores in the 200 to 460 range. Its smallgroup workshop methods seem transferable to all stu-dents. Its fifteen- to twenty-student recitation sectionsmust, however, be regarded as labor-intensive. More-over, it would seem that any successful program for sig-nificantly increasing the student success rate with cal-culus will be just as labor-intensive.

Calculus is with us now. A Calculus for a New Cen-tury will evolve from our present calculus, yet at toomany colleges and universities an insufficient teachingstaff is currently assigned to calculus. Its "cash cow"status with many university administrators is shameful.

Through student tuitions and in some cases throughformula funding, calculus enrollments generate consid-erable funds for college and university coffers. With ex-penditures only for the teaching of large course sections,all too frequently taught by part-time faculty, with helpfrom graduate teaching assistants, and with supplieslimited to a little chalk, perhaps an overhead projector,and some transparencies, the net profit from calculusteaching can be considerable. Moreover, at many uni-versities, graduate teaching assistants tend to pay theirown salaries by paying their own tuitions and throughformula funding generated by their own course registra-tions.

No calculus, old or new, can be effectively taughtwithout sufficient teaching staff to allow regular and ex-tensive feedback from students. Homework and quizzesmust be graded line by line. Tests to effectively mea-sure students' grasp of concepts in calculus should notbe "multiple choice" tests. (Ever. Advanced Placementcalculus exams have their essay poriions.)

Like no other course at the freshmen-sophomorelevel, calculus concepts build with each successive class.A student who gets behind in the first few classes al-most certainly will be lost for the entire course. Early

and regular student feedback is thus essential for thecourse. Assessment of each student's grasp of conceptsmust begin early in the course and continue frequentlythereafter. Conferences with students, often extensive,should be held regularly.

There is a need for teachers of calculus to instill intheir students the ability to communicate scientifically,or mathematically. To at least some degree, then, cal-culus teachers are teaching writing. The circling of an-swers indicated as (a), (b), (c), or (d) on a multiplechoice quiz will not stop a student from writing suchmathematical garbage as 3 x 4 = 12 1. = 11. + 4 =15/3 = 5, which just as frequently occurs on studentpapers as in calculator instruction books.

Only close scrutiny of student homework and quizzes,essay questions on quizzes, classroom discussion of con-cepts, and student classroom or office presentations canassist in meeting an effective mathematical communica-tion objective for the teaching of calculus. The very na-ture of the calculus, new or old, suggests that it shouldbe labor intensive.

If mathematics faculty have failed to communicatethe need for effective class size for calculus to universityadministrators, it may well be a major cause of the in-adequacy of the old calculus. We should not necessarilycall for small sections, but for a sufficient teaching staffto be assigned to calculus to allow for regular feedbackfor students.

If we have allowed class size in calculus to increasebeyond that which is pedagogically sound, then thisColloquium has given us a new opportun'..y, a new be-ginning. We must communicate the new enthusiasm fora new calculusperhaps a lean and lively calculuswc of support by mathematics faculty, by facultyfrom client disciplines, and by academic administrators.Since the new calculus will evolve from present calculus,however, effecti a class size for calculus classes shouldbe communicated as an imperative for the present.

JOHN D. FULTON is Professor of Mathematics andHead of the Mathematical Sciences Department at Clem-son University. He is a member of JPBM Committee forDepartment Chairs, and Chair of the MAA ad hoc Commit-tee on Accreditation. He received his Ph.D. in mathematicsfrom North Carolina State University.

48 RESPONSES

Don't!Richard W. Hamming

NAVAL POSTGRADUATE SCHOOL

I have attended a numbetr of similar conferences, de-voted a lot of time to worrying about the calculus as itis currently taught, have thought long and hard aboutthe topic, and decided that action is worth far morethan endless talk. As a consequence, I have written abook that I thought would meet my standards of whatthe calculus course should be for the future. There aremany other possible books besides the one I wrote, butat least it is something to go on rather than endlesstalk.

Listening to all the talk has caused me to compile alist of "don'ts:"1. Don't try to optimize the calculus course. We are

trying to provide a mathematical education, and theoptimization of individual courses leaves too much 8.to fall between them. We should view calculus aspart of a student's total mathematical education,as a means to an end and not as an end in it-self.

2. Don't look to the past. Our society has passed re-cently from a manufacturing society to a service so-ciety, and our teaching should be directed towardsour students' future needs, not to our past activi- 9.ties.

3. Don't try writing a textbook by a committee. Alltoo many of our texts are written by uninspired sec- 10.

and and thi,2 rate minds, and it is foolish to expectstudents to respond well to them. You need, morethan anything else, an inspiring book. The particularcontents are of less importance.

4. Don't think that you can move by small steps fromwhere you are to where you want to be. We areat present at a local optimum, as can be seen bythe fact that we have essentially only one text. Thevarious books that are widely used so resemble eachother that even the proofs are the same. Anyonewho knows the least about the calculus theory of op-timization must realize that any small step from the 12.

local optimum will be a degradation. To get to abetter relative optimum you must move a large dis-tance.

5. Don't think that discrete and continuous mathe-matics are separate topics. Any competent math- 13.

ematician is well aware of the fact that the Rio-mann zeta function and prime numbers are closelyrelated.

6. Don't think that calculus is only the development oftangent and area problems. It is also used widelyto get new identities from old ones and to han-dle generating functions that arise just below thesurface of combinatorial problems. The conver-gence of these generating functions is of no impor-tance.

7. Don't neglect complex numbers as we now do; theyare basic for the unification of various parts ofmathematics as well as being essential in many ar-eas.

Don't think that the old mechanical problems are ofinterest to the current crop of studentsthey are not!Both probability and statistics are of much more in-terest, are more useful, and can provide much moreinteresting problems. Furthermore, those engineerswho claim that mechanics is more important to themthan are probability and statistics are simply livingin the past.Don't think that money can buy the changes youwant: you must use persuasion. The "New Math" isa perfect example of what not to do.Don't present mathematics as a fixed, known thingfor all eternity. Present the changing definitionsthat we actually have, the various attitudes to-wards mathematics, and the philosophy of mathe-matics. We Ire educating students, not just trainingthem.

Don't emphasize the cbing of algorithms. A friendof mine in Computer Science recently said to me:"Why think when you can program?" Adjusting thisto the teaching of calculus: "Why teach the studentsto think when you can train them to follow algo-rithms?"

Don't start abstractly and move to the definite andconcrete; rather start with the definite and exhibitthe process of abstraction, extension, and general-ization. They are the heart of mathematics. Youcannot get motivation otherwise.Don't teach huge classes of 500 students. Education

11.

6

HODGSON: EVOLUTION IN THE TEACHING OF CALCULUS 49

appears to require more personal contact. I hear thatthe large classes are for reasons of economy. I sus-pect they are also to let 19 professors escape teachingcalculus. No wonder so few students major in math-ematics!

14. Don't continue to act as if mathematics was the ex-clusive consumer of the calculus courses. If you wantreality to respond, you must pay far, far more atten-tion to the vast sea of user's needs.

RICHARD W. HAMMING is Adjunct Professor of Math-ematics at the Naval Postgraduate School in Monterey, Cal-ifornia. He worked at Los Alamos during the war, and from1946 to 1976 at Bell Telephone Laboratories. He is active inthe Association of Computing Machinery, IEEE, a'id variousstatistical societies. He received his Ph.D. in mathematicsfrom the University of Illinois.

Evolution in the Teaching of CalculusBernard R. Hodgson

UNIVBRSITE LAVAL

Plutiit la tete bien faite gue Hen pleineMontaigne, Essais

One of the most remarkable features of this Collo-quium is the huge number of attendees (above 600),much more, we are told, than the organizers originallyexpected. While such a level of participation may beinterpreted as reflecting the extreme intensity of an ac-tual "crisis in calculus," it can more simply be seen asan indication of a greater awareness among the math-ematics community of the inn ssibility of keeping cal-culus teaching essentially unchanged, as it has been forso many decades.

The fact that computers, micro-computers, and evenhand-held calculators are compelling changes in theway all mathematics, and especially calculus, is beingtaught, has been advocated over the years by variouspeople. But time now finally seems to be ripe for realcollective action to be initiated.

It might be instructive to recall briefly a few publica-tions and meetings that have taken place since 1980 thathave contributed to efforts to modify the way computerscience is being perceived in influencing both mathe-matics and its teaching. Although by no means exhaiis-tive, the following list nevertheless reveals the actualsituation as an evolution which started slowly among afew enthusiasts and has progressed steadily so that nowit concerns a great many people in various countries.

1980: Publication of the book Mindstorms: Children,Computers and Powerful Ideas, by Seymour Pa-pert. The computer is presented as an "object-to-think-with."

1981: In a paper published in the American Mathemati-cal Monthly (88 (1981) 472-485), Anthony Ralstonargues for the consideration of a separate math-ematics curriculum for computer science under-graduates, beginning with a discrete mathematicscourse rather than calculus.

The paper "Computer Algebra" (by Pavelle, et al,Scientific American, Dec. 1981) makes symbolicmanipulation systems known to the general (sci-entific) public.

1982: A distant early-warning signal by Herbert Wilf:"The disk will a college education" (Amer. Math.Monthly 89 (1982) 4-8).

1983: Proceedings of a Sloan Foundation conference cen-tered around Ralston's thesis on the balance be-tween calculus and discrete mathematics: The Fu-ture of College Mathematics (Springer-Verlag).

1984: Some sessions of ICME-5 (Adelaide) devoted tothe teaching of calculus and the effects of sym-bolic manipulation systems on the mathematicscurriculum.

NCTM 1984 Yearbook on Computers and Educa-tion.

1985: NCTM 1985 Yearbook (The Secondary SchoolMathematics Curriculum) includes papers relatedto the issues raised by symbolic computation.

A symposium is organized by ICMI (The Interna-tional Commission on Mathematical Instruction)in Strasbourg on the topic: The Influence of Com-puters and Informatics on Mathematics and its

50 RESPONSES

Teaching (Proceedings published by CambridgeUniversity Press, 1986).

1986: The Tu lane Conference: Toward a Lean and LivelyCalculus.

A symposium tales place in Tunisia on the in-fluence of computer science on the teaching ofmathematics as it relates especially to developingcountries. (Proceedings published by the Inter-national Council for Mathematics in DevelopingCountries.)

1987: The International Federation for Information Pro-cessing organizes a working conference in Bulgariaon the subject: "Informatics and the Teaching ofMathematics."

This Colloquium: Calculus for a New Century.

1988: The programs of both the AMS-MAA annualmeeting and ICME-6 meeting (Budapest) con-tain many activities related to computers and theteaching of mathematics (and especially of calcu-lus).

A recent event has definitely served as J.,. catalyst inaccelerating the evolution suggested by the precedingmilestones, namely the commercial availability of hand-held calculators with graphic or symbolic capabilities(like the Hewlett-''ackard 28C, the Casio fx7000G, orthe Sharp EL 5200). In spite of several inherent lim-itations and their often awkward usage, these deviceshave already had a considerable effect among teachersof mathematics: if many of these teachers still believedthey could safely ignore Wilf'.s early-warning signal of1982, a vast majority now realize that these calculatorsare here to stay, that they can only become cheaper andmore powerful over the years, and that students will beexpecting to use them.

In addition, symbolic manipulation systems, reservedto a small group of specialists only a decade ago, havenow become sufficiently widespread that both mathe-maticians and users of mathematics are aware of theirusefulness. These systems have created a situationwhere a mere technician, with little mathematical back-ground, could work on problems which, because of theirmathematical sophistication, would escape the exper-tise of today's typical engineer. What sense will thistechnician make of the "answer" (numeric, symbolic, orgraphic) produced by the computer? How can such atechnician appreciate the validity and limitations of themathematical model being used?

This raises forcefully the difficult question of mini-mal competency in mathemzit.:s for engineers and other

users of mathematics. One is reminded here of a sim-ilar situation actually occurring in statistics, where al-most anyone can use commercially-available packageslike SAS or SPSS to painlessly have the computer printpages and pages of output in all forms (tables, piecharts, bar graphs, etc.), often without having the inter-pretative abilities to thoughtfully use this information.(And here enters the consultant statistician!)

But things have evolved greatly ovel the last tenyears in the teaching of statistics. Instead of concen-trating on tricks for the calculation of various statis-tical parameters, teachers of statistics now stress thedevelopment of a sense of appreciation and judgment.Exploratory data analysis is quite typical of the shift inapproach, where the calculator or the computer plays acentral role in working with the data.

Althoug'. ',he inertia of the system is far greater inthe case of calculus, a similar change will occur. It iscompelled on us by the wide availability of calculatorsand numeric, symbolic, or graphical software which ourcustomer departments are quite eager to use. Even ifno clear relationship has yet been identified between,say, procedural skills developed by lengthy hand manip-ulations of algebraic expressions and the understandingof the underlying algebraic concepts, it seems beyonddoubt that a shift will take place from purely computa-tional to more complex interpretative abilitiesin otherwords from calculation to meaning. More than ever, theadage of the bonhomme Montaigne prevails, and devel-opment of mathematics judgtnent greatly contributes tothis "tete bien faite."

A lot of people in different places are now gettingtheir feet wet in trying new approaches to calculusteaching. Attendance to this Colloquium indicates thatthis corresponds to a real need. We are now in a phasewhere experiments need to be performed, evaluated,and communicated to others. Identification of new cur-ricula and production of related materials is a a:lcultand unrewarding task. But only such efforts can pro-duce, as was wished by Robert M. White in his keynoteaddress, a calculus that is no longer a filter but a pumpin the scientific pipeline.

BERNARD R. HODGSON is Professor of Mathematics atUniversite Laval (Quebec). Besides his research interests inmathematical logic and theoretical computer science, he isan active member of the Canadian Mathematics EducationStudy Group and regularly teaches courses for elementaryschool teachers. He received his Ph.D. degree from the Uni-versite de Montreal.

KNIGHT: DREAMING OR PLANNING? 51

Collective Dreaming or Collaborative Planning?Genevieve M. Knight

COPPIN STATE COLLEGE

The halls of the National Academy of Sciences echoedthe voices of 600 plus persons who gathered to reflecton the current status and future direction of Calculus!Messages were loud and clear that today's profile of cal-culus compares poorly with the world of reality. If onemoves away from the innovational uses of disks, sym-bolic algebra, and calculators not much is going on. Thefuture complexion of calculus must mirror with zest themagnitude in which science and technology are chang-ing.

The presentors spoke of urgency:For the calculus reform to consider the effort a na-tional collaborative one (however, the effort must begenerated at local levels and coordinated throughnetworking).For collegiate mathematics to be less isolated and tolink to other subjects, to professional users externalto the classroom, and to secondary school mathemat-ics.For an in-depth analysis of the practice of teachingcalculus that focuses on purpose, content, method-olocv, and assessmentbeyond two-hour tests and afinal examination.For the keepers of calculus to "do something" aboutwhat is and and what should be.For the teaching arena to capture the beauty of thesubject with meaningful instructional activities cen-tered around students' understanding and needs.After several speakers, I began to become concerned

that too much time v. .:being allocated to talking about"issues." Flooding my mind were the reflections gath-ered from reading the background papers, from personalinteractions with colleagues, and from my own twentyplus years of teaching calculus. As a practitioner I wasready to engage in an intellectual mental brainstormembracing reality with the host of talent in the meetingroom. At this junction the voices representing NSFchimed softly, "We are listening to the mathematicssciences community and here is our proposal. Reachbeyond the "talking stage" and initiate the process todevelop this new calculus curriculum."

Lunch time was alive with groups of voices buzzingabout the points of view presented during the morningsession. As I circulated among the crowd listening tobits and pieces of conversations, I began to sense that

the initial steps had commenced and the mathematicssciences community was generating the seeds for curric-ula reforms in calculus.

Speakers at the afternoon session artfully integratedthe morning discussions to reflect their positions as"clients" and "administrators." The images of calcu-lus we project to users and others are pictures thatare fuzzy and out-of-focus. The Colloquium challengedmathematics professors to stop, to reflect, to regroupand to enter into the world of technology and the 21stcentury.

The picture of calculus now takes on many eiffer-ent forms for the individuals who attended this confer-ence. I leave with a commitment to share my notes withcolleagues back home. In addition, I'll shepherd somefine-tune analysis by all departments whose studentsare required to take the calculus sequences. At least 30different topics, issues, and concerns were voiced dur-ing the conference. I urge readers to generate from thefollowing list questions to be answered by their mathe-matical science faculty:

Purpose and sincerity of calculus reformManagement of resource and materialsCalculus textbooksFacultyPlacement and assessmentDiscrete mathematicsMethodologyComposition of calculus topicsUse of technologyDemographic dataDrop in supply of human resourcesThe collective dream can become a reality. What-

ever we are now calling calculus will not survive in thenew century. Mathematicians must assume the respon-sibility for what will replace it.

GENEVIEVE M. KNIGHT is Professor of Mathematicsand Director of Mathematics Staff Development Programs,K-8, at Coppin State College. She received a Ph.D. de-gree from the University of Maryland at College Park. Herresearch interests are in non-traditional approaches to theteaching and learning of mathematics in an urban schooldistrict.

52 RESPONSES

CalculusA Call to ArmsTimothy O'Meara

'UNIVERSITY OF NOTRE DAME

This colloquium will either be a watershed in math-ematics education or a flop. Which it is depends en-tirely on the kind of follow-up that will be taken ingenerating interest in the mathematical community atlarge.

My initial reaction during the first few hours ofthe Colloquium was one of bewilderment and dis-appointment. Based on the announced title of theevent, I subconsciously expected to be presented witha brand new calculus on a platter. In actual fact, Iheard a lecitation of the problems facing us in mathe-matics and science education, apocalyptic predictionsfor the year 2000, philosophical messages from oursponsors in science and engineering, and a call formobilizing all sorts of forces in our society. Self-analysis and self-criticism were conspicuous by their ab-sence.

My moment of truth came at the cocktail hourat the end of the first day. First, I realized thatour real concern was with mathematics education atthe college level, not with calculus. The word cal-culus was just a snappy way of putting it all to-gether. More importantly, I realized that Calculus fora New Century is far from being a final productitis still a movement. We are in process. The nextstep in the process must be an awakening, followedby an involvement of the entire mathematics commu-nity.

We are all aware of the crises in mathematics andscience education in our country today: a severe short-age of mathematicians; few decent teachers in ourschools; alarming drop-out and failure rates in manyof our colleges; problems with the pipeline; depen-dency on foreign talent; a disgraceful inability to nur-ture mathematicians on our own soil and in our ownculture. In fact, we are so accustomed to this recita-tion that we have been lulled into accepting it as im-mutable.

This was the starting point of the Colloquium. Itwill have to be repeated and repeated as the processcontinues. Those with a mercenary turn of mind willbe interested to know that calculus accounts for half abillion dollars of business each year.

On the apocalyptic side, we were reminded thatpush-button calculus is just around the corner. So whyteach students how to find one volume of revolutionafter the other. The calculator will surely tell us all.Will calculus become strictly utilitarian? Need its in-ner workings be known to any but the elite? If calculusis just a skill, should it be taught in our universities? Iwould hope not. Will discrete mathematics replace thecontinuous?

Mathematics will continue to flourish; but where willits creativity come from? From our economists, our bi-ologists, our engineers? Will our mathematicians besufficiently flexible and imaginative? I hope so, but Iam not convinced.

Interestingly enough, little was said of the intrinsicbeauty of mathematics. Epsilons and deltas were men-tioned in whispers. Mathematics as part of liberal edu-cation for the new century was not mentioned at all. I

find that alarming.Mathematics education at the college level will have

to be stripped of rote. Concepts will have to beemphasized. Research mathematicians will have toview teaching as an honorable part of their lives.And this will have to be transmitted by example totheir doctoral students who will become the next gen-eration of professors. All of us will have to con-vey our enthusiasm for our subject, not only to eachother as we now do, but also to our deans who con-trol the purse strings, and most of all to our stu-dents.

The next step in the process is up to us. If the sur-prisingly large turnout at the Colloquium is any indi-cation, then I think we are on our way.

TIMOTHY O'MEARA is Kenna Professor of Mathemat-ics and Provost of the University of Notre Dame. In histenth year as Provost, he maintains his research interests inalgebra and number theory. He has published three books inthese areas; a fourth, The Classical Groups and K-Theory,coauthored with Alex Hahn, will be published by Springer-Verlag in 1988. He received his Ph.D. from Princeton Uni-versity.

PRIESTLEY: SURPRISES 53

SurprisesWilliam M. Priestley

UNIVERSITY OF THE SOUTH

On a "typical" final examination in a calculus cours^last year, a student who was given permission to pushbuttons intelligently on the latest hand-held calculatorcould easily have ensured himself of a passing grade.Yet the failure rate among college calculus courses issurprisingly high, and surprisingly few euch courses in-corporate any use of a computer. At the same time, astudy of enrollment trends leads to a prediction that thenext century will find more students studying calculusin secondary schools than in colleges and universities.

These unwelcome surprises from the conference "Cal-culus for a New Century" call for concerted effort toavoid a fin de siecle crisis in the teaching of calculus.For some of us who teach wrong mathematics in liberalarts colleges, however, a pleasant surprise came out ofthe conference as well. This was the response of a smallbut significant number of participants who endorsed asa partial remedy our most cherished forlorn hope andlost cause: the insistence upon clear writing in Englishby students of calculus.

These bewildered students have been so ill condi-tioned by re drill problems whose answers areso transparent as to require no explanation that theythink the proper response to every mathematics prob-lem is to dispose of it as quickly as possible withoutone word of explanation to the intended reader of theirwork. The reader, of course, is their dedicated instruc-tor (whose role will later in life be played by their exas-perated boss) whom they confidently expect to rewardthem highly if the correct answer can be found scribbledanywhere at all in their chaotic ramblings.

The most overlooked shortcoming in the teaching ofmathematics is the failure of teachers to insist that theirctudents justify their answersif not with complete sen-tences, at least with a few suggestive English phrases.In the case of an optimization problem in calculus, forexample, it is surprising how much good is done by ateacher who demands that students

Never put an "equals" sign between unequal expres-sions, andPepper their computation with the proper use of asmall giossary of words like let, denote, if, then, so,because, attain, and when.

Students understand the theory behind the techniqueof optimization if and only if they can carry out thesedemands.

But the list of worthwhile changes to bring to the cal-culus is never-ending. A dean and professor of English,who was surely sympathetic with the idea of writing,admonished conference participants not to try to teachwriting. She had already heard about too many otherthings to do. Surely almost everyone agreed with her.

The major problem my students have at the outset ()fa calculus course, however, is that they don't (or can't)learn from reading the textbook. No one needs to doa statistical survey to know that this is true generally,and teachers who give tests that calculators can passprobably don't expect their students to be able to readmathematics.

How can this problem be solved? Recall how New-ton solved a famous problem and discovered the calcu-lus: he calculated the area beneath a curve between 0and 1 by attacking instead the problem of finding thearea between 0 and x. The technique of solving a hardproblem by attacking in its place a still harder problemthat ought to have been impossible is one of the mostsurprising methods learned in mathematics.

The solution to the hard problem of getting studentsto read the textbook is to attack instead the impossibleproblem of teaching them how to write. What wouldhappen if all of us who instruct should insist to ourstudents that we really expect them to learn to write?What would happen if we told our classes that a stu-dent cannot learn to think like a mathematician withoutlearning to write like a mathematician? Some studentsmight actually learn, grow up and study analysis, andattain the background needed to teach calculus to allthose high school students of the next century.

As for the rest, they may never learn to write. Butif they try to learn how to write, they will have thebiggest surprise of their mathematical lives. They willlearn how to read.

WILLIAM M. PRIESTLEY is Professor of Mathemat-ics at the University of the South and author of a calculustextbook written for liberal arts students. He has puHishedpapers in analysis) participated in an NEH-sponsored sum-mer seminar on Frege and the foundations of mathematics,and is interested in using the history of mathematics to pro-mote the better teaching of mathematics. He received hisPh.D. degree in mathematics from Princeton University.

RESPONSES

Calculus for a PurposeGilbert Strang

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

The following comments are not so much about whatwas said in Washington, as about questions that weremomentarily raised, and then dropped. Some of theunspoken or barely spoken needs are fundamental tosuccess in the classroomand the success of this wholeinitiative.

One is the student's need for a clear sense of pur-pose, and for a response that encourages more effort.Actually the instructor has the same need. Our ques-tions to the panel representing the "client" disciplinesmight have been combined into a single question: Whatis calculus for? If the course is to be recognized asvaluable and purposeful by students, that has to be an-swered. The afternoon audience was asking the rightquestion!

It may seem astonishing that we who teach the sub-ject don't know everything about that question. Ofcourse it is not at all astonishing. First, there are toomany answers. No one can be familiar with all theuses of calculus. But that is more than the studentsare asking, and more than they could be told. Stu-dents need to have some clear purpose. There is asecond reason why we don't answer wellwe are verymuch inside the subject, teaching it but not seeing

Calculus is a language, but what do we have to say inthat language? There are many texts that explain the"grammar" of calculusthe rules of the 7anguagebutthat is not the same as learning to speak it. We needideas to express, things to say and do, or it is a deadlanguage.

I was struck that in Tom Tucker's list of groupsthat have to contribute to a reform, authors werenit mentioned. It was an oversight, but I thinkthey are at least as important as publishers. I amconvinced that the textbook itself is absolutely cru-cial in explaining the purpose as well as the rules.That is the job of the book, and I think it can bedone.

I was fascinated by the observation that calculuscourses reflect so little of the last 100 yearsalmostnothing since Riemann. Students cannot fail to seethat in the biographies and to draw conclusions. But

new ideas have developed, as well as many Ile ap-plications. The instructor and the book are rt ..on-sible for making time for new ideas in the class-room.

My last comments are in a diffee.ent directionaboutcomputers. It is one thing to believe, as we do, that theywill come to have a tremendous part to play in calculus.It is quite another thing to see clearly what that partwill be.

I can see one effect, which may not be centralbut will make a big difference. Lynn Steen men-tioned a recent study that revealed that in more thanhalf of the couiies, horneworks are not graded (oreven looked at). In other words, the student getsno response. That zero is worse than any grade.To work well without recognition is a lot to ask.We ask it because of pressures of time and of stu-dent numbers. Those are pressures that the com-puter is made for. I believe we will approach (slowly)a homework design in which the computer does thetime-consuming part and we do tne thought-producingpart.

One difficulty is obvious, but not yet discussed. Thecomputer is too quick. If we ask a direct questiona definite integral, or a system of linear equationsit answers immediately. The human part. is reducedto input of the problem. The student becomesandknows itthe slowest and weakest link in the system.That is frustrating, not educating. This difficulty willbe overcome, and I hope there will be publicity forsuccesses (even partial successes) in finding the an-swer.

I hope the other difficulty will also be overcometomake this course not a barrier but a door. We need firstto see clearly where it leads.

GILBERT STRANG is Professor of Mathematics at MIT.He is the author of the textbooks Linear Algebra and Its Ap-plications and Introduction to Applied Afathcmatics. An ear-lier book on the finite element method reflected his researchin analysis and partial differential equations. His advanceddegrees are from Oxford University and the University ofCalifornia at Los Angeles.

67

WATKINS: YES, VIRGINIA ... 55

Yes, Virginia ...

Sallie A. Watkins

AMERICAN INSTITUTE OF PHYSICS

Yes, Virginia, there :s a Santa Claus. And yes,fledgi;.4g physicist, there 'Ls a gift in the making for you.Thoughtful: ret.,..erneil persons are designing a new cal-tolus curse for your generation. It will be lean, lively,anki troe. I'm going to guess some of its other charac-teristikz.

If you learn integration, it won't be to build char-acter; you will use calculators and computers as natu-ral tools in the course; you will learn to solve diffe:7-ential and difference equations; you will develop 0,epower and know the satisfaction of mathematical proof;you will learn to make valid approximations, to domodeling; you will see where all of this is going, be-cause there will be a steady supply of reai-life appli-cations; you will solve related problems as homeworkto oe turned in for grading; you will see a symbioticrelationship between continuous and discrete mathe-matics in your course. Your calculus will be a richercourse f.t,an today's, but your text will have a mass ofless tn.. 3.4 kilograms. There you have it, in a nut-shell.

But there remains a burning question which will con-cern us both, Virginia: How will the lean and livelycalculus affect the field of physics? We physiciststalk about "the calculus-based introductory physicscourse"and we mean it. Calculus is the mainstay ofphysics.

Only yesterday, the world we physicists were ableto study was a perfect one. We neglected flIction; wetalked about physical phenomena in cast approxima-tion; we generally did violence to reality so as to arriveat solutions in closed form.

Take Newton's second law of motion. In pointof fact, F does not equal ma. Nor does F equalmirdz + b-ajdz + lea except in first approximation.

We live and move and have our being in a nonlinearworld. Suddenly we h./Ae the mathematical capability

of treating that world as it is. Shall we have a burningof the physics bool:s and charge our physicists to begin,the disciplink. anew?

Resetach in the learning theory of physics has shownthat students are hampered by the baggage of a worldview that insists on the existence of friction. Wouldthese students have an easier time with a course inwhich they could feel more conceptually r.t. home, eventhough the mathematics were messier?

But would something be lost if we were to redophysics to match the nonlinear world it studies? IfGalileo and Newton had had the capability of han-dling the mathematics that we have today, would ourphysics be the beautiful, simple, clear structure that itis?

I submit that one aspect of the unique character ofphysics as a discipline is that it empowers its practition-ers to abstract the simple out of the complex, to sensewhich features of complexity can be suppressed or ig-nored without loss of validity, and to construct sweepinggeneralizations.

Yes, students of the new lean and lively calcuhls willbe ecuipped to deal with the nonlinear world as, indeed,it presents itselfbut if they limit themselves (and us)to this level of information, we will find ourselves in alose-lose situation.

Not everything that can be done should be done,Virginia.

SALLIE A. WATKINS is Senior Education Fellow of theAmerican Institute of Physics, on leave from her positionas Dean of the College of Science and Mathematics at theUniversity of Southern Colorado. Her field of research is thehistory of physics. Dr. Watkins received a Ph.D. in physicsfrom the Catholic University of America.

Cu

REPORTS

r. 1

MA-R7 1 C l "PicrtVT- RES-Po/1Jc E-S c(.:--e--

i.t_e. st-i cYl. AA,,c,L-t- c_o, 1,\ be GIL 0_-.

-iti-li-bve. CCt 1 CtAlitA 11rk "frtA CNO117

. .

Pt 1"-S LO ein Cf_ ..

C/71.14) IA e-v,. --C---cc,.(,(/.. OA I 1...c co In & A e.. reA c.

titi o r i_.0 k i te___i v_i-e_. 1 k. c_ tiA,cj A CA-ti v.1-1-1A x--LS efx-v-t-L. co. t,,, v.1,-1_,..1, I c---

{-i,e,,,_,-] i-i--: 1..1tr.D v Q.__.k)...C.Ll

/14 (.0.12k, r.--i:--_evci-5 --L 0,1,N. 1 A --I.2f -(-- (2_4

Si--1/_A_e- t4.-1-s. ni e7 ti e_ck.r a_ 3--,Le,y-> e- C!---7.7---1,.. 1,-\c c f i,, isL, e_j c4 l cut LAAS ----44

Co vti6-t-- IAA e,1/4. --i-__ oi- bwr 6 e-ic -F- we,. i ves i ges C-.,-1--0 1 y e.... 6 Ol iii\ ck.tAe. L-,-\. ALi- ckfriS

t-'1 IA...e._ st-p. w or- I. on 1,v\ oci-e-1- '° Is . al-71

of. tAg G- cc,, A U 1 C_S

17 0-. . .

C9c.

GeAA. 0...-y-ekrCD LA-A. 1,t-

tici

- C cyv \ vve_ co utArke_ uu I ti,\ r-Dc_ick vioe,o.s.?

Cti\ovx^ s 1 0 (...0 IL?,

____ C,_C 1 vvo 4. ,2_ IA\_..0 ZArC_A- -0 , .

Calculus for Physical Sciences

First Discussion SessionJack M. Wilson and Donald J. Albers

We began by considering the following agenda of is-sues:

Should numerical techniques be taught as part of cal-culus?Should topics from linear algebra be included?Probability and statistics.What role for statements or proofs?Should mathematics teachers be required to havemore of a background ix physical science?Should discipline examples be included? If so, how?Role of drill and practice.Role of problem formulation and conception.Should there be formal instruction in problem solvingtechniques?Is there a special relationship between physics andcalculus?Visualization of solutions. Graphics.Making physical sense of solutions.Relevance to the workplace.Should we do everything, or a few things well?What's in? What's out?The TA problem: language, time, training, motiva-tion.Should calculus be taught as an umbrella course orpartitioned into various courses for various majors?How do we handle large sections of calculus?Is (should) technology be the driving force behindthe new calculus?After initial discussion to consolidate these diverse

issues, the group prioritized the major issue] as follows:A. Should we do everything in calculus or a few things

well? What should be left out? What should beadded?

B. What should be the role of pro( r?C. Is (should) technology (be) the driving force behind

the new calculus?D. Visualization of solutions, graphics, and making

physical sense of solutions.E. Should numerical techniques be taught as part of cal-

culus?The absence of the other issues fiom this priority list

means that either group members were in agreement onthe issue or that they did rot view the issue as contro-versial.

We first discussed what items should be excludedfrom calculus and what should be added. There wasgeneral agreement that we could not develop a detailedlist in the allotted time, but could examine a short listof possibilities.

Several participants suggested that a zero-based bud-geting approach be employed. There was consensusthat we should reduce the amount of time spent onclosed-form integration techniques. Some suggestedthat trigonometric substitutions provide students witha valuable review of trigonometry. Others argued thatpractice with integration methods help to teach pattern-recognition skills.

The suggestion that work with volumes of solids ofrevolution and the computation of centroids be elimi-nated did not get much suppo-t. Participants empha-sized the need for applications such as those in whichthe integral is seen as a limit of sums.

Intuition, Calculation, and ProofThe elimination of epsilon-delta proofs in presenting

limits was supported by all participants except the highschool teachers, who said they must teach such ideasbecause of the presence of epsilon-delta questions onAdvanced Placement examinations.

A reduction of time on derivative calculations wassuggested by our group. The presence of calculators andcomputers that easily and quickly compute derivativeslends support to this call for reduction.

Our group also recommended the addition of morework with qualitative examples and exercises. For ex-ample, students might be asked to construct the graphof the derivative of a function given the graph of thefunction.

The group next considered the place of numerica!techniques, including some elementary Ropects of nu-merical analysis. There was general agreement that acareful selection of strategically-placed numerical meth-ods should be included, but that they should not simplybe grafted on to a standard text.

The question of the ,ole of proof in a first course incalculus was sharply debated. A few argued that someformal proofs are essential to attract strong students t,omathematics. A few mentioned that client disciplinesoften apply pressure on ca,, ulus instructors to mini-mize proofs. Some suggested that intuitive proofs beincluded. All agreed that concept building is a funda-

60 REPORTS

mental goal. To that end, it is essential to give clear,concise statements of theorems, and that changes in thehypotheses be explored to help motivate theorems.

TechnologyThe group felt that the question "Is calculus reform

driven by the new technology?" in some way sub-sumed other questions on numerical techniques, on thestatement-proof approach, and on developing a physicalor graphical sense of solutions. This topic brought onan animated discussion, but there was a surprising con-sensus that calculus is being driven by the technologyand that it may not be a bad thing. Although the useof computers in calculus teaching is very low, it was feltthat the use of new technology by faculty is growing.

Several individuals noted that the computer allowedthem to present old tonics from a better and more easilyunderstood perspective. There was also a discussion ofthe powerful graphic presentations now possible and theability of the computer to explore qualitative featuresof complicated systemsa form of intuition building.

The availability of powerful symbolic manipulationprograms, such as those provided by the HP-28C, willchange the terrain of calculus instruction. Skills onceimportant will now be available with the punch of abutton. It will still be necessary to teach some of theseskills, but how_much and in what way remain open ques-tions. This is one area that may free time for teachingother kills which now may be more important.

There was considerable discussion of the difficulty ofretraining faculty. Students are coming to class expect-ing the faculty to have certain skills that they may notpossess. This problem is as severe, or even more se-vere, at the college and university level as it is at thesecondary level.

It was also noted that the computes has radicallychanged physics research and engineering practice, butthat these changes have not yet been fully reflected inthe teaching of these subjects.

ImplementationIt was generally agreed that calculus courses are un-

likely to change much until new textbooks and otherinstructional materials T produced. As a first step,the group recommends the formation of a blue-ribbonwriting team made up of teachers of calculus from alllevels. The team might have an advisory board com-posed of individuals from client disciplines in academeand industry. The first task of the writing team wouldbe the creation of a draft syllabus for the new calcu-lus. This syllabus would then be revised through review

processes that might include meetings of focus groupsat local, regional, and state levels, by professional or-ganizations of teachers of mathematics, as well as atmeetings of client discipline organizations. The revisedsyllabus could then serve as the basis for new calculustexts.

It is likely that several individual effo ts toward thecreation ,..f new textbooks could use this syllabus as aplace to start. Established authors might be influencedto include ideas from this syllabus.

It was also suggested that a sourcebook of applica-tions from client disciplines be produced in order toheighten the appreciation of calculus by both studentsand teachers of calculus.

On the technology side, calculator manufacturers areurged to produce devices that are very user-friendly.Such efforts are likely to increase their acceptance byteachers and their use in calculus instruction.

JACK M. WILSON is Professor or 2hysics at the -Uni-versity of Maryland, College Park, and Executive Directorof the American Association of Physics Teachers. He is alsoco-director of the Maryland University Project in Physicsand Educational Technology and has published frequently inchemical physics, educational physics, computers in physicseducation, and public policy issues. He received his Ph.D.degree from Kent State University.

DONALD J. ALBERS is Associate Dean and Chairmanof the Department of Mathematics and Computer Science atMenlo College. He has served as Editor of the College Math-ematics Journal and as Chairman of the Survey Committeeof the Conference Board of the Mathematical Sciences. Heis co-editor of Mathematical People. He is currently Chair-man of the Committee on Publications of the MathematicalAssociation of America.

Second Discussion SessionRonald D. Archer and James S. Armstrong

This session, which consisted of a well-balanced mixof mathematicians and physical scientists, identifiedseveral major issues dealing with calculus content andinstruction. Foremost among them was the need forcontinuing dialogue between mathematics faculty andphysical science faculty.

The calculus course should be designed by mathe-maticians for mathematics majors, both pure a id ap-plied, and coordinated whenever possible with m3mbersof the physical science faculty. Whereas it is important

7

ARCHER AND ARMSTRONG: CALCULUS FOR PHYSICAL SCIENCES 61

for mathematicians to consider the needs of the scien-tist, conflicts should be resolved in a manner consistentwith the goals of the calculus course.

Although all participants agreed that applicationscan be motivating, some cautioned against distortingcalculus with an exaggerated emphasis on physical ap-plications. Other participants expressed uneasinesswith their own ability to discuss certain applicationsoutside their own discipline, especially since their stu-dents often lack the necessary background to fully ap-preciate these applications.

TechnologyConcerning the impact of technology on calculus in-

struction, there was strong consensus on several aspectsof this issue:

Technology is here and we must deal with it.Paper-and-pencil drill must precede student use ofelectronic devices.These devices should ultimately nhance students'understanding of the complexities and intricacies offie calculus.the proper balance between the use of electronic de-vices and traditional methods has yet to be deter-mined.Caution must be exercised to avoid adding topics toan already over-crowded calculus curriculum just be-cause technological advances allow teaching certaintopics more quickly and efficiently.A need exists for a well-publicized national clearinghouse for calculus software.

Core ConceptsIn thinking about a core for calculus courses, the

group conceded that it was probably inappropriate todevelop a curricular outline in one morning withoutadequate time for reflection. Nevertheless, there waswidespread support for incorporating the following fourmajor concepts as identified in the recent publicationToward a Lean and Lively Calculus:

Change and stasis.Behavior at an instant.Behavior in the average.Approximation and error bounds.

There was also strong support for including formalproofs in the study of calculus, including epsilon anddelta proofs. Most participants felt, however, that ep-silon and delta proofs should be delayed until the sec-ond semester. Participants urged that it is vitally neces-sary to know your audience rnci their background beforelaunching into the more sophisticated concepts.

Important pedagogical considerations related to theteaching of calculus were di'w issed at length. It wasnoted that the flavor of the calculus course should em-phasize:

The spirit and natural beauty of mathematics.The way mathematicians do mathematics.The intuitive dimension of mathematics.

For example, it is highly instructive for students tosee the teacher develop the solution to a problem fromscratch.

TextbooksMuch of the concern about calculus instruction cen-

ters on current textbooks. Weaknesses in current cal-culus textbooks include too much "plug-and-chug," toomany topics, excessive use of highlighting and summa-rizing sections, and too many "template" word prob-lems.

New textbooks, written to support calculus curricu-lum reforms, should minimize highlighting and, in fact,encourage students to read the mathematics. There is astrong feeling among mathematicians that the studentmust know how to read mathematics before they canwrite good mathematics. New textbooks should alsoinclude physical application problems carefully chosenfrom other physical science disciplines.

Homework should be assigned, collected, and gradedat frequent intervals. Examinations and homeworkshould reflect the course objectives. There appears tobe significant faculty resistanc. fn the use of commondepartmental examinations. Nevertheless, close super-vision must be exercised over the preparation of exam-inations by less-experienced teachers.

Serious concern was expressed over the lack of 1-

phasis on a cohesive mathematics-science curriuarthread throughout the K-12 curriculum. This makesit extremely difficult for college faculties to use the nat-ural connections between mathematics and the physicalsciences in order to motivate the study of mathematics.

Good teaching must be rewarded in the same mannerthat good research is rewarded. The teaching of calculuscan be exciting to both the teacher and the students, ifwe appropriately harness the new technologies, interactwith our colleagues, and e-.tablish a reasonable core.

RONALD D. ARCHER is Professor of Chemistry at theUniversity of Massachusetts, Amherst. He has served ashead of his de}.-rtment and serves as Chair of the Anvil-.can Chemical Society Committee on Education. He is ChiefReader for Advanced Placement Chemistry for the Edu-cational Testing Service. An active research chemist, Dr.

62 REPORTS

Archer received a Ph.D. degree from the University of Illi-nois.

JAMES S. ARMSTRONG is a Senior Examiner of Math-ematics at the Educational Testing Service. His primary re-

Calculus for Engineering Students

First Discussion SessionDonald E. Carlson and Denny Gulick

Among the 30 participants in our session nearly ev-eryone spoke at one time or another, and showed thediversity in types of departments and academic institu-tions from which they came. The participants includedhigh school teaLie.s, college-university teachers, and arepresentative from a well-known publishing firm. Mostwere mathematicians; a few were engineers.

SyllabusMost of the discussion centered on the topics of the

calculus syllabus and the problems identified with stu-dents in calculus. Among the suggestions for inclusionin the calculus courses were:

real-life applications and modeling,heavier use of approximation methods,computer graphics,symbolic manipulation, anddiscrete mathematics.Real-life applications and models would tend to make

more meaLingful the concepts presented in the calculus.Not only are approximation methods important fromthe standpoint of all students of the calculus, but alsothey can be very fruitful in motivating certain aspectsof calculus, especially if observational data provide thebasis for these approximations.

Computer graphics can give not only a visual imageof various concepts, but also a much more realistic im-age of functions such as exponentials and polynomials.In addition, symbolic manipulation, which is only nowbeginning to come into the classroom, could minimizethe tedium of routine calculatirns.

Finally, we recognize the recent effort to incorpo-rate discrete mathematics ' , the first years of collegemathematics. In all likelihJod discrete and continuousmathematics should be intermingle'. How sho wld it beeffected?

sponsibility is the Advanced Plactment Program in Calcu-lus. Previously, he was an Associate Professor of Mathemat-ics at the United States Military Academy at West Point,New York.

If there are to be new items added to the calculuscourse, it is agreed that some topics in the already too-full syllabus would have to be eliminated. Althoughour session did not dwell on which topics should beeliminated, the question of including only some of thetechniques of integration arose. Also, it was suggestedthat perhaps some geometric applications of the integralcould be replaced by applications more indigenous toengineering and physics. Finally, should less clat.3 timebe devoted to series?

At the same time certain participants emphasizedthat the fundamental concepts in calculus must remainin .:-.e course, along with at least a certain amountof drill work to develop manipulative skill. We mustminimize the tedium of working problems where nothought at all is necessary. A basic question is the fol-lowing: What would be an appropriate blend betweengeometrically-motivated concepts, definitions and theo-rems, relevant applications, and rigorous proofs?

StudentsThe second major topic centered on the calculus stu-

dent: diverse backgrounds, work ethic, and retention ofcalculus concepts.

We are all aware that. high school preparation in pre-calculus topics varies greatly. In addition, ever morestudents come to college having already encounteredsome calculus. Should colleges and universities meetthe students "where they are," giving Clem a specialcalculus course? Ir mathematics, how should collegesand universities interface with the high school?

Several members of the session indicated that stu-dents nowadays do not seem to be highly motivatedto study (calculus). Is that because the course is notpackaged well? Or because the students have improperbackgrounds? In order to have a real impact on thecalculus curriculum, the "work ethic" of the calculusstudent will need to be changed.

Many students have difficulty retaining the methods

HAINES AND BOUTILIER: CALCULUS FOR ENGINEERING STUDENTS 63

and the concepts of the calculus. Can effective reten-tion be accomplished primarily through routine prac-tice and by graded homework exercises? Should it beaccomplished through exemplary applications? Proba-bly the answer lies in a mix of t e two. Again, beforethe calculus curriculum can be really Successful, we willneed to find an appropriate mix.

In conclusion, the participants noted that nothingthat we suggest can have much effect on the calculuscurriculum unless

publishers are receptive to innovative texts and soft-ware;communication between mathematics depa:tmentsand engineering and science departments is fostered;academic institutions demonstrate their concern forimproved undergraduate education and reward ef-forts to that end.We hope thrt the present symposium will hc:p give

new vigor to the calculus curriculum.

DONALD E. CARLSON is Professor of Theoretical andApplied Mechanics at the University of Illinois at Urbana-Champaign. He is active in mathematics education in Illi-nois and is educated in both engineering and mathematics.He received his Ph.D. degree in applied mathematics fromBrown University.

DENNY GULICK is Professor of Mathematics at theUniversity of Maryland. He has served as chairman of theundergraduate mathematics program at the University ofMaryland and is co-author (with Robert Ellis) of a calculustext.

Second Discussion SessionCharles W. Haines and Phyllis 0. Boutilier

The six major issues that were Identified by ourgroup are1. How do we develop mathematical maturity so that

students are better able to adapt to new situationsin the future?

2. Personal computers and sophisticated hand calcula-tors will soon be in the hands of many, if not, all ourstudents. How do we make the best use of diem?

3. Many of the engineering disciplines are seeking ad-ditional topics in calculus a. a changed emphasis inthe first two years.

4. Student success rate in the first year is affected h,algebra and trigonometric skills, life styles of the stu-dents, and faculty or teaching assistant enthusiasmfor the course.

5. Suitable textbooks for some anticipated changes arenot widely available.

6. And finally, where do we go from here?The issue of mathematical maturity arose from a dis-

cussion of particular topics that might or might not bekept in a calculus course for the future, and from, therealization that we cannot predict what will be needed.Thus, one of the primary aims of the mathematics cur-riculum, along with othr r parts of the curriculum, isto teach students how tc, learn by developing the fun-damental skills and mathematical maturity it takes tolearn on their own now and in the future. Some partic-ular suggestions (not meant to be exhaustive) are:

Get students comfortable with the concept of func-tions and families of functions.Encourage inferences from functions concerning theirgraphs and vice versa.Make much more use of the conceptual approach,which lies between pure skills and pure theory. Il-lustrate with geometrical proofs and convincing orplausible arguments. Pure theory probably is notdesirable in an engineering calculus course.Allow time to linger over concepts, to consider con-sequences, ctc.Do examples and then generalize.Emphasize relationship between symbols and ideas.Develop the ability in students to read and then doexamples.Develop the material concerning topics such as se-ries, differentials, optimization problems, substitu-tions, and others in light of the above.Develop two- and three-dimensional concepts thor-oughly, then extend to n-dimensions.

Computers and SoftwareThere was an overwhelming consensus that we must

address the proper use of the newest generation of handcalculators and PC software in the calculus course, asthey are already here and students will be using themeven if we, as faculty, don't. Some criteria ,Ind issues forconsideration when employing these technologies are:

They can't be introduced as an add-on; they mustbe integrated properly into the course.Whea developing examples, make sure the proposerdoes the examples to make sure they work, as somecurrent programs don't do what they say they do.

0

64 REPORTS

Concepts must be developed and understood first,before being used on a calculator or a computer. Thisapproach can be used to enhance and expand an un-derstanding of a concept or procedure, but we can'treduce mathematics to a "black box approach."The emphasis should always beare the answers -ea-sonable?This technology can assist with placing more empha-sis on geometrical concepts and intuition.The idea of challenging students to "crack" softwarewas proposed, as it is an interesting way of havingthem learn the concept in order to "get around it."The issue of repackaging the first two years of engi-

neering mathematics was discussed only briefly, as therewere only a few representatives from engineering. Someof the pressures for changing the first two years comesfrom the diverse needs of the different engineering cur-ricula. Discrete mathematics, linear algebra, numericalconcepts, probability and statistics were mentioned ascandidates for incorporation into the first two years.

Questions and concerns these topics raise are:J. What topics do we cut from the current courses?2. Are the engineering disciplines willing to add another

quarter or semester of mathematics?3. Should there be a common first year calculus for all

engineering disciplines, with the second year moretailored to each department's need?

4. Should some foreign engineering educational systemsbe studied to gain insight into other ways of ap-proaching this problem? (This could also be ex-panded to cover other issues identified in this report.)

5. The use of the hand calculator and PC software alsoneeds to be considered here in light of "repackaging."

6. "Lean" calculus may not be appropriate for the en-gineering disciplines.

Student PreparationThe issue of rr any students starting college under-

prepared in algebra or trigonometry drew many com-ments. Consensus st. -med to be that pretesting andplacement programs can be reasonable and effective.With more minority students and re-entry students be-ing recruited for c.reers in engineering, placement pro-grams for calculus and 7recalculus mathematics will bedesirable.

Students should not be forced to repcat what they a1ready know, as boredom will . filict the class, nor shouldthey be in a calculus class where they are "down thedrain" in the first two weeks due to lack of skAils inalgebra and trigonometry. Beyond carefully-designedplacement programs, further reducing attrition in the

calculus sequence must address other factors such asthe students' study habits and mot;vation and facultyattitudes toward the course.

The large number of over-sized textbooks, most ofwhich differ only by the number of exercises or by thecolor of the graphics, drew many comments. Mostparticipants would like to see new lively texts but noconsensus was reached regarding the content of thesenew texts. The mathematics community will be waryof adopting "non-standard" texts and publishers wantreasonable expectation of sales before publishing a newtext.

Follow-up ActivitiesMost of the participants put great emphasis on post-

conference activities. Those activities should include:Local campus seminars by participants to inform andexcite his or her colleagues in mathematics and en-gineering.District or regional workshops to exchange ideas andexperiences.Information distributed L.ationally through as manyvehicles as possible to info. . the engineering com-munity of this national initiative, as very few en-gineers attended this Colloquium. At the nationallevel two societies come to mind, ASEE and SIAM.Other discipline-specific national societies should beinformed. Involvement of enginee.ing faculty is nec-essary for success.Curriculum development, experimentation, assess-ment, follow-up, feedback, i.e., a closed loop whichif shared through a newsletter compiled and editedby a national committee could lead toward a bettercalculus course.A national committee to oversee and collect ideas,initiatives, experiments, etc., and to edit and dissem-inaLe the material to mathematics faculty and otherinvolved persons including secondary school facultythroughout the country. This committee needs am-ple financial support over a five to ten year period.

CHARLES W. HAINES is Associate Dean and Professorof Mechanical Engineering at Rochester Institute of Tech-nology. He teaches mechanical engineering and mathemat-ics, has published in the areas of engineering analysis anddifferential equations, and is active in several professional so-cieties. He received his Ph.D. degree from Rensselaer Poly-technic Institute.

PHYLLIS 0. BOUTILIER is Professor of Mathematicsat Michigan Technological University. She has served as As-sistant Department Head and as Chair of the Mathematics

78

ERDM AND MALONE: CALCULUS FOR ENGINEERING S,TUDENTS 65

Division of the American Society for Engineering Educa-tion. Currently she is Director of Freshman Mathematicsat Michigan Technological University. Boutilicr received anM.S. degree in mathematics from Michigan TechnologicalUniversity.

Third Discussion SessionCarl A. Erdman and J.J. Malone

After discussing a wide variety of topics relatedto calculus instruction. the participants in our groupdecidedby majority votethat there were four areasof major concern, which are presented below withoutregard to priority.

Student BackgroundMany students arrive at college not prepared to take

calculus. Placement tests can help remedy some of theproblem, but they are not a cure. Calculus coursesshould not be split into fast and slow tracks as a wayof- responding to student differences. It is preferableto have students take -re-calculus courses even thoughthis may be resisted by students (and their parents).

If calculus is taught on the secondary level, it mustfollow a standard syllabus such as AP. The quality ofthis course is very important, since students may beturned on or off to mathematics from this experience.Indeed, the seventh and eighth grade experiences inmathematics may have already fixed student attitudes,a major problem which needs attention. Clearly, entryinto a high school calculus course should not be at theexpense of adequate exposure to algebraic skills. Thereis some suspicion of high school courses which are notsimilar to AP courses.

Syllabus IssuesThe present calculus syllabus is overloaded; there is

just too much material in the courses. However, thereis no agreement as to what might be left out. We needsome mechanism for deciding on what can be omitted.All of this is tied to a quantiiy- versus- quality agree-ment.

It is worth noting that only one-third of the partici-pants felt that there was a strong need to change calcu-lus in a dramatic fashion; many felt that we currentlydo a good job in calculus instruction. This may bea reflection of the group's makeup; all 1.-..tt one person(excluding the session leader) were calculus teachers,

and they generally came from schools with good calcu-lus programs. However, there were several commentsconcerning the need to have more time to deal withproblem solving. A question was raised as to whetherthere was too much emphasis on training rather thanon education.

There v.,s a feeling that the engineering communitywanted some discrete mathematics topics introduced,but probably did not want to give up much in the wayof traditional topics. However, it wasn't clear that "dis-crete mathematics" meant the same thing to engineersas it means to mathematicians. There is a need here todefine terms.

TechnologyThose who saw a need for considerable revision in cal-

culus often cited the need to incorporate new technolo-gies as a motivating force. Computers can be used to de-velop intuition, to improve the pedagogy of the course,and to influence the choice of topics to be taught. Thesethoughts seemed to be broadly accepted.

There seemed to be agreement on the benefits offreshmen having (or having access to) a personal com-puter. However, there was no consensus as to whetherthe College should provide personal computer facilitiesor whether students should be required to buy them.

Goals and ImplementationWhat is the essence of calculus and how should we

teach it? Certainly, it is not the repetitive working ofroutine problems. Some of the current calculus discus-sion should be directed toward a better definition of theobjectives of calculus. The role of ideas or concepts asopposed to the role of techniques needs to he addressed.Some participants believed it was possible to teach con-cepts without using e (5 techniques.

»srveys of final examinations indicate there are fewquestions of the "state and prove" type. Is this a re-flection of the goals of the course? It was noted thatthere is a need to stress understanding. But, when thisis done, students often complain that the course is tootheoretical. It appears that calculus instruction doesno do a good :ob with either concepts or applications.An interesting question was posed as to whether engi-neers really care if their students have seen proofs incalculus.

It was strongly felt that there should not be separatesequences for each of the various engineering disciplines.This wasn't practical, and the course and educationalobjectives were better served with a variety of studentsin the course.

66 REPORTS

Other IssuesThe issue of class size and the use of TA's was raised

as an area of much less concern than were the previ-ous four areas. It was noted that horror stories aboutnon-English-speaking TA's abound and that some ofthem are true. But it was also said that some TA'sare very good and that having a professor as instructoris no guarantee of good teaching. Some feelings wereexpressed that a valuable personal touch was lost withsections of large size.

Concerns with pressures produced by the engineer-ing curriculum wet.: also discussed. It was thought thatengineering schools press for too hectic a pace; thatthey are forcing a compression in the mathematics pre-sentation. One cause is the fact that most engineeringschools will not openly admit that engineering is, infact, no longer a four-year program.

Several participants suggested that a mathematicsdepartment resident in the engineering college made co-operation much easier and led to good curriculum plan-ning. If the only mathematics department is in the arts

Calculus for the Life SciencesWilliam Bossert and William G. Chinn

Biologists complain that the mathematics curriculumis designed to meet the specifications of physicists andengineers and that our concerns and the needs of theirstudents are ignored. Is this true? Are their needs formathematical training really different and if so, howare they different and how could they be better metwith changes in the mathematics curriculum? Severalobservations are frequently made by biologists.

Mathematics has been less central to the life sciencesthan the physical sciences. Perhaps this was true dur-ing the development of biology when description wasmore important than theory, but it is certainly not thecase today. The vast majorit if articles in some impor-tant biological journals from ..inical medicine to ecol-ogy present mathematical models or the application ofmathematical technique. The biology major in mostcolleges requires physics and physical chemistry coursesthat themselves need fundamental mathematical ,,rain-ing. Many biology educators refuse to recognize the im-portance of mathematical training, however, and allotonly two semesters in the crowded biology curriculumfor it.

and sciences college, campus politics can make cooper-ation much more difficult.

The final point was that engineering faculty neededto provide the mathematics faculty with examples of thekinds of applications they would like to see incorporatedinto the calculus.

CARL A. ERDMAN is Associate Dean of Engineeringand Assistant Director of the Texas Engineering Experi-ment Station at Texas A &M University. He has been Headof Nuclear Engineering at Texas A&M, a faculty memberat the University of Virginia, and a research engineer atBrookhaven National Laboratory. His Ph.D. is from theUniversity of Illinois at Urbana.

J.J. MALONE is Sinclair Professor of Mathematicsat Worcester Polytechnic Institute. He has also taught atRockhurst College, University of Houston, and Texas A&M.His areas of research include group theory and near rings.He received a Ph.D. degree from Saint Louis University.

Within mathematics, calculus is less central to biol-ogy than are other subjects. Although there are manyimportant applications of the calculus in all areas of bi-ology, many fundamental biological researcil resultswhich should be included in college biology coursesdepcnd on applications of modern algebra, particularlycombinatorics, of probability, of statistics, and of linearalgebra. (Some illustrations are given in the backgroundpaper of Simon Lev;1.) If we do not encourage youngbiologists to take more than two or three semesters ofmathematics in college, should they all be devoted tothe calculus?

For the life sciences, training in formulating prob-lems is as important as training in solving them. Toooften calculus courses present, illustrative problems thatare carefully selected to show off the technique. In biol-ogy the problems that students face even in elementarycourses are either not well formulated or do not yieldto simple techniques. Perhaps in engineering the bestpart of one's activity is applying well-studied modelsand their associated techniques to new situation, butthat is not true for biology. Students must learn to ab-

BOSSERT AND CHINN: CALCULUS FOR THE LIFE SCIENCES 67

stract simpler formal descriptions from problems thatthey cannot solve, and need to be shown how to de-pend more on qualitative and numerical methods thanon those traditionally taught in the calculus.

If the mathematical needs of the life sciences do differin these ways, and others, what should be done aboutit? A first thought is to start from scratch and designa new course in college mathematics specially for biol-ogists. In two or three semesters we could package allof the broader range of mathematics that biologists feelare needed, taught with a problem first and qualitativestyle.

This is not easy to do. Although some have tried itand several textbooks labeled "mathematics for the lifesciences" are available, the efforts are not altogethersuccessful. A simplistic critique holds that Illey aremerely expositions of the same topics of traditional pre-calculus and calculus courses, perhaps reordered andwith biological illustrations, and do not involve a newselection of topics and a "problem first then technique"attitude that might be required. Also, they regularlyachieve breadth by simply being longer and placingmore demands on our students.

There are some strong reasons for avoiding specialdiscipline-oriented calculus courses. First, many stu-dents take this first course in college mathematics be-fore their career directions are set. Many liberal artscolleges do not allow students to select a major until af-ter the freshman year when the course would normallybe taken. Second, the value of a mathematics course asa liberal arts experience is compromised if it is taughtwith a narrow disciplinary focus.

Third; and most important, there is a danger thatthe specialization of the calculus course will obscure thegenerality of the mathematical way of thinking that sep-arate disciplines have come to mathematics to gain. Itis useful for students to see that the mathematical mod-elling process and even some specific models transcenddisciplinary boundaries.

Separate courses may be justified when resources per-mit, and when disciplinary education can be more ef-ficiently begun in the calculus course. Student mightalso be usefully taught in separate sections that recog-nize their differences in previous experience arid com-mitment to further training. Care must be taken, then,that the courses do not lose the conceptual and aestheticessence of the mathematics.

It might be better to depend upon significant changesin the calculus curriculum, such as will surely come fromthis conference, rather than striking out on our own. Ifso, what changes would be particularly important to usthat could also be desirable for other "client" fields and

hence possible to be adopted? Some first thoughts are:1. More illustrative ,txamples from the life sciences.

This obvious improvement is not a problem. Math-ematics educators are regularly asking for teachingexamples from applied fields.

2. More mathematical modelling. More time should bespent developing models, not just to apply knowntechniques for drill, but to introduce new topicswhich are required because of the model and not thereverse.

3. More linear algebra. Many applied fields deal withsystems of differential equations which can be taughtefficiently only after students can deal with matrices.This topic is also important to biology as the basisof multivariate statistics, which is too often left un-til late in the mathematical curriculum, or worse tocomputer packages.

4. More qualitative and numerical methods. The lim-ited presentation of numerical methods in elementarymathematics courses is a puzzle to most biologists.Taylor's theorem should be exploited early and of-ten in demonstrating nun- lrical solutions of dieren-tial equations and optimization problems. Matricesare regularly taught by the presentation of importantnumerical algorithms. This might be another goodreason for having some linear algebra in our calculuscourse: as a good example.

5. More applications of probability and statistics. Leastsquares rarely appear in calculus courses althoughit may be more appropriate there than in a goodstatistics course.There is no disagreement on the value of more bio-

logical illustrations in calculus courses. We simply needto generate more catalogs of appropriate problems likethe "Mathematical Models in Biology" complied by theMAA some years ago. These could enrich the calcu-lus course greatly, since the life sciences at the currenttime depend on numerous realistic, yet simple modelsthat may be more approprial.e to elementary calculuscourses than are those from physics or engineering.

There should be better communication between lifesciences and mathematics faculty at a college to pro-vide this input. Perhaps team teaching of the calculuscourse or specialty sections of adjunct courses in sep-arate disciplines could present the biological relevanceof the calculus. Convincing students of the relevancerequires more than adding biology illustrations to amathematics course. Biologists must increase the useof ,nathematics in their own courses.

In general, changes which broaden the range of top-ics and deal with real problems, perhaps in qualitativeor numerical ways if they are the only ones available,

68 REPORTS

would be desirable for biologists. In return we must un-derstand that as mathematics becomes more central tothe stndy and advancement of the life sciences, we can-not relegate it to the remains of the biology curriculum,with a priority lower than chemistry or physics.

One year of college mathematics is simply not suf-ficient, not for the study of organs, cells, organismsor populations and communities of organisms, and cer-tainly not for the study and practice of medicine. Biolo-gists will not be satisfied with the calculus curriculum,or the mathematical programs that biologists, to thedismay of many mathematicians, lump under the nameof calculus, until enough time is allowed for it to betaught properly.

__WILLIAM BOSSERT is Professor of Science, with teach-

ing responsibilities in biology and applied sciences at Har-vard University. His t reas of research work include functionof the mammalian kidney and rapid evolution in animal pop-ulations. In the past year he designed an advanced calculuscourse for biology concentrators. He received a Ph.D. degreein applied mathematics from Harvard University.

WILLIAM G. CHINN is Professor Emeritus of Mathe-matics at the City College of San Francisco. During 1973-1977 he served on the U.S. Commission on MathematicalInstruction and in 1981-1982 he was Second Vice Presidentof the Maeiematical Associanon of America.

Calculus for Business and Social Science Students

first Discussion SessionDagobert L. Brito and Donald Y. Goldberg

Mathematics courses in American colleges and uni-versities serve a diversity of students, programs, and in-terests and are provided by a diverse collectio" of insti-tutions of higher education. Their traditional functionshave been to provide a set of conceptual and computa-tional skills and to serve as a screen for the allocationof scarce positions in various programs, both at the un-dergraduate and graduate level. Advances in computingtechnology as well as development of new mathematicshave raised questions about the appropriate choice ofconcepts and skills which fulfill the first role. Changesin the composition of the undergraduate student bodyhave raised questions about the second.

DiversityPrograms in business and economics vary widely to

meet the needs of students who have a diversity of math-ematical preparation and career aspirations. Some un-dergraduate business programs are open to all students,others find it necessary to limit admission. Many degreerequirements include a calculus course, some requiresuccess in a calculu3 course (or sequence of courses) foradmission to the program.

Graduate management school curricula, in the tradi-tion of the "case method," have little or no need for for-mal mathematics. However, case method schools, andeven a few law schools, may requ: e a calculus course

for admission. Other curricula are more quantitative intheir focus and require a higher level of mathematicalsophistLation. These quantitative schools have morestringent mathematical requirements such as optimiza-tion theory and econometrics, which may use differen-tial equations, linear algebra, and other advanced math-ematics.

The demand for mathematical training by economicsprograms varies similarly. The traditional economicsmajor may require a calculus course for the bachelorsdegree but very often calculus is not a prerequisite forany specific courses in the economics department. Stu-dents who plan to do graduate work in economics, orto enter a quantitatively oriented business school, areencouraged to study a substantial amount of mathe-matics, often equivalent to a strong mathematics minoror even a mathematics major.

Other undergraduate economics majors, for whomthe degree is terminal or who plan to attend law schoolor a non-quantitative business school, find no needto take mathematics beyond the minimum required.A substantial number of students enroll in economicscourses as part of their general education or as a re-quirement for another major; therefore, economics de-partments arc reluctant to impose a calculu.. prerequi-site for most courses. There is, however, a consensusthat training in calculus is very useful.

in other areas of social science, some statistical meth-ods courses are calculus-based, others are not. Othersocial science and business courses which use discrete

60

BR:TO AND GOLDBERG: CAT.CULUS FOR BUSINESS AND SOCIAL SCIENCE STUDENTS 69

mathematical techniques may require a prerequisitemathematics course, but not necessarily calculus.

Functions of Calculus CoursesOne difficulty encountered by the discussion group

was identifying the variety of interrelated functions ofthe calculus courses. One function, a systemic one, isas a screening mechanism for admission into degree pro-grams. Other functions identified in the discussion areto teach particular calculation skills, to inculcate math-ematical maturity, and to provide significant experiencewith mathematical models.

The screening function is often resented by instruc-tors of business calculus or calculus for social scienc...:courses. Yet with this burden come obvious benefitsto mathematics departments: large enrollments in the"short" calculus can be "bread and butter" for many de-partments. Use of calculusor indeed of discrete math-ematics, of principles of economics, or other coursesasan allocation mechanism is not necessarily inappropri-ate.

Participants in the discussion, principally from stateand community colleges, recognized the screening func-tion of their courses. Brc 'der social questions of theappropriateness of using courses as specific screeningdevices and implications for gender and ethnic diver-sity in the professions were not raised in the discussion.

Basic calculus skillsmuch of the algebra and geom-etry prerequisite, manipulation of functions and simplediffe-entiation and integrationwere seen as essentialto the calculus course itself. The advent of the super-calculators certainly has lessened the need for exten-sive technical competence, especially for stutlents tak-ing a "short" callulus course; yet the group agreedthat hands-on pencil-and-paper experience with somece.1..:ulus techniques is a necessary condition for stu-dents to understand the fundamental concepts. Forquantitatively-oriented students in the social sciences,the traditional mainstream calculus course can be ex-pected to provide in the future, as it has in the past,foundations for further study of mathematics.

The development of some mathematical maturity instudents was identified by many participants as a keyihnction of calculus courses in which management stu-dents and others enroll. One participant referred to thedevelopment of a "reading knowledge" of mathematics;as an example, understanding the notion of a differen-tial equation, even without the techniques to solve one,was viewed as a worthy goal of an introductory course.

Another example was that a glimpseeven with-out masteryof an important and subtle mathemati-cal structure can aid the development of mathematical

maturity. Understanding the nature of mathematicalabstraction and developing confidence in using the con-cepts of functions were viewed by others as importantsteps in the ability to use mathematics profitably.

The one function of the calculus courseindeed ofany introductory mathematics coursewhich was mostenthusiastically endorsed by the group was the devel-opment of skills and understanding in mathematicalmodelling. It may be unclear whether "modelling"is an identifiable skill to be taught, but all agreedthat it is essential for students to be able to representparticular problemsin business, economics, or otherdisciplinesin the language of mathematics. It whsclear to all that the increasing power and availabilityof calculators and computers, whatever their implica-tions for skills instruction, heighten the need for studentcapability in modelling.

Why Calculus?One question frequently asked by students, social sci-

entists, and many mathemarcians, is "Why calculus?"As noted above, some business and social science pro-grams require a calculus course for the degree but notas a prerequisite for any course. One participant, foina state college, helped develop a course tailored to theneeds of her institution's business program: the demandwas not for calculus but for a mastery of the notion ofnumerical functions, particularly exponentials and log-arithms, sequences, matrix algebra, and introductionsto linear programming and probability.

William Lucas, of the Claremont Colleges and NSF,rejected the notion of calculus as "the mathematicsof change." Continuous change, yesbut discretechanges, especially those of human beings and hu-man organizations may be modelled using new discretemathematics, much of it being developed outside ofmathematics departments.

Gordon Prichett of Babson described the Quantita-tive Methods course at his institution, an undergrad-uate business college. The course begins with a dis-crete approach to fundamental algebra skills in the con-text of linear modelling. Difference equations, differen-tial equations, and some calculus techniques are con-sidered; a follow-up course introduces some statisticalconcepts, using the integral. The Babson course ex-ploits classroom computers, linear programming pack-ages, and symbolic manipulations; another notable fea-ture is the requirement of homework collected daily.

For the RecordThe discussion group, with few exceptic Is, was corn-

70 REPORTS

posed of college an: high school mathematics faculty.As "providers" rather than "users' of calculus coulees,the group sensed a need to learn from social scientistswhat, in fact, their students require in their mathenlics courses. Some unresolved questions and commeifollow:

Has there been significant research to determinewhich mathematics skills and concepts are used inbusiness and social science programs?Will calculus reform efforts be directedin term ifgrantsto social science calculus courses, or only tothe mainstream course for physical science students?One participant suggested that grant funds be allo-cated to development areas on the basis of studentenrollments. (It was noted, in this regard, that themainstream calculus mass may be too large to move;change may be more likely on the periphery.)The discussion group participants urged calculus re-

form leader: to acknowledge the importance of the non-physical science course in general education, in prepar-ing students for non-science profeosions, and in educat-ing many of the nation's future leaders.

DAGOBERT L. BRITO, Peterkin Professor of ?oliticalEconomy at Rice University, has written and edited workson arms races and the theory of conflict, on public finance,and on the economics of epidemiology. Formerly he wasChairman and Professor of Economics at Tulane University.He received his Ph.D. from Rice University.

DONALD Y. GOLDBERG is Assistant Professor ofMathematics at Occidental College. His research interestsinclude algebraic coding theory and combinatorics. He hasparticipated in reading Advanced Placement Calculus exam-inations since 198U. Be received his Ph.D. in mathematicsat Dartmouth Caege.

Second Discussion SessionGerald Egerer and Raymond J. Cannon

The ;ussants were primarily drawn from the fieldof mathematics. A variety of viewpoints were presented,reflecting the diverse nature of the participants' insti-tutions and their personal experiences.

it quickly became clear that there is not one idealcalculus course for business and social science students,since the mathematics requirements at represented in-stitutions varies from one to four semesters. Thereare also wide differences among student backgrounds,

mathematical preparation, and motivation. However,.cipants did agree on the importance of a wide range

hematics departments need to establish motcollaboration with other disciplines. Suggested

points of departure include interdisciplinary seminars,joint teaching of courses, consultation regarding courseprerequisites, and the revision of the content of coursesof mutual interest.

Participants further agreed that mathematical tech-niques should be routinely and extensively used in eco-nomics courses so that the formal mathematical require-ments become genuine prerequisites. This is becauseeconomics is becoming as mathematical as physics, as acursory perusal of the major journals makes clear. Theneed for =the* tics in business studies, while not sointense, remains nonetheless real, e.g., inventory con-trol, production theory; operations research, statisticsand probability

It was generally felt that the mathematics commu-nity should not be overly defensive in its view of theneed for a minimum level of mathematical sophistica-tion on the part of business students. Sonic of thebenefits accruing A. these students as a result of suchcurses include increased analytical ability and hencegreater acceptability by graduate schools or improvedcareer opportunities. Furthermore mathematics coursescan provide students with an important opportunity toconstruct, articulate, and interpret formal models in thecontext of their own discipline.

Nonetheless, there was a lack of consensus as tohow 1,r.st to serve the needs of these business students.Should, for example, their calculus courses begin with areview of needed algebraic techniques or with a discus-sion of rates of change, with the algebra to be cover-d asneeded? Should the mathematics be presented in a con-crete context (such as the theory of the firm) or shouldit be presented more "purely" so that its universality isst 'essed? (The value of UMAP modules as a source ofapplications was noted by several participants.)

Some concern was expressed about the general at-mosphere in which learning takes place. While societyis broadly appreciative of ti-,e results of technology, itis at best indifferent to the underlying basic scientificactivity.

Discussion then turned to the effect which comput-ers and calculators are likely to have on classroom in-struction. Questions were raised as to whether "blackbox" technology (for integration, as an examplr) is anacceptable substitute for understanding analytical con-cepts and procedures. What is the relationship betweendeveloping computational skills on the one hand and

YOUNG AND BLUMENTHAL: CALCULUS FOR COMPUTING SCIENCE STUDENTS

conceptual understanding on the other?Finally, concern was expressed that instructors, be-

cause they feel uneasy discussing applications outsidetheir own field, might omit a certain amount of other-wise instructive and helpful material.

GERALD EGERER is Professor of Economics at SonomaState University. He has worked both as a government andas a corporation economist in London, and thereafter as

an academic economist at several universities in the UnitedStates. He received a doctorate from the University ofLyons, France.

RAYMOND 3. CANNON is Professor of Mathematics atBaylor University. He has served on the faculty of the Uni-versity of North Carolina at Chapel Hill, where he receivedan award for inspirational teaching of undergraduates. Heis actively involved with the Advanced Placement programin calculus. He received a Ph.D. degree from Thlane Uni-versity.

Calculus for Computing Science StudentsPaul Young an Marjory Blumenthal

According to Hamblen [1], over 25,000 bachelors de-grees were awarded in 1983 to students who majoredin some form of computer science, information science,or computing technology. Hamblen estimates that thistotal will have reached over 50,000 students per yearby 1988. This is about twi.ie the combined total formathematics, statistics, chemistry, and physics.

Of these 50,000 students, there are no reliable esti-mates of how many are in technically-oriented programsfor computer science and computer engineering, whichtypically require calculus as a prerequisite or at least asa corequisite. Nevertheless, accrediting organizationsfor undergraduate programs in both computer scienceand computer engineering require calculus as a part ofaccredited programs.

In addition, a variety of recommendations, by vari-ous ACM committees have included calculus as part ofthe recommended course sequence for computer sciencestudents. Thus, it can be expected that in the imme-diate future there will continue to be be significant de-mand from computer science students for calculus as a"service" course.

In spite of this, it is sometimes difficult to pinpointthe exact calculus topics which are valuable for com-pletion ,a the typical coi7:puter science program. Obvi-busly students :,aking numerical analysis must be well-grounded in most of the topics covered in introductorycalculus and linear algebra courses. But numerical anal-ysis is not a required portion of all undergraduate (oreven graduate) programs in computer science.

Students taking courses in simulation and perfor-mance evaluation must similarly be we.: trained in the

,nalytic methods required for series analysis, statistics,and queueing thec.y. But, again, performance evalua-tion and modeling courses are not required of all com-puter science undergraduates. Students taking coursesin computer graphics and image analysis must be pre-pared to handle the underlying analytic techniques foranalysis of two- and three-dimensional bodies hi space.And generally, as computer science becomes more ex-perimental, it can be expected that increasingly sophis-ticated statistical techniques will be employed in com-puter science, and these should depend on knowledge ofthe underlying analytical techniques.

While it is possible for many computer science un-dergraduates to pass through their .ntire undergradu-ate curriculum seeing little or no use of analytic tech-niques, and to use no such techniques when employedafter graduation, academic computer scientists gener-ally see enough analytic applications, and potential formore such applications, that they are reluctant to allowundergraduates to complete undergraduate programs incomputing with no exposure to analytic methods.

Furthermore, students planning graduate careers incomputer science are more likely to see the applicationof such techniques in simulation and modeling, 5 ,ier-formance evaluation, in image processing and graphics,in statistics, and occasionally in analysis of algorithms.Hence, most of the better programs in computer sciencenow require calculus. This requirement is reinforced bythe common belief that calculus should be part of theuniversal culture for all scientists and engineers.

Finally, many computer scientists regard their disci-pline as requiring the same sort of mathematical skills

P,3

72 REPORTS

and abilities as required for mathematics generally.Hence any course which enhances, or checks, students'"mathematical maturity" is often welcomed.

Still, it is safe to say that means of the analytictechniques (e.g., solving differential equations) taughtin the current standard calculus sequence have little orno direct bearing on the core areas of computer science,including programming languages and compilers, soft-ware and operating systekk. , architecture and hardware,and perhaps even theory and analysis of algorithms.(In the latter area students occasionally use integrationto place upper- or lower-bounds on the running timesof algorithms, and sometimes use generating functionsto produce solutions to recurrence relations which arisenaturally in the analysis of algorithms.)

Purpose of CalculusGiven the above, it follows that computer scientists

are not looking for particular collections of applica-tions skills for computer science students tat :ng cal-culus. Fundamental understanding of the underlyingconcepts is what is important for those students.

Thus, what is important in calculus for computer sci-entists is mastery of fundamental concepts, the abilityto perform basic symbolic manipulations, and above allthe ability to use analytic models in real applications,to know what arkalytic techniques are applicable, andunderstanding how to use them. "Plug and chug" forfast applications in some particular application domainis seldom, if ever, useful for computer science studentsin these programs.

This analysis does not imply that current approachesto teaching calculus are satisfactory for computer sci-ence students, but it does imply that what may be goodfor computer science students ;nay well be good forquite a variety of other students as well:1. Students should be taught, and must understand un-

derlying concepts and intuitions.2. Students s'aould be given problems that involve mod-

eling real-world problems analytically and solvingthem. Analytic, clmed-form solutions should bestressed when appropriate, but numerical approxi-mations should also be explored so that students gainan understanding both of the relationship of discrete,approximate solutions to continuous processes andof the notion of modeling large finite processes us-ing continuous models. Integration, differentiation,and series summations all provide examples wherethe interplay between the discrete and the continu-ous should be apparent.

3. Series summations also provides an example wherethere is an opportunity to teach induction, recursion,

and closed-form solutions of finite summations in thecontext of the calculus sequence. Recursion and itsrelation to induction is so fundamental to computerscience that these topics must be taught in calcu-lus tour. es designed to meet the needs of computerscience students.

Using ComputersThe panelists also discussed the role of programming

and of computer aids in teaching calculus. It was gener-ally believed that it was impractical, and indeed objec-tionable, to require programming ability as a prerequi-site to calculus. It was felt not only that programmingcannot reasonably be required of all students taking thecourse, but that in fact packaged mathematical softwarewill often be the only c Imputational technique that willever be needed by many students. (Editorial comment:The panel did not discuss what could be achieved if anintroductory computer science course were required asa prerequisite to the cs culus.)

In view of the fact that calculus is seldom required asa prerequisite to any standard undergraduate computerscience course except numerical analysis, this seems un-fortunate. It is surely true that a calculus course taughtto students IA Ito are proficient in programming could bedesigned so that it simultaneously enhanced the stu-dents' understanding of the relationship between anal-ysis and computing and their understanding of 5asicmathematical principles.

While use of certain software packages (e.g., thosewhich display convergence of rectilinear approximationsto the area under a curve or convergence of tangentialapproximations to a derivative) may provide useful vi-sual tools in helping all students understand analyticalmethods, such tools clearly do not address underlyingissues connecting analysis to computer science.

Discrete MathematicsAlthough calculusnot discrete mathematicswas

the topic for panel discussion, it was clear from the gen-eral discussion that, with respect to computer science,the problem of teaching discrete mathematics for com-puter science majors was of more concern to the panelists than the problems of teaching calculus. Computerscientists typically believe that discrete mathematics,including mathematical logic at various levels, elemen-tary set theory, at least introductory graph theory, andabove all combinatorics at all levels are more importantto computer scienr than the calculus course.

Those mathematics departments that have failed tooffer such courses for computer science studeliA have

YOUNG AND BLUMENTHAL: CALCULUS FOR COMPUTING SCIENCE STUDENTS 73

frequently found their computer science departmentsteaching these courses. Indeed, so.ne computer scien-tists believe that such courses are better taught by com-puter scientists who are more familiar with the appli-cations, or equally typically, many computer scientistsbelieve that the more elemen' ry topics often coveredin discrete mathematics courses are best cone direct,yin the context of the computing applications, where theunderlying concepts (say of graphs, sets, and proposi-tional logic) are both easily motivated and simply de-fined.

One view of elementary discrete mathematics sup-ported by the panelists was that much of this material,e.g., elementary graph theory, propositional logic; ele-mentary set theory, etc., might be better taught in highschools rather than in the universities, and might morereasonably be taught in the schools than the calculus.One way to encourage such a development might be formathematicians to ds-velop an advanced placement testin discrete mathematics.

While some panelists questioned the ability of themathematics community to influence the schools andthe Educational Testing Service, others pointed out that30,000 university mathematicians having two profes-sional societies with faculty who had to read advancedplacement tests should be influential in getting such areform instituted in the schools and the EducationalTesting Service.

It was also believed that placing the more elemen-tary parts of discrete mathematics in the schools, andemphasi.3ing the relationship between discrete and con-tinuous mathematics in the calculus sequence, (e.g., inthe study of limits of sequences) would not satisfy com-puter scientists' need for training in more difficult topicsin discrete matuematics, but there was little agre menton what was important.

One view expressed was that the appropriate discretemathematics for mathematicians to teach for computerscientists is the analysis of (nonanalytic) algorithms.But computer scientists like to teach this themselves,regarding it as integral and basic to computer science.What is needed is help in essentially mathematical top-ics, including probability theory and statistics, combi-natorics, and mathematical logic.

There was little belief among the panelists that thissort of discrete mathematics could be integrated intothe calculus curriculum, but there was recognition thatsome of it, for example elements of statistics and per-haps generating functions, could at least be introducedin the calculus sequence, so that students would at leastunderstand the applicability of analytical methods tothese areas when they meet these topics later in their

careers as ccrnputer sc ientists. In the long run, mathe-maticians should be alert to the possibility that discretemathematics mry replace calculus as a basic mathemat-ical prerequisite for computer science students.

A Enal SuggestionThe panel also discussed, not entirely successfully,

the question of finding suitable computer science exam-ples out de of numerical methods to integrate into theelementary calculus curriculum. The difficulty here, un-like for more traditional felds like physics, economics,and some fields of engineering, reflects both the pan-elists' inexperience with various subfields of computerscience, and the fact that applications are scatteredthroughout various subfields of computer science whichare often not part of the core of the discipline.

Nevertheless, there was general agreement that com-puter scientists, students from most other client disci-plines, and indeed prospective mathematicians are notwell served by calculus courses Nub:eh do not success-fully integrate real problems int the course. Wordproblems, problems from client disciplines, and prob-lems that require students to think about the meaningof their solutions are all important in calculus courses,and generally not well treated in current courses.

To help with this problem, it was suggested that aproblem bank with motivating problems from a vari-ety of client disciplines could be developed. Problemscould be tailored and classified by client discipline, byrelevance to standard course topics, and by degree ofdifficulty. Ideally, a data base of such problems couldbe maintained and distributed "on-line" via computernetwork. Such a resource cold(' even be continuallyupdated. One way to launch such a data base wouldbe through a special panel, perhaps federally-funded.The panel could generate potential problems and ex-amples, discuss their suggested problems and exampleswith colleagues from the client disciplines, and revisethem based on feedback from colleagues ana students.

Reference

[1.] Hamblen, John. Computer Manpower, Supply and De-mand by States. Quad Data Corporztion (Tallahassee,FL 32316), 1984.

PAUL YOUNG is Chair of the Computer Science Depart-ment at the University of Washington in Seattle. He 1. .'ice-Chair of the Computing Research Board and has served asChair of the National Science Foundation's Advisory Sub-committee for Computer Science. 17.e is an editor for Leveral

C:

74 REPORTS

research journals and publisbes regularly in theoretical com-puter science. He received his Ph.D. degree in mathematicsfrom the Massachusetts Institute of Technology.

MARJORY BLUMENTHAL is the. Staff Director for the

NAS/NRC Computer Science Technology Board. Sbe hasresearched and written about computer technology and ap-plications. Formerly with General Electric and the Congres-sional Office of Technology Assessment, she did her graduatework in economics and policy analysis at Harvard.

Encouraging Success by Minority StudentsRogers J. Newman and Eilecn L. Poiani

If minority students are to participate in Calculus fora New Century, then all those involved in and respon-sible for its instruction must be concerned about whatis happening in the present dec!e.

e.12he.large and increasing minority population (in thisreport, "minority" is defined to be Black, Hispanic, orNative American) in the United States offers a criticalnational resource for the mathematical sciene_J. Demographic data show that by the late 1990's, one-thirdof American elementary school students will be of His-panic orig i.

At the same time, the proportion of minority stu-dents entering and persisting in higher education is slip-ping. According to data from the American Council onEducation Office of Minority Concerns, between 1980and 1984, Bla.:k undergraduate enrollment declined by4%, while Black enrollment at the graduate level fellby 12%. During this -!riod, Hispanic enrollment inhigher education increased by 12% while American In-dians/Alaskan nat;ves experienced a 1% deciine. Morethan half (54%) G: the Hispanics enrolled in higher_ed-ucation attended two-year institutions. At the sametime, the proportion of Black and Hi :panic full-time col-legiate faculty remained about the Jame (4% and lessthan 2%, respectively).

Among the measures of "success" in mathematics asperceived by the public is perfc:rnance on the Scholas-tic Aptitude Test and the American College TestingProgram. Students from most minority groups showedimproved scores in 1987 when compared with those in1985.

ninth mental IssuesTo munch a disoession of encouraging success in

mathematics by --dnority students, the following issueswere raised:1. There is no .noticeable difference in the mathemati-

cal performance of minority students when compared

with majority students with comparable mathemat-ical background. The differences are caused by dis-parities in background preparation.

2. Many teachers have low level of expectations for mi-nority-student achievement.

3. Teachers, counselors and school administrators pro-vide inadequate encouragement for minority studentsto pursue solid mathematics courses, such as AlgebraI and II, as well as additional senior-level mathemat-ics courses.

1. Small numbers of minority students aspire to careersin mathematics or related fields, both on the under-graduate and-the graduate levels. Indeed, there hasbeen a decline in enrollment of minority students inthe calculus, and consequently among mathematicsmajors and in all other programs leading profes-sional careers in the mathematical sciences. Thereis a real need to attract high achieving minority stu-dents to mathematics-related majors.

5. Teachers need to foc.is on minority student strengthsand to emphasize the need for academic discipline.Students must be encouraged to work hard. Mostlow income minority students do not have the familysupport system or tradition of exposure to highereducation necessary to promote success.

6. Student:. must be able to read well in order to makefull use of textbooks and notes.

7. We need to identify mathematically-talented stu-dents at an early age and nurture them through ele-mentary and secondary school.

Pre-College PreparationThe group discussed several areas in which work

should be done to address these issues. WI began witha list of tivities to improve pre-college mathematicalpreparation of minority students:

Develop rapport with school systems for the purposeof encouraging more minority students to take at

0,

NEWMAN AND POIANI: ENCOURAGING SUCCESS BY MINORITY STUDENTS 75

least two years of algebra and at least one advancedmathematics course in the 3tnior year.Provide enrichment opportunities, especially for stu-dents from low income backgrouhds, including .:ul-tural programs and field trips to science centers, tech-nical industries, and college campuses.Develop summer enrichment programs for minoritystudents by collaborative initiatives among collegesand universities, school systems, and funding agen-cies.Share pedagogical ideas about teaching mathematicsin a way that makes it more inspiring and appealingand makes the image of mathematics more positive.Use television and related media to promote math-ematics. For example, perhaps Bill Cosby could bepersuaded to include a mathematician, or at leastpeople using mathematics, in the scripts of this pro-grams.-Mathematics professional organizations should coop-( rate to support initiatives which identify and nur-ture talented minority students in grades K-12.

Collegiate StudentsThe major part of our discussion concerned colle-

ginte-level-mathematics. Everyone agreed that it is es-sential to maintain standards of quality and high levelof expectations for all students in mathematics courses.In addition, it is especially important to identify andmake use of minority student strengths while teachingmathematics.

When students have difficulty in matheniatics, it issometimes (but not always) caused by poor preparation.Here are some suggestions fol encouraging success incalculus among students who enter college with poorpreparation in mathematics:

Provide the opportunity for personal attention sothat instructors get to know each student as an indi-vidual and can follow his or her progress in course-work. This is easier to achieve in small class set-tings; it will require innovative approaches if thelarge lecture- recitation model cannot be modified.Enforce regular class attendance to complement thecontinuity of the calculus content. Motivation mustbe reinforced among poorly-prepared students.If "Desk-Top Calculus" is to become the norm, finan-cial resources will be needed to enable low incomestudents, many of whom are minority students, toacquire personal computers.When students with good preparation in mathemat-

ics fall behind in calculus, it is usually for lack of moti-vation. Here are some suggestions for ways to promotesuccess among such students:

Expose students to reasons why calculus in itspresent and future forms serves as a critical entreeinto majors that lead to higher paying technical andprofessional careers.Integrate the resources of the personal computer withthe teaching and learning of calculus.Invite guest lecturers (e.g., successful alumni/alum-nae, Jr colleagues from client disciplines) to addressthe usefulness of calculus in other fields. (Note par-ticularly the resources of the BAM and WAM pro-grams of the Mathematical Association of America.)Provide a network of "mathematics mentors" to helpmotivate those who need support and to guide thosealready committed to the mathematical pipeline.Develop interactive approaches to the teaching andlearning of calculus.

Adult LearnersAdult learners form an increasing percentage of those

studying calculus. They bring to the classroom an in-tellectual maturity and curiosity which demands moredepth of understanding about the relationship betweenmathematics and the real world than most textbooksprovide. New materials must be developed to meet thisgrowing and im?ortant need.

The percentage of adult learners is especially highin two -y :ar colleges, a group of institutions that wasnot specially addressed at the Colloquium. Perhapsit should have been. Many graduates of two-year col-leges often do not pursue higher education, so calculusbecomes their last mathematics course. The need., ofthese .1stitutions and their students require consider-ation well beyond the brief time that we were able toprovide.

Increasing the involvement and success of minoritystudents in calculus for non-mathematics major pro-grams as well as for the mathematics major should bea national priority. Otherwise our nation will miss theopportunity to make full use of the potential of all ofits students.

To encourage continued discussion of these impor-tant issues, our group recommends a national confer-ence of educators and mathematicians at all levels todiscuss issues surrounding the encouragement of minor-ity success in mathematics.

ROGERS 3. NEWMAN is Professor of Mathematics atSouthern University at Baton Rouge, where he served asdepartment chair for twelve years. He is President of theNational Association of Mathematicians and served as Deanof the College of Sciences and Humanities at Alabama State

76 REPORTS

University. He received a Ph.D. degree from the Universityof Michigan.

EILEEN L. POIANI is Professor of Mathematics at SaintPeter's College and Assistant to the President for Planning.A former Governor of the New Jersey Section of the Math-

Role of Teaching Assistants

Bettye Anne Case and Allan C. Cochran

The discussion group was concerned with the use ofteaching assistants (TA's) in a "lean and lively" cal-culus. Unsure of the exact nature of such a course,we made an attempt to consider some of the ma-jor issues of calculus instruction by TA's and othernon-regularfaculty, and to project possible effects ofchange.

Realities of the current situation, like a bad back,will continue to plague us whether or not we changethe content or teaching of calculus. These grim reali-';es include a lack of money, poorly prepared and scarcefaculty, inadequate facilities, inadequate preparation ofstudents, tension between research, teaching and servicedemands, and more.

Much of the discussion of this group reflected theproblems expressed in Teaching Assistants and Part-time Instructors: A Challenge [1]. The Foreword tothat work began the discussion and seems appropriatehere:

The dilemma of the beginning professor in our publish-fast-or-perish academic world is whether to devotetime almost entirely to research or to put effort inteaching. Graduate teaching assistants must walk,even more, a thin line to acquit their teaching dutieseffeztively and responsibly and enji y teaching whileefficiently pursuing studies and research. Discoveringhow to help than is our goal.

In most mathematics departments there is a se-rious effort to help graduate students teach effec-tively, but the number of regular faculty members in-volved in this effort it necessarily small. Those in-volved faculty members mr.j not find much agree-ment among their colleagues on these matters, andthey find even less information on helpful activities.Compounding the nroblem is thy current diversityof college teachers who are not in the professorialranks. noth graduate departments with graduateteaching assistants and two- and four year college de-partments may have part-time and temporary teach-

erratical Association of America, she is founding NationalDirector of the Women and Mathematics (WAM) program.She is currently President of Pi Mu Epsilon and chairs thÛ.S. Commission on Mathematical Instruction of the Na-tional Research Council. Dr. Poiani received her Ph.D.from Rutgers University.

ers. In graduate departments they teach the samelevel courses as graduate teaching assistants and aredrawn from graduate students in other disciplines,uneergraduates, moonlighters who are employed ingovernment, industry, high schools or other colleges,and those who are sometimes called "gypsy schol-ars."

The role of teaching assistants (and other non-regularfaculty) in the tear. mg of calculus (lean and lively ornot) is of serious concern. A recent survey of collegesconducted by the Mathematical Association of Amer-ica Survey Committee shows the percentage distribu-tion by type of faculty for all single-insti actor sectionr,[2] :

Mainstream Non-MainstreamFaculty Type: I II III I II

FT ProfessorFT InstructorPart-timeTeach. Assist.

70% 73% 82%9% 14% 10%6% 4% 3%

15% 9% 5%

47%13%

13%

25%

15%

12%23%23%

More teaching by non-professorial 1:Ficulty is indicatedwhen the survey population is restricted IA, .1epartmentshaving graduate programs [3]:

Individual Calculus Classes (< 65) Taught By:

Tenure Other Fulltime: Pt-timeTrack PhD Non-PhD TA Instr.

GP1-2GP 3MA-TAMA

46%61%74%

73%

2%1%

2%6%

41% 6%23% 7%3% 10%0% 13%

The categories GP1-2 and GP 3 stand for insti-tutions with doctoral programs classified according

00

e.CASE AND COCHRAN: ROLE OF TEACHING ASSISTANTS 77

to [4]. MA-TA stands for masters-granting depart-ments with teaching assistants, while MA stands formasers- granting departments without teaching assis-tants.

Many of the views expressed in this discussion groupreflect a wide range of constraints on the participantschools, including many shared problems:

It might be Pasier to effect change with teaching as-sistants than ar.ross the "hard-core" regular faculty.At the least, TA's would not present unsolvable prob-lems. One reason is that only the best teaching as-sistants teach calculus.The exploitation of "gypsy scholar" and "permanen4visitor" instructors is deplorable on both ethical 7.ndhumanistic grou'ds. No one had realistic solutionsfor this problem.There is tension between academic freedom and"what should be taught" which may be heightenedby changes in the calculus.There will be a lot of expense in training peoplefor a new mode. A. TA present in our group de-scribed current computer applications in his calcu-lus class; his supervising professor pointed out thatthis TA only teaches one section to allow time forpreparation. A 50% drop ih productivity wouldbe difficult (impossible) to handle in many depart-ments.There were no good ideas for how to retrain fac-ulty.

4. Hand-held calculators are feasible, but real computerexperiences for all calculus students would not befeasible in a short or medium time frame at n )stschools.

The need for gradual cllanges, for pilot sections andprograms, was repeatedly mentioned. (Esamples:Syracuse tried computer use in calculus in honorssections first; Clemson has three sections of calculususing the HP28Cfurnished this one time at reducedcost.)The need for sharing and keeping resource files fromsuccessful classroom experiences was mentioned as agreat help in producing good instruction.A large amount of time is needed for preparation, andthere is a need for lead time in preparing demonstra-tions.

Discussion of departmental "coping devices" pro-duced a variety of anecdotes and opinions: classesover size GO (but under 100) having a grader butno recitations were often felt, given a suitable room,to work well. Lecture-recitation mode-100 to 700studentswas considered as a last resort by most par-

ticipants. However, one member of the group relateda case where a change to smaller sections broughtmore complaints -id less enrollment. SuLsequently,a change back to the lecturer-recitation mode wasmade.

Another "coping device," uniform departmental testsand syllabi, were blamed as the cause of "chug-and-plug" courses. Some felt that poor student perfor-mances over the years forced an easier course. Thesuggestion was made that, in fact, the major copingdevicethe common syllabusdid not go far enoughin prescribing word problems and simple proofs.

The single recurrent theme of this discussion groupwas that teaching by teaching assistants and part-time instructors, although a very signifiLant problemat schools with graduate programs, would not be the,or even a, major hindrance to changes in the teaching orcontent of calculus. A generally positive attitude washeld concerning teaching by these people, along withconcern for the dual demands of being a student andbeing a teacher.

Reference

[1) Case, Bettye Anne, et al. "Teaching Assistants and Part-time Instructors." MAA Notes, 1987.

[2] Anderson, R.D. and Loftsgaarden, Don 0. "PreliminaryResults from a Special Calculus Survey." MAA SurveyCommit t :a.

Lase, Bettye Anne, et al. Analysis of partial returns fromthe "1987 Survey of the Committee on Teaching Assis-tants 1,nd Part-time Instructors." Mathematical Associ-ation of America, to appear.

[4] P.urg, Donald. "CEEP Data Reports: New Classifica-ticli of Graduate Departments." Notices of Amer. Math.Soc. 30:4 (1983) 392-393.

(31

BETTYE ANNE CASE is Associate Professor of :,a,...di-ematics at Forida State University where she directs theteaching assistants. She chairs the CUPM Subcommittee onthe Undergraduate Major and is Director of the Project onTeaching Assistants and Part-iime Instructors of the Mathe-matical Associatior of America. She received a Ph.D. degreein mathematics frGai the University of Alabama.

ALLAN C. COCHRAN is Professor and Vice Chairmanof Mathematical Sciences at the University of Arkansas atFayetteville, where he directs the instructional program. APast Chairman of the Oklahoma-Arkansas Sect;on of theMathematical Association of America, he received a Ph.D.degree in mathematics from the University of Oklahoma.

8 D.

78 REPORTS

Computer Algebra Systems

First Discussion Session

John W. Kemeny and Robert C. Es linger

The discussion group participants represented a va-riety of experiences with computer algebra systems(CAS). They ranged from individuals with no first-handpersonal experience to authors of some of the currentsystems. At least one -third of the pa,t.icipants were cur-rentiy engaged in curricular activities using CAS. (A listof the projects represented in the group is included atthe end of this report.)

Amidst a lively discussion, the group reached agree-ment on several major issues. These included recog-nition of several forces for change in the approach tocalculus instruction. Future students will be enteringcollege with increasing experience with technology, inparticular with graphing calculators and microcomput-ers. With this experience they will tend to see much ofthe current instructional material as being out-of-date,and to some extend irrelevant in today's environment.

Pressure for change will be exerted, many believe,by faculty in allied departments. They will see the con-tent of the current calculus course as inappropriate fortheir students who operate in a computer-oriented atmosphere. In addition to these technological forces, thegroup felt we need to be sensitive to the pedagogicalimplications of research in cognitive psychology.

Many in the grc.tp were concerned about skepticismof their mathematical colleagues toward the use of CASin calculus instruction. It was noted that the math-ematical community was, in general, very conservativein their choice of textbooks and reluctant to supplementtextual material with their own notes or assignments.This reluctance is incongruent with the accepted needfor experimental text material to support change in cal-culus instruction. Unfortunately, there was no agree-ment on the mechanisms to develop these materials.

This discussion group benefitted especially from thecontributions of several calculus text authors. In par-ticular, the authors noted that the content of currenttexts reflects market pressures of conservative facultyinterests in including specific topics. In this context,the group agreed that they could not select specific al-gebraic skills, for example, that would be rendered ob-solete by the use of CAS.

Black Box SyndromeThere were also issues on which participants voiced

disagreement. A primary point of dispute could be char-acterized as the "black box syndrome." Some partici-pants were concerned that students would use mechani-cal systems without understanding underlying concepts.Others argued that these systems allow students to con-centrate on higher-level reasoning instead of on lower-level manipulative skills.

For example, on-going projects were described inwhich solutions to word problems were broken downinto appropriate steps by students using CAS. It wasalso pointed out that these systems facilitate studentexploration, thereby leading to discovery learning.

Disagreement also arose over the complexity of theuser interface with CAS. Some felt that the difficulty inlearning these systems was sufficient to inhibit their useby many faculty and students. It was argued ,ha;, fac-ulty were reluctant to abandon traditional approachesto teaching for uncertain results with a system thatrequired a substantial investment in time in order tomaster. There was also concern that students' mathe-matical matuiity upon entering calculus was inadequateto use these systems productively. In contrast, partici-pants who were using CAS indicated that initial train-ing in its use was not a significant barrier.

In summary, participants in this CAS discussiongroup concluded that the mathematical community isin the experimental phase of developing effective use ofCAS in the teaching of calculus. Some expressed frus-tration with the lack of a clear statement either iden-tifying problems with current practice in calculus in-struction or outlining objectives for change. However,the following major issues were identified by the group:

Student preparation for calculus.Balance between conceptual understanding and cal-culation.Relevance of iculus to allied fields.Utilization current technologies.Resistance to change.Evaluation of experimental methodologies.

CAS ProjectsTryt, Ager, Stanford University: Build'ing an on-line

calculus instructional system which uses REDUCE asan "algebra engine" and interactive graphics as an ex-pository device. The project is directed toward pre-

90

7,ORN AND VIKTORA: COMPUTER ALGEBRA SYSTEMS

college calculus, but covers all material in the AP cal-culus syllabus.

George Andrews (with Michael Hen le), Oberlin Col-lege: Supplementary materials for calculus sections us-ing Maple. Each exercise is designed tc fit into the tra-ditional curriculum and provide a deeper understandingof some aspect of the calculus.

Dwayne Cameron, Old Rochester Regional SchoolDistrict: Using muMath in honors algebra II, precal-culus, and calculus classes to a limited degree.

J Douglas Child, Rollins College: Macintosh Inter-face to Maple. Experimental lab course to better under-stand (and improve) student problem-solving processes.

Joel Cohen, University of Denver: Teaching a junior-senior level course in symbol manipulation for appliedmath-maticians, physicists, and engineers, using a num-ber of computer algebra systems; also experimetingsome with Maple in calculus.

Abdollah Darai, Western Illinois University: UsingMACSYMA for the first time in a first-year calculuscourse.

Franklin Demana (with Bert Waits), Ohio State Uni-versity: Developing a precalculus text that makes sig-nificant use of graphing calculators or computer-basedgraphics.

John S. Devitt, University of Saskatchewan: UsingMaple as an electronic blackboard.

Harley Flanders, University of Michigan: Developingsoftware specifically for teaching calculus (and precalcu-lus). Includes symbolic manipulation, plane and spacegraphics. Classroom testing.

Richard P. Goblirsch, College of St. Thomas: Haveoffered calculus with numerical emphasis using comput-ers; have begun experiments with SMP.

Edward L. Green, Virginia Polytechnic Institute:Employ IBM PC in cz.:culus sequence; have partiallydeveloped graphics packages.

Don Hancock, Pepperdine University: Developingmaterial for a calculus course using MACSYMA. Thecurriculum will emphasize the interplay between dis-crete and continuous ideas.

Alan Heclombach, Iowa State University: UsingFlanders' Microcalc for Calculu,. using muMath inMaster of School Mathematics Program.

M. Kathleen Heid, Pennsylvania State University:Projects using symbol manipulation programs:1. Algebra with Computers. Continuing and pending

NSF projects using muMath and programming inhigh school algebra; examines the numeric, graph-ical, and symbol-manipulative connections.

2. Computer-based general mathematics, using mu-Math for 10th and 11th grade students.

3. Applied calculus course using muMath and graphi-cal programs to refocus course on applications andconcepts. (Report in upcoming issue of Journal forResearch in Mathematics Education.)

4. Algebra I with muMath, focuses on word problemsusing the symbol manipulator to perform routineprocedures prior to student mastery.Jame° F. Hurley, University of Connecticut: Com-

puter laboratory course using TrueBasic to graph anddo numerical experimentation directly tied to conceptsof calculus, and to motivate them.

John Kenelly, Clemson University:1. Freshman calculus section covering regular depart-

mental syllabus. Students are issued individualHewlett-Packard HP 28C calculators.

2. State grant for 160 secondary mathematics teachersto eceive and be trained in the use of Sharp EL5200super-scicntific graphic calculator. Study of cu^.ricu-lum implications of graphing calculator technologyare part of the course.Andrew Sterrett, Denison University: Using Maple

in two sections of calculus.

JOHN W. KENELLY is Alumni Professor of Mathemat-ical Sciences at Clemson University. He has been Chair ofthe College Boards' Advanced Placement Calculus Commit-tee and he now chairs their Academic Affairs Council and isDirector of the Adve..t.ced Placement Reading for the Educa-tional Testing Service. A member of the Board of Governorsof the Mathematical Associati-n of America, he also chairsthe MAA Committee on Placement Examinations. He re-ceived his Ph.D. degree from the University of Florida.

ROBERT C. ESLINGER is Associate Professor of Math-ematics at Hendrix College. Until this year he served asChairman of the Department of Mathematics and Head ofthe Division of Natural Sciences. He was Governor of theOklahoma-Arkansas Seeion of the Mathematical Associa-tion of America and currently serves as a national Council-lor of Pi Mu Epsilon. He received a Ph.D. in mathematicsfrom Emory University.

Second Discussion SessionPaul Zorn and Steven S. Viktora

The discussion section began with a simulated dem-onstration of the computer algebra system SMP. Next,a large number of participants described computer-oriented calculus projectssome involving CASat avariety of institutions, large and small.

0 10 A.

80

Computer Algebra Systems (CAS) already affect theteaching of calculus and will likely play an even greaterrole in the future. The defining feature of CAS is asymbol manipPlator. Most systems do arithmetic, op-erations with polynomials, linear algebra, differentialequations, calculus, abstract algebra, and g' apps. Thegroup discussed how CAS are used now and \ hat effectthey may have on the teaching of calculus in the future.

A surprisingly large number of colleges and universi-ties aiready use computers to aid in teaching calculus.Most approaches seem to fall into one of two categories.Some institutions use what might be called special pur-pose packages. Individual programs, whether locally orcommercially pr_duced, are typically used only for cer-tain topics, not for the whole calculus course.

-Other institutions use "full service" CAS programsfor a wide range of tasks. A common characteristic ofthese courses is an emphasis on problem solving. Ex-ploratory projects, realistic applications, and modelingare stressed.

After participants described computer-oriented proj-ects at their own institutions, discussion turned 5 moregeneral issues: problems, strategies, and prospects forthe future. Four main themes and issues emerged:

1. The need to defineand agree upongoals andobjectives for calculus instruction. No clear consen-sus exists on what calculus is or should be. Perfectagreement is impossible, but it should be possible toagree upon a small finite number of acceptable calculuscourses.

Among the questions: Which particular traditionalcalculus skills and topics are really essential? More gen-erally, what balance should be struck between routinealgorithm performancedifferentiation, antidifferentia-tion. etc.and "higher-order" activitiesproblem-solv-ing, mathematical rxperimentatic.i, and understandingof concepts? Shall we proceed conservatively (tinkeringwith the present standard course) or radically (design-ing a new course from the ground up)?

There was no systematic effort to answer all of thesequestions. It was agreed, however, that future calculuscourses will differ significantly, not just in details, frompresent courses. Developments in computing will forcesuch changes.

2. How will computer algebra systems and otherforms of computing drive, or be driven by, changes incalculus instruction? Discussion centered on both cur-ricular and pedagogical matters. Among curricular ef-fects of computing, the following were cited:

Computers handle routine operations; hence, timecan be spent on better things.

REPORTS

More realistic applications are possible when compu-tations are cheap.Approximation and error analysis are important anduseful dimensions of calculus, but they are compu-tationally expensive. Cheap, easy computing solvesthis problem.Some traditional topics and methods arose histor-ically from the high (human) cost of computation.Cheap machine computation render- them obsolete.Computing may save time in the curs....ulum. On theother hand, present calculus syllabi are crowdeditwould be a mistake simply to add more material.Whether we like it or not, the calculus curriculumwill change. Hand-held machines, if nothing else,will force this; too many traditional exercises becomeinane when perforriied-by computer.The effects on calculus pedagogy of modern comput-

ing are just beginning to be felt, and so are hard topredict confidently. They might include:

A more active, experirnental attitude of students to-ward mathematics, supported by less painful manip-ulations. Conjectures can be made and understoodas part of the process of mathematical proof.With computers, mathematical objects can be rep-resented graphically and numerically as well as alge-braicallg. This should lead to a deeper, more flexibleunderstanding of "function," for example.Computers permit a larger sheer number of examplesand exercises to be worked out. This could speedstudents' development of .nath, ,,,atical intuition.Computers could support more quolitative reasoningin mathematics. E.g., students could see concretelythat polynomials grow more slowly than exponer tialfunctions.With computing to handle details, students can carryout multi-step problems without foundering in cal-culations. This should foster more effective problem-solving.

3. New approaches to calculus require new materi-als. books, software, problem sets, etc. Who will writethem, and with what rewards? How will re-inventionof the wheel be avoided? Many participants reportedfinding standard text and exercise materials unsuitablefor computer-aided courses. Writing suitable materialis difficult and tirro-consuming. It was agreed that co-operation is essential in developing and sharing prob-lem materials. NSF was mentioned as a possible sourceof support for distributing materials outside the usualcorr.ilercial channels. Meetings of pee. )le involveu incomputer projects would also help spread the word, andbuild morale. The Computer Algebra Systems in Edu-cation Newsletter, based at Colby College, is a start.

92

BUCCINO AND ROSENSTEIN: OBJECTIVES, TEACHING, ASSzSSMENT 81

4. It is widely agreed that calculus courses shouldbecome more conceptual. Clearly, computing can helpsupport this goal. But what problems do such changesraise, and how will we solve them? CAS's can do muchof what we traditionally teach. If CAS's do handle suchtasks, what will students do? Will their algebra skillsatrophy, or perhaps, aided by better motivation andmore varied experience, even improve? How will moreconceptual matters be tested? Can a more conceptualcalculus be taught successfully in large, bureaucraticsettings?

No agreement emerged on such questions. It wassuggested, though, that we mathematicians should notapologize for asking for more resources to solve theproblems attendant upon improving calculus instruc-tion, any more than our natural science colleagues dofor requiring their version of laboratory resources.

The group -.-:as unable to agree on, or even discuss atlength, every issue raised. Some open questions raisedduring the discussion follow:1. How will the calculus syllabus be affected by CAS?

What topics should stay, and which should go? Whatshould be added?

2. How will changes in college calculus affect the wholeprecollege curriculum, not just the Advanced Pli.,e-

ment program?3. What is the "right" machine (if there is one)?4. What ,.lakes a "friendly" CAS? How do we ensure

that CAS are easy for students to use?5. Should programming be taught? If so, how much?6. Where do hand-held calculators fit in? Present cal-

culators are not yet as powerful as CAS, but they arerelatively inexpensive and convenient.

PAUL ZORN, Associate Professor of Mathematics at St.Olaf College, is currently a visiting professor at Purdue Uni-versity. He is co-leader of a project, supported by the Na-tio .al Science Foundation and the Department of Educa-tion, to integrate numerical, graphical, and symbolic corn-putin:, into mathematics. Dr. Zorn received a Ph.D. degreefrom the liniver-'4y of Washington.

STEVEN S. VIKTORA is Chair of the Mathematics De-partment at Kenwood Academy, Chicago. Previously hetaught in teacher-training colleges in Ghana for five years.He served as President of the Metropolitan MathematicsClub of Chicago, and is currently an author for the Univer-sity of Chicago School Mathematics Project. He received anM.A.T. in mathematics from the University of Chicago.

Objectives, Teaching, and Assessment

First Discussion SessionAlphonse Buccino V George Rosenstein, Jr.

This group unanimously agreed that calculus teach-ing needs improvement. Opinions varied on the exis-tence and degree to crisis. Despite the differences therewas clear enthusiasm regarding the need for the "Cal -'ulus for a New Century" program.

ObjectivesThe objectives of the calculus sequence must be

reformulated within the context of the objectives forthe undergraduate mathematics major and of under-graduate education generally. Within this framework,we identified several :ategories of objectives, includ-ing:

Transcendent purposes: communicating the power,excitement and beauty of calculus.

Understanding: calculus as a tool for modeling real-ity.

Skill: symbolic manipulation.

Development of faculties: geometric intuition and de-ductive skill.

Rite of passage: calculus as a talent filter formathematics majors and students in other disci-plines.

Generic skills: writing, reading, and note-taking.There was strong agreement that the higher-order

cognitive objeca es of insight and understanding shouldbe emphasized.

These higher-order objectives are strongly relatedto skill development and manipulation. In any re-examination of the objectives of calculus, the rela-tionship between objectives at different cognitive levelsmust be emphasized. Additionally, the responsibilityof mathematics and mathematicians to,iard the other

82 REPORTS

disciplines that utilize calculus must be considered andclarified. Finally, calculus courses need to be fit intoa context that includes goals for majors and for gen-eral education, as well as the preparation of studentsfor calculus.

TeachingThe participants agreed that whatever objectives

might emerge for a particular course, students ought tobe cognizant of them. Daily interaction (e.g., quizzes,"board problems," graded homework) provide valuableinformation for the student about the objectives ofthe nurses and the teacher's expectations, and for theteacher about the progress of the class and of individu-als. Feedback in this form must not be merely routinedrill.

Instructors are often too distant from their studentsand often are unaware of serious problems or signifi-cant successes iriividuals may be experiencing. Conse-quently, a climate of sensitive awareness with substan-tial feedback should characterize calculus classes. Reg-ular accountability and close scanning to assess statusand progress should occur.

There was some discussion of the role of textbooksin teaching calrilus. To a large extent, books de-fine both the content and style of presentation inour standard courses. In particular, textbook prob-lems seem to stress drill and template applications,often relegating "interesting" problems, should theyoccur, to the end of a list of forty or fifty prob-lems.

Although courses appear to depend on the textbook,it was noted that class time is frequently spent re-peating the material in the text. Many believed thatthis repetition is peculiar t mathematics. Some par-ticipant:. reported that students would read the bookif the instructor made his or her expectations clearand endeavored to enforce them. Participants gener-ally agreed that teaching students to r,:ad the text wasworthwhile.

Writing was another generic skill that teachers be-lieved was worthy of class time. Writing for clarityenhanced students' higher-order thinking skills, forcedstudents to consider their results, and increased stu-dents' appreciation for the problems of writing for thebenefit of others. Participants mentioned journals andsummaries as post-learning activities and brief "writeeverything you know about ..." exercises as introduc-tions to topics.

AssessmentIn discussing assessment, the group focused on test-

ing to assess student progress and achievement. Thereis a substantial need for consciousnes, raising, facultydevelopment, and technical assistance in the construc-tion and administration of good tests based on currentknowledge about assessment.

However, there is also a substantial need for re-search and development on assessment and testing toadvance the state-of-the-art, especially with referenceto the higher-order cognitive objectives.

Although some members of the group questionedthe accuracy of the profile of present tests as 90%manipulation and template problems, there was gen-eral agreement that tests frequently do not ask stu-dents to perform higherorder conceptual tasks. Withsome prodding, the group produced several kinds ofquestions that seemed to require more than routineskills

Assessment of transcendent objectives probablycould lot be fit into a grading scale, but should bepart the evaluation of any course. These objectivesdo not fa most teaching evaluation procedures either,nor, apparently do some other higher -level objectives.Several members of the group expressed the view thatthese evaluations thereby interferer with true teachingeffectiveness.

What To DoOne major omission from the Colloquium program

was discussion of research and development on theteaching and learning of calculus. Development of aprogram providing support for coordinated researchprojects and activities is essential. In addition, thevalue of natural experiments that continually occur toimprove calculus teaching and learning can be enhancedthrough support of such things as clearing houses, com-munication, reporting, and travel. Research and devel-opment should include investigation of experiences andpractices in other nations.

Dialogue with colleagues in other disciplines shouldoccur. This can be a grass-roots effort involving suchsimple things as brown-bag lunches. Efforts pyramid-in upward to an integrated and synthesized perspectiveon calculus and its relation to other disciplines are alsoneeded.

Course and classroom testing is a major problem forall college teaching, and not just calculus. Instructorssimply are not trained in even the rudiments of test

CURTIS AND NORTHCUTT: OBJECTIVES, TEACHING, ASSESSMENT 83

construction and administration. Each campus shouldaddress this problem and provide necessary technicalassistance to improve the situation.

Partici:sants saw advanced technology as an oppor-tunity for re-examining the objectives of the calcu-lus. Because advanced calculators and recent com-puter software trivialize many of the computationalskills on which many courses seem to be based, we willsoon be forced to confront the inadequacies of thesecourses.

ALPHONSE BUCCINO- is Dean of the College of Edu-cation at the University of Georgia. Previously he servedin several management positions in the Directorate of Sci-ence Education at the National Science Foundation, and asChairman of the Mathematics Department at DePaul Uni-v,Asity. He received a Ph.D. in mathematics from the Uni-versity of Chicago.

GEORGE M. ROSENSTEIN, JR., is Professor of Math-ematics at Franklin and Marshall College. He received hisPh.D. from Duke University in topology. His current re-search interest is in the history of American mathematics.He is presently the Chief Reader for the Advanced Place-ment Examination in mathematics.

Second Discussion SessionPhilip C. Curtis, Jr. and Robert A. Northcutt

Our group discussed these three topics from two per-spectives: proposed changes in orientation of the calcu-lus course from one devoted primarily to technique toone devoted to conceptual understanding, and increas-ing .utilization of computing technology in instructionboth in calculus and in preparatory subjects.

There was general agreement that a change of em-phasis in the teaching of calculus is a desirable goala change from a technique-oriented course with appli-cations to one where the major emphack was on un-derstanding major concepts. There wa J consensus,however, on the extent to which this n w goal could beseparated from the current goal of technical mastery.

If emphasis on concepts is to be achieved, however,there are several necessary ingredients which are notnow present. First, there has to be general agreementon the part of the teaching staff that this change isdesirable and that individual efforts in this directionshould form an important part of a faculty member's

professional life. Teaching techniques would then needmuch more attention and support than is now the case.

Secondly, the orientation of the textbooks Twistchange to include much more of an emphasis in bothtext and exercises on the understanding of conceptsand their applications to problems. Central to this suc-cess will be high-quality feedback from the instructor,teaching assistant, and homework reader. Understand-ing and communication of ideas on the part of the stu-dent will not be achieved without this feedback.

The similarity to the problems encountered in theeffective teaching of writing is inescapable here. Al-though technique-oriented questions will probably notdisappear from examinations, concept-oriented ques-tions must form a more important part than is now thecase. Both students and faculty should be well awarethat this is the orientation of the course and of its as-sessment.

It ;s clear that this change in orientation will notoccur quickly; it will be difficult and mistakes will bemade. It should be realized from the outset that ifthis change is to be successful it will be an evolutionarydevelopment rather than a revolutionary one.

It is inescapable that technology in the form of in-creased computing power will play an increasingly im-portant role in the teaching of calculus. It already doesto some extent and this role will increase rapidly. Thechallenge will be to manage the role of technology in away that deepens the mathematical understanding andcapabilities of students rather than just replace whatthey now do by paper and pencil.

The opportunities are many. understanding thefunction concept should be greatly enhanced by taegraphical capabilities of the new calculators. Deriva-tives as rates of change and areas as limits of sums canbe easily visualized where this vp.. --'y imperfectly re-alized before. More realistic problems can be confrontedwhere parameters are not carefully chosen that solu-tions can be given in closed firm.

The role calculators will play in the elimination ofsymbolic calculation at this point is not as clear. Al-gebra is the language in which mathematical problemsare pk -,ed. Correctly stating the problem, organizing itin such a way that it can be attacked, and finding aroute to a solution are always the more difficult partsof problem solving. It is certainly conventional wisdomthat at this juncture technical mastery forms an impor-tant part (,r insight. The challenge will be to utilize thetechnology in such a way that will preserve and deepenthe insight, but lessen the burden of tedi,, is algebraicand numerical calculation.

Our challenge will be to manage the changes pro-

84 REPORTS

posed, rather than to let the many forces outside math-ematics dictate the directions. Historically, we have notdone a good job in either the implementation of new di-rections or the necessary follow-up. "New math" is theclassic example. The necessity to carefully plan, realis-tically implement, and train teachers are major obsta-cles.

T.raining faculty through better apprenties/mentor:ograms, giving appropriate rewards for successful

teaching, and increasing recognition of the professionalaspect of calculus will help. Bad practices in classpresentation, assessment, and materials must be elimi-nated. The mathematics profession must take an activerole in this management process. Calculus is a part ofthe over-all fabric of mathematics and as such, instruc-tion prior to calculus and after calculus must reflectan awareness of what is happening in the calculus, andwhy.

Many students do not now develop an overview es-sential to understanding calculus. Technology shouldallow instructors to broaden objectives in this area.The function concept, change, qualitative behavior, andglobal insights can become more accessible for students.Problems In background review, remediation, and prac-tice can be addressed using calculators and computers,but this alone will not solve the problems of apathy, mo-tivation; or attitudes. Students a190 must bear respon-sibility for the process of their learning. Good teachingrequires both an active teacher and an active student.

The introduction of technology into the calculusclassroom will not be devoid of problems. Technologyhas a good side in its utilization; but misuse can polar-ize many of our present faculty, instill a false expecta-tion in problem solving, and may give rise to a host ofmaterials so specialized as to make faculty resistant toinnovation.

Finally, there should be a conscientious effort on thepart of the mathematics profession to involve secondaryschool teachers in the design and in the implementa-tion of any proposed changes. It should be the goalof all concerned to narrow the spectrum of preparationof students for calculus to an acceptable and accessiblerange. In the future, an increasing number of the betterstudents will be receiving their single variable calculusinstruction in the secondary schools. Consequently, aclause, understanding of the desired goals of calculusinotruction is essential at all levels.

PUMP C. CURTIS, JR. is Professor of Mathemat-ics at the Uni tersity of California at Los Angeles. He hasbeen active :or several years in the University of California's

state-wide programs on admissions and articulation with theschools. A former Fulbright fellow, Dr. Curtis held severalvisiting faculty positions and served as chair of his depart-ment. He received a Ph.D. degree from Yale University.

ROBERT NORTHCUTT is Professor of Mathematicsat Southwest Texas State University. Formerly Chairmanof his department, he is the Texas Section G:vernor ofthe Mathematical. Association of America. He received hisPh.D. degree from the University of Texas at Austin.

Third Discussion SessionLida K. Barrett and Elizabew J. Teles

It is nn secret that calculus, as it is now taught, isnot understood or appreciated by most students. Themathematical power and beauty of calculus that math-ematicians know and relish cannot be delivered to stu-dents in the context of our present calculus course. Ourgroup identified this problem as the crucial issue in cal-culus.

There were basically two parts to our discussion. inthe eginning part, the group identified five issues ofimportance. The first centered around the fact thatstudents do not understand or appreciate calculus, butsee it as a selection of rules and problems which theymust memorize in order to be allowed to proceed. Cal-culus is therefore seen as a filter, or a rite of passage.According to Robtrt White, it is time for this to end:calculus must become a pump, not a filter. . order forit to become this pump, mathematicians mus marketcalculus as they know it both to students and to clientdisciplines.

The second issue is that the content of calculus is seenas rigidly controlled by a large number of forces. Amongthese are client disciplines, accreditation boards, de-partmental expectations, and t'cx600ks. Accurate curlrent information on the needs of client disciplines mightyield a different perspective, perhaps even support formore teaching of concepts and less manipulations. '11,-actual specifications of accreditation boards should ticdetermined. Faculty instead of the textbook should de-termine the syllabus and the course.

A third constraint is the quality of instruction: teach-ing by increasing numbers of part-time faculty and grad-uate students; the inability of the faculty to communi-cate with students; and the insistence of most univer-sities that all tenured faculty carry on research. It wasgeneral./ agreed that institutional rewards of rank :endsalary, in most settings, are greater for mathematical

9'6

BARRETT AND TELES: OBJECTIVES, TEACHING, ASSESSMENT 85

research than for quality teaching, curriculum develop-ment, or research on teaching.

In addition, international teaching assistants as-signed to recitation sections often fail to communicatewith students, not only because of their accents but alsobecause of their inability to comprehend the underlyingmeaning of the questions. Lastly, even though calculusteachers often give lip service to the problem-solvingapproach, it is often not the technique actually used inthe classroom.

Articulation among all levels of mathematics is a veryimportant issue since success in calculus depends on thestudent's previous mathematical experience. In orderthat calculus be taught for understanding and appreci-ation, precalculus and other high school courses mustsupport this endeavor. Of particular importance is ar-ticulation among high schools, two-year colleges, andfour-year colleges and universities on the calculus courseitself.

Finally, the role of technology in the Calculus fora New Century was explored. It was readily agreedthat technology should play an important role but thatits use should be carefully constrained. There is justas much danger in the improper use of ,.ethnology asthere is in ignoring it altogether. Technology must bea facilitator rather than an appendage.

To address these issues will require changes first ofall in the reward structure for faculty, supported by ad-ministrations and departments. Changes will also beneeded in teaching methods brought about in part bythe new technology. Other new methods could includegreater use of a historical perspective, more student mo-tivation for problems by ensuring that they understandthe question before they are asked to respond, and theteaching of differentiation and integration rather thanthe manipulative skills of differentiating and integrat-ing. Furthermore, long-term commitment to teachingmust be encouraged. The primary resource neededfaculty timemust be made available and faculty effortmust be rewarded.

In the second half of the session, the group exploredobjectives, teaching, and assessment. Many suggestionswere m.scussed in the short time available, and consen-sus was reached in many areas.

Students must be taught how to think about newproblems as well as how to solve template problems.An alliance must be built with client disciplines to aid

not only in the development of courses but in the de-velopment of support for courses. Teachinb style andpresentation of material must convey enthusiasm forknowledge and an appreciation of the beauty of cal-culus.

To support better teaching, the group as a wholerecommended a newsletter-type publication that wouldcontain course outlines, teaching techniques, reports ofexperiments, and examples of new types of problems(e.g., open-ended problems and problems that use mul-tiple techniques).

Of great importance to good teaching is a well-organized assessment program. Assessment was seen asneeded in a variety of waysassessment of students asthey enter courses (i.e., placement, high enough as wellas low enough), assessment of curriculum, assessmentof faculty and teaching, and assessment of the accom-plishments of the course. Whether a course meets itsobjectives is often difficult to determine. New courses,however, must be assessed by new standards, and notby standards established for old courses.

The primary objective must be marketing of the newcalculus as a gateway to future study in the majority ofother disciplines. Some skills needed in client disciplinesare changing, contact with these disciplines must be es-tablished in order to appropriately reflect their changesin our courses. However, it is up to mathematicians toconvince others that a lean and lively calculus will meettheir needs. It is important at this time that mathe-maticians become proactive rather than just reactive.

LIDA K. BARRETT is Dean of the College of Arts andSciences at 1Vlississippi State University. Previously, she wasAssociate Provost at Northern Illinois University and headof the Department of Mathematics at the University of Ten-nessee. President-Elect of the Mathematical Association ofAmerica, Dr. Barrett received a Ph.D. from the Universityof Pennsylvania.

ELIZABETH J. TELES is Associate Professor of Mathe-matics at Montgomery College, Takoma Park. She is Chair-man of Maryland-Virginia-DC Section of the MathematicalAssociation of America and currently is completing a Ph.D.in Mathematics Education at the University of Maryland.She has published papers on assessment testing, on use ofcomputers in mathematics, and on gifted middle school stu-dents.

ISSUES

cltir4f5,/

i

e- f'--TA-RTICIPA-NT RES-ForvsQs 03

LLestiavN IA) 1A A.+" CAA-Le. `me- -1) r.:On (iP &_1 fri43-ALe.LKLA. e,ss e_..s I v)______Cal c& ikstYlAcf

Av.swe-ISli.e._ tow pri ore-t-. cl i il e.t 10 La.

WAR:Me tiv.a.-1- Ccd CO vm Infl Lk. 1,..i. / ..

-a-nAe. a 0 es lel t± 9 eA- -t-e)-1 LA-A-e j ep ro L-"oti aYi -4::-kr

c,r, -py-es -t--1 9 e 21 ))e-11,%, Coy) Le/rin eeh ::C.-

''..'

eA 12 0 1/\_-k- f,e_s G., L,0,,,, v,--, cii, e- 1:::--.A.i- LS - ...AL

I/ t..,t a c-f-4,.. 7.z-"X it-,s tru. c_4--; CY) Lir trtAre 11-a__ 1 #v 1 f, L

Si-EA M4/.4- LC tl,e_ I( lot ct.c.k.. b cyX 1

T1-Ne- e1:{1-6vtn ri- 'HD 1/,%a_124_ 1-(- An a i 1, Ct--

.

.

-h2 etLt 07 e_e6) Ce_, - -,- -

1' 0 0 Irv% cwt. e stz,, , 1-e,vi-f-s -1--rt,P2--i k..,_:_c_ct t cry

C 0 Li tA_s i th i b e. .4 1, / e t A , 1 i I A .5 -tr. t-t ci-o 13 adv.. .611

5't-u ol_e^.).-E-s - co ti e,Nr. cAf Li/ -644-+' 72'

.

GaG1-1

t-/N Cei.3_L___A.t.4-- l'e-iNe%. 0. t-dll,

w

l'A_e o c C 6 aIY erge aid eAet c_----JT.

. 11 Gk $e C t-tx le' e.vvt- -4-e-7C t--..b 0 0 ks I GI-

The. c Go^ r t c-LA. tAA 1,1.1 0 Gt-t-a ockek, o los 0 ( e_i-e_.

1),-. e_ S C ( et) e-1 A 1 Crx-tA/1 ---- c-L- se-t"- t -e-)

i..

-:.

,17-44

1,-,A6 14 -1)6...SS p c,( :

Calculus Reform: Is It Needed? Is It Possible?Gina Bari Kolata

NEW YORK TIMES

Calculus, the "gateway to all areas of science and en-gineering," in the National Science Foundation's words,and the mathematics course most dreaded by hun-dreds of thousands of undergraduate students, is chang-ing. Mathematicians agree that because of advances incomputer technology, calculus cannot remain the samecourse that it has always been. And there is a growingfeeling within the mathematics and science communitythat the course is desperately in need of revitalization.So the real question is whether calculus will change hap-hazardly or whether its alterations can be planned andits subject matter and teaching invigorated.

The National Academy of Sciences and the NationalScience Foundation hope initiate a national debateover how calculus can and E ald be changed. In orderto start the debate, the Na.ional Science Foundationwould like to invest about $2 million a year for the nextseveral years in conferences, workshops, and demonstra-tion projects. The program bzgan with a Colloquiumon the future of calculus teaching, hell in Washing-ton D.C. on October 28-29, 1987. The colloquium issupported by the Sloan Foundation and sponsored bythe National Research Council of the National Academyand the Mathematics Association of America.

An Unprecedented EffortThe new effort to revitalize calculus is almost un-

precedented in educational circles. It would affectmathematics departments, all of which teach calculus.In fact, calculus is the overwhelmingly dominant math-ematics course taught in colleges. It would affect themore than half a million students who enroll in calculuscourses each year. It would affect textbook publishers,who invest substantial resources developing and pro-moting their calculus books, and it would affect otherscience departments who use the high failure rate in cal-culus courses as a means to eliminate weaker studentsand who frequently design their own courses aroundwhat students should have learned in calculus.

The revitalization of calculus, however, is most em-phatically not another instance of a "new math"theill-fated attempt to change the teaching of mathemat-ics at the elementary level. Bernard Madison, a mathe-matician from the University of Arkansas and calculusproject director for the National Research Council, says

that this effort will be carefully coordinated and willinvolve everyone whose life will be changed by a newcalculus course.

Mathematicians and educators havelearned their lesson from the newmath.

Mathematicians and educators have learned their les-son from the new math. Changes cannot be imposed onpeople and they have to occur gradually. "This is notjust a curriculum issue," says Madison. The entire in-frastructure of calculus must be changed. "We have tolay the groundwork for the project and give it the kindof visibility and prestige that is unquestionable. This isa political and social process."

Is Calculus Irrelevant?The movement to reinvigorate calculus began sev-

eral years ago with challenges to the very existence ofcalculus. Some mathematicians said that calculus is ir-relevant for today's students. It should be replaced bydiscrete mathematics, the sort used in computer sci-ence.

But others argued strongly against this extremeview, saying that calculus is of central importance, al-though it may no longer be the course it should be.Several meetings were held, including one sponsored bythe Sloan Foundation that took place at Tulane Uni-versity in January of 1986. The MI:me conference re-sulted in a now well-known collection of papers called"Toward a Lean and Lively Calculus" and published byThe Mathematical Association of America.

At the same time, the National Research Councilthrough its Board on Mathematical Sciences and itsMathematical Sciences Education Board was examin-ing the state of university mathematics in general, andfinding it wanting. Madison explains: "I've been look-ing at various aspects of mathematics instruction since1980, and I knew there was something wrong. Mathe-maticians in the early days thought they were just beingmistreated and not given enough money." Fewer andfewer students were selecting mathematics as a major

1.00

90 ISSUES: INNOVATION

and even fewer were going on to get Ph.D.'s in math-ematics. Yet more students than ever before were tak-ing mathematics courses because other departments re-quired them. If mathematics. courses were truly excit-ing, more students might be lured to the department.

"We have a faculty that is relatively inactive as pro-ducing scholars. Most are primarily teachers," saidMadison. Many are uninspired by their subject mat-ter and fail to inspire their students. The NationalResearch Council is now putting together an agendato reinvigorate all of college and university mathemat-ics and it considers the calculus project a specialandcrucially importantcase.

"We have a faculty that is relativelyinactive as producing scholars. Mostare primarily teachers."

Last fall, the NSF requested funds from Congress tosupport a revitalization of calculus and now the agencyexpects that Congress will appropriate those funds. Re-formers believe that the groundwork is laid for a change.But no one expects that change will be easy and no oneexpects that the impact of a new calculus course willbe confined to mathematics departments. Calculus isnot just an isolated course. It is essential for other sci-ences, including physics, chemistry, biology, computerscience, and engineering. Many colleges and universitiesrequire that students majoring in business, psychology,and other social sciences take it as well.

A Dor rinant CourseOf all the mathematics courses offered at universities,

calculus is the most well known and most often taken.The best data on enrollment in college and universitycalculus courses derives from a survey conducted everyfive years under the auspices of the Conference Board ofthe Mathematical Sciences. According to Richard An-derson, a mathematician at Louisiana State Universitywho directed the 1985 survey, more than 600,000 stu-dents took a calculus course at a college or university inthe fall of 1985the most recent data. If anything, evenmore students are taking it today. More than 40,000 in-structors teach calculus. "It's an absolutely dominantcourse," Anderson says.

For many students, their grade in calculus will de-termine whether they go on to study mathematics orscience. But, ai least in large universities, fewer thanhalf of all students who enroll in an introductory calcu-lus course complete it with a grade of C or above. And

many students who pass do so only after repeating thecourse. Ronald C. Douglas, a mathematician and deanof physical sciences at the State University of New Yorkat Stony Brook, says that at his university as many as20 to 25 percent of students in introductory calculuscourses are taking the course for the second, third, oreven fourth time.

Many students who pass do so onlyafter repeating the course.

Because calculus is considered a difficult course, itfrequently is used as a "filter" other departments useit as a way to eliminate of less-than-stellar students."Math departments often complain that physics orchemistry departments don't want to kick out the stu-dents so they figure they'll just send the students over tothe math department. Thrtt will weed them out," saysRonald Graham, director of mathematics and statis-tics research at AT&T Bell Laboratories in Murray Hill,New Jersey. Any changes in calculus, then, will changedepartments' abilities to use the course in this way.

It is even said that medical schools use calculusgrades to distinguish among their applicants. Douglassays that he has often heard from students who moanthat if their grade in calculus is not changed to an A,their dreams of becoming a doctor will be for naught.

Those leading the movement to revitalize calculusare often asked, however, why they want the course tochange and what, in fact, they mean by change. Cal-culus has been around for hundreds of yearsit beganwith Isaac Newton and has been developed by someof the greatest mathematicians that ever lived. It de-scribes such fundamental processes as motion in physicsand diffusion in biology. Science students through thecenturies have studied it. It is "a monument to the in-tellect," according to John Osborn of the University ofMaryland. What is there to change?

"What a subject is as part of thediscipline is often quite different fromwhat it is as a part of education."

First of all, says Kenneth Hoffman of MIT, it is ina sense misleading to talk of changing calculus. "Eachbranch of mathematics is whatever it is and each branchis to some extent unchanging," Hoffman explains. Cal-culus is calculus and no one imagines changing themathematics itself. However, Hoffman continues, "what

101

KOLATA: CALCULUS REFORM 91

a subject is as part of the discipline is often quite dif-ferent from what it is as a part of education. Whatcalculus is as a part of mathematics is quite differentfrom what we present to 19-year-old kids. We havecome to realize that the way we look at calculus from adisciplinary point of view is not necessarily the way weshould look at it when we teach it."

So the real question for mathematicians and educa-tors is: What, if anything, is wrong with the way cal-culus is being taught and v:hat can be done to improveit?

Reasons for ChangeThe first point that the advocates of a new calcu-

lus make is that what is being taught as calculus todaybears little resemblance to the course 20 years ago. Onereason for the change is that the group of students tak-ing calculus is different and the course has been adaptedin response.

During the past 20 to 30 years, college education hasbecome more commonplace. Now the masses, ratherthan just the wealthy and privileged, go to college, andtheir high school mathematics education frequently iswoefully inadequate. To accommodate these students,calculus courses were made less demanding. No longerare students asked to do proofs, for example, only towork out simple problems that are exactly like worked-out examplesiin their textbooks.

Different departments at colleges and universitieshad their own reasons for wanting students to take cal-culus, and everyone has a say in the course material."Every user of calculus got a word in and calculus be-came taught so that the average student could learn it,"says Lynn Steen of St. Olaf College in Northfield, Min-nesota. "There was a major change in the philosophyof textbooks that no one had planned."

"We no longer ask students tounderstand."

"We no longer ask students to understand," saysDouglas. "Now it is manipulation, pure and simple,"meaning that students are just plugging numbers orsymbols into formulas. The calculus tests rei.lect this."What you test is what the students learn," Douglaspoints out. "In fact, we have a great deal of difficultyusing class time on anything else besides what will beon the tests and we have abandoned testing anythingbut manipulation."

Reducing Rote LearningNearly everyone who /las thought about an agenda

for a revised calculus concludes that the course mustinclude a renewed emphasis on mathematical conceptsand understanding. "When you use calculus, you can'timagine that anyone will ever give you a problem likeone on an exam," Douglas says. "What you might en-counter will be a conceptual underpinning or you mightencounter a problem where the whole purpose would beto turn it into the kind of problem that occurs on anexam." Any student who just learned by rote to pluginto formulas would be lost in the real world.

"But," Douglas continues, "it's even worse than that.We now have computers and even hand calculators thatwill solve these calculus problems. There are now handcalculators that would get a B in most calculus courses.What we're teaching is not only the wrong thinginthat it is not what students will usewhat we're teach-ing is obsolete. It is like spending all your time in ele-mentary school adding and subtracting and never beingtold what addition and subtraction are for."

Mathematicians who are working to revitalize cal-culus feel quite strongly that the rote learning must goand that students should learn to rely on computers andhand calculators to do routine calculations. In place ofthe time now spent working out problems by hand, stu-dents should learn conceptswhat mathematics is allabout.

Neglect of TeachingBut, the revisionists argue, more than just the sub-

ject matter of calculus must change. "There are linksthat we cannot deny and cannot ignore between whatis taught, who teaches, and who is taught to," saysBernard Madison, of the National Research Council.And calculus teaching is also in drastic need of reform.

The crisis in calculus teaching began in the late1960s, according to Douglas. "For a number of reasons,the resources, effort, and money put into teaching calcu-lus have fallen over the past 15 years," he says. Duringthe 1970s mathematics departments found themselvessqueezed for funds at the same time as mathematics en-rollments increased dramatically. Students enrolled inthese courses in greater numbers, they were more likelyto want to major in the physical, biological, or socialsciences, and they were more likely to have to take cal-culus.

So, to deal with the influx of calculus students, uni-versities and colleges increased the number of studentsin their classes. Now it is common for calculus classesto be taught in lecture sections of 250 or more students.


Once or twice a week, the students break up into recita-tion sections, of size 30 or 40 in many schools, where ateaching assistant goes over the work with them. "Insome places, the recitation sections are now larger thanthe classes used to be," Douglas remarks.

"The resources, effort, and money putinto teaching calculus have fallen overthe past 15 years."

One of the problems with this method is that muchof the teaching of calculus is relegated to teaching assis-tants. These graduate students may not be particularlyinterested in the course and may have difficulty commu-nicating even if they try. As many as half of all teachingassistants are not U.S. citizens and they frequently haveonly a rudimentary command of English.

Teaching assistants, says Graham, often "are notnatural teachers. Many are not even math majors." Inother cases mathematics departments press undergrad-uate students into service. "I have seen sophomoresteaching to freshmen," says Graham. But hard-pressedmathematics departments that have far too few facultyto handle the hoards that take calculus courses feel theyhave no choice but to use teaching assistants.

Even when a bona-fide Ph.D. mathematician teachescalculus, the course has been made so instructor-proofby textbook writers that there is almost no opportunityfor the mathematician to introduce any concepts thatmight reflect the beauty and excitement of the field."At big universities, they standardize things and youteach by a syllabus," says Robert Ellis of the Universityof Minnesota. "You don't need a mathematician to dothis. It requires at most 5 minutes preparation to teacha class, and usually a mathematician can do it off thetop of his head."

Calculus requires that you know thematerial from the entire high schoolmathematics curriculum ... that's nottrue for most students.

All too often, the mathematicians who teach calculusregard it as a boring burden. Typically, says Madison,"we will go to the classroom, teach section 3.4, go home,and mow the yard." Teaching calculus has become rou-tine and mindless for the professors.

Although other departments have had some successwith impersonal large lecture sections combined with

smaller sections taught by teaching assistants, "at leastin calculus, this did not work," Douglas says. Calculusis different--ft cannct be taught in an impersonal en-vironment, according to Douglas. "Suppose you comeinto another sort of course, say freshman psychology,"Douglas says. "You start from scratch, you don't buildon anything. And you don't need to have masteredwhat you learned one week to go on to the next week'smaterial. But calculus requires that you know he ma-terial from the entire high school mathematics curricu-lum. Already, that's not trim- for most students. Andthe other problem with calculus is not just that it buildsbut that you have to keep up. If you did not do wellon one test, the chances are you will not do well onthe next.. There is no other course on the freshman-sophomore level like it."

Personal InvolvementTo do well in calculus, most students need the per-

sonal involvement of a professor or teaching assistant,according to Douglas. "They need to have homeworkassigned and they need to have homework collected andreturned. If you are going to teach calculus well, youhave to have someone reading papers, grading them,and returning them. That doesn't save money."

The way it is now, all too many students have nopersonal involvement. Homework may not be assigned,and even if it is assigned, it often is not collected. Stu-dents let the course slide, thinking they will learn thematerial at the last minute before an exam. "There isno feedback," Douglas says. "When students get thesense that no one is personally interested in what theyare doing in the course, they put very little effort in.They put in what the/ think is the minimum work nec-essary, and, of course, they often judge wrong."

The revitalized calculus ... "will bemuch more difficult to teach and willrequire much better preparedteachers."

A new calculus, which emphasizes understanding andconcepts and in which no student can slip by learningto plug into formulas by rote, will require more moneyfor teaching staff and it also will require that the staffput more effort into teaching and more time into givingstudents feedback. The revitalized calculus, Madisonpoints out, "will be much more difficult to teach andwill require much better prepared teachers."

Yet some say that the new calculus might be effec-tively taught even with the resources now available and

1.03

KOLATA: CALCULUS REFORM 93

even with the cost-cutting large lecture sections. "Thereare ways to teach better in large sections," says DonaldBushaw, who is vice-provost for instruction at Washing-ton State University. "It's just a matter of practical ad-vice, making the best use of the lecture and small groupmethod. There is a good deal of conventional wisdomthat is not being followed everywhere." Douglas pointsout that if calculus were better taught, fewer studentswould have to repeat it and this would also save money.

Textbook StagnationStill another issue for the calculus reformers is to en-

courage the publication of a new kind of textbook. Thecurrent crop of textbooks would be all but useless forthe new course. But developing new textbooks meansgoing against an entrenched industry that seems verycontent to go on the way it always has. Fourteen newcalculus books are scheduled to be published in the fallof 1988, according to Jeremiah Lyons, an editor at W.H.Freeman and Company, and none are innovative.

Lyons explains that the textbooks have evolved tosatisfy so many constituencies that publishers have gonebeyond the bounds of reason. "The books publishednow are really our own fault," Lyons says. "We havegone well beyond an adequate number of exercises andworked-out examples." In addition to what many feelare bloat and expensive books, the textbook publish-ers supply supplementary material to the instructors atno additional cost. This includes a solutions manualthat consists of worked-out solutions to every problemin the text and computerized test banksa computer-ized compendium of as many as 3000 questions so thatinstructors can easily give weekly quizzes, without hav-ing to make them up, and even have the quizzes printedfor them.

Educators "talk a great game ofinnovation, but if we move onestandard deviation from the mean,they don't use our books."

For publishers, innovation involves risk. Educators,says Lyons, "talk a great game of innovation, but if wemove one standard deviation from the mean, they don'tuse our books." If publishers were to experiment with anew sort of calculus text, they would be wary of invest-ing the usual amoant If money in it. For example, saysLyons, to take a chance on a new sort of text, publisherswould want to forego the usual 1100 pages of text byproviding far fewer worked-out examples and exercises.

Ai

They would like to save money on illustrationsthetypical calculus book now has a $75,000 art budgetbyusing computer-generated art, simplifying the illustra-tion program, and not using a second color. And theywould like to get rid of the $50,000 to $100,000 budgetfor all supplementary materials for instructors. "If welower the cost and reduce the basic investment, we cangamble more," Lyons says.

Links to Science CurriculaYet even if the textbooks change, there is still the

problem of the responses of other departments to a newcalculus. Any changes in calculus will have a ripple ef-fect in other sciences, particularly physics and engineer-ing. Courses will have to be revamped and re-thought,which further complicates efforts to change the mathe-matics curriculum.

In physics, says Edward Redish, a physicist at theUniversity of Maryland, "the physics curriculum is tiedvery, very closely to math." Physics students usuallytake calculus at the same time as they take introducto.physics and the two courses are coordinated so that asstudents learn a technique in calculus, they use it inphysics.

Finally, there must be some way to evaluate whethera revitalized calculus is, in fact, more effective. Mathe-maticians say that it is even diglcult to evaluate the ef-fectiveness of the cookbook, standardized calculus thatis being taught today. No one has ever done a nationalsurvey to determine such basic things as failure rates,pass rates, or even how many hours a week studentsspend in calculus classes. "We need that kind of data,"says Anderson, for without it it would be impossible toeven think of comparing a new calculus to the old.

Among the issues that NSF and the calculus reform-ers would like to see debated on a national level arewhat sorts of changes in calculus would best suit otherscience departments and how those changes could beaccomplished. So far, those who have thought aboutthe issues have reached no consensus.

Conflicting AdviceEven when scientists from other departments do

want to see calculus changed, they do not always agreeon what changes should be made to better serve theirmajors. Redish would like to see more stress placedon approximation theory and on numerical solutions todifferential equations. He would like to see a great dealmore emphasis on methods of checking qualitatively tosee whether a number that comes out of a computer isapproximately what would be expected.


Biologist Simon Levin of Cornell University wouldlike t,o see more emphasis on qualitative analysis andless on computations; he would especially like to sechis students introduced to partial differential equations,particularly as they describe diffusion.

Gordon Prichett, dean of the faculty at Babson Col-lege, a business college just outside of Boston, says thatalthough business students routinely take calculus, thecourse for them is, strictly speaking, unnecessary. So,he says, "if we are going to teach calculus in the busi-ness curriculum at all, it would be for breadth and forproblem-solving ability."

Advocates of the Status QuoRedish, Levin, and Prichett, however, want to see

calculus changed. Not everyone does. Some argue thatthe course is a classic that has withstood the test oftime and that is crucially important for the other sci-ences. Most physicists, according to Redish, are happywith the status quo. Redish believes that this system,in which physics courses are developed around the cur-rent calculus courses, distorts physics and cheats stu-dents of a feeling for what physics, to say nothing ofmathematics, is all about. But, Redish cautions, "I amnot typical. I'm out here at the edge trying to pull myimmense department. Most don't want to think aboutchange. 'If it ain't broke, don't fix it' is an attitude Isee a lot."

Most don't want to think aboutchange.

Anderson, who has presented the idea of revitaliz-ing calculus to forums of engineers and other scientists,says that the engineers and scientists frequently wereuninterested in change. "They wanted students to havethe same math that they had had," Anderson says."They thought of it as good training and they wouldsay, 'What was good enough for me is good enough formy students'."

"What was good enough for me isgood enough for my students."

Still others just do not want to be bothered. Cal.culus, they say, is a bread-and-butter course for mathe-matics departmentsit is one of those courses that keepmathematics departments in business. But it is not re-ally worth the time and trouble to radically change it.

And besides, any student who cannot pass calculus asit is currcntly taught ought to re-think his or her plansto become a scientist anyway.

Others are concerned that if a committee starts tin-kering with calculus, the course is likely to becomeworseless useful and less meaty. Ellis of the Universityof Minnesota thinks that it would be impossible to reallyteach mathematicsas opposed to routine formulasto poorly-prepared and poorly-motivated students. Thenew calculus would be a much harder course than thecourse that is taught now, and it is not clear what couldbe done about the even higher failure rates that mightresult. Ellis also notes that a true change in calculuswill have to have widespread support. "You can't im-pose this from above. It has to be up to the professors,"he says.

The new calculus would be a muchharder course than the course that istaught now, and it is not clear whatcould be done about the even higherfailure rates that might result.

Nelson Markley, the chairman of the University ofMaryland's mathematics department, worries that iftoo much is done with a computer or hand calculator,the students will never really learn what the calculationsmean. "I tend to think a great deal about the analogiesbetween mathematics and music," he says. "You can'tlearn to play the piano by going to recitals. You canerr in the direction of going trio far from actual calcu-lations." So far, the revitalization of calculus "is notsomething that I'm enthusiastic about," Markley says.

But, Madison and others point out, change is cominganyway with the increasing sophistication of computersand calculators that do calculus problems. "The truthis that things are going to change," Madison remarks."It is just a matter of whether we want to control thatchange and make it happen in a positive way or whetherwe will let it happen haphazardly. The only excusefor the argument for no change is that controlling thechange is impossible. I've never accepted that."

GINA BARI KOLATA is a science writer for The NewYork Times. Formerly, for more than a decade, she coveredbiology and mathematics for Science, the official journal ofthe American Association for the Advancement of Science.

1.03

CIPRA: CALCULUS INSTRUCTION 95

Recent Innovations in Calculus InstructionBarry A. Cipra

ST. OLAF COLLEGE

Good teaching doesn't come easily, and good teach-ers are rarely satisfied with the job they've done in theclassroom. Even if calculus were not the linchpin ofcollege mathematics, instructors would probably keepon tinkering with the course. But because of its promi-nence in the undergraduate curriculum, calculus is thefocus of much educational innovation. This brief ar-ticle will describe some recent efforts to improve theway that calculus is taught. Since an exhaustive surveywould require several volumes, w, -mly sample a few ofmany themes.

Let the Computer Do ItMuch of the current innovation in calculus instruc-

tion is centered around use of computers. As computershave become cheaper, smaller, friendlier, and also morepowerful, many people have begun exploring their pos-sible applications to the calculus curriculum. The com-puter as tedium-reliever, as expert draftsman, as superblackboard, and even as teacher, are some of the possi-bilities being looked at.

There is widespread agreement that calculus coursestend to be excessively "technique-" and "skill-oriented,"with corresponding agreement that calculus ought to bea more "concept- and 'application-orierted" subject. "Iwould say the problem is that most calculus instructionfocuses on computational details," says John Hosack ofColby College. That is, Hosack explains, a student canpass the course by carrying out standard algorithmswith little understanding of ideas. "We think that cal-culus should reorient itself toward concepts and appli-cations."

The hypothesis at Colby and at a number of othercolleges is that Computer Algebra Systems, such asMACSYMA, Maple, or SMP, will render obsolete thecomputational emphasis of traditional calculus instruc-tion. A typical Computer Algebra System can carryout all the routine steps of an algebra or calculus prob-lem, including formal differentiation and integrationprecisely the skills that are at the core of the currentcurriculum. Most of these systems also house numericalequation-solving routines and superb graphics capabil-ities, two more topics traditionally treated in calculus.There is little doubt but that a Computer Algebra Sys-tem could do very well on a typical calculus exam; it

L

would certainly make fewer mistakes than the students.Indeed, Herbert Wilf wrote in 1982 of one system (mu-Math), calling it "the disk with the college education."

A Computer Algebra System could dovery well on a typical calculus exam; itwould certainly make fewer mistakesthan the students.

Colby College, the University of Waterloo, Ober-lin College, St. Olaf College, Denison University, Har-vey Mudd College, and Rollins College are among theschools that have received grants from the Alfred P.Sloan Foundation or the Find for the Improvement ofPost-Secondary Education (I ..PSE) to experiment withComputer Algebra Systems in calculus. Some projectsinvolve software development. Doug Child of RollinsCollege, for instance, is adapting Maple for use on theMacintosh and writing an interface for what he calls an"interactive textbook." But the main thrust is towardrethinking the calculus curriculum to take advantage ofthese powerful programs.

Two replacements have been suggested for the tradi-tional computational emphasis in calculus. One is "ex-ploratory computation," in which the student looks ata number of related examples, such as graphs of succes-sive Taylor polynomial approximations to the sine func-tion. The computer does all the unpleasant stuff, thepoint being to illustrate vividly some concept ormoreambitiouslyto have the students discover concepts forthemselves. According to David Smith of Duke Univer-sity, by the time students had seen the 19th-degree ap-proximation to sin z, they were asking if you couldn'tjust let the degree go to infinity. He adds that sequencesand series went very smoothly thereafter.

Every teacher's dream is to have students who willplay with the homework, modifying problems to see howthe answers change, and posing the dangerous questionWhat if? But this goal gets lost in the press of routineexercises. Moreover, translating that ideal into prac-tice is not an easy thing to do. According to Paul Zornof St. Olaf College, it's hard to convey what you meanby "experimental math," especially when students have

1 CIS


preconceptions about the type of problems they are sup-posed to do. Zorn suggests easing the students into thepractice. "One way to do it is to give progressively moreopen-ended problems." Some instructors are now op-timistic that, with machines shouldering the algebraicburden, students will be more receptive to the idea ofopen-ended problems.

Every teacher's dream is to havestudents who will play ...modifyingproblems ...and posing the dangerousquestion What If?

The other suggested alternative to the traditionalemphasis on technique is the inclusion of more realisticand complex applicationsproblems that are not artifi-cially tailored for simplicity (textbook arc-length prob-lems are frequently derided) but that require a num-ber of separate steps for their complete solution. BruceChar and Peter Ponzo of the University of Waterloocite the development of such multi-step, multi-conceptproblems as a major component of a Sloan grant there.The idea, Ponzo says, is to test the student's abilityto put together a "long-winded" solution to a problem.The computer will handle the mundane calculations;the student will concentrate on setting up the problemand deciding on the sequence of formal operations thatwill lead to the solution.

One of the material goals of the Waterloo project isto compile problems into a "symbolic calculation work-book." Ponzo and Char both acknowledge that thishas turned out to be rather riore difficult than theyhad anticipated. Ponzo originally thought they wouldget hundreds of problems, but says "finding problemsof that ilk ain't easy." The goal now is several dozen, ofwhich an initial batch of five was published last April.

Another goal of the Waterloo project is to createcomputerized tutorials in calculus and to develop "user-friendly" software that will allow individual instructorsto create their own tutorials in a nonprogramming, text-editing environment. The Waterloo project uses Maple,which was developed at the University of Waterloo inthe early 1980s. ("Maple" is not an acronym; it stemsfrom Canada's national symbol, the Maple leaf.) Maple.is considered one of the easiest of the Computer Alge-bra Systems to learn, but Char, one of the group whocreated Maple, says that future work will look at mak-ing Maple even more user-friendly. "You still have togo through at least an hour of training in order to useMaple," he says, noting that this is a barrier to somecalculus teachers who feel they don't have the time.

Complexity and limited access are two factors lim-iting the use of Computer Algebra Systems in calculusinstruction. Some faculty are themselves reluctant tolearn the special grammar and vocabulary of the sys-tems. Others are concerned with the amount of classtime required to instruct students in use of the systems;an hour spent on computer syntax is an hour not spenton the integral.

"I don't even know how to log onto a computer," saysAlvin White, at Harvey Mudd College, who has other,more serious objections as well. "The more computerpower we have, the less the students know what they'redoin;; . .. The infatuation with computers moves thestudent further and further from thinking and creating.The promise we were given by the calculator people isthat we can spend more time on the underlying ideas.But in my experience, the time is spent on showingthem more buttons to push."

"The infatuation with computersmoves the student further and furtherfrom thinking and creating."

Proponents of the Computer Algebra Systems, how-ever, claim that explaining the systems does not requirean inordinate amount of class time; students tend topick up what they need to know either from handoutsor from each other. Paul Zorn, who has taught an SMP-based calculus course for several semesters, says that hespends "maybe half a lecture right at the start" givingan overview of the system. A big surprise, he adds, hasbeen "how little frustration students seem to experienceusing the computer." The paradigm of giving a com-mand and getting an answer comes very quickly. "Itdoesn't seem to be a big distraction."

Wade Ellis, Jr. of West Valley College in San Joseechoes this observation. Ellis and a colleague, Ed Lodi,wrote a computer activity book called Calculus Illus-trated, which they used last spring at West Vall.,fy Col-lege with a group of 15 students. Ellis describes thestudent reaction as being more or less "isn't this whatwe're supposed to do?"a change from the "why dowe have to use computers?" reaction of five years ago."Now everyone knows they have to use computers," El-lis says.

A BASIC DisagreementFor most proponents of Computer Algebra Systems,

an important ease-of-use feature has been the removalof programming as a requirement for use of the sys-tems. Herb Greenberg of the University of Denver,

1.07

CIPRA: CALCULUS INSTPUCTION 97

which has developed its own instructional softwarecalled "Calctool," says that students should not beaware of programming, only of the mathematical ap-plications. "We are not teaching programming, and thestudents are not doing programming," Greenberg saysof the program there.

George Andrews of Oberlin College notes that about20 years ago he participated in developing an experi-mental text called Calculus: A Computer-Oriented Pre-sentation, which he tried in class but "bailed out" whenhe found that "the programming got in the way." DavidSmith of Duke-University adds that enrollment at Dukein a supplementary "calculus and the computer" course,which involves programming, is way down, from 100 aproximately five years ago to a recent class of 12. Smithfeels that the dwindling enrollment may be due in partto students' reluctance to take on extra work.

Jim Baumgartner of Dartmouth College disagreesvehemently with this anti-programming outlook. Hedisputes the idea that progremming detracts from theideas of the course. "If you choose the problems cor-rectly, they're simple, they're short, and they teach theessence of programming." Dartmouth emphasizes pro-gramming from the very beginning. "What we're doingis building a base for further down the line," Baum-gartner says, adding that later courses assume that stu-dents can program in 71-ueBASIC. (BASIC is particu-larly popular at Dartmouth.)

Many students (Baumgartner estimates 70%) cometo Dartmouth with Macintoshes, and these are net-worked throughout the campus. The mathematics de-partment gives each student a TrueBASIC disk withshort demo programs, the idea being to give them some-thing simple that they ca modify, together with prob-lems and solutions.

James Hurley of the University of Connecticut alsoreports success with TrueBASIC programming in cal-culus there. "We give them something they can uselater on," Hurley says. The programming component,however, is done only in special sections of "enhanced"calculus, which include a one-hour computer lab eachweek. Hurley adds that a hig rason for their success isthe use of 1heBASIC, to which they switched in 1985.The Computer Science deportment, he says, had com-plained abou" their use of old versions of BASIC, whichoften encourage "sloppy logic" and poor programminghabits.

The programming/anti-programming "dispute" islargely a comparison of apples and oranges; presumablythere ;F room fo- ..oth in the calculus diet. Neveeheless,

,se is a major selling point of the Computer Al-Lorn.. "If I had an extra hour per week," says

Paul Zorn, "I'd still use SMP rather than 3 languagelike 1heBASIC, and do something else."

The Super CalculatorAccess may turn out to be a more serious limita-

tion than complexity, at least until Sun work stationsshow up on discount at K-Mart. Complexity is primar-ily a software problem, and thus amenable to program.ming improvements; but access is essentially an eco-nomic problem having to do with the cost and numberof the machines required to run the software. Accordingto John Harvey of the University of Wisccnsin, smallercolleges have an advantage in this regard over the largerstate schools: accommodating a few hundred studentson a Computer Algebra System is economically feasible;accommodating several thousand students is not.

Harvey, who is project director for a Texas Instru-ments-funded grant to develop calculator-based place-ment exams for the MAA, believes that powerful pocketcalculators will be the dominant innovation at largeschools. It's not unreasonable, he says, to ask a stu-dent to buy a $100 calculator for three semesters ofcalculus. Harvey anticipates having experimental sec-tions of calculator -based calculus at Wisconsin by thefall of 1988.

Powerful pocket calculators will be thedominant innovation at large schools.

John Kenelly of Clemson University is a strong ad-vocate of the new generation of pocket calculators, par-ticularly the Casio 7000, which primarily does graph-ics, the Sharp 5200, and the Hewlitt Packard HP 28C,which does graphics and some symbolic manipulation.Kenelly, who has been called "the 28C salesman of themath community," had the HP 28C on secret loan fromHewlitt Packard for six months prior to its release inJanuary, 1987. In the summer of 1987, Kenelly taught athree-week course to 16 teachers of advanced-placementhigh school calculus, funded by South Carolina. Par-ticipants were each given a Casio 7000 calculator, thepurpose of the course being for the teachers to writecurricula for use of the calculator in their AP courses.

Clemson is embarking on a pilot program to beginusing the HP 28C in freshman and sophomore engi-neering mathematics courses, from first-semester calcu-lus through differential equations, matrix algebra. andengineering statistics. Hewlitt Packard is loarhg Clem-son some one hundred 28C's, which the mathematicsdepartment will in turn issue each semester to students


who are signed up for the sections that will experimentwith use of the calculator.

In 1987-88, six faculty members will teach one courseeach, "playing" with the 28C at appropriate "momentsof opportunity," according to Kenelly. These same sixhope to spend the summer of 1988 rewriting the syllabifor their courses to incorporate the 28C in a substantialway, and then teach experimental sections from thesesyllabi the following year. The summer of 1989 is thenprojected for special early training sessions for TA's andother faculty members. In the fall of 1989, all sectionsof engineering calculus will be using the 28C.

Kenelly observes that the faculty is excited by theproject: "We have a colleague or two who think we'regoing to bring up a bunch of button-pushing dead-heads, but the bulk of the community is behind it."

Kenelly adds that Clemson also has a grant for thecoming year to train local high school teachershe an-ticipates a total of 160 participantsin curricular usesof the Sharp 5200 graphics ,lculator. (This is Kenelly'sthird calciilPtor, and he predictssafely enoughthatthe electr,,iics industry will keep coming out with more.The 5200 currently sells for about $100, and Keneilyforesees the price coming down to around $50.) Kenellydescribes the 5200 as "very friendly," especially in itsmatrix capabilities; he says it will invert a 10x10 ma-trix in about 15 seconds. It also has a "solve"" keyfor finding zeros of functions, which Kenelly considersa "must" these days.

The Super BlackboardIn a curiously opposite extreme from the economi-

cally small pocket calculator, the computer as "superblackboard" is another technological innovation thatholds promise for large universities as well cs small col-leges. Much of what an instructor writes in chalk on ablackboardespecially graphscould just as easily bedone on a computer screen, if only the screen were largeenough to be seen by more than the first two rows ofstudents. But special equipmenteither new attach-ments that feed the computer screen's output into anoverhead projector or large monitors posted around theclassroomcan overcome that handicap. Using suchdisplays an instructor can, for instance, draw an in-stant and accurate graph of the sine function on, say,the interval [jr, id, and superimpose graphs of the firstseveral Taylor approximations.

David Smith and David Kraines of Duke Universityare among the proponents of the computer as a superblackboard. With funds from Duke and the Pew Memo-rial Trust, they have equipped two classrooms with com-puters and display equipment (monitors). Smith and

Kraines have written about their project in an arti-cle for The College Mathematics Journal. Their arti-cle is partly about the technical aspects of installationand display (and also about security from theft), but italso addresses the question of getting people to use theequipment.

"Ease of operation is the key to moreuse of computers in teaching."

"Ease of operation is the key to more use of comput-ers in teaching," they write. "Only a few instructors willmake any substantial effort to plan a computer demon-stration, especially if they must set up the computer.... We have not hounded our colleagues as aggres-sively as we might have, but several have found uses forthe classroom computers, sometimes in ways that mightnot have occurred to us."

Herb Greenberg reports that the University of Den-ver is planning to try out the Kodak Datashow, a devicethat transfers a computer screen display to an overheadprojector. John Kenelly adds that Hewlitt Packard isworking on a system to feed the infra-red signal fromthe HP 28C into the Kodak projector.

The Super TutorWhile the "super blackboard" uses the computer as

a prop for the classroom instructor, a project under wayat the Institute for Mathematical Studies in the SocialSciences at Stanford University is looking to replace theinstructor altogether! A group headed by Tryg Agerunder the supervision of Patrick Suppes began in 1985to develop an intelligent tutor that will give interactivelessons in calculus by checking the validity of a student'sreasoning as he or she works through the steps of a com-plicated problem or proof. The goal of the project is notactually to do away with the teacher, but rather to de-sign a course "that could be taught with supervision by

teacher who is mainly playing the role of instructionalmanager," according to an interim report.

Ager argues that a student's time with the computeris spent more efficiently than in traditional instruction."You don't have the sort of low-intensity activity likesitting in lectures, and the frustrating activity of do-ing homework" with the long turn-around time betweenlectures, he says, adding that the computer doesn't gettired of giving feedback.

The project, which is funded by NSF's Office ofApplications of Advanced Technology, supplements theComputer Algebra System REDUCE with a system of"equational derivations" dubbed EQD. "The idea that


students construct complete solutions to complicatedexercises dominates our pedagogical model," accord-ing to the interim report. The report cites the use ofREDUCE and EQD as one of three themes, the othertwo being the development of interactive instructionalgraphics and the development of an interactive theorem-proving system. The latter is based on work of Suppes'group on proof logic, which was tried out earlier in anadvanced course in axiomatic set theory. Ager calls it"a terribly difficult problem" in artificial intelligenc .

A student's time with the computer isspent more efficiently than intraditional instruction.

The project is aimed at eventual application in high-school calculus instruction; Ager says they would like tobegin tests next Fall (1988). The reason for targetinghigh schools, Ager explains, is that the majority of highschools (Ager estimates 75%) have only a handful ofstudents ready to take calculus, too few to meet school-board requirements for minimum class size.

Rural high schools especially are often too small tooffer calculus. Tom 'Ricker, of Colgate University, par-ticipated in another solution to this problem, involving13 students in five rural schools in upstate New York:'Ricker taught them calculus over the telephone! Theclass, which "met" one hour a week in 1985/86, wassponsored by the state of New York, which providedthe conference line and some fancy electronic writingtablets. Each location could speak and write at anytime, except that Ticker's electronic pen took prece-dence.

'Ricker says that several other states are mandatingsimilar programs. South Carolina, for instance, is re-quiring high schools to offer advanced placement in atleast one subject. John Kenelly says that South Car-olina is considering the same system that 'Ricker usedin New York, 'Jut adds that they are looking down theroad toward even more sophisticated, computer-basedcommunication systems.

Should High Schools Teach Calculus?The quality.of high-school calculus instruction--and

mathematics instruction generallyis of concern tomany at the college level. It is traditional for collegeprofessors to bemoan the poor preparation of their stu-dents, but the problem in mathematics seems to beacute. Indeed, The Underachieving Curriculum is thetitle of the American ref art of the Second International

Mathematics Study, which gathered data from eighth-and twelfth-grade level classes in 20 countries aroundthe world. According to t: e report, "the mathematicalyield of U.S. schools may be rated as among the lowestof any advanced industrialized country taking part inthe Study."

One of the culprits pointed to by the Study is theexcessive repetition of material from one math courseto the next: "I didn't learn much this year that I didn'talready know from last year. Math is my favorite class,but we just did a lot of the same stuff we did last year,"a fourth-grader named Andy is quoted as saying.

The U.S. pattern is based on a valid educational the-ory that Jerome Bruner calls the "spiral curriculum,"in which topics are introduced early and then revisitedrepeatedly in progressively more complex forms. How-ever, the Study points out that practice has not livedup to theory: "The logic of the spiral curriculum hasdegenerated into a spiral of almost constant radiusacurriculum that goes around in circles."

"The logic of the spiral curriculum hasdegenerated into a spiral of almostconstant radiusa curriculum thatgoes around in circles."

The problem in high schools is pertinent to collegecalculus courses, because many students in high schoolsthat do offer calculus are getting what John Kenellytersely refers to as "crap calculus," in the expectationthat these students will take "real calculus" in college.Schools are teaching "trashy" calculus, Kenelly says,and kids are skipping fourth-year "real" math to takeit, so that mother and father can brag about it down atthe country club.

Don Small, in a paper prepared for a January 1986calculus workshop at Tulane University, writes, "Thelack of high standards and emphasis on understandingdangerously misleads students into thinking they knowmore than they really do. In this case, not only is theexcitement [of learning calculus] taken away, but an un-founded feeling of subject mastery is fostered that canlead to serious problems in college calculus courses."

Indeed, the problem is considered serious enoughthat a joint letter endorsed by the MAA and the NCTMwas sent out in 1986 to secondary school mathematicsteachers nationwide, urging that calculus in high schoolbe treated as a college level course and only be offered tostudents who have a full four years' preparation in alge-bra, geometry, trigonometry, and coordinate geometry.The re :ommendation favored by the MAA and NCTM

110


is that high school calculus students take university-level calculus, with the expectation of placing out ofthe comparable college course.

Advanced-placement high schoolstudents actually learn calculus betterthan college students, even after"ability level" is factored out.

According to Kenelly, who has a long involvementwith the calculus AP exam, advanced-placement highschool students actually learn calculus better than col-lege students, even after "ability level" is factored out.John Harvey points out one possible explanation: highschool students get nearly 180 hours of instruction, insmaller classes, and oftentimes with more careful in-struction.

Should Colleges Teach Calculus?While it is easy and in some ways satisfying to point

the finger of blame at high schools, there is concernabout the quality of calculus instruction at the collegelevel as well. Large lectures taught by disinterested fac-ulty, and classes taught by inexperienced TAs with min-imal supervision, are among the problems cited. Manycritics feel that the computer will become part of theproblem rather than a solution. And there is nearly uni-versal disgust with the current gargantuan textbooks.

"Getting the faculty back in calculus isthe most significant thing teat shouldbe done these days."

"Getting the faculty back in calculus is the most sig-nificant thing that should be done these days," saysRobert Blumenthal of St. Louis University. Blumen-thal took over as lower-division supervisor in the math-ematics department at St. Louis University in 1986.The department was faced with declining enrollmentsin upper-level courses, due, it was determined, to poorteaching in the lower-division courses. Looked at moreclosely, the department found most of the complaintsconcerned non-permanent facultyTAs and part-timeinstructors. Blumenthal's solution: have only full-timefaculty teach calculus.

"You need a department that feels calculus instruc-tion is important and deserves the attention of the fac-ulty. It's our bread and butter," Blumenthal says.

In addition to moving away from TAs and part-timefaculty, Blumenthal eliminated common finals in calcu-lus, which used to be multiple choice. Blumenthal wasalso unhappy with the textbook being used at the time.He feels that current calculus texts are too long and toowordy, and offers an interesting explanation: "Thesetexts are written more to correct the deficiencies of theinstructor than to help the student." Consequently hewas "delighted" to find that Addison-Wesley has reis-sued the "classic" second edition of Thomas, which orig-inally appeared in the 1950s. His department changedto Thomas last year.

There is nearly universal disgust withthe current gargantuan textbooks.

Morton Brown of the University of Michigan isadamant on the issue of classroom size. The mathe-matics department at Michigan tried large lectures forseveral years, but abandoned them in 1985 in favor ofsmaller classes of 30-35 students that meet four timesa week with one instructor. "In math it's more like alanguage," Brown says. "You have to get in there andfind out what the student is doing." Most of Michigan's40-50 sections of calculus each semester are taught byTAs, and most of the rest are taught by younger fac-ulty, but Brown feels this is preferable to the lecture-recitation format. "We think large lectures provide verypoor teaching," he says.

Peter Ponzo describes the opposite experience at theUniversity of Waterloo. About 700 students at Water-loo take first-year calculus in lecture sections of 120-140students that meet three times a week. These sectionsare supplemented by a two-hour "problem lab" of 30-40students which meets with the professor and some grad-uate students. Ponzo says the, make a conscious effortto put the best teachers into calculus. The departmentused to have small classes, some taught by graduatestudents, but about ten years ago it was decided thatgraduate students are not good teachers for first-yearstudents. Graduate assistants, Ponzo says, "like to im-press students with how much they know. They tendto make things far more complicated than they are."

Should TAs Teach Calculus?The role and effectiveness of graduate teaching assis-

tants is the focus of a study by the MAA Committee onTeaching Assistants and Part-Time Instructors, chairedby Bettye Anne Case of Florida State University. Withsupport from FIPSE, the committee surveyed nearly


500 departments with graduate programs to ascertainpatterns of present practice and issues. A preliminaryreport was published in 1987, and the committee plansto publish a "Resource Manual" in 1988 which will con-tain additional data along with models of training pro-grams ,nor TAs and part-time instructors.

The first survey found that the majority of doctoral-granting departments offer training programs, .7.1ereasthe majority of master's-granting programs do not.Members of the committee (all of whom are fromdoctoral-granting departments) consider training andsupervision of TAs to be of paramount importance.

Committee member Thomas Banchoff of Brown Uni-versity writes, "If the TA gets the impression that theprofessors themselves, especially those in charge of thecourses TAs teach, care about teaching effectively, thenthey are likely to develop well themselves and pass ontheir experience to those coining after them. On theother hand, if the faculty is perceived as uninterestedin teaching or in working with students, ;t is this atti-tude that the students will take to their own TA jobsand pass on to the new graduate students."

The training of foreign TAs is of particular con-cern. Several schools have tightened their languageproficiency requirements. At The Ohio State Univer-sity, for instance, international students must be cer-tified as having oral communication skills in English.This is done by the University, but the mathematicsdepartment then makes its own, independent judgmenton whether to place the student in a classroom or holdhim or her out for the first year as a grader.

Beginning in 1985, students are given four quarters tobecome certified, and based on experience in 1986/87,the department now requires that international stu-dents come in the summer as a precondition for sup-port. According to Harry Allen, former chair of grad-uate studies for the mathematics department at OSU,all but one of the international students who came inthe summer were certified by the end of the followingspring, whereas of those who came in the fall, only acouple were certified.

Allen feels that the program at OSU has been highlysuccessful, and points to a "drastic decline" in the num-ber of complaints going to the undergraduate chair-man's office over the past few years. Moreover, theforeign TAs themselves are positive about the training,in part, Allen says, because "the people who are run-ning it are doing it the right way."

Should Anyone Teach Calculus?Most college mathematics teachers learn their trade

by the example of others and through their own expe-rience; few have any background in educational theory,and most have little interest in it. Stephen Monk, of theUniversity of Washington, is one of the few. Monk de-scribes himself as a Piagetian, after the Swiss psycholo-gist Jean Piaget, who is well known for his studies of theintellectual development of young children. Monk wasmoved by Piaget's stress on students' learning throughtheir own activity. Students of calculus must proceedfrom their own intuition, he says, adding that calcu-lus, which is usually taught as a formal subject, can beunderstood in the way that arithmetic is understood.

Consequently, Monk is keenly interested in the groupdynamic of the classroom and thc roleor function, ashe discriminates itof the teacher. "I was mildly flat-tered when I was asked to teach a large lecture course,"he writes in an article in Learning in Groups. "Therequest indicated that I had joined a circle of compe-tent teachers in my department who could give the clearexplanations that a large lecture demands; who are suf-ficiently well organized, patient, and self-assured to getthe subject across to a large, diverse audience; and whoare free of the idiosyncrasies of style that are charm-ing in a teacher with a few students but disastrmis ina teacher with many .... Looking back to that time, Iview my attempt to express my teacherly impulses asa lecturer in a large course as roughly parallel to at-tempts to express sensitivity for the less fortunate ofour society as a county jailer."

"Collective work is a key ingredient tointellectual growth."

Monk has worked with psychologist Donald Finkel, ofthe Evergreen State College, on the idea of using small"learning groups" in the classroom with the teacherdistributing worksheets and then acting as a roving"helper" rather than as "the expert." "The evidencethat collective work is a key ingredient to intellectualgrowth surrounds us," they write. "Yet to judge bythe typical college course, most teachers do not believethat it is either appropriate or possible to foster theseimportant processes in the classroom." They call thisattitude in which the teacher shoulders all responsibilityin the classroom the "Atlas complex."

David Smith, of Duke University, has also embarkedon another daring innovation in his calculus classes:writing assignments. "failure to read and analyze in-structions prevents students from getting started on aproblem, and their ability to understand a solution pro-cess is related to their ability to explain in English what


they have done," he writes in an article in SIAM News(March 1986). Smith now requires, on at least oneProblem from each assignment, that students explainin complete sentences how they set up the problem,the sequence of operations, and their interpretation ofthe result. His exams, which are wide-open, take-hometests (he even allows collaboration!), require writing onalmost all problems. Typical instructions: "Evaluateeach of the following six limits, if possible. State in asentence or two what your technique is. If the limit failsto exist, say why."

A few of the students at Duke were really hostileto the idea of writing, according to Smith. They didit to get the grade, but thought he was being unfair.Most, however, were intrigued with the idea. Smith de-scribes their course evaluations as being cautiously posi-tive. Faculty were also interested but somewhat nervousabout the time commitments and their own competencein grading students' work. Smith acknowledges th..t hewas able to handle the time commitments (upwards ofone hour per paper) in part because he had fewer stu-dents last year. However, he also points out that a writ-ing center workshop at Duke helped him learn what tolook for. One piece of advice he found helpful: Look forkey features in a paper; if these aren't there, don't tryto read it, ask for a rewrite!

The Rise (and Fall) of Discrete MathWhile calculus is traditionally considered the math-

ematics course for first-year students, discrete mathe-matics has been proposed as an avant-garde alternative.Many of the modern usages of mathematics, particu-larly in computer science, are based not on the analysisof continuous phenomena, but rather on combinatorialand recursive principles. Engineers need calculus rightaway, computer scientists don't, the proponents of dis-crete mathematics argue. Why not teach people whatthey need to know?

The answer may be that it just doesn't work verywell. Two schools that tried putting discrete mathe-matics into the first year, Dartmouth College and theUniversity of Denver, have taken it back out.

In 1983 the mathematics department at DartmouthCollege proposed a new curriculum in which a four-quarter sequence of calculus through uifferential equa-tions was reduced to three, and a new course in discretemathematics, which could be taken at any time after thefirst calculus course, was offered. "As it turned out, itwas a disaster," says Jim Baumgartner.

The main problems were with the discrete mathe-matics course and the second-quarter multivariate cal-culus course. The discrete mathematics course hadn't

been placed in the sequence, and it therefore attracted atremendous variety of students, with anywhere from oneto five courses of mathematics in their backgrounds. Itwas also hard to know what to teach, Baumgartner says,so the course varied widely from quarter to quarter, de-pending on who taught it. He adds that the ComputerScience Department at Dartmouth dropped the courseas a requirement, which also hurt. Finally, the Mathe-matics Department was hoping that the discrete math-ematics course would make things easier in upper-levelcourses by introducing proofs, induction, and so forth,but it didn't seem to be doing that. Discrete mathe-matics at Dartmouth, Baumgartner summarizes, was a"total failure."

Multivariate calculus in the second quarter was alsoa mistake, according to Baumgartner. There was a"sophistication problem," he saysthe students justweren't picking it up. (He notes, however, that ad-vanced students, who took it as their first course atDartmouth, did pretty well.) Students disliked the mul-tivariate course intensely (more than usual, Baumgart-ner says), and the majority disliked the discrete math-ematics course and found it quite difficult.

The third-quarter Differential Equations course wasthe most successful, according to Baumgartner. Its onlyweak point was the introduction of series here ratherthan in a prior calculus course. In the new programinfinite series will move back to calculus and the differ-ential equations course will get more of .a linear algebraslant.

Dartmouth's new four-quarter mathematics sequenceis largely a return to the original program: two quar-ters of calculus, including series-and a chunk of matrixalgebra, a quarter of multivariate calculus with linearalgebra, and a quarter of differential equations. Discretemathematics is being dropped discreetly.

At the University of Denver, Herb Greenberg andRon Prather, who is now at Trinity College, used aSloan grant to experiment with a combination of cal-culus and discrete mathematics. Their idea was toteach a "discrete structures and calculus" course in thefirst quarter, differential calculus in the second quar-ter, and integral calculus in the third quarter. The ex-periment began in 1983, and was abandoned two yearslater. The department, which combines mathematicsand computer science, now offers a three-quarter calcu-lus sequence and one quarter of discrete mathematics.

"Student evaluations were negative and somewhatsurprising," Greenberg says. Students did not find thediscrete structures course at all easy (Greenberg reti-tled the course "Destruct Creatures"), and, more im-portantly, they did not see the relevance of it. Even

113


the computer science majors, Greenberg says, said ithad nothing to do with the computer science they werestudying, and called the course "a complete waste oftime."

Greenberg was also disappointed by the lack of"carry-over" from discrete mathematics to calculus; forinstance, students seemed to have no idea that proofby induction was a useful technique outside of the firstquarter. The reason may simply be students' lackof experience in high school (and earlier) with non-computational mathematics. "They weren't matureenough to understand it," Greenberg says.

Enthusiasm vs. RealismMany other individuals around the nation are doing

innovative work in calculus instruction. Their effortsare in large part self-rewarding: they impart to theirstudents their own excitement at doing and learning

calculus. "The success of innovations depends predom-inantly on the enthusiasm of the person doing it," saysFrank Morgan, now of Williams College. Programs andfunding to improve calculus instruction will certainly in-crease in the coming years, and new ideas will continueto be discussed.

Morgan, however, seasons his enthusiasm with a con-cluding note of realism: "Any system you come up with,after awhile the students figure out how to get throughit with the least amount of work."--

BARRY A. CIPRA is a mathematics writer, teacher, andeditor. The author of several research and expository papersin number theory, and of the calculus supplement Misteaks,he has taught at Ohio State University, M.I.T., and St. OlafCollege. He received his Ph.D. degree in mathematics fromthe University of Maryland.

Some Individuals Involved with Calculus Projects:

TRYG AGER, Institute for Mathematical Studies inthe Social Sciences, Ventura Hall, Stanford University,Stanford, California 94305.HARRY ALLEN, Department of Mathematics, OhioState University, Columbus Ohio 43210.GEORGE ANDREWS, Department of Mathematics,Oberlin College, Oberlin, Ohio 44074.TOM BANCHOFF, Department of Mathematics, BrownUniversity, Providence, Rhode Island 02912.JIM BAUMGARTNER, Department of Mathematics,Dartmouth College, Hanover, New Hampshire 03755.ROBERT BLUMENTHAL, Department of Mathematics,St. Louis University, St. Louis, Missouri 63103.MORTON BROWN, Department of Mathematics, Uni-versity of Michigan, Ann Arbor, Michigan 48109.STAVROS BUSENBERG, Mathematics Department, Har-vey Mudd College, Claremont, California 91711.BETTYE ANNE CASE, Department of Mathematics,Florida State University, Tallahassee, Florida 32306.BRUCE CHAR, Computer Science Department, Univer-sity of Waterloo, Waterloo, Ontario, Canada N2L 3G1.DOUG CHILD, Mathematical Sciences Department,Rollins College, Winter Park, Florida 32789.WADE ELLIS, Department of Mathematics, West ValleyCollege, San Jose, California 95070.HERB GREENBERG, Department of Mathematics, Uni-versity of Denver, Denver, Colorado 80208.JOHN HARVEY, Department of Mathematics, Univer-

sity of Wisconsin, Madison, Wisconsin 53706.JOHN HOSACK, Mathematics Department, Colby Col-lege, Waterville, Maine 04901.JAMES HURLEY, Department of Mathematics, Univer-sity of Connecticut, Storrs, Connecticut 06268.ZAVEN KARIAN, Department of Mathematical Sciences,Denison University, Granville, Ohio 43023.JOHN KENELLY, Mathematics Department., ClemsonUniversity, Clemson, South Carolina 29631.STEPHEN MONK, Department of Mathematics, Univer-sity of Washington, Seattle, Washington 98195.FRANK MOILGAN, Department of Mathematical Sci-ences, Williams College, Williamstown, Massachusetts01267.PETER PONZO, Department of Mathematics, Univer-sity of Waterloo, Waterloo, Ontario, Canada N2L 3G1.DONALD SMALL, Mathematics Department, Colby Col-lege, Waterville, Maine 04901.DAVID SMITH, Department of Mathematics, Duke Uni-versity, Durham, North Carolina 27706.ANDREW STERRETT, Department of Mathematical Sci-ences, Denison University, Granville, Ohio 43023.Tom TUCKER, Department of Mathematics, ColgateUniversity, Hamilton, New York 13346.ALVIN WHITE, Mathematics Department, HarveyMudd College, Claremont, California 91711.PAUL ZORN, Department of Mathematics, St. Olaf Col-lege, Northfield, Minnesota 55057.

Calculus for Engineering

Kaye D. Lathrop

STANFORD LINEAR ACCELERATOR CENTER

The engineering with which I am most concernedis research and development engineering, usually con-ducted by individuals with at least a master's degree.Research and development problems usually require ex-tensions of existing technology, and indeed, part of theresearch and development effort is directed toward find-ing new ways of solving engineering problems.

The individuals who conduct this research must beextremely competent applied mathematicians. Theymust be able to derive the equations that describe thesystems with which they are working and they mustbe able to extract solutions from these equations withenough ease to permit parameter, surveys and sensitiv-ity analyses. If the equations they derive are not thosefor which standard solutions are available or for whichusual numerical methods apply, they must be capableeither of developing suitable approximate solution tech-niques or of approximating the equations to be solved torender them tractable. These people will have program-ming support, access to supercomputers and access toexperts in appropriate disciplines, but they will oftenbecome the expert in the particular class of problembeing addressed.

i'or most students who expect to dograduate work in the physical sciences,the conventional' college calculus courseis not sufficient.

For the people described above, calculus is a funda-mental part of their mathematical training. For thesepeople, and indeed for most students who expect to dograduate work in the physical sciences, the conventionalcollege calculus course is not sufficient. Even the spe-cial four quarter (or three semester) courses sometimesoffered for thnse majoring in engineering or the physicalsciences are at the same time neither broad enough nordeep enough.

Chickens and EggsIn addition to this problem of insufficiency, calcu-

lus suffers from a chicken and egg, problem. Appliesphysics and most engineering is best taught using cal-culus and differential equations as a known subject, but

most freshmen students don't speak these languages.At the same time calculus is not learned except byworking examples. Since the best examples are thosewhich motivate the student, these are most likely tobe those selected from the students' field. To some ex-tent this latter need is recognized by the offering of spe-cial courses tailored to disciplinescalculus for businessmajors or calculus for mathematicians. Although suchcourses do not resolve the paradox of needing to knowcalculus to learn the subject, they do lead to the pro-liferation of first level classes, some of which might aswell be called "calculus for those who don't want to takecalculus."

Calculus should be part of acurriculum of computationalmathematics.

The real need is for a calculus curriculum to provideearly a solid base of understanding and practical toolsfor subsequent engineering and physical science courses,and to provide later another coverage of calculus, bothdeeper and broader. I happen to believe that such arepetition of subject matter also greatly increases pro-ficiency and understanding.

Another form of the chicken and egg problem occurswithin mathematics itself. There is a tendency in sometreatments of calculus to emphasize the proving of the-orems and the pathologies of functions. Especially forengineering students, this more abstract information isbest reserved for second courses in calculus. Even thenthis type of information is best illustrated by examiningit in the context of examples relating the significance ofthe result obtained to practical applications.

Problems with Problem SetsIn addition to coordinating the pace of students'

mathematical development with their engineering orphysical sciences development, it would be most desir-able to train them simultaneously in both areas. Oneway this might be done is by teaching calculus usingproblem sets and illustrative examples from engineer-ing and physical sciences applications. This is not anovel idea. There has been a trend in this direction,and I think it should be accelerated.

1.15

LATHROP: ENGINEERING 105

It is clearly more work to prepare such educationalexamples and care must be taken to choose problemsand illustrations appropriate to either basic or advancedlevels of the students, but the payoff should be bettermotivation and clearer understanding of both subjects.Older texts seem to use predominately abstract exam-ples of functions, and to spend inordinate amounts oftime integrating and differentiating special functions,finding areas and volumes and so on, with little regardfor the relevance of these acts. This state of affairs couldbe improved.

Every student in calculus should bedeeply exposed to the derivation anduse of numerical approximationmethods.

In a related vein, many calculus texts use problemsthat can be solved by following recipes, and typicalhomework exercises involve a couple of hours of repet-itively following the recipes. In general, more timeshould be devoted to making the ideas clear and lesstime devoted to rote learning.

In particular, calculus, engineering, and physicscourses should spend proportionately more time on thetopics of problem formulation and problem solving. Toooften students' only exposure to the thinking that goesinto the process of formulating and then solving a prob-lem is the verbalization of the process by the instructorin the classroom. When one has mastered the solu-tion of a particular class of problems, particularly ifthis mastery was gained some long time ago, there is atendency to present the problem formulation and solu-tion as a smooth, seamless, effortless process, forgettingthe struggle, the trial and error, and the dead ends thatare pursued before success is obtained.

As students progress to more advanced courses, pro-portionately more time should be spent on the tech-niques of problem solving. Examples in texts shouldinclude case histories discussing the reasoning that goesinto a variety of approaches. Classroom lectures shouldinclude periods when experts are confronted with new(even unsolnd) problems and asked to attempt a solu-tion and describe what they are thinking as they do so.This sort of thing can be a close approach to the actualresearch process.

Using ComputersMany treatments of calculus, especially in books that

have been revised over a period of years, scarcely rec-ognize the use of computers. This is a serious omission

on at least two counts. Present day computer programsprovide extremely powerful symbol manipulation, in-cluding the capability of integrating and differentiating.The student should become aware very early of the ca-pabilities of these programs.

More fundamentally, every student in calculus shouldbe deeply exposed to the derivation and use of numeri-cal approximation methods. In the fields of .engineeringand physical sciences, almost all solutions are obtainedby numerical methods. The successful professional willuse computational evaluations far more often than thetechniques of classical calculus. The understanding ofcalculus is extremely important and the language of cal-culus is a useful shorthand, but once the applicableequations are derived, the greatest challenge becomesthe determining of numerical solutions.

That is not to say that all the classical analysis thatis taught is of limited use. Indeed, the validation of nu-merical approximationsthat is, the experimentationnecessary to show that a computer code produces cor-rect solutionsis a discipline in its own right that usu-ally involves comparison of evaluations of analyti: solu-tions, where such are available, with solutions producedby numerical approximations.

The successful professional will usecomputational evaluations far moreoften than the techniques of classicalcalculus.

Both analysis and approximation are useful, but thecurriculum should combine them synergistically ratherthan treat them disjointly. In my view, calculus shouldbe part of a curriculum of computational mathemat-ics, with the initial course emphasizing numerical differ-entiation and integration and with subsequent coursescovering first ordinary differential equations and thenpartial differential equations.

GeometryFor research and development in engineering, clear

visualization of the problem geometry and a clear un-derstanding of the problem solution as a function of keyproblem parameters are imperative. I believe the cal-culus curriculum should emphasize analytic geometryand combine it early with the use of computer graph-ics displays. CAD systems modeling three-dimensionalsolids are commonly available to the engineer. Studentsshould become familiar with these powerful tools thatcan describe all geometric aspects of the actual systemand permit viewing from any desired vantage point.

106 ISSUES: SCIENCE, ENGINEERING, AND BUSINESS

As students begin to generate numerical solutions,they should at the same time become familiar with thetools available to display these solutions. They shouldalso be trained in the techniques of processing masses ofdata to extract significant information. Visual detectionof anomalies is an extremely important and powerfultool, both to detect errors in solution techniques and todiscover new effects.

As computers make possible the solution of mornand more complex problems, the importance of prop-erly sifting the output becomes greater and greater. Inhigh energy physics, for instance, terabits of experimen-tal data are repeatedly mined with s-ohisticated toolsto extract new physical effects. Monte Carlo evalua-tions of theoretical models gives equal amounts of datato be similarly sifted.

Differential EquationsThe more advanced part of the calculus curriculum

should concentrate on solutions of the most importantclasses of the differential equations of engineering andthe physical sciences. In the last twenty years an enor-mous body of knowledge has been accumulated aboutthese equations and about their practical means of so-lution.

In my view, a coherent treatment of solution tech-niques does not exist, even though "black box" com-puter algorithms exist for ordinary differential equa-tions and are beginning to emerge for partial differen-

'vial equations. Use of these solution techniques is ex-tremely important in much research and do telopment inengineering and the physical sciences. Hence studentsshould be trained in these areas, being given a com-bination of classical analytical methods and numericalmethods.

SummaryMy suggestions for calculus for engineering and phys-

ical sciences can be summarized concisely: more math-ematical training should be offered the undergraduate,and more of that training should be computer oriented.In particular, I'd recommend that universities

Make the calculus curriculum broader (numericalmethods) and deeper (two, perhaps three full yearsthrough partial differential equations).Closely integrate the calculus curriculum with theengineering and physical sciences curriculums.Emphasize computational mathematics as well asconventional analysis.

KAYE D. LATHROP is Associate Laboratory Directorand Head of the Technical Division of the Stanford LinearAccelerator Center in Stanford, California. He has servedas Associate Director for Engineering Sciences for the LosAlamos National Laboratory, and as Chairman of tilt. Math-ematics and Computation Division of the American NuclearSociety.

The Coming Revolution in Physics InstructionEdward F. Redish

UNIVERSITY OF MARYLAND

The current introductory physics curriculum hasbeen highly stable for almost thirty years and is nearlyuniform throughout the country. It is closely linkedwith the introductory calculus sequence in that the or-dering and content of the physics courses are stronglydetermined (sometimes inappropriately) by the stu-dents' mathematical skills, and in that the calculuscourse frequently uses examples from physics.

Despite its apparent stability, there are indicationsthat three revolutions are beginning to make an impacton the curriculum: the explosion of new knowledge andmaterials in physics, new insights into the process oflearning gleaned from studies in cognitive psychology,

and the power and widespread availability of the com-puter.

I suggest that these changes, with the computer act-ing as a lever on the first two, are making inevitable arevolution in the way we teach physics. These changeswill require associated changes in the calculus sequencewhich will have to carry the burden of introducing awider variety of I lathematical techniques to scientistsand engineers than is currently the case. These newtechniques include numerical and approximation tech-niques, the study of discrete equations, and pathologicalfunctions.

REDISH: REVOLUTION IN PHYSICS 107

Physics and CalculusPhysics and mathematics have been close colleagues

throughout their history. Newton's development of thetheory of mechanics was intimately tied to his inventionof the calculus. The understanding of the theory of elec-tromagnetism was achieved when Maxwell completedhis set of partial differential equations. The mathemat-ical studies of transformation theory and tensor analysisat the end of the nineteenth century, when applied toMaxwell's equations, led Einstein to develop his specialand general theories of relativity. This close interplaycontinues today in a variety of forefront research, in-cluding the development of string theory and the theoryof chaos.

Explosion of new knowledge, ...newinsights into the process of learning,...and the power ...of the computer...are making inevitable a revolution inthe way we teach physics.

The close connection between physics and mathe-matics is nowhere more obvious than in the structureof the undergraduate physics majors' curriculum. Thestandard curriculum covers a variety of subjects includ-ing:

mechanicselectricity and magnetismthermodynamics and statistical mechanicsmodern physics and quantum mechanics.

These subjects form the foundation of fundamen-tal material on which more specialized subjects, suchas condensed matter physics, atomic and molecularphysics, nuclear physics, plasma physics, and particlephysics, are built.

The standard approach to these subjects is a spiral.They are first covered in an introductory manner in asurvey course in the freshman and sophomore year, re-done in individual courses in the junior and senior years,and finally covered one more time in graduate school.Both the spiral character and the specific ordering oftopics in the physics curriculum is primarily controlledby the mathematical sophistication the student is as-sumed to have.

The control imposed b3 mathematics is very rigidand even influences the detailed order of presentationin individual courses. For e.-ample, the first semesterof physics is usually taken with first year calculus as acorequisite. In teaching mechanics, nearly every text-book author introduces the subject by considering the

physical example of motion in a uniform gravitationalfield, with Newton's Laws of motion presented at a laterstage. The primary reason for this is that the formercan be solved algebraically without the use of differen-tial equations and is an appropriate place to introducethe concept of derivative. The latter requires a morecomplete understanding of the concept.

From the point of view of the physics this is highlyinappropriate. The uniform gravitational field is a veryspecial and peculiar case. Putting it first gives it a pri-macy which is both undeserved and misleading.1 Manyother examples could be given, including the delay ofCue presentation of the harmonic oscillator, the placingof electrostatics, the study of normal modes of oscilla-tion, the treatment of quantum mechanics, etc.

On the other hand, physics content plays a significantrole in the calculus sequence. Many standard calculustextbooks make heavy use of physics problems as ex-amples and to help motivate students concfning thereal-world relevance of the material presented.

Both the spiral cLeracter and thespecific ordering of topics in thephysics curriculum is primarilycontrolled by the mathematicalsophistication the student is assumedto have.

Only a relatively small number of students actuallymajor in physics. However, in most universities thesame introductory physics course taken by majors isalso taken by engineers, chemists, and mathematics ma-jo:s. Almost all physical scientists trained in Americanuniversities take a physics course of the standard type.The structure and content of introductory physics withcalculus therefore has a profound implication for thetraining of all our (hard) scientists.

Curricular StabilityCalculus-based physics for scientists and engineers

is taught in a variety of formats and conditions, butit is almost always found in the format of a three orfour semester (occasionally three quarters) introductorysequence with calculus as a corequisite (or occasionallywith one semester as a prerequisite). For majors, thisintroduction is followed by the spiral described above.

The physics curriculum was stabilized in a semi-formal sense in a series of conferences and articles [3],[19] in the late '50's and early '60's. At that time, a


mini-revolution took place in the style of physics teach-ing, shifting emphaeis towards concept and understand-ing and away from development of technical skills.

The content of this course is very stable. Despite theexistence of dozens of introductory physics texts andthe yearly publication of many more, their approachesdiffer only in detail. A recent survey of ten of the mostpopular first-year physics texts by Gordon Aubrecht [1]strongly confirms this.

The widely-held view among physiciststhat physics is the most cumulativeand mathematical of the sciences...leads to a strong tendency to teachphysics semi- historically.

This stable content is a result of the widely-held viewamong physicists that physics is the most cumulativeand mathematical of the sciences. Even scientific revo-lutions such as quantum mechanics and special relativ-ity arc viewed by most professional physicists as exten-sions rather than replacements of previous theories. Inthe research forefront, techniques from older models of-ten coexist with current dogma. This leads to a strongtendency to teach physics semi-historically.

The present introductory curriculum gives the stu-dent an overview of the basic techniques of those partsof historical physics which still survive from the period1600 (Galileo) to about 1940 (nuclear physics of fissionand fusion). The content from after 1916 (the Bohrmodel of the atom) is usually slim, since going furtherrequires treading on the slippery ground of quantummechanics. Quantum mechanics is a less intuitive sub-ject than the others, and one which relies heavy on themathematics of differential equations and matrices. Itis usually suppressed in the introductory course.

A Revolution Is BrewingThis characterization of the physics curriculum may

lead one to project long-term stability in the wayphysics is taught at the introductory level. Nonetheless,there is evidence on the horizon of a major revolutionin the teaching of physics that is driven by revolutionsin three major areas of relevance: physics research, ed-ucational psychology, and available technology. JackWilson, Executive Director of the American Associa-tion of Physics Teachers (AAPT) refers [21] to thesethree drivers of change as "the three C's:"

Contemporary

CognitiveComputer.

Predicting the future is a chancy business under anycircumstances, but predicting major changes in any hu-man activity is a long shot. As I discuss the impactof each of these revolutions, the reader should bear inmind that these views are necessarily speculative andidiosyncratic.

Contemporary PhysicsAs a result of the perceived cumulative structure of

physics, the current introductory course has a strongoverlap with the courses taught at the turn of the cen-tury. (The changes agreed upon thirty years ago dealtprimarily with style rather than content.) For example,the material covered in 21 of the 23 chapters in RobertMillikan's turn of the century book [16] is containedin current texts.2 According to Aubrecht's survey, onlyabout 20% of current introductory texts have chapterson the physics of the past 50 years and those only spendabout 5% their chapters on that material.

John Rigden, AIP Director of Physics Programs andformer editor of the American Journal of Physics statedthe issue compellingly in one of his editorials [18]:

Great-grandparents and grandparents and parentstook ... the same physics course as contemporary stu-dents are now taking. ...No new information appearsin a new edition cf a physics textbook.

This wouldn't be a problem if physics were a staticfield. It is not. In the past thirty years we have seenan explosion of new understanding and power in a va-riety of subfields of physics, ranging from the discoveryof the substructure of protons and neutrohs, to hightemperature supercondctivity, to the discovery of thethree-dimensional structure in the clustering of galax-ies. These discoveries cover all possible scales of length,time, and mass.

The excitement and vitality ofcontemporary physics is not conveyedin the current introductory course, norare the student's skills developed in amanner appropriate for the newphysics.

There have even been major breakthroughs in fieldslong thought to be well understood. Current develop-ments in Newtonian mechanics are evolving into a the-ory of chaotic behavior which may lead to a change in

119

REDISH: REVOLUTION IN PHYS:CS 109

our way of viewing physics as deep as that produced bythe discovery of quantum mechanics.

The excitement and vitality of contemporary physicsis not conveyed in the current introductory course, norare the student's skills developed in a manner appropri-ate for the new physics. Recently, however, Lhe leadersof both the research and teaching communities are be-ginning to take a broad interest in including contempo-rary physics in the introductory courses. This interestis displayed in two recent conferences held in Europeand the United States [2], [14].

Cognitive SciencePsychological studies of what students know and how

they learn are producing fundamental changes in thetheories of learning and education. The few studies thathave been done on physics students indicate that thereare profound difficulties in the way we teach physics[5], [15], [17]. The traditional assumption that the stu-dent is a tabula rasa on which new descriptions of theuniverse may be written appears to be false.

Students bring to their first physicscourse well- formed yet often incorrectpreconceptions about the physicalworld.

Students bring to their first physics course well-formed yet often incorrect preconceptions about thephysical world. The present structure of introductoryphysics does not deal with this well. A number of pre-and post-course tests taken at a variety of universitiesshow that a course in physics does little or nothing to-hange the average student's Aristotelian view of theuniverse [10], [11]. As we learn how students learn andchange their views from "naive" to "expert," we candesign our courses so as to facilitate this transition.

Computer TechnologyThe immense growth in the power of high-tech tools

in the past thirty years has had a profound impact onthe way professional physics is done. The most power-ful and influential of these tools is the computer. Morecomputational power is packed into a desk-top com-puter the size of a breadbox than was available in thelargest mainframes thirty years ago.3

Approximately 75% of contemporary research physi-cists use computers. At the time our current "stable"curriculum was designed, the number was more like 5%.

(These numbers are based on informal surveys and con-sultation of the research literature.) The result is thatthe computational skills required by the professionalmust all be learned at the graduate level.

Certainly many students learn to program as under-graduates, and indeed --one are already superb pro-grammers by the time enter college. But program-ming as taught in high schools, in computer science de-partments, and learned on one's own to write gameswith is not the same as learning to do physics with thecomputer.

The physicist who wants to do physics with a com-puter needs a wide variety of skills, not all of themnumerical. Estimation skills are essential, not only forthe experimentalist who must have a good idea of therate at which data must be taken, but for the numeri-cal analyst who must have a reasonable idea of what anappropriate discretization is. An important skill is tobe able to tell when to do an exact analytic calculation,when to approximate, and when to use the computer.This skill requires substantial experience.

Computers are necessary tools at the undergraduatelevel as part of a student's training as a professional.But in addition, the availability of the computer opensimmense opportunities for the physics teacher. If thestudent has the computer available from an early stage,a larger class of problems and subjects may be intro-duced, avoiding the previously imposed mathematicalconstraints.

Alternative CurriculaAt the University of Maryland, members of the

Maryland University Project in Physics and Educa-tional Technology (M.U.P.P.E.T.) have been investigat-ing the impact of the computer for the introductoryphysics curriculum since 1984. Students are taught touse the computer in their first semester, and its presencehas permitted a number of interesting modifications ofthe standard curriculum:1. Newton's law can be introduced in discrete form as

the first topic in mechanics. This allows the studentto think about the physics before becoming involvedin special cases (uniform accelerations) or mathemat-ical complexities (calculus, vectors).

2. A wider variety of projectile motions can be studiedthan is possible when only analytic techniques areused. These include motion with air resistance, in afluid, with electromagnetic forces, etc. This gives abetter balance of analytic and numerically solvableproblems and relates better to the real world than acurriculum with only idealized solvable models.

I 2, 0


3. Non-linear dynamics can be studied. This permits usto teach fundamental concepts of numerical physics,to introduce some contemporary topics, and, byshowing examples of chaotic systems, to make bet-ter connections between mechanics and statisticalphysics/thermodynamics than was previously possi-ble.

4. Even first-year college students can develop enoughcomputer power to do creative independent workon open-ended projects whose answers may not beknown. This can provide an exposure to how scienceis really done.

5. Flexible, powerPal, pre-prepared interactive pro-grams can permit the student to investigate the so-lution of a wide variety of problems in a fraction ofthe time it would have taken with pencil and paper.This can permit even mediocre students to developan intuitive feeling for problems previously accessibleonly to the best students.Other uses of the computer have been made by other

groups, including the development of Socratic tutorialprograms [4], the mods:' rization and organization ofthe physics curriculum into a Keller-plan no-lecture for-mat (Project PHYSNET, Michigan State University),and the extensive use of microcomputer-based labora-tories [20], [6]. Although these applications have a sub-stantial effect on the physics curriculum, they do notproduce obvious and significant modifications of theway the physics and mathematics curricula interact. Iwill therefore not discuss them here and leave the inter-ested reader to seek out the references.

In addition to the Maryland project, new interest inthe impact of computers on the teaching of physics isindicated by the large number of universities introduc-ing upper-class courses in computational physics and anumber of new textbooks taking that point of view.

"No new information appears in a newedition of a physics textbook."

A recent text by R.M. Eisberg and L.S. Lerner [7]provides a fairly standard introduction that includesa number of numerical examples and problems.4 Awidely used advanced text, Computational Physics by S.Koonin [13] contains numerous programs and numericalexamples. The first truly computer-based introductorytext is about to appear, written by Harvey Gould andJan Tobochnik [9].

The AAPT leadership has called for a review of thecurrent curriculum [12] and has been funded by NSF to

organize conferences and workshops to discuss the shapeof a new curriculum. While it may be a few years be-fore a generally acceptable formula is found, and beforeenough colleges and universities get enough computerresources for the change to be broadly accepted, thesmell of revolution is definitely in the air.

Implications for MathematicsIf we presume that the postulated revolution in

physics teaching does in fact take place, and if we alsopresume that the service calculus courses are to con-tinue to play a strong and relevant role in introduc-tory physics, then the emphasis of the course shouldchange somewhat. Although all of the cubjects I pro-pose adding to the traditional course are taught some-where in the mathematics curriculum at many majoruniversities, we must operate under the understandingthat the primary service sequence for scientists is thethree or four semester sequence in calculus. Given theheavy schedule of most scientists in the first and secondyear of their college, it is not possible to require thata significant number of additional mathematics coursesbe taken in the underclass years.

We must operate under theunderstanding that the primary servicesequence for scientists is the three orfour semester sequence in calculus.

Therefore, to make the introductory course more rel-evant, I propose that as the mathematical needs of sci-entists become broadened, especially by an expandedinteraction with the computer, the content of the intro-ductory mathematics service sequence be broadened toinclude topics which do not formally fall under the topic"calculus." However, for simplicity, I will continue torefer to this sequence as "the calculus course."

Specifically, I would like to see the traditional calcu-lus course extended to include more emphasis on:

practical numerical methods;qualitative behaviors;approximation theory;study of discrete systems;pathological functions.

Many individual mathematics teachers idiosyncrat-ically include some discussion of one or more of thesesubjects in their courses; but most traditional calculustextbooks ignore them entirely.

REDISH: Mr:VOLUTION IN PHYSICS 1.

Numerical MethodsPractical numerical methods form the heart of com-

puter approaches to real-world problems, yet these areconsistently ignored in :he traditional introductory se-quence. The problems of numerical integration and dif.ferentiation are suppressed in favor of extensive discus-sion of how to differentiate and integrate large num-bers of special cases, despite the fact that the real-worldproblems most scientists face will almost certainly haveto be treated numerically a large fraction of the time.

Even when practical rules, such as Simpson's rule ofintegration, are mentioned, almost never is there anydiscussion of where they are appropriate, how they canbe improved, or the fact that there may be better al-ternatives. Practical solution methods for differentialand integral equations are rarely discussed in the stan-dard calculus sequence. Methods such as power seriessolutions, useful in proving general results but not veryhelpful in solving a new equation, are strongly empha-sized. A better balance needs to be struck. The ques-tion of stability and improvability of methods shot.:dalso be mentioned.

As discussions of numerical methods replace some ofthe discussions of analytic ones, a good understandingof the qualitative behavior of functions becomes evenmore important than before. Some of this is alreadypresent in traditional discussions of the theory of differ-ential equations and the phase plane; but in the absenceof discussions of numerical methods, their relevance isobscured.

By approximation theory I mean the process of how'to understand the relevance of approximation schemesto specific situations, and of how to develop newschemes. If students are not even aware of the possibil-ities, for example, of the optimization of power series,they will be unable to seek out references on their own.Some broad introduction to the variety of practical andpowerful methods available should be given.

Discrete EquationsSince students are usually introduced to discrete

equations in the context of approximations to contin-uous equations in a numerical analysis, physics, or en-gineering course, they develop the mistaken impressionthat a discrete equation is a poor relation of the con-tinuous one, and that any differences between themis a "failure" of the discrete equation. Discrete equa-tions are themselves the relevant mathematical modelsin a number of circumstances, such as iterated functionproblems. They have their own unique set of character-istics.

Some of these discrete equations lead naturally to thec ussions of functions whicl were considered "patho-logical" by mathematicians and scientists for manyyears, and which were thought to be irrelevant to physi-cal phenomena. This has turned oat not to be the case.For exa.nple, a discussion of the Cantor set and relatedfractals would be very valuable. Usually these are leftto more advanced courses, such as topology, and there,their real-world applications tend to be ignored.

Functions which were considered"pathological" by mathematicians andscientists ... were thought to beirrelevant to physical phenomena. Thishas turned out not to be the ease.

Anyone who proposes the addition of new materialto a course currently packed to bursting with a super-fluity of material has some obligation to identify whatshould be removed to make room. I would be happier tosee less discussion of out-moded analytical techniqueswhich have limited applicability. I also -believe thatsome room could be made by not analyzing the verylarge number of analytical examples usually treated.Specific cuts will of course have tc., be decided by thecommunity through extensive discussion.

Despite these proposed changes, I suspect that alarge majority of physics teachers feel, as I do, that thecalculus sequence provides a strong base for the fun-damental mathematical skills of all physicists; that theprimary emphases should be retained, including the em-phasis on the study of smooth, non-pathological func-tions, on analytic techniques and theorems, and aboveall, on the rigor and structure of mathematical thought.

Notes

I. One of the few texts which reverses the traditional orderis the classic Feynman Lectures in Physics [8] based onthe teaching of Nobel Laureate Richard P. Feynman.

2. Although the content is similar, Millikan's emphasis issomewhat different from modern texts.

3. A contemporary student would probably feel more com-fortable with the explanation "a breadbox is about thesize of a dezktop computer" than with the one given here.When was the last time you saw a breadbox?

4. This text may have come a bit before its time: it in-cluded a number of problems appropriate for the pocketprogrammable calculator. It is already out of print.


References

[1] Aubrecht, Gordon. "Should there 'de twentieth-centuryphysics in twenty-first century textbooks?" University ofMaryland preprint.

[2] Aubrecht, G. (ed.) Quar' Quasars, and Quandries.Proc. of the Conf. on the Teaching of Modern Physics,Fermi lab, April 1986 AAPT, College Park, MD, 1987.Bitter, F., et al. "Report of conference on the improve-ment of college physics courses." Amer. J. of Physics 28(1960) 568-576.Bork, A. Learning with Computers. Bedford, MA: DigitalPress, 1981.Champagne, A.B.; Klopfer, L.E.; Anderson, J.H. "Fac-tors influencing the learning of classical mechanics."Amer. J. of Physics 48 (1980) 1074-1079.

[6] De Jong, M.; Laymai., J.W. "Using the Apple II as a lab-oratory instrument." The Physics Teacher, (May 1984)293-296.Eisberg, R.M.; Lerner, L.S. Physics, Foundations andApplications. New York, NY: McGraw-Hill, 1981.Feynman, R.P.; Leighton, R.B.; Sands, M. The Feyn-man Lectures in Physics. Reading, MA: Addison-Wesley,1964.Gould, H.; Tobochnik, J. An Introduction to ComputerSimulation: Applications to Physical Systems. New York,NY: John Wiley. (In press.)

[10] Halloun, I.A.; Hestenes, D. "The initial knowledge stateof college physics students." Amer. J. of Physics 53(1985) 1043-1055.

[11] Halloun, I.A.; Hestenes, D. "Common sense conceptsabout motion." Amer. J. of Physics 53 (1985) 1056-1065.

[12] Holcomb, D.F.; Resnick, R.; Rigden, J.S. "New ap-proaches to introductory physics." Physics Today, (May1987) 87.

[3]

[4]

[5]

[7]

[8]

[9]

[13] Koonin, S.E. Computational Physics. Menlo Park, CA:Benjamin/Cummings, 1986.

[14] Marx, G. (ed.) Chaos in Education. Balaton, Hungary,June 1987. (To appear.)

[15] McDermott, L.C. "Research on conceptual understand-ing in mechanics." Physics Today 37 (1984) 24-32.

[16] Milliken, Robert A. Mechanks, Molecular Physics, andHeat. Boston, MA: Ginn and Company, 1902.

[17] Reif, F.; Heller, J.I. "Knowledge structure and problemsolving in physics." Educational Psychologist 17 (1981)102-127.

[18] Rigden, John S. "The current challenge: Introductoryphysics." Amer. J. of Physics 54 (1986) 1067.

[19] Verbrugge, F. "Conference on introductory physics cours-es." Amer. J. of Physics 25 (1957) 127-128; "Improv-ing the quality and effectiveness of introductory physicscourses," ibid, 417-424.

[20] Wilson, Jack M. "The impact of computers on the physicslaboratory." Int. Sum. Workshop: Research on PhysicsEducation, La Londe les Maures, France, June 26-July13, 1983. Proceedings, p. 445.

[21] Wilson, Jack M. "Microcomputers as learning tools."Inter-American Conference on Physics Education, Oax-tepec, Mexico, July 20-24, 1987. (Proceedings to ap-pear.)

EDWARD REDISH is Professor of Physics and Astron-omy at the University of Maryland, College Park. For-mer chairman of his department, he is creator and prin-cipal investigator of M.U.P.P.E.T.the Maryland Univer-sity Project in Physics and Educational Technology. He isa member of the Nuclear Science Advisory Committee andChairman of the Advisory Committee for the Indiana Uni-versity Cyclotron.

Calculus in the Undergraduate Business CurriculumGordon D. Prichett

BABSON COLLEGE

If there is widespread common concern that a corecourse in the standard undergraduate business schoolcurriculum is irrelevant and unneeded, it is a concernover the calculus. Altl. )ugh there has been, for twenty-five years, a stalwart cadre of supporters of calculus inthe business core, there has been little common agree-ment on either the level or the topics which are criticalto the success of such a core course. To gain a clearinsight into the appropriate role of calculus in the busi-

ness curriculum in the 1990s, it is important to sketchthe evolution of the present role played by calculus inthe life of a business student.

The PastIn 1963 Richards and Carso [3] reported that only 4

of 71 AACSB (American Assembly of C- ,giate Schoolsof Business) schools responding to a ,aestionnaire re-quired differential calculus. Tull and Hussain [4] in a

, 0

PRICHETT: BUSINESS CALCULUS 113

1966 report on AACSB schools show 38 out of 84 re-quiring differential calculus. Something seems to haveoccurred in the early sixties that suddenly convincedschools of business that calculus was important to theircurriculum.

We should recall the tempestuous climate of edu-cational concern of the early sixties, the emergence ofmany new mathematics incentives, and the widespreadcriticism of any curriculum that did not have "strong"mathematical underpinnings. It may be worth notingthat the Harvard Business School introduced a calculusrequirement for entering students in the early sixtieswhich survived only until the late sixties!

Two questions are critical to our present concerns:1. Why was calculus suddenly inserted into the busi-

ness curriculum?2. Who designed the appropriate calculus course for

this specialized educational track?In the business school reports of the 1950s, a few

schools like Carnegie Mellon and the Sloan School ofMIT emerged as models of futuristic curriculum em-phasis. The background ambience of engineering andeconomics departments within both these schools, cou-pled with strong positions taken up by specific apostles,made one semester of calculus appear to be minimalmathematical background for management students.

The push toward calculus was further supported byworks like Samuelson's Foundations of Economic Anal-ysis. Some, like Kemeney, Snell and Thompson, decidedthat business applications made it preferable to teachfinite vs. continuous versions of the same concepts andtools, but they were fighting two battlesthe generalbattle for more emphasis on mathematics, and a battlefor a variant approach which few of their fellow mathe-maticians endorsed.

Many business calculus courses ...aretaught by mathematicians ... unfamiliarwith the general business curriculum....This marriage of an orphaned courseto a distant curriculum leads one totoday's concerns.

The schools that took calculus most seriously ex-pected students to do a full year, and they were contentto let existing service faculty in mathematics depart-ments teach it. 'ilany business calculus courses beganand still reside in mathematics departments and aretaught by mathematicians, a great number of whomare unfamiliar with the general business curriculum.

Since business schools had not utilized calculus in thepast and were not with many of the applicationsof calculus beyond those in economics, course designwas left to mathematicians and textbook authors. Thisled to a calculus course designed in a wilderness sepa-rating pure mathematics from a very specialized use ofmathematics. This marriage of an orphaned course toa distant curriculum leads one to today's concerns.

The PresentThere is not a standard introductory calculus course

in today's business curriculum. Although AACSB re-quirements do not explicitly state that calculus is anecessary prerequisite for an accreditable school, anyschool not offering calculus in their core would mostlikely com_ under criticism for the exclusion. It shouldalso be noted, however, that of the approximately 1200business programs in the United States, only 237 areaccredited by the AACSB. Many nonaccredited schoolsoffer no calculus requirements, many claiming (justifi-ably) that their students are far too weak in algebrato undertake even a superficial calculus course. Busi-ness schools do not attract a great many quantitativelystrong students, so weak arithmetic and algebraic skillsplague teachers of the business calculus course.

Teaching business calculus today is, tomost, a thankless task.

The present calculus course offered by those schoolsrequiring calculus is frequently referred to as a watered-down version of calculus, and looked at with distainby many pure mathematicians. One must realize thatmuch of what is taught in the standard calculus formathematics students has evolved from studying thepathologies of special functions or special situationswhich enjoy an important role in the life of a mathe-matician, but which almost never occur in the life of apractitioner in management.

A typical business calculus course would be selectedfrom the following menu, in which parentheses surroundtopics that are usually optional and often omitted:

1. Review of algebra and sets2. Formulas, equations, inequalities, graphs3. Linear equations and functions (Applications:

Break-even analysis; Linear demand functions)4. Systems of linear equations and inequalities (Ap-

plications: Supply and demand analysis; Intro-duction to linear programming)


5. Exponential and logarithmic functions6. Mathematics of finance7. Introduction to differential calculus

a. Limits and continuityb. The difference quotient and definition of the

derivativec. The simple power rule d(xn)/dx = nxn-1d. The derivative of [f(x)]"e. The product and quotient rulesf. The derivatives of ,exponential and logarithmic

functionsg. Maxima and minima of functionsh. Maxima and minima applicationsi. (Sketching graphs of polynomials)j. (Relative rates of change)k. (The chain rule and implicit differentiation)1. (Calculus of two independent variables)

8. Integral calculusa. Antiderivatives: The indefinite integralb. Integration by substitutionc. Integrals of exponentialsd. The integral of (mx + b)-1e. Area and the definite integralf. Interpretive applications of areag. (Improper integrals)h. (Numerical integration)i. (Integration by parts)j. (Differential equations)

A quick look at the preceding outline should shockany teachers sensitive to the fact that they have onlyone semester to bring a class of students with a widevariety of backgrounds and quantitative abilities to acommon level of proficiency in most of the above topics.Teaching business calculus today is, to most, a thanklesstask. Add to this the pressure today to perform wellon teaching evaluation surveys, the philosophy used bymany schools that calculus is a primary sieve to weedout poor students, and the general discrediting of theintegrity of the course by most mathematicians, andone wonders with whom the course is staffed in anycollege or university. Clearly th_ design and role of thecalculus course in the business curriculum needs reviewand restructuring.

The FutureTo understand why so much must be taught in one

semester, one must have a better understanding of +hescope of the entire business curriculum. As with engi-neering students, most business students have almostno room for electives in their program. As more and

more emphasis is being placed on broadening the lib-eral arts base of the business curriculum, more coursesare attached to the front end of the curriculum, but fewcourses yield space elsewhere in the curriculum.

At present most business students are required totake three quantitative methods courses: Business Cal-culus or some equivalent mathematics course; Statistics(this course is essentially descriptive and inferential,requiring little probability theory); and InformationSystems (a course involving elementary programmingand computer applications). Electives include Manage-ment Science (applications of operations research), Fi-nite Mathematics (a course that might better be taughtin the spirit of an applied discrete mathematics course),and further courses in Management Information Sys-tems. Little physical or life science is required of ortaken by business stn -lents.

Calculus is not a true curricularprerequisite to studying in anundergraduate business program.

From this :t is clear that calculus is not a criticalprerequisite for any of the technical courses in the stan-dard business curriculum. In fact the only courses thatrequire calculus, excluding case by case instances, aresome introductory economics courses. In most circum-stances the applications of calculus in these courses arewhat many term as "toy problems" and these applica-tions could easily be presented without relying on thecalculus. Hence calculus is not a true curricular pre-requisite to studying in an undergraduate business pro-gram. (Recall the decision of Harvard Business Schbolto drop the calculus requirement.)

A New CourseFor years calculus has been used by colleges and uni-

versities as a credentialing hurdle, as a prerequisite toassure sufficient "mathematical sophistication." Thisremains the primary justification for maintaining a fullsemester calculus course in the business curriculum.It is time to design a new introductory quantitativemethods course to measure this vital prerequisite. Thetreatment of calculus should be embedded in a coursewhich will assure that all business students have suf-ficient mathematical sophistication, receive content ofreal educational value, and encounter the mathematicsand quantitative problems of management education.

The design of such a course should be inspired bythe goals of teaching clear . _asoning, critical and an-alytical thinking, and good organizational and writing

1,`"b

PRICHETT: BUSINESS CALCULUS 115

skills. The course must assure that students who com-plete it have sufficient mastery of fundamental arith-metic and algebraic skills, and can apply the use of cal-culators and computers to obtain and process quantita-tive information. In addition, this course must addressspecific mathematical techniques frequently applied tosolve problems encountered in management education.Above all the underlying intent of such a course is toimpart educational challenge and value to first-year stu-dents of management, with utility and relevance as acatalyzing rather than driving force of the course.

Such a course may be properly called Foundationsof Quantitative Methods in Management; a suggestedcourse outline might be:

1. Linear equations and functions (Applications:Break even analysis; Linear demand functions)

2. Introduction to linear programming techniquesa. Problem formulations and applicationsb. Graphical solutionsc. Computer solutions using a computer package

3. Exponential and logarithmic functions4. Mathematics of finance

a. Use of the calculator5. Introduction to differential calculus

a. The difference quotient and definition of thederivative

b. The simple power rule, d(x")/dx = ne-1c. The derivative of [f (x)]"d. The product and quotient rulese. The derivatives of exponential and logarithmic

functions (Applications: Response functions;Marginal analysis in business and economics)

f. Computer computations using a package suchas MACSYMAMaxima and minima of functions (Applica-tions: Inventory control models)

6. Integral calculusa. Antiderivatives: The indefinite integralb. The integral f zncix,n 0 1c. Integral of exd. Area and the definite integrale. Numerical integrationf. Computer integration using a program such as

MACSYMAInterpretive applications of the integral (Appli-cations: Marginal analysis)

h. Difference and differential equations (Applica- 3.

tions: Rumor spreading models)

g.

g.

Exclusion and InclusionsMuch material that has been prevalent in the past

has been excluded from the above outline.

1. Limits: They should be taught in the context ofthe definition of the derivative, not in general

2. Continuity (define only)3. Curve sketching4. Chain rule and implicit differentiation; treat case

by case5. Calculus of two independent variables6. Improper integrals: Discuss in the statistics course

if probability is introduced as an integral7. Techniques of integration

In place of these omitted topics, special emphasis isgiven to certain topics not highlighted in the past:

1. Linear programming: Problem formulation, withgraphic and computer solutions

2. The use of programs such as MACSYMA to com-pute derivatives, integrals, and solutions to com-plicated equations

3. Numerical integration4. Interpretations of the derivative and integral5. Concept of optimization6. Difference and differential equations with applica-

tions

An Agenda for the 1990'sSince the object of the above outline for a quantita-

tive methods course in the business curriculum is thesame as the object of recent suggestions by Graham[6] for changes in mathematical preparation for science,viz., "to help create new mathematics curricula thatare interesting, sophisticated, innovative, and well in-tegrated with the applications of mathematics," I willfollow the example of Steen in a recent article on math-ematics and science (7) and suggest that the followingagenda is worth starting on:

1.

2.

Firm prerequisites should be established for studentsentering the quantitative methods course for busi-ness. A core course should not be used as a sieve.Students should be required to make up deficiencies,or not be admitted to the -..urriculum.Only teachers who arc familiar with the business cur-riculum and appreciate the role of quantitative meth-ods in business should teach business mathematics.Mathematics departments must work more closelywith the curriculum for which they are a service de-partment.The design and _maintenance of a good- businessmathematics program should be justly rewarded both

116 ISSUES: SCIENCE, ENGINEERING,_AND BUSINESS

professionally and financially. Ways must be foundto reward quality performance and innovation withina mathematics department in pedagogical areas notclassically considered mathematically pure.

4. The chief objective of a core course in quantitativemethods for managers should be the teaching of crit-ical and analytical skills. Clear reasoning, a senseof confidence in one's ability to solve mathematicalproblems, and well-developed organizational and ex-pository skills are critical to success in managementeducation in the 1990's.

5. Any core course in quantitative methods for man-agers must incorporate the use of computers and cal-culators. To exclude computers and calculators fromsolving quantitative problems in the 1990s would beequivalent to attempting solutions to trigonometricproblems fifty years ago without tables.

6. The quantitative methods course should be linkedto other courses in the business school curriculum.Understandable and believable reasons why skillslearned from mathematics transfer directly to skillsneeded in management must be included in everyquantitative core course.

7. Teaching and evaluating writing and organizationalskills should be an integral part of teaching any quan-titative methods course. The expository skills of mostcollege students are dismal; it is time to correct theproblem, not point the finger. We must embracewriting across the curriculum as a primary initiativeof the 1990's.

Calculus for the Biological SciencesSimon A. Levin

CORNELL UNIVERSITY

Any consideration of the teaching of calculus to biol-ogists must recognize that calculus is just one of a vari-ety of mathematical tools that increasingly are provingimportant to the biologist, and that therefore the teach-ing of calculus must be viewed within the context c abroader mathematics education. The relevant math-ematical tools will vary depending on the biologicalspecialty. For example, for the ecologist or geneticist,statistics and experimental design are integral parts ofthe curriculum; but for the biochemist, such courses areconsidered a luxury.

However, needs change, and often over very shorttime scales. For example, in the last decade, comput-

References

[1] F.C. Pierson, et al. The Education of American Busi-nessmen. McGraw-Hill, 1959.

[2]` A.A. Gordon and J.E. Howell. Higher Education forBusiness. Columbia University Press, 1959.M.D. Richards and R. Carso, Jr. Mathematics in Colle-giate Business Schools. Monograph C-10, SouthwesternPublishing Company, January 1963.

[4] D.S. Tull and K.M. Hussain. "Quantitative methods inthe school of business, a study of AACSB member insti-tutions," AACSB Bulletin, January 1966.E.K. Bowen. "Mathematics in the undergraduate busi-ness curriculum," Accounting Review, October 1967, 792.W.R. Graham. Not. Am. Math. Soc. 34 (1987) 245.L.A. Steen. "Mathematics education: A predictor of sci-entific competitiveness," Science 237 (July 17, 1987) 251-252, 302.

[8] E.K. Bowen, G.D. Prichett, and J.C. Saber. Mathemat-ics: With Applications in Management and Economics,Sixth Edition. Richard D. Irwin, 1987.

[3]

[5]

[6]

[7]

GORDON D. PRICHETT is Vice President for AcademicAffairs Dean of the Faculty of Babson College in BabsonPPer. (Wellesley), Massachusetts. Previously, he was Chair-man of the Department of Mathematics at Hamilton Col-lege, Chairman of the Department of Quantitative Methodsat Babson College, and Secretary-Treasurer of the NortheastSection of the Mathematical Association of America.

ing has become an essential part of any scientist's train-ing. F. _'ability and combinatorics, which within bi-ology were once primarily the province of populationgeneticists and ecologists, have grown in importancefor molecular biologists interested in sequence analy-sis. Even topological methods are finding applicationswithin the core areas of biology. Such potential forchange in the areas of mathematics that may prove use-ful argues for the value of training biologists broadly inmathematical methods, rather than restricting atten-tion to narrow and classical areas.

There is a tradeoff, however. The typical biologycurriculum, especially within areas such as molecular

LEVIN: BIOLOGICAL SCIENCES 117

biology and biochemistry, is packed tight with requiredcourses in biology and the physical sciences. There islittle slack, and little room for flexibility. A year ofmathematics is recognized as being essential, but it isunlikely that most biology programs can afford to addadditional mathematics as a general requirement. Thus,new topics can be introduced only if one is willing toeliminate others, or to sacrifice depth for breadth.

Needs change, and often over veryshort time scales.

My argument is that some such compromise is es-sential, and that the way to achieve it is to emphasizeconceptual understanding at the expense of analyticalcomputational ability. This is a painful, exchange, be-cause understanding in mathematics cannot be achievedwithout repetitive drill and problem solving. Nonethe-less, it must be recognized that the biologist's needs aredifferent in kind than the physicist's or engineer's. Thebiologist needs to understand the role of models in biol-ogy and medicine; to know what mathematics is, whatit can do, and what methods are available; and to knowwhere to go to obtain deeper capabilities.

Mathematical ModelsModels in biology serve as pedagogical tools, as aids

to understanding, and as rough approximations thatguide treatment or experimentation. On the otherhand, although there are exceptions to the general rule,mathematical models rarely serve as devices for exactprediction in the way that models can in physics. Thusthe most important mathematical problems facing thebiologist are conceptual, involving the proper formula-tion of a model, rather than analytical.

Some examples should strengthen this argument. Aphysician, in prescribing a drug regimen, should under-stand the importance of the underlying kinetics, themeaning of the half-life of the drug within the body,and the notion that the system eventually will reach anapproximate steady-state, although the plasma level ofthe drug may fluctuate substantially between dosages.This will alert him or her to the importance of testingplasma levels at a fixed point in the dosage cycle, and tothe necessity of establishing quantitative relationshipsbetween dosage levels and steady-state plasma levels.

There is sufficient variati a among individuals, how-ever, that generic models cannot be completely reliable,even when standardized by weight or age. Steady-state

levels must be determined empirically. The model, how-ever, has performed an invaluable role in suggestingwhat needs to be measured. More generally, identifyingwhat biological quantities need to be measuredthe re-fractory time of a neuron following firing, the bindingconstant of a ligand in relation to a particular substrate,the heritability of a genetic character, the diffusion rateof a particular compound across a membrane, the max-imal rate of increase of a bacterial populationis oneof the most common uses of models.

Because of the difficulty of controlling individual andenvironmental parameters, the biological model moretypically is used to describe an idealized situation. Forexample, the Hardy-Weinberg conditions, which specifythe genotypic frequencies to be achieved in a populationin the absence of any selective differences among indi-viduals, describes an idealization that may represent anexcellent approximation under some conditions, and avery poor one under others.

The most important mathematicalproblems facing the biologist areconceptual, involving the properformulation of a model, rather thananalytical.

Similarly, the Lotka-Volterra model, simplisticallydescribing the interaction between predator and preyspecies, represents the biological analogue of the fric-tionless pendulum. The structural instability of thesystem makes it totally inadequate in describing anyreal biological interaction; but as a pedagogical tool, themodel is sufficient to illustrate how a particular mecha-nism can lead to inherently oscillatory behavior. Often,the role of the model is simply to abstract and isolatea piece of a larger system, in the recognition that it isdifficult or impossible to achieve such isolation in anyreal system, no matter how controlled.

The preliminary conclusions are that the teachingof calculus must be integrated with the teaching ofother mathematical topics, such as analytic geometry,dynamical systems, linear algebra, and cc ,nputationalmethods. Furthermore, considerable attention shouldbe paid to discussion of the proper role of mathematicalmodels, to the conceptual insights that can be achievedthrough the qualitative theory of dynamical systems,and to an appreciation of how analytical approaches in-terface with computational ones.

The availability of' high speed` computers has ren-dered archaic the practice of !quipping the student with


a slide rule; and similarly, it has reduced the necessity oftraining students to be facile in the clever manipulationof integrals. Indeed, there still is value in introducingthe student to the slide rule and to log tables, in or-der to provide a perspective on logarithms and powers;similarly, there is considerable value to introducing thestudent to the techniques of integration, and to providesome drill. However, such techniques cannot be viewedas ends unto themselves, but must take their place inan integrated and balanced curriculum whose primarygoal is to teach the relevance of mathematics in biolog-ical and medical applications.

Calculus for BiologistsSetting aside the importance of teaching discrete

mathematics, which is outside the charge of this partic-ular report, I turn attention next to the key ingredientsof a two-semester sequence in calculus for biologists.In so doing, I do not distinguish numerical methods asa separate topic, since such methods are taught mosteffectively when integrated with mathematical applica-tions. Assuming some background in analytic geometryand complex numbers, I propose that the essential in-gredients are:

1. Introduction to limits, rates, and the differential cal-culus

2. Differentiation rules and formulas3. Curve plotting and the theory of maxima and min-

ima; Taylor's theorem4. Introduction to first-order differential equations5. Introduction to the integral calculus6. Integration rules and formulas7. Integration and the solution of differential equations

a. Solution of linear equationsb. Methods for nonlinear equations

8. Areas, averages9. Introduction to vectors and matrices, with eigenval-

ues10. Partial differentiation: Taylor's theorem for several

variables, and the theory of maxima and minima11. Systems of differential equations

a. Solution of linear systemsb. Qualitative theory for nonlinear systems

12. Differential equations of higher order13. Partial differential equations, especially of parabolic

type14. Other topics, as time permits, including infinite se-

ries, volumes, and areasNaturally, there is no single way to teach this mate-

rial, and some-aspects-of the above outlinefor exam-ple, the preferende for teaching integration as motivated

by the solution of differential equationsare mattersof personal taste. Other substantive choices also havebeen made in the above syllabus, going beyond thoseidentified in the previous section.

For example, the inclusion of parabolic partial dif-ferentlal equations as an essential topic is motivatedby the widespread applicability of concepts of diffusionthroughout biology. On the other hand, other than theuse of Fourier analytical methods for time series anal-ysis, infinite series play little role in the training of abiologist. The Taylor expansion is important as a con-cept; however, it need not be viewed within the contextof infinite series, since in applications it almost alwayswould be truncated after the first few terms.

Biological ExamplesMost of the topics listed are standard for a basic cal-

culus course, but their presentation to biologists can beimproved considerably by the development of biologicalexmples. One should build on models that are fairlysimple in biological content, and that offer the potentialfor addressing a variety of mathematical issues withinthe same biological framework.

Often, the role of the model is simplyto abstract and isolate a piece of alarger system, in the recognition thatit is difficult or impossible to achievesuch isolation in any real system, nomatter how controlled.

For example, various methods for graphing the re-lationship between the amount of a ligand bound to asubstrate and the dosage level are used by pharmacol-ogists for discovering the mechanisms underlying bind-ing, and for distinguishing among competing bindingmodels. So-called allosteric models postulate that thereceptor molecule undergoes a conformational changewhen a single drug molecule is bound to it, and that thischange affects binding and dissociation rates for othersites on the receptor molecule. Depending on whetherbinding thereby is facilitated or inhibited, this is knownas positive or negative cooperativity. It introduces aparticular nonlinearity into the model, representing thefact that binding rates change with concentration. Byexamining binding curves directly, and various trans-forms such as double reciprocal plots, one can use qual-itative information regarding curvature and extrema todistinguish among the various hypotheses.

12°


Such examples are likely to be closer to the heartof the biologist than are the usual examples from en-gineering, although the latter should not be neglectedcompletely. It is an unfortunate fact that most begin-ning biology students know very little biology, and maynot be as strongly motivated by biological examples asone would hope. Examples therefore must be kept assimple as possible, so that the focus is on learning themathematics rather than on learning the biology.

Furthermore, it is important that the student beclear in knowing whether the particular mathematicslearned is of general relevance and applicability, orwhether it is particular to the example used to moti-vate it. The importance of maintaining this distinctionis an argument for presenting ideas fairly abstractly, us-ing biological examples for occasional motivation.

OptimizationThe theory of maxima and minima is important to

biologists not only as an aid to graphing, but also in re-lation to fundamental biological and management con-cepts: principles of optimal design, maximal rates ofincrease of populations, optimal treatment regimes inmedicine or optimal harvesting regimes in fisheries, andevolutionary adaptation. Most of these are best treatedwithin the later sections of the course, embedded withina dynamical systems description. The change of popu-lations through evolutionary processes is the single mostimportant organizing principle in biology. The funda-mental philosophical problem is to understand the rela-tionships and distinctions between the process of adap-tation and the possible attainment of optima.

It is an unfortunate fact that mostbeginning biology students know verylittle biology.

The simplest models of gene frequency change as-sume that there is a fixed selective value associatedwith each genotypic combination. For the dynamicalsystems appropriate to these assumptions, it is possibleto demonstrate that the average fitness of all individu-als in the population increases monotonically with time,tending asymptotically to a local maximum. Thus, suchmodels can be used to illustrate a variety of concepts:maxima and minima, local versus global extrema, theasymptotic behavior of ordinary differential equations,multiple steady states, linearization, and stability.

More complicated models, incorporating the evolu-tionary tradeoffs and constraints that are familiar to

the biologist, provide opportunities for illustrating theuse of Lagrange multipliers. Other such examples comefrom fisheries or epidemic management, where there isan explicit economic or effort constraint. Similar mod-els appear in behavioral biology, where time or energymust be allocated to tasks such as foraging for food,grooming behavior, or other activities, and in physio-logical ecology, where the principles underlying a plant'sallocation of resources to growth and reproduction, orto roots and shoots, is of fundamental interest.

Dynamical SystemsDynamical systems have a pervasive influence in the

understanding of biology, and it is hard to think of anarea where they are not an essential part of basic in-struction. In biochemistry and pharmacology, the dy-namics of enzyme-substrate and drug-receptor associa-tions are among the most basic of concepts. Chemother-apy models, even with simple first-order kinetics, giverise to systems of differential equations representing theflow of the chemicals through a network of compart-ments. Formally identical models are applicable to theflow of materials and energy through ecosystems. Thuseven the linear theory is of direct relevance to a widevariety of problems, spanning the biological spectrum.

Indeed, the most basic model of drug dynamics is ofthe form &T/ER= I /ex, where I is the dosage rate andk is the per unit rate of elimination. The physician isfamiliar with the fact that this model predicts a steady-state level I/k, that there is a characteristic half-lifedetermined by k, and that this half-life determines thetime to rid the body of the drug if I is suddenly setto zero, or the time to reach a new equilibrium if I isabruptly changed from one value to a new one.

Thus this model has immediate applicability and rel-evance, and furthermore serves as a starting point forthe investigation of such extensions as allowing I tovary periodically (as would be the case for any dosageregimen other than a continuous intravenous), or allow-ing for an explicit delay representing the time it takesfor the drug to make its way into the plasma. An-other direction for extension is to consider the multi-compartment systems that are more appropriate todrug dynamics, and thereby to account for the dis-tributed delays associated with the gradual entry of thedrug into the plasma.

The consideration of nonlinear dynamical systemssubstantially broadens the range of applications, andprovides a framework for the discussion of a variety ofimportant concepts: steady states, linearization, tran-sient dynamics, asymptotic stability, bifurcation, peri-odic solutions, and chaos. As advanced and esoteric

130


as some of these topics may seem, they all are findingapplication in biological investigations. Understandingthese concepts undoubtedly is of more value to the bi-ologist than is the ability to make specific analyticalcomputations.

The change of populations throughevolutionary processes is the singlemost important organizing principle inbiology.

Qualitative shifts in the physiological behavior of anindividual, from one basin of attraction to Another, maycorrespond to a shift from a healthy to an unhealthystate. In some cases, this may correspond to a break-down of a regulatory mechanism, and the transitionfrom a homeostatically maintained stable equilibriumto a fluctuating or even chaotic dynamic. In other cases,the healthy state may exhibit carefully regulated peri-odicity, as in cardiovascular dynamics, or in the dailycircadian rhythms of the body. Qualitative changes inbehavior may be associated with internal changes, suchas the breakdown of normal regulatory mechanisms, orwith external changes, such as the intake of a toxin.

Attention to dynamic phenomena is equally impor-tant in other areas of biology. For the neurobiologist,the sustained repetitive and sometimes chaotic firingin networks of neurons represents a behavior that canbe understood only within the context of the mutualexcitatory and inhibitory interactions within the net-work, and the transition from normal to more erraticbehaviors can be understood in terms of bifurcationstracking changes in critical internal and external pa-rameters. The genetic control of development similarlyis mediated through networks of regulatory pathways.

For the ecologist or epidemiologist, periodic phenom-ena in the dynamics of populations or disease long havebeen central subjects for investigation, and the varioushypotheses regarding the mechanisms underlying themhave been well studied within the context of dynarn-ical systems theory. Indeed, even simpler nof.ons as-sociated with transient behavior have been of criticalimportance.

An Example from EpidemiologyAgain, it is wnrthwhile to illustrate with an example.

One of the and classical models of the trans-mission dynamics of infectious diseases assumes thatthe host population is of constant size, and broken into

three categories: susceptible (S), infectious (I), and re-covered (immune) (R). Individuals move among theseclasses according to certain rules, including the possibil-ity of losing immunity and returning to the susceptibleclass. Because the time scale of a particular epidemic isassumed to be short compared to the change in popu-lation density, it is assumed that births balance deaths.The usual resultant model takes the form

dS/dt = b kSI bSdI/dt = kSI bI vIdR/dt = vI bR

Here b is the birth (=death) rate, v is the recoveryrate of infectious individuals, and kSI, the incidencefunction, represents the assumption that the rate thatnew infections occur is proportional to both the numberof susceptible individuals and the number of infectives.Naturally, any of the assumptions can be modified, andso the above model again serves as a starting point forinvestigation, and as a way to introduce concepts.

Because population size is constant (it needn't bein more general models, such as those that incorporatedisease-induced mortality), it is convenient to treat S,and R as fractions, so that S + I + R = 1; this assump-tion is implicit in the particular formulation above, andallows reduction of the apparent 3-dimensional systemto two dimensions.

The simplest concept that emerges from the systemabove is that of the threshold transition rate. It is easilyseen, because S cannot exceed unity, that no outbreakcan occur unless k > b + v; that is, if k < b + v, thenumber of infectives will decrease monotonically, what-ever the initial values. This simple concept is of greatimportance to disease management, because it gives ameasure of the amount by which the transmission of adisease must be reduced in order to prevent outbreaks.

By similar reasoning, one can demonstrate that nooutbreak can occur unless the proportion of susceptiblesexceeds a threshold value (b + v) /k. Thus, vaccinationstrategies aimed at reducing the susceptible populationbelow that threshold can succeed in the control of thedisease. Finally, when the threshold condition is ex-ceeded and outbreaks are possible, the above model pre-dicts that the population will settle down to a situationin which a stable fraction (b + v)/k of the individualsare susceptible.

Thus, as with some of the other examples mentioned,this example has sufficient reality to motivate the im-portance of the underlying concepts. More complicatedversions, for example those that incorporate a latentperiod, or different susceptibility classes, or seasonality,

131


can be used to introduce more complicated dynamics;and equally importantly, to demonstrate how modelsare used in the investigation of biological phenomena,and in the management of applied biological problems.

Diffusion ProcessesFinally, the above outline makes a case for the early

consideration of partial derivatives, and for some discus-sion of partial differential equations. In defense of thelatter recommendation, I make reference to the ubiqui-tous nature of diffusion phenomena in biology. Diffusionmodels have been used successfully to describe the flowof heat and materials within animals and plants, themovement of solutes in ecosystems, the spread of par-ticulates from smokestacks, and the passive spread ofanimals and plants. Early applications of these mod-els addressed the spread of species introduced into newhabitats, the rates of spread of advantageous genes, andthe chemotactic movement of bacteria.

Fundamental work in neurobiology has used diffusionmodels to describe the transmission of neural impulses,and basic models in genetics have used diffusion approx-imations for stochastic processes to determine the genefrequencies to be expected under the primary influenceof random genetic drift. Diffusion-based models havebeen fundamental to the understanding of geographicalgradients and patterns in the distribution of terrestrialand oceanographic species, and in the frequencies of dif-ferent genetic types. Finally, some of the most originaland stimulating models of how development takes placehave been structured on a discussion of pattern forma-tion in systems involving chemicals that react with oneanother while diffusing through a medium. Thus, thereis considerable motivation for including at least an in-troduction to these concepts in the basic calculus coursefor biologists.

Demonstrating RelevanceAlthough I made specific recommendations concern-

ing topics to be included in a one-year calculus coursefor biologists, my fundamental recommendations con-cern the objectives of such a course. The primary goalsof teaching biologists mathematics should be to teachthem how models can be used in biology and medicine,and to introduce them to a broad range of basic con-cepts even at the expense of training that makes stu-dents facile in performing calculations.

Too often, the biologist takes calculus as a freshman,and has forgotten all that he or she learned long beforegraduation from college, primarily because the issue ofrelevance has never been addressed. Our goal in teach-ing mathematics should be to demonstrate relevance,by our choice of topics and examples, by our method ofpresentation, and by our emphasis on concepts.

The primary goals of teachingbioloeits mathematics should be toteach them how models can be used...even at the expense of training thatmakes students facile in performingcalculations.

As biologists so taught realize the importance ofmathematical concepts, such concepts will become bet-ter integrated into biological investigations, and ulti-mately into the teaching of biology. This has occurredin some instances, for example in the teaching of enzymekinetics, population genetics, and population ecology;but there is considerable potential for improvement.

Our goal should be to provide biologists with a bet-ter appreciation and understanding of mathematics andmodels, and thereby to move the subject in the direc-tion of more quantitative rigor. In my opinion, the fail-ure to do this in the past has been because we haveassumed that the needs of biologists are identical tothose of physicists and engineers, and that biologistsare somehow mathematically more backward and lessquantitative.

We need to recogn. the different nature of the sub-ject, and to tailor oui courses so that they are of max-imal use to the biologist, and can convince the biolo-gist of the need to think mathematically and rigorously.Only then will we have met the challenge facing us.

SIMON A. VIININ is Charles A. Alexander Professorof Biological Sciences at Cornell University in Ithaca, NewYork. He is President of the Society for Mathematical Bi-ology, and is Director of the Center for Environmental Re-search and of the Ecosystems Research Center at Cornell,and a member of the Center for Applied Mathematics. Heis a member of the Commission on Life Sciences and of theBoard on Basic Biology of the National Research Council.

The Matter of AssessmentDonald W. Bushaw

WASHINGTON STATE UNIVERSITY

The content of a calculus course, and even the spiritin which it is taught, do not exist in isolation. Theyhave external associations with prerequisites, coursesfor which calculus itself is prerequisite, applications, theZeitgeist, career plans of students, numbers and qualityof students, and all sorts of other pressures and con-straints; about these the aspiring reformer of calculuscan do little. But things can be done about the choiceof instructional materials, technological aids, and modes-of instruction.

The content and spi.it of any course, calculus for ex-ample, are also comm,,nly associated with many kindsof evaluations: of students who want to enter the course,of students in the course, of students who have com-pleted the course, of texts, of teachers, of the contentand spirit themselves.

Any ...reform should anticipate theneed to keep the evaluation aspects ofthe new calculus in harmony with itsintended content and spirit, andinclude ways of meeting that need.

A reform of calculus that ignores evaluationthatattempts to put the new wine of a "lean and lively calcu-lus" into an old bottle made of evaluation techniques de-veloped, perhaps inadequately, for old coursesis verylikely to perish. Any such reform should anticipate theneed to keep the evaluation aspects of the new calculusin harmony with its intended content and spirit, andinclude ways of meeting that need.

For this reason, it is important that anyone consider-ing calculus reform be aware of some information aboutacademic evaluation. The subject is large, and the allot-ted space is small, so this treatment is perforce sketchy.The most useful part of the essay may be the list ofreferences, which provide openings into pertinent partsof the literature.

A Few DefinitionsTalk of evaluation and assessment is very much in

the higher education air these days, and rightly so. Theterminology, however, is muddy. In this essay the term

"assessment" will be taken as generic, and will be meantto apply to any systematic procedure for judging thequality of an educational process or of the outcomes ofthat process.

It will be useful for our purposes to identify threemajor forms of assessment:

In-course Testing: In-course or final assessment ofstudent progress;Teaching Evaluation: Assessment of the quality ofthe educational process, usually by direct observa-tion in courses;Outcomes Assessment: Assessment of student abil-ities, attitudes, accomplishments, etc., presumablyacquired through some specific educational experi-ence or experiences.These three categories of assessment are fuzzy sets.

For example, a final examination, surely a form of in-course testing, can also be considered a form of out-comes assessment. It can also be regarded as a type ofteaching evaluation: when a whole class's performanceon a final examination is disappointing, we tend to askourselves what we did wrong; and examination resultsin multi-section courses are sometimes used quite delib-erately as measures of the effectiveness of the instructorsinvolved.

In-course TestingThe most common type of assessment is in-course

testing, which uses many devices: graded homework,quizzes, examinations, oral presentations, team proj-ects, term papers, etc. It provides a base for grading; itprovides students with information about "where theystand;" it can provide the instructor with valuable cluesabout individual or epidemic student difficulties; and,when properly done, it can provide experiences of muchintrinsic educational value. Good advice on this subjectis contained in books like those by Eble [6], McKeachie[14], and Lowman [13]; more technical information ap-pears in such books as that by Gronlund [11].

Teaching EvaluationThe purpose of teaching evaluation is to obtain infor-

mation about the quality of instruction in a particularclass, or by a particular instructor. The two purposes

BUSHAW: ASSESSMENT 123

for ft:Aching evaluation most often mentioned are theidentification of areas where improvement is in orderand the provision of part of the base for personnel de-cisions.

Teaching evaluation is not limited to the use ofquestionnaires to be filled out by students. "Stu-dent evaluation" can be, and usually is, an impor-tant element in teaching evaluation, but the latter canalso include systematic evaluation by colleagues, self-evaluation (or, better, self-reporting), and assessmentof student progressnot to mention subjective judg-ments by administrators, or the venerable supposition,still unfortunately current in some quarters, that in-struction is satisfactory unless there is clear contraryevidence, e.g., an exceptionally large number of studentcomplaints.

Justice simply cannot be done to some"central ideas" ... by using onlyelementary mathematical symbolism ormultiple-choice responses. Writing isoften the natural vehicle.

Teaching evaluation is discussed seriously in a largenumber of articles and several recent books, includingthose by Centra [4] and Se ldin [16] and that edited byFrench- Lazovik [9]; see also the recent article by Mc-Keachie [15]. A report on teaching evaluation addressedspecifically to instructors of postsecondary mathematicsis in preparation by a committee of the MathematicalAssociation of America chaired by the author of thepresent paper.

Outcomes AssessmentThe broadest form of assessment is outcomes assess-

ment, which is usually done for purposes of student cer-tification, program evaluation and improvement, or ac-countability to some monitoring agency or to the public.

Placement testing, as a form of assessment of out-comes of previous educational experience, really belongshere, and in some schools is the most prominent form ofoutcomes assessment. Many of the things that will besaid about outcomes assessment or assessment in gen-eral therefore apply also to placement testing, whichnevertheless will not be emphasized in this paper.

Outcomes assessment can be performed in manyways through a variety of generally available instru-ments: many placement examinations, GRE generaltests and subject tests, the Fundamentals of Engineer-ing Examination (formerly EIT), the LSAT, the MSAT,

the COMP syste:n of ACT, and the Academic Profilejust now being launched by ETS. There are also locallyproduced written examinations as well as senior theses,oral examinations, surveys of graduates, analysis of jobplacement data, and various special ways of assessingwriting proficiency.

Until recently outcomes assessment has not been con-spicuous in higher education, except in areas relating toplacement, licensing, testing for admission to graduateand professional schools, and the likeprimarily for thebenefit of the individual student.

But in very recent years, partly in response towidespread dissatisfaction withor at least uncertaintyaboutthe quality of higher education, there has beenan upsurge of interest in "assessment" as a vehicle forprogram improvement or institutional accountability.This is often expressed in state mandates or in crite-ria newly adopted by accrediting agencies. "A year ortwo ago, only a handful of states had formal initiativeslabeled 'assessment.' Now, two-thirds do" (Boyer [3]).Useful information about assessment in this sense iscontained in the volumes edited by Ewell [8] and Adel-man [1].

Basic PrinciplesAlthough the three major types of assessment are

usually discussed separately, the literature presents anear-consensus on certain principles that apply to all ofthem, principles based in part on research and in parton practical experience and good common sense. Hereare some of them:

1. Before any program of assessment is developed,that which is to be assessed should be clearly identified.

In- course testing should really be designed to mea-sure attainment of class objectives; teaching evaluationshould not be practiced without some reasonable un-derstanding (shared by assessors and assessed) on whatcharacterizes effective teaching; and outcomes assess-ment should presuppose some standards related to stu-dent competency.

An important corollary to this principle is that em-phases in in-course tests should agree with emphasesin the course syllabus. Tests in a possible new calcu-lus course that is more sharply focused on central ideasand on the role of calculus as the language of scienceshould themselves be focused on the same themes. Stu-dents can learn quickly to give short shrift to "centralideas" and "roles" if it is known that "central ideas" and"roles" do not come up at exam time. Likewise, instruc-tors in such a course should be evaluated on their ability

I ,...1 0 4..-

124 ISSUES: TEACHING AND LEARNING

to teach those things that are considered most impor-tant in the course, and in a manner compatible withthe intended spirit of the course. Outcomes assessmentshould be conducted under the analogous requirement.

2. Ideally, an. assessment program should be multidi-mensional.

For in-course testing, for example, one test is hardlyever enough. Even if tests were infallible, as they neverare, the common practice of giving several tests in acourse would be methodologically sound as well as hu-mane. Studies show that when there are several testsbefore a final examination, performance on the final it-self is improved. But there are various kinds of writtentests, and also such potential assessment devices as as-signed homework, separate essays, oral exams or inter-views, evaluations of student notebooks or journals, andold-fashioned observation of classroom participation.

The common view that mathematics should perhapsbe exempted from the currant call for "writing acrossthe curriculum" is questionable. Courses such as the"lean and lively calculus," where there is to be increasedemphasis on ideas and decreased emphasis on merememorization of procedures, will be natural settings fortesting systems that rely not merely on problems withnumerical answers or multiple-choice clue out alsoon writing.

If writing about mathematics is to be an n..portantelement of assessment, then it should be an importantelement of the rest of the course. Justice simply cannotbe done to some "central ideas" of the calculus by usingonly elementary mathematical symbolism or multiple-choice responses. Writing is often the natural vehicle,and its use in a calculus class will be both a contribu-tion to the improvement of writing skills generally andan opportunity to use writing as an effective device forlearning. See for example Griffin [10), Herrington [12),or Emig [7].

In any case, a mix of reasonable testing methods canbe expected to give more valid evaluations, and a richereducational experience, than any one of them alone.

With obvious modifications, the same remarks ap-ply to teaching evaluation and outcomes assessment,where exclusive reliance on one assessment instrumentor practice is almost always inadequate and is stronglydiscouraged by specialists in these fields.

3. Assessment instruments should be. designed withcare and if possible checked for validity and reliaba-ity. When commercially produced instruments are avail-able, they are often betterand cheaper, all thingsconsideredthan local productions.

There is probably no need to dwell on the first of

these observations; any experienced teacher knows howeasy it is for something to go wrong in the production ofa test, and how much damage can be done by a defectivetest.

To my knowledge, there now exist no generally avail-able examinations that come anywhere near matchingthe vision of calculus emerging from the Tulane confer-ence. This does not mean that there will not be any;indeed, the production of such an examination mightbe a good way of redefining that visionof implicitlydefining its objectivesfor instructors and writers.

Let there be no facile talk about theevils of "teaching to the examination."If an examination is well crafted,...then "teaching to" it can be a verygood thing.

And please, let there be no facile talk about the evilsof "teaching to the examination." If an examination iswell crafted, and is complemented by other means of as-sessment in accordance with Principle 2, then "teachingto" it can be a very good thing. In general, performancecriteriaand a good test should be a faithful expres-sion of performance criteriashould provide guidancefor the performers as well as for those who are respon-sible for evaluations of performance.

4. Finally, ways should be sought to get somethingout of assessment besides ratings or scores of some kind.

It is a commonplace of the literature on teachingevaluation that teaching evaluation should be coupledwith a system of faculty development opportunities, sowhen the instructor is found to need improvement insome respect, there is a resource through which thatimprovement can be accomplished effectively. As sug-g.eted earlier, outcomes assessment is often part of asyn,em of program review and planning and may leadto program improvements attained by diversion of re-svurces or by some other means. Even in-class test-ing, except perhaps at the very end of a course, shouldWhelp both student and instructor identify weaknessesand should provide a base for more effective teachingand learning afterwards.

All three of the major types of assessment that havebeen identified, and many of the specific methods of im-plementing them that have been mentioned, with corre-sponding methods of follow-up, might arise in the devel-opment of a new or substantially redefined course such

130

HUGHES: WOMEN UNDERGRADUATES 125

as the "lean and lively calculus." Carefully selected as-sessment methods should be expected to contribute inimportant ways to the educational experience and alsoto the review and continuing improvement of the course.Thus anyone who is expected to benefit from advice onwhat forms the new calculus might take and on how itshould be taught can also be expected to benefit fromgood advice on the assessment activities that go withit.

[1]

ReferencesAdelman, Clifford (ed.). Assessment in higher education:issues and contexts. Washington, DC: Office of Educa-tional Research and Improvement, Department of Edu-cation, 1986.

[2) Astin, Alexander W. "Why not try some new ways ofmeasuring quality?" Educational Record 63 (Spring 1982)10-15.Boyer, Carol M., et al. "Assessment and outcomesmeasurementa view from the states." AAHE Bulletin(March, 1987) 8-12.Centra, John A. Determining Faculty Effectiveness. SanFrancisco: Jossey-Bass, 1982.Donald, Janet G., and Sullivan, Arthur M. Using Re-search To Improve Teaching. San Francisco: Jossey-Bass,1985.Eble, Kenneth E. The Craft of Teaching: A Guide ToMastering the Professor's Art. San Francisco: Jossey -Bass, 1976.Emig, Janet. "Writing as a mode of learning." CollegeComposition and Corn, .: ..linv:ion 28 (May 1977) 122-27.Ewell, Peter T. (ed.) Assessing Bducational Outcomes.San Francisco: Jossey-Bass, 1985.

[3)

[4)

[5)

[6)

[7)

[8]

[9) French-Lasovlk, Grace. (ed.) Practices That ImproveTeaching Etmluation. San Francisco: Jossey-Bass, 1982.

[10) Griffin, C. Williams. (ed.) Teaching Writing In All Dis-ciplines. San Francisco: Jossey-Bus, 1982. See espe-cially Barbara King, "Using writing in the mathematicsclass: Theory and practice," 39.44.

[11) Gronhnd, Norman E. Measurement and evaluation InTeaching, 4th ed. New York: Macmillan, 1981.

[12) Herrington, Anne J. "Writing to learn: Writing acrossthe disciplihes." College English 43:4 (April 1981) 379-87.

[13) Lowman, Joseph. Mastering the Techniques of Teaching.San Francisco: Jossey-Bass, 1984.

[14) McKeachie, Wilbert J. Teaching Tips: A Guidebook forthe Beginning College Teacher, 7th ed. Lexington, MA:Heath, 1978.

[15) McKeachie, Wilbert J. "Instructional evaluation: Cur-rent issues and possible improvements." Journal ofHigher Education 58:3 (May/June 1987) 344-350.

[16) Zeldin, Peter. Changing Practices In Faculty ESan Francisco: Jossey-Bass, 1984.

DONALD I3USHAW is Vice Provost for Instruction andProfessor of Mathematics at Washington State Universityin Pullman, Washington. He has served as chair of theCommittee on the Ur.dergraduate Program in Mathematics(CUPM) and as a member of regional accreditation teams.Bushaw chairs a committee on Evaluation of Teaching of theMathematical Association of America, and is a member orchair of three other MAA committees.

Calculus Reform and Women UndergraduatesRhonda J. Hughes

BRYN MAWR COLLEGE

One of the goals of the current initiative to reformthe calculus curriculum in colleges and universities is tomake science-based careers more accessible tl women.Nevertheless, any program for reform should reflect, evi-dence that women appear to do as well in calculus, as itis now taught, as do men. There is much to be learnedfrom the relative success of women in calculus and incollege mathematics, and this paper will examine ev-idence for that success, offer some explanations for it,and explore pedagogical possibilities for preserving thatsuccess in the midst of major reform of calculus as we

now know it.For most women, enrollment in calculus represents

survival of their interest in mathematics despite over-whelmingly negative cues from our culturewomen incalculus are already going against society's expecta-tions for them. Statistics indicate that for most youngwomen, interest in mathematics wanes in the adolescentyears, and that the trend of avoiding mathematics con-tinues throughout high school, accompanied by lowerSAT scores in mathematics and reduced expectationsof success in mathematical and scientific endeavors [1,

1 :? 6

126 ISSUES: TEACHING AND LEAR14ING

3]. There is some recovery in college (where calculus isusually taken), and then the attrition recurs in the grad-uate years [2]. Any program for calculus reform mustbear in mind this complicated picture, and take intoaccount the strengths and needs of women students.

The present interest in calculus reform was gener-ated by a variety of concerns, including the widespreaduse of computers and the instructional possibilities theypresent, the realization that other topics might betterserve the needs of undergraduates, and the failure ofmany calculus courses to capture the interest of stu-dents. Nevertheless, it is also a reflection of a gen-eral soul-searching among members of the mathematicscommunity.

There is considerable evidence, bothanecdotal and statistical, that womenare doing well in calculus.

A study of Undergraduate Programs in the Mathe-matical and Computer Sciences (1985-86) showed thatthe number of computer science degrees in 1984-85 hasmore than tripled since 1979-80, while the number inmathematics and statistics was up only slightly fromthe level in 1980, and below the level in 1975 [4]. Ina recent report on graduate mathematics enrAmentsby the Conference Board of the Mathematical Sciences(CBMS), it was noted that in 1986, 45.8% of graduatestudents enrolled in mathematics (at the top thirty-nineinstitutions) were foreign, as compared with 19.6% in1977 [5]. The report states that "it is clear that weare failing to attract many of the very best potential[American] mathematicians." The number of Ph.D.'sawarded to U.S. citizens has fallen steadily from 774 in1972-73 to 386 in 1985-86, while the relative percentagedropped from 78% to 51% [6]. In addition, the 1984David Report warned that the community faces seriousand urgent problems of revitalization and renewal, inthe face of inadequate federal funding [7]. It is only nat-ural that this pressing need for self-examination shouldspawn a movement to rethink and improve the mannerin which we teach calculus, presently the focal point ofmathematics education at the college level.

For those of us who have been concerned for yearsabout the failure of mathematics to attract youngwomen and minorities, this climate of reform and theaccompanying flurry of activity is somewhat ironic. Inthe CBMS report, the committee cites a letter fromone American-born graduate student stating that theforeign graduate students form close-knit groups, and

he has few students to talk with [5]. This form ofisolation is particularly damaging in the sciences, butit is by no means a new experience for women andminority graduate students. Indeed, many women Ihave known have experienced the same isolation as theabove-mentioned American male. Thus the enthusiasmof the entire mathematical community for this issue ismost welcome. As a consequence, calculus may undergorejuvenation in a manner that will attract more youngpeople, regardless of sex or ract , to careers in mathe-matics, science, and science-based fields.

Women Succeed in CalculusJ:*

There is considerable evidence, both anecdotal andstatistical, that women are doing well in calculus (thecriteria I use to determine that a student is "doing well"are grades and continuation in mathematics or sciencecourses). Indeed,. in 1936 46% of baccalaureate math-ematics majors were women [2]. If calculus were anobstacle for- women, one would expect the percentageol women majors to be somewhat lower. Moreover, theattrition that occurs at the graduate level, reflected inthe fact that (in 1986) 35% of mathematics master'sdegrees, and only 17% of Ph.D.'s in mathematics wereawarded to women [2], can hardly be attributed to dif-ficulties in calculus. A casual survey of calculus gradesfgpyomen and men in several calculus courses taught ata variety of institutions over a ten-year period yieldedthe following data:

Grade Women Men % Women % Men

A 50 79 30.67 19.85B 55 106 33.74 26.63C 34 112 20.86 28.14D 18 57 11.04 14.32E 6 44 3.68 11.06

The data are from twenty-three first-year calculussections at Harvard University, the University of Ohio,'ifts University, Temple University, and Villanova Uni-versity; the instructors were both women and men. Thedata supports in a variety of ways anecdotal impres-sions that women do well in calculus: more -than halfof the women received A's or B's, and the failure ratefor women is extremely low. In fact, the women appearto do better than the men, anough the larger pool ofmen in the sample may account for the latter's higherfailure rate and lower grades. A thorough investigationalong these lines would be most informative.

There are several possible explanations for the phe-nomena observed in this modest experiment. The **lostobvious one has already been mentioned: those women

1 7

HUGHES: WOMEN UNDERGRADUATES 127

who get to calculus are already somewhat motivated,having survived educational and cultural influences thatmay have been less than encouraging of their mathe-matical and scientific interests. The low failure ratefor women convincingly supports this hypothesis. An-other explanation is offered by the fact that (in 1983) al-most 60% of college freshwomen who intended to majorin mathematics reported "A" averages in high school,while the corresponding figure for men is about 45%[1]; thus, based on high school performance, the womenin college mathematics courses are as strong as (if notstronger than) the men.

Even the brightest women oftenexhibit a marked lack of self-confidence, and are disproportionatelydiscouraged by setbacks. ...For thisreason, encouragement and support incalculus ...may fuel women studentsfor the road ahead.

A discouraging postscript is that despite their mo-tivation, ability and successful performance, even thebrightest women often exhibit a marked lack of self-confidence, and are disproportionately discouraged bysetbacks; the lack of encouragement at earlier stagesseems to take its toll. For this reason, encouragementand support in calculus are vital elements in counter-acting the damage that may have already been done,and may fuel women students for the road ahead.

Teaching MethodsThe Methods Workshop at the 1986 Tulane Calculus

Conference offers several goals for Instruction in Cal-culus that involve modifications in both the mannerin which calculus is taught, and in the course content.One of the difficulties cited by this Workshop in teach-ing calculus effectively is that calculus is a "stepchild"in many departments, with large enrollments and lim-ited attention from tenured faculty [8]. These courses,given low priority by departments or institutions, arefrequently taught by TA's, instructors, or non-tenuredfaculty. The language difficulties of some foreign TA'smay exacerbate this problem [5].

In my own case I benefitted significantly from whatmight have been the failure of a department to givehighest priority to the teaching of calculus. Indeed,both of my calculus teachers were women, both un-tenured, and one of them gave me considerable en-couragement and individual attention (the classes were

smallabout twenty, as I recall). Alas, for the remain-der of my mathematical education (nine years worth),I did not have another woman teacher.

Junior faculty often have greater rapport with stu-dents than do older faculty, and thirc'an benefit women,as well as men. As only about 6% of tenured facultyin the mathematical sciences at four-year colleges anduniversities are women [1], as opposed to 15% of allfull-time faculty [4], a move towards having calculustaught by higher-ranking faculty significantly reducesthe chance of a woman teaching calculus. In fact, pro-vided TA's can speak English, and have some teachingexperience, I do not feel that their use as calculus in-structors is bad per se [9].

The reformed calculus would be a streamlined course,more conceptual, more relevant to real-world problems,and taught in a more open-ended, probing fashion thancurrent versions [8]. These aims are difficult to fault;women students are certainly equal to the challengeof the "new calculus," and should benefit from thesechanges as much as men students.

A Case StudyHowever, as far as women are concerned, changes in

content may be less crucial than changes in methodol-ogy. As women progress along the path towards moreadvanced mathematics, we know that they fall by thewayside. Wide-scale change could undo some of thesubtle positive influence of a successful experience infirst-year calculus. Therefore, given the conflicting goalsof smaller classes with more individual attention, andthe presence of tenured faculty, I feel there is more tobe gained by both women and men students from thepersonal attention smaller classes afford. (An ?rest-ing point, made by one of my colleagues, is that whenclasses are too small, there may be only one or twowomen, resulting in feelings of isolation by the womenstudents. Perhaps the optimum class size for women issomewhat larger than one might expect.)

When classes are too small, there maybe only one or two women. ...Perhapsthe optimum class size for women issomewhat larger than one mightexpect.

A women's college provides a convenient setting forconsidering the effect of various programs on womenstudents. Many of the suggestions made by the Tulane

138


Methods Workshop are already in place at my institu-tion, with varying degrees of success. In spite of oursmall size, we still face many of the same problems in-herent in a larger institution. (I spent ten of my mostformative years ata state university, and have experi-enced some of those difficulties first-hand.)

For us, a large calculus class has at most sixty stu-dents. We teach both large and small calculus sections,a small section having a maximum of thirty students.There is at present only one course to meet the needs ofaspiring mathematicians and scientists, as well as thoseof students fulfilling a mathematics requirement. De-spite the administration of placement exams, a widevariation in preparation of the students still occurs inthe first-year course. The Methods Workshop statesthat "it is essential to have an effective diagnostic andplacement program," and that the success of the pro-posed calculus curriculum hinges on this [8]. We havefound this goal elusive, for a variety of reasons: (i) over-prepared students, (ii) underprepared students, and (iii)ABT's (All But Trigonometry).

Students in the first category have already had cal-culus in high school, in some cases the equivalent of oneyear, but they have not received advanced placementcredit. Well-meaning advisors who are not mathemati-cians often project their own uneasiness about math-ematics Onto the students, encouraging them to takecourses for which they are overprepared. If the studentis really able, she should be strongly urged to try thenext course, for the presence of such students makesthose who have never had calculus (a small minority ofeur students) quite uneasy, with good reason. ( I do notremember th. being a problem at a state institution.)

Underpr ted students are usually identified by theplacement. exam and enrolled in precalculus, but someare only marginally underprepared, or may merely lackconfidence. We usually send these students to calculus,but keep an eye on them for the first few weeks. Thethird category is a subset of the underprepared student,and there are usually several such students each year.

A precalculus course is usually inappropriate, andquite boring for most of these students; we do not wantthem to sit through a course waiting for the last three orfour weeks of trigonometry. In order to deal with thisproblem, we have devised a series of non-credit mini-courses, taught by a graduate or advanced undergradu-ate student, that treat trigonometry, exponentials andlogarithms, and other topics in separate two-week mod-ules. If these are carefully coordinated with the calcu-lus sequence, a student has the opportunity to brushup on material she feels uncertain about, or becomeacquainted with the basics of trigonometry before it

is introduced in calculus. (The logistics here are dif-ficult, for trigonometric functions appear fairly early inthe syllabus). In institutions with teaching assistants,the minicourses should be easy to organize. Of course,the student doing the teaching should have some ex-perience, as well as solid evidence of being an effectiveteacher. Even though we have a mall student body, weare fortunate in usually having such students available.

Another advantage of manarmble class size is thatpapers may be regularly graded, providing the valuablefeedback recognized by the Tulane Workshop as cru-cial [8]. We employ large numbers of mathematics andscience students to grade papers for us. For a class ofsixty students, it is not uncommon for homework to becollected twice a week and be thoroughly graded. Sincehomework is generally graded by the student graders,I usually give quizzes (I don't count them) so the stu-dents have some idea of how I grade, and what typesof questions I think are important:, before the actualexams.

The difference between "feedback" and"support" is like the differencebetween "eating" and "dining."

A reasonably successful means of helping studentsin calculus, as well as actively involving mathematicsmajors in reviewing courses they have already taken,is our Math Clinic. This operation is run solely byundergraduate students, on a voluntary basis. They areavailable a couple of nights a week to help students withquestions about their mathematics courses. Althoughthe faculty is occasionally tempted to intervene, we havenot done so for many years. Directorship of the Clinicpasses from one student to another, and they do anadmirable job of keeping the operation afloat.

ConclusionThese details are offered in the hope that they give

some picture of how calculus is presently managed ata small women's college. Roughly 30% of our studentsmajor in mathematics and science, and most of thempass through our calculus course. Nevertheless, farmore effective than any of these structural aspects ofour course is the patience, encouragement, and supportI see my colleagues offering their students. While this iseasier to do at a small college, I have vivid memories ofdedicated professors trying to do the same thing withclasses of one or two hundred students.

While the MAA report frequently mentions the im-portance of "feedback," there is virtually no mention of

1.?

MALCOM AND TREISMAN: SUCCZSS FOR ALL 129

offering support to students who need it, perhaps be-cause the latter is far more difficult to "package." How-ever, the difference between "feedback" and "support"is like the difference between "eating" and "dining." Ifmathematics as a profession is to recover from its cur-rent malaise, and the vast untapped resource of womenand minorities is to be realized to the fullest extent, wemust allow our students to dine on the fruits of math-ematics. A "lean and lively" calculus, thoughtfully im-plemented with the needs of students in mind, couldwell contribute to this goal.

Acknowledgement. The author gratefully acknowl-edges the assistance of Lori Kenschaft, Executive Direc-tor of the Association for Women in Mathematics, forsupplying the data in references [1: lnd [2], and to mycolleague Kyewon Park for careful , and thoughtfullyreading this paper, providing data on calculus grades,and offering numerous helpful suggestions. The authoralso thanks Professors Bettye Anne Case, Joan Hutchin-son, Anthony Hughes, Anne Leggett, Paul Melvin, Ju-dith Roitman, Alice Schafer, and Ms. Jaye Talvacchiafor informative conversations. Above all, she thanksPatricia Montague, her first calculus teacher.

[1]

References

Women and Minorities in Science and Engineering, Na-tional Science Foundation, January 1986.

[2] Educational Information Branch. US Department of Ed-ucation; data for 1985-86 academic year.

3] "Girls and math: Is biology really destiny?" Education

Calculus Success for All Students

Life, New York Times, August 2, 1987.[4] Undergraduate Programs in the Mathematical and Com-

puter Sciences, The 1985-86 Survey, MAA Notes Num-ber 7. Washington, D.C.: Mathematical Association ofAmerica, 1987."Report of the Committee on American Graduate En-rollments." Conference Board on Mathematical Sciences,December, 1986."Report on the 1986 survey of new doctorates," Noticesof the American Mathematical Society 33 (1986) 919-923.Edward E. David, Jr., "Toward renewing a threatenedresource: Findings and recommendations of the ad hoccommittee on resources for the mathematical sciences."Notices of the American Mathematical Society 31 (1984)141-145.

[8] Douglas, Ronald G. (ed.) Toward a Lean and Lively Cal-culus. MAA Notes Number 6. Washington, D.C.: Math-ematical Association of America, 1986.Teaching Assistants and Part-time Instructors: A Chal-lenge. MAA Notes. Washington, D.C.: MathematicalAssociation of America, 1987.

[5]

[6]

[7]

[9]

RHONDA HUGHES is Associate Professor and Chair ofMathematics at Bryn Mawr College, Bryn Mawr, Penn-sylvania, and President of the Association for Women inMathematics. She has been a member of the Institute forAdvanced Study at Princeton, New Jersey, a Fellow of theBunting Institute of Radcliffe College, and has lectured andwritten extensively on functional analysis and perturbationtheory. She is currently a member of the MAA Committeeon the Participation of Women.

Shirley M. Malcom and Uri Treisman

AMERICAN ASSOCIATION FOR THE ADVANCEMENT OF SCIENCEUNIVERSITY OF CALIFORNIA AT BERKELEY

Consider these facts about student achievement inmathematics:

In 1985 according to the National Research Council,31,201 doctorates were awarded by U.S. universities;689 or 2.2% of these were awarded in mathematics.Of the 689 Ph.D.'s, 15.4% went to women and 34.7%to non-U.S. citizens with temporary visas.

Not one American Indian or Puerto Rican receiveda doctorate in mathematics in 1985; 7 Blacks and 5Mexican Americans received degrees for a total of 12

doctorates (1.7%) awarded to members of minoritygroups under-represented in science and engineering.

Between 1982 and 1983, 2,839 master's degrees wereawarded in mathematics, 953 or 33.6% to womenand 23.2% to non-resident aliens. Minorities-68Blacks, 48 Hispanics, and 6 American Indiansreceived 4.3% of the degrees.

in 1983, 11,470 bachelor's degrees were awarded inthe mathematical sciences &.J to Blacks, 27 toAmerican Indians, and 253 to Hispanics. Under-

1 4 0

,s


represented minorities received 7.9% of bachelor's de-grees in mathematics.In 1985, 23 American Indians, 173 Blacks, 41 Mexi-can Americans, and 31 Puerto Ricans received doc-torates in all natural sciences and engineering fields.The 268 minority Ph.D.'s (2%) in natural sciences

and engineering had successfully negotiated the calcu-lus barrier, as did those persons who received master'sand bachelor's degrees in these fields. As the minor-ity proportion of our school age population increases toover a third, our concern must be with the hundreds ofthousands of capable minority students who are felledby the calculus hurdle.

Those of us who are concerned about increasing thediversity of persons involved in science and engineer-ing in this country and in broadening the pool of talentavailable to an economy based increasingly in scienceand technology must also be concerned about the di-rection of efforts by the mathematical community toreform the calculus.

Hundreds of thousands of capableminority students [are] felled by thecalculus hurdle.

As the base for science and engineering, mathemat-ics holds a unique position among the science and en-gineering fields; as the base for commerce and trade,mathematics holds a unique position in business and fi-nance; and as the critical filter to an increasing numberof careers for women and minorities, mathematics alsoholds a unique position in the movement for social andeconomic equity for these groups.

It is critical that the larger social context of educa-tion, the economy, demographics, and national need befactored into what might otherwise be considered justan attempt by the discipline to reform itself in responseto a changing intellectual context.

Social Context for Calculus ReformMathematicians have been concerned for some time

about problems that hay, appeared within the disci-pline:

The increasing image of mathematics departments asprimarily "service departments" in most institutions;The decreasing proportion of students who declarean intention to major in mathematics;The decreasing number of bachelor's, master's, anddoctorate degrees in mathematics;

The decreasing proportion of U.S. citizens amongthose receiving Ph.D.'s in mathematics;The increasing proportion of remedial course workbeing offered in college mathematics departmentsand the corresponding decrease in upper-level coursework in mathematics.These and other indicators have led to serious discus-

sion both within and outside of the mathematics com-munity. The link between these issues and equity con-cerns have been the subject of discussion at some of thehighest policy levels. These included discussions by theMathematical Sciences Advisory Committee of the Na-tional Science Foundation and by the 1987 AmericanAssociation for the Advancement of Science (AAAS)Congressional Seminar, "Reclaiming Human Talent."

Guess Who's Coming to College?If one starts to look at the changing population

trends in this country, concern about social equity is-sues in relation to higher education becomes a nationalcall to action. Lynn Arthur Steen cites many of thesetrends in his recent Science article [1] "Mathematicseducation: A predictor of scientific competitiveness,"including:

The declining number of 18-24 year olds, the propor-tion from which the college age population is largelydrawn. The nearly 30% decline between now and theend of the century will take place just at the timelarge numbers of teachers retire and the baby boomecho produces a 30% increase in the size of the schoolage population. Who will teach these children?

As the critical filter to an increasingnumber of careers for women andminorities, mathematics also holds aunique position in the movement forsocial and economic equity for thesegroups.

The increasing proportion of this school age popula-tion who are members of minority groups. Accord-ing to Harold Hodgkinson [2], by around the year2000 "Americans will be anation in which one ofevery three of us will be non-white." If we add tothese groups those others within the society who havetended to be less well served by tl-,! educational sys-tem (e.g., women and persons with disabilities), weare looking at over two-thirds of the school popula-tion.

1.41

MALCOM AND TREISMAN: SUCCESS FOR ALL 131

In an issues paper included in The Condition of Ed-ucation [3], Phillip Kaufman pointed out that a declinein the 18-24 year old age group does not automaticallytranslate into a de-line in college enrollment. He citedas evidence the fact that the projected decline for the1980's did not take place. As a matter of fact, collegeenrollments increased in the early 1980's due largelyto the increase in college going rates by 18-21 yearolds (mostly among whites) and increased enrollmentby women, partictiarly older women.

Women increased as a proportion of all college stu-dents, from 49.9% in 1978 to 52.9% in 1985 and ac-counted for 63.3% of the increase in college attendancebetween 1978 and 1985. While the proportion of part-time students has remained fairly constant over the1978-1985 time period (around 35% of all college stu-dents), women's share of part-timers increased.

Minority college-going rates have remained stableduring this time; in the case of Blacks, the rate has ac-tually declined. American Indian, Black, and Hispanicstudents are more likely to be found in two-year collegeprograms than are white students.

Our national need for scientifically andtechnically-trained talent requires thatwe succeed in addressing the issue ofmathematics as a gateway or barrier.

Despite the counter-example that Kaufman cites forthe nonlinear relationship between 18-24 year old cohortsize and college enrollment, he concludes that the pro-jected declines were merely postponed. He suggests thatwhile national declines may be moderate, the effects willlikely be different on different types of institutions. Hefurther suggests that prestigious institutions with largeapplicant bases and low-priced community colleges areless likely to feel the enrollment declines, while all otherinstitutions are likely to experience considerable losses.

The growing minority proportion of the college agepopulation must be enabled to grow at least propor-tionately among the college attending population or thecontinued existence of many institutions (or the via-bility of departments within those institutions) will beseriously jeopardized.

Mathematics: Gateway or BarrierAs critical as the question of who is coming to col-

lege is the question of the skills that they will bringwith them. At present, large proportions of American

Indian, Black, and Hispanic students leave high schoolunder-prepared to pursue quantitatively-based fields ofstudy in higher education, having neither taken the ap-propriate courses nor obtained the requisite skills to en-ter a calculus sequence in college. Young women also lagbehind young men in high school mathematics course-taking; but this is largely at the highest level of courses,such as trigonometry and calculus.

Early in the 1970's Lucy Sells coined the phrase"critical filter" to describe the overwhelming effect thatmathematics preparation has on the career aspirationsof women and minorities, especially in limiting theirparticipation in careers in science and engineering. Al-though-it-was-her- interest.in barriers-to-careers-in-sci--ence and engineering fields that led Sells to examine thehigh school mathematics preparation of women and mi-norities entering the University of California at Berke-ley and the University of Maryland, she quickly dis-covered that inadequate mathematics preparation alsolimits participation in other fields as well, such as busi-ness, architecture, and the health professions.

The trend is clearly toward an increasing numberof majors that require âlculLs as an entry ticket: ifyou cannot obtain that ticket, an increasing numberof fields are closed to you. The implications of failinggo far beyond grade point average or eligibility to playsports. And the implications for our national need forscientifically and technically- 'rained talent requires thatwe succeed in addressing the issue of mathematics as agateway or barrier.

Students Failing CoursesSuccess in college-level calculus obviously depends

on the past preparation of students as well as on theamount of productive effort that students spend on theircourse. The Professional Development Program (PDP)at the University of California at Berkeley has demon-strated that "good" backgrounds in mathematics and"smarts" are necessary but not sufficient conditions forsuccess in calculus courses.

Minority students can be successful incalculus.

By looking at Asian students at Berkeley who weresucceeding, PDP staff could see that Black and His-panic students were employing unproductive behaviorsin their calculus courses: studying alone without feed-back from peers, studying for much less time than wasneeded to master the material; passing over materialthat they did not understand; and failing to ask for help.

142

ISSUES: TEACHING AND LEARNING

With each passing day, such students found themselvesdeeper and deeper in the hole.

Cuatrast this with the performance seen after inter-vention when capable minority students replaced unpro-ductive study behaviors with productive ones. Thesesame minority students succuded. Berkeley's PDPshows that minority students can be successful in calcu-lus. Other examples, such as the extensive participationin AP Calculus programs at mostly Hispanic GarfieldHigh,School in East Los Angeles, reinforce the lessonsfrom Berkeley. There are no inherent barriers to minor-ity student success in calculus.

Achieving ProficiencyThe key to the success of Berkeley's PDP program

is the discovery that one can promote high levels ofachievement among Black and Latin calculus studentsby adapting techniques being used by Chinese under-graduates. A significant number of Chinese studentsachieve proficiency in mathematics largely through theirwork in informal study groups. These groups providestudents the opportunity to discuss with other studentstheir mathematics homework and their performance ontests and quizzes.

These discussions accomplish a number of importantobjectives: they provide students with critical feedbackabout the accuracy and quality of their work, and theyforce students to explain to the satisfaction of others,how complex proofs were derived or how difficult prob-lems were solved.

The process of forming clear explanations was thekey to the success of these groups: one cannot explaina difficult concept to another unless that concept is wellunderstood.

The PDP Mathematics Workshop attempts to createan instructional setting where students would be forcedto talk with others as they did mathematics. The work-shops enroll 15-20 undergraduate calculus students whoare all members of the same calculus lecture section.Workshops meet for two hours per day, twice a week.Since workshop participants are in the same class, theyare assigned the same homework, must prepare for thesame tests, and rust struggle with the same material.This common experience allows them- to share a com-mon foundation for the work that they are assigned inthe workshops.

The focus for this work is a worksheeta collectionof difficult, challenging problems that test skills andconcepts that students must master if they are to besuccessful in later work in calculus. When students ar-rive at a workshop session, they are given the day's

worksheet. Students work alone at first and then areencouraged to share the results of their labors with fouror five others.

Those who h.ve successfully completed problemsmust explain to others how a solution or proof was de-rived. Listeners challenge what they hear and critiquewhat has been presented; if the explanation is clear (andcorrect), others in the group will repeat what has beensaid until each person can replicate the steps to a suc-cessful solution or until a better, more elegant means ofworking with the problem has been discovered.

A workshop leaderwho is usually a graduate stu-dent in mathematics or physics (or some other quan-titative field)is responsible for both the creation ofthese worksheet problems and for monitoring the workof students in the workshops. The leader walks aboutthe meeting room and observes the groups closely, pay-ing particular attention to the content of conversationsstudents are having.

These conversations provide important insights intothe students' thought processes: they provide a windowthrough which one can observe how well or how poorlyimportant concepts have been understood. When stu-dents appear to be "on target" they are left alone withtheir deliberations; when they appear to flounder, theworkshop leader can intervene. Once on target, stu-dents returh, to their labors.

The process of forming clearexplanations was the key ...one cannotexplain a difficult concept to anotherunless that concept is well understood.

The workshops provide students with the opportu-nity to practice skills that they are expected to exhibiton tests and quizzes; moreover, their practice occursin the presence of a skilled mathematician who is ableto provide them with instant feedback on their efforts.Thus, bad habits can be quickly dealt with and goodhabits can be immediately strengthened and reinforced.

Expectations of Compete/ I,An additional, critical feature of the workshop pro-

gram's success is that there is no hint of a "remedial"focus in any of the work that students do. Studentsare encouraged to see themselves as competent and ascapable of achieving high standards of academic excellence. Significantly, the performance of workshop stu-dents since the program's inception more than justifiesthis assumption of competence.

143

MALCOM AND TREISMAN: SUCCESS FOR ALL 133

Black students at Berkeley (and in most predomi-nantly white universities) are at greater risk of aca-demic failure and are more likely to drop out of col-lege than any other comparable group of undergradu-ates. It is particularly. significant, therefore, that since1978 approximately 55% of all Black students enrolledin the workshops have earned grades of B- or betterin first year calculus, while only 21% of non-workshopBlacks earned comparable grades. The mean calculusgrade point average of Black workshop students was 2.6(N=231); the comparable mean for Blacks not in theworkshop was 1.9 (N=284).

During this period, only 8 workshop students in 221failed calculus. Comparable failure rates for Blacks notin the program were substantially higher (105 of 284failed the course). Significantly, the program has haddramatic impact on students with a weak foundation inmathematics: the mean grade in calculus among poorlyprepared Black workshop students (students with SATMath scores between 200 and 460) was four-tenths of agrade point higher than that of non-workshop studentswho entered the university with strong preparation inmathematics (SAT Math scores between 550 and 800).

There are no inherent barriers tominority student success in calculus.

Participation in the workshops was also associatedwith high retention and graduation rates. Approxi-mately 65% of all Black workshop students (47/72) whoentered the university in 1978 and in 1979 had gradu-ated or were still enrolled in classes as of spring semesterof 1985. The comparable rate for non-workshop Blackstudents who entered the university in those years was47% (132/281). Of particular significance was the factthat 44% of the workshop graduates earned degrees ina mathematics-based field such as engineering, environ-mental design (architecture), or one of the natural sci-ences. Comparable, findings have been reported for His-panic workshop students who appear to have persistedin the university and earned grades in mathematics atrates that are almost identical to those reported forBlack workshop students.

Courses Failing StudentsOther critical issues in the success of students in

college-level calculus include the nature of the courseand the nature and quality of the instruction. Therehas been a tendency to blame lack of success almost

exclusively on the student. This has been especiallytrue for students who are non-traditional. While lackof background or lack of effort may account for manyproblems, we must at least consider the possibility ofthe courses failing the student.

If we look at programs at the precollege and collegelevel that promote success in science and engineeringby minority and women students (including mastery ofthe calculus) we see a number of characteristics emergewhich suggest lines of inquiry that should be explored inany re-examination of the intellectual underpinnings ofcalculus. Unless the restructuring starts with a commit-ment to results that serve all students who are intellec-tually capable of mastering calculus, the effort may onlyhave been an interesting intellectual exercise doomed tofailure.

Strategies That SucceedResearch and experience with special programs tell

us that there are no inherent barriers to success in math-ematics, science, or engineering by women, minority, ordisabled students if they are provided with appropriateinstruction and support systems:

High expectations by students and their teachersthat the student will succeed. There is no "presump-tion of failure."Presence of a support structure and safety net.Bridge programs ready the students for the rigors ofcollege-level work, especially in mathematics. Diag-nostic testing identifies problems which are addressedin summer programs. Problems in class performanceare not allowed to snowball but are handled throughimmediate feedback, referral, and tutoring.Capable and appropriate instruction that links math-ematics to science and engineering. Where the rela-tionship between a science experiment or a designproblem and the mathematics is made clear, stu-dents seem to perform better and are more highlymotivated. Too often the trend has gone in thewrong directionnot only a separation of mathemat-ics from the hands-on activities by faculty in math-ematics, but also a substitution of mathematics forhands-on experience and practical understanding byfaculty in the sciences.Purpose of instruction is to enable students to suc-ceed, not to weed them out to reduce the numbersin highly-competitive programs.Cooperative learning, peer tutoring, and other formsof instruction are utilized, including the use of tech-nology where appropriate.

144


A willingness to take students from where they are 7.to where they need to be.The key to success is to make sure that the learning 8.

environment can work for them; that the real goal ofinstruction is learning; that the best possible teaching isavailable at this entry level; that content is tied as mushas possible to real-world problems as they are likely tobe encountered, utilizing the tools that are likely tobe available; and that the institutions and departmentsare committed to providing the resources necessary tosupport learning.

One can address the problem of poor precollegemathematics preparation by raising entry standards,by working with precollege-level instructors to improvetheir content-base and instructional abilities, or by ac-cepting the need to provide remedial instruction on atransitional basis until reforms of the entire system takeeffect. Many institutions have chosen the first option,preventing contact with the under-prepared. Unlesschanges are instituted now, this option may soon beunavailable.

Recommendations for Action1. Keep in mind the results of research on what it takes

to help women, minorities, and disabled students suc-ceed in mathematics.

2. Start restructuring by considering the needs of thosestudents who form the overwhelming majority of thefuture talent pool for science, mathematics, and en-gineering, even though they have been inadequatelyutilized in the past.

3. Include in the discussions people who have experi-enced success in teaching calculus to significant num-bers of students from these groups.

4. Recognize the need to build a system of precollegeinstruction that supports the reforms of calculus (in-cluding content, skills, and pedagogy).

5. Promote dialogue on these issues among mathe-maticians, mathematics educators, and mathematicsteachers.

6. Promote interaction of mathematicians with facultyfrom engineering, physical, biological, and social sci-ences as well as non-science fields that require calcu-lus, as a source of real-world problems to be includedin instruction as well as allies in reconnecting math-ematics to the other content areas.

9.

10.

Work to change the image of ,mathematics from thatof a difficult and abstract subject.Work to change the image of mathematicians andother people who use mathematics extensively to in-clude the widest variety of people possible. Be par-ticularly conscious of the need to show examplesof people working in groups or as teams to solvemathematics-based problems.Study the structure and promote replication of ef-fective models such as the Professional DevelopmentProgram of the University of California at Berkeley,and successful efforts at women's colleges and minor-ity institutions.Implement new models which address the serious re-cruitment and retention issues for mathematics ma-jors, especially among women. When interventionprograms have been instituted to increase participa-tion of members of under-represented groups in sci-ence and engineering, they have been found to beequally effective and valid for all students.

References

Steen, Lynn Arthur. "Mathematics education: A predic-tor of scientific competitiveness." Science 237 (17 July1987) 251-252, 302.Hodgkinson, Harold L. All One System: Demographics ofEducation. Institute for Educational Leadership, Wash-ington, DC, 1985.Kaufman, Phillip. The Condition of Education. Centerfor Statistics, Washington, D.C., 1985.

SHIRLEY M. MALCOM is Head of the Office of Op-portunities in Science of the American Association for theAdvancement in Science (AAAS). A biologist by training,Malcom has served as a Program Manager for the NationalScience Foundation and as Chair of the National ScienceFoundation Committee on Equal Opportunities in Scienceand Technology.

PHILIP URI TILEISMAN is Associate Director of theProfessional Development Project at the University of Cal-ifornia at Berkeley. for which work he recently receivedthe $50,000 Dana Foundation Award for pioneering workin higher education. He received a Ph.D. uegree in mathe-matics education from the University of California.

HARVEY: PLACEMENT TESTING 135

The Role of Placement TestingJohn G. Harve

UNIVERSITY OF WISCONSIN

Had I, about 30 years ago, been placed in a state ofsuspended animation when I completed my introduc-tory calculus courses and had I been revived this year,I would be as much at home with the present entry-levelcollege mathematics curriculum as I am presently.

It is true that as a student I took only two 3-creditintroductory calculus courses that were preceded by a 3-credit analytic geometry course and that today I teach asequence of three 5-credit introductory courses for en-gineering, physical sciences, and mathematics majors.However, the instructional techniques I used and theprimary expected outcomesa repertoire of analyticalskills and techniquesare about the same.

In the intervening 30 years analytic geometry, vectorcalculus, inultivariable calculus, and elementary differ-ential equations have joined the standard core curricu-lum and account for most of the increase in the timedevoted to introductory calculus. But, even as 4 gradu-ate student, I discovered that these topics were alreadycovered in introductory ci 3r courses at some uni-versities. Thirty years has changed surprisingly little inintroductory calculus.

Thirty years has changed surprisinglylittle in introductory calculus.

r. co.-,trast, the introductory curriculum in manymathematically related areas has not been static. Af-ter a stint in suspended animation I r.Q .1d not et enrecognize introductory college courses in plzy..ics, chem-istry, or electrical engineering. In physics these courses

inclucte the elements of quantum mechanics nnd et.ementary particle theory; in chemistry they irJudu newknowledge about the structure of atoms and ..he interac-tion of molecules. And electrical engineering has beentransformed into computer engineering where there isless an emphasis on power and motors and more oncomputer technologies.

It may be unfair to make these comparisons or toconclude that introductory mathematics courses needchanging. After all, what was true 30 years ago inmathematics is still true. Although new knowledgein physics and chemistry has altered our conceptionof those worlds and computer technologies have com-pletely changed the way electrical engineering is done,

mathematics describes the world as well today as it didthen. Perhaps the introductory college mathematicscurriculum may not need changingonly fine tuning,as Sherman Stein's colleagues suggested ([17], p. 167).However, most participants in the 1987 Tulane Confer-ence to develop curriculum and teaching methods forcalculus at the college level concluded otherwise ([17]),and I believe that many college faculty agree with them.

In addition, it seems to me that this is a pro-pitious time to initiate changes intended to improveboth our entry-level courses and instruction in thosecourses. The cost of computers seems to declinesteadily; presently, I can do many complex tasks ona computer costing less than $4,000 that could onlybe done on a computer costing $400,000 in 1977 and$4,000,000 in 1967. In addition, calculators are becom-ing portable computers, are increasingly able, and arevery inexpensive. (At present the Casio fz-7000G, theSharp EL5200, and the Hewlett-Packard HP-28C canbe purchased for $55.00, $69.99, and $175.00, respec-tively.) These tools make it possible for us to forsaketables, point plotting, and routine computation in fa-vor of teaching concepts, problem solving, and moredetailed arid more realistic applications.

Confronting DiversityIf I had been in state of suspended animation, there

are a few things that I would not recognize. Amongthem are mathematics placement testing, remedialcourses, disproportionately large enrollments in precal-culus courses, and the (limited) use of calculators andcomputers. Two of these changesremedial coursesand precalculus enrollrgrntsare ones that most col-legiate mathematiciat ew with disfavor. The use ofcalculators and computers is a change that many of uswould like to encourage. Mathematics placement test-ing may help us to accomplish these g' als.

Colleges and universities now admit amuch more diverse group of studentsthan they did, say, in the 1950's.

Colleges and universities now admit a much morediverse group of students than they did, say, in the

148

136 IssuEs! TEACHING AND LEARNING

1950's. A majority of our students do continue to resem-ble the college-intending high school senior of the 50's;such students are white, from middle-income families,have completed a college preparatoty curriculum thatincludes at least three years of high school English, ayear of algebra, a year of plane geometry, and possiilly,a second year of algebra and trigonometry, and entercollege in the fall of the year they graduate from highschool. But we also have a large group of students whocome from minority groups; by the year 2000 they willcomprise at least 30% of public high school enrollment([1], p. 39). In addition, we now have a heterogeneousgroup of non-traditional students. Twenty-nine percentof all students enrolled in four-year institutions ([16], p.80) are part-time students. Other characteristics of ournon-traditional students are shown in Table 1, adaptedfrom Boyer ([1], p. 50).

Characteristics: Part-Time Full-Time

Age: 25 or older 67% 13%Dropped out since entering college 58% 16%Employed full-time 59% 4%GPA of at least B 61% 55%

Non-Traditional Students at Four Year Institutions

TABLE 1

As the entering college population became more di-verse it became difficult (and often impossible) to placestudents accurately in mathematics courses by using ex-isting data such as high school transcripts or grade pointaverages. Data about high school mathematics coursesalso proved to be unreliable for students who had re-cently graduated from high school since these coursesvaried widely in content. As a consequence, it was notpossible to conclude Lhat they had provided adequatepreparation for college-level courses. Moreover, non-traditional students often remembered little about themathematics courses they had previously taken.

Faced with rising course dropout and failure rates,many colleges and universities introduced mathemat-ics placement tests. To help colleges and universitiesdevelop their placement programs, the MathematicalAssociation of America (MAA) established a Place-ment Testing (PT) Program in 1977 administered bythe Committee on Placement Examinations (COPE).The growth of MAA's PT Program is probably a goodindicator of the overall growth of mathematics place-ment testing in the United States; Table 2 indicates thenumber and kind of schools who subscribed to the PTProgram since 1980.

Year TotalTwo Year Four YearColleges Colleges Universities

1980 1291981 2261982 2031983 295 36 125 1341984 327 50 107 1461985 321 59 131 971986 301 62 110 961987 379 72 143 119

MAA Placement Testing Program, 1980-1987

TABLE 2

Effectiveness of Placement TestingPlacement testing programs do seem to be effective.

I base this conclusion on several kinds of information.

Many institutions have well established placementtesting programs that seem to work well. For exam-ple, the University of Wisconsin at Madison has aplacement program that is more than 20 years old.We use the scores from three tests as the sole sourceof the data we use to place students in our entry-level mathematics courses: intermediate (i.e., highschool) algebra, college algebra, trigonometry, engi-neering calculus, and business calculus.

Second, a recent survey of subscribers to the PT Pro-gram indicated that 93% of the responding institu-tions used at least one of the PTP tests for placementduring the previous year ([3], 1987). Other kinds ofinformation were also used:

1. The number and kind of previous mathematicscourses (62%),

2. Grades in previous mathematics courses (58%),3. SAT quantitative score (39%),4. ACT mathematics score (37%),5. SAT qualitative score (17%),6. High school rank in class (17%),7. High school grade point average (14%),8. ACT verbal score (8%).

If these data are representative of all colleges anduniversities that have a placement testing program,then it is clear that placement test scores are themost heavily used among the factors most usuallyconsidered in making placement decisions.

Several studies document the effectiveness of place-ment programs. In a study conducted by the Ameri-can Mathematical Association of Two-Year Colleges,it was concluded that the PT Program tests BA/1B

147


and SK were "useful placement instruments" [14]. Ina study conducted at the United States Coast GuardAcademy [13] it was shown that the PTP test CR wasmost highly correlated with grades cadets receivedin the introductory calculus and analytic geometrycourse.

Finally, in a study conducted at Tallahassee Com-munity College [2] it was shown that seven of eightstudents who disregarded the placement advice, butonly nine of 37 students who followed the advice,failed to complete intermediate or basic algebra witha grade of C or better.From evidence such as this, I conclude that present

placement testing programs are successful in helpingcolleges and universities to place students in entry-levelmathematics courses. However, these placement pro-grams are based upon present high school and collegecurricula, the instructional techniques used in teachingthose curricula, the technologies presently utilized, andpresent college and university entrance requirements.Thus, any noticeable change in any one of these areas=7 make it necessaryat least, very importanttochange the placement testing programs.

Changing Mathematics CurriculaAny change in the introductory calculus courses will

necessitate a change in both college prccalculus coursesand in the high school courses we normally consider asprerequisites for calculus: algebra, geometry, and pre-calculus. This seems clear to me when I examine theproposals for change in the report of the Tulane Confer-ence [11], especially when I consider the history of the"new math" revolution that sprang from the report ofthe College Board's Commission on Mathematics [7].

Two sets of recommendations made by Tulane Con-ference participants deal with changes in the content ofand instruction in introductory calculus. Even thoughthese two sets were authored by disjoint groups of con-ference participants, th..lre is remarkable agreement be-tween them. On the surface the changes suggested bythese two groups may seem minor and would not ap-pear to require a noticeable change in the curricula weteach or the ways we teach it. However, I believe this isnot true even if the recommendations made about theuse of technologies are disregarded.

Both groups suggest that a rev nlized introductorycalculus is one (a) in which students see a broader rangeof problems an problem situations, become more pre-cise in written and oral presentations, and better de-velop their analytical and reasoning abilities, and (b)

from which students gain a better understanding of con-cepts, develop a better appreciation of mathematics andits uses, and learn better to use mathematics resourcematerials ([11], pp. vii-ix, xvi).

We are not teaching many ... higher-order skills: the problem-solvingabilities we teach to and expect fromstudents are, generally, ones requiringroutine application of procedures andtechniques.

These goals involve both the teaching and the learn-ing of higher-order thinking skills and an improvementin students' problem-solving abilities. At present weare not teaching many of these higher-order skills: theproblem-solving abilities we teach to and expect fromstudents are, generally, ones requiring routine applica-tion of procedures and techniques.

If calculus is revitalized along the line suggested bythe Tulane Conference, then students will need to enterthose courses with a better (i.e., higher order) under-standing of and ability to apply the algebraic, geomet-ric, and analytical concepts they have already learned.Thus, it will be necessary for placement tests to changeso as to determine if students have met these new pre-requisites for calculus since at present most placementtests examine primarily low-level skills.

For example, my analysis (see Table 3) of thetest items on the MAA's PT Algebra and Calculus-Readiness Tests revealed that all of the items testedlow-level skills: recall of factual knowledge, mathemat-ical manipulation, and routine problem-solving.

Algebra Calculus-Level of Thinking Readiness

Recall factual knowledgePerform math. manipulationsSolve routine problems

0 822 10

10 6

Classification of PT Program Test Items

TABLE 3

A revitalized calculus will also require changes in thecourses prerequisite to it. The changes needed appearto be those already endorsed by the National Council ofTeachers of Matheraatics [15] and outlined by The Col-lege Board [4], [5]), and so it seems possible that high

148


school college preparatory courses will change. How-ever, we will also need to chaLige entry-level collegecourses so that they, too, reflect the changed prereq-uisites for calculus.

Changing TechnologiesAt the 'Wane Conference the Content Workshop rec-

ommended the immediate use of calculators in calculusand envisioned that a symbolic manipulation calculuscourse would eventually become the norm. It is clearthat if calculators are going to be used in entry-levelcollege mathematics courses, then students should beable to use them when taking placement tests and thatthe placement tests should be designed for calculatoruse.

If calculators are going to be used inentry-level college mathematicscourses, then students should be ableto use them when taking placementtests.

With the financial support of Texas Instruments,the MAA Calculator-Based Placement Test Program(CBPTP) Project is presently developing a calculator-based version of each of the six present PT Programtests. The project has progressed far enotigh to yieldsome results:

When a calculator is used during testing, some itemson present placement tests must be replaced;It is possible to design items that test mathematicscontentnot just calculator skills;Use of a calculator can permit expansion of the test-ing of both higher-order thinking and problem solv-ing skills.The three present CBPTP Project test panels are

basing their development of "calculator-active" itemson the assumption that students will have a scien-tific, non-programmable calculator like the Texas In-struments TI-30. When it can be assumed that studentsin entry-level college mathematics courses actively usegraphics calculators like the Cublo 7000G or the SharpEL5200, or graphics and symbolic manipulation calcu-lators like the Hewlett-Packard HP-28C, then a nextgeneration of placement tests will be needed that accu-rately reflect both the way that mathematics is beingtaught and learned and the prerequisites of the coursesin which students are placed.

Test AdministrationAs soon as present technologies are better incorpo-

rated into education, they will affect not only the con-tent but also the administration of placement tests. Forone thing, it will be easy to produce parallel or equiv-alent versions of placement tests. To anticipate thisability, for the past three years COPE has been work-ing on the distractor analyses for each item on each testin order to describe each multiple-choice response witha formula. This makes it possible to develop computerprograms that can, within parameters specified by thedistractor analyses, randomly generate parallel itemsand parallel forms of each of the PT Program tests.

It is already possible to administer placement testsusing computers. Indeed, the College Board has devel-oped the Computerized Placement Tests [6] that usesa limited, fixed pool of test items; the tests are fairlyexpensive to use. However, it seems unlikely to me thatlarge colleges and universities will use computer admin-istration of their placement tests because they need totest large numbers of students simultaneously and test-ing sessions are held only infrequently.

Prognostic TestingIn their most recent report, the Carnegie Founda-

tion for the Advancement of Teaching related that thefirst problem they encountered was the discontinuity be-tween schools and colleges ([1], p. 2). According to theCarnegie report, school and college educators work inisolation from each other, students find the transitionfrom high school to college haphazard and confusing,and there is a mismatch between faculty expectationsand the academic preparation of entering students. Allof these claims are probably true and are thready influ-encing college entrance and placement testing.

Prognostic testing is one way ofsmoothing the transition from highschool to college.

Prognostic testing is one way of smoothing the tran-sition from high school to college. In a prognostic test-ing program, college-intending high school juniors aregiven a placement test; based upon the score on thattest, the junior student is given a "prognosis" of whatmathematic.: course he or she would enroll in if he orshe takes no additional mathematics courses during thesenior year and if he or she performs as well on the col-lege placement test that is used to place students intoentry-level mathematics courses.

149


Prognostic mathematics testing originated at OhioState University (OSU) in 1977 when students in oneColumbus area high school were tested. In 1985 theOhio Early Mathematics Placement Testing (EMPT)Program tested 65,217 students in 80% of 01 do's pub-lic, private, and parochial high schools. Until 1986 theEMPT Program only advised students whether theywould be placed in a remedial course. Prom 1979 to1985, the percentage of the OSU freshman class withremedial placement dropped from 43% to 25% [9]. TheEMPT Program, directed by Bert Waits at OSU, is aprogram of the Ohio Board of Regents; the results ofEMPT testing are used by all state-supported collegesand universities in Ohio.

The results from the Ohio EMPT Program are soencouraging that several states, colleges, and universi-ties 'have started or are planning to start similar pro-grams. Similar programs exist in Illinois, Louisiana,and Oregon. The University of Wisconsin at Madisonis planning to initiate a program in 1987. Several state:,colleges, and universities have contacted COPE for in-formation on prognostic testing.

Changing Entrance RequirementsOne response to large remedial enrollments and the

disparity between faculty expectations an student aca-demic preparation is to raise entrance requirements. In1982, admission requirements were being changed or re-viewed by the public higher education systems in 27states [18]. This change is probably overdue; at present,only 67% of all colleges and universities require anymathematics course for admission ([1], p. 88). How-ever, most students presently entering our institutionsalready exceed any mathematics entrance requirementwe might reasonably impose. Consider these two cases:

Only 67% of all colleges anduniversities require any mathematicscourse for admission.

The Univeraity of Wisconsin at Madison is a re-stricted admissions university; at present it requiresentering students to have completed two years of highschool mathematicsone year each of algebra and ge-ometry. In order to graduate, Wisconsin students mustcomplete an additional mathematics course if they didnot have two years of high school algebra and a yearof geometry when they were admitted. Table 4 showsfor the years 1985 and 1986 the number of high school

courses our entering freshmen reported they had takenand the entry-level courses in which they were placed.

CollegePlacement

Years of H. S. Mathematics<2.5 <3.5 <4.5 >4.5

Inter. Algebra 1985 98 281 452 691986 85 313 397 30

College Algebra 1985 2 209 1468 5801986 0 318 1561 473

Trigonometry 1985 0 11 185 1511986 0 14 225 129

Calculus 1985 1 17 672 9321986 0 14 X25 750

Course Placement of Wisconsin Freshmen

TABLE 4

As is easily seen from these data most University ofWisconsin freshmen report that they had taken at leasttwo and one-half years of mathematics in high school.Yet in each of 1985 and 1986 a majority of the studentswho had taken at least three and one-half years of highschool mathematics were placed into interm-diate orcollege algebra.

Ohio State University is an open admissions univer-sity. However, data from OSU are similar to those fromWisconsin. In 1986, 206 OSU freshmen had taken lessthan two years of college preparatory in 'hematics, 478had taken at least two but less than three years, 1615had taken at least three but less than four "ears, and4280 students had t:' :n four or more years of collegepreparatory mathematics.

Increasing the mathematics entrancerequirement may not necessarilyeliminate the need for remedialmathematics courses.

The placement procedures at OSU assign st,udentsto one of five placement levels indicating readiness forcalculus, for precalculus mathematics, or for remedialmathematics. More than 60% of the students withless than four years of college prepa. atory mathemat-ics, about 54% a the students with thr' e years of col-lege preparatory mathematics, and a; out 86% -: thestudents with less than three years of college prep, Ira-tory mathematics had remedial mathematics placementscores [10].

150


These two cases indicate that increasing the mathe-matics entrance requirement may not necessarily elim-inate the need for remedial mathematics Courses or foran effective placement testing program. However, in-creasing the entrance requirement may reduce the num-ber of students who end up in remedial mathematicscourses.

Conclu3ion1. Present placement programs successfully help place

students in entry-level mathematics courses.2. A revitalized calculus will make necessary new place-

ment tests that assess higher-order thinking skillsand problem solving abilities.

3. A revitalized calculus will make it necessary to re-vitalize both high school and college mathematicscourses that are prerequisite to the calculus.

4. Placement tests will need to be revised in order toprovide for the use of new technologies in learningand teaching mathematics.

5. Prognostic placement testims of high school juniorshas shown some encouraging results and should beexpanded.

6. Raising college entrance requirements will not changethe need for an effective placement testing program.

References

[1] Boyer, E.L. College: The Undergraduate Experience inAmerica. New York: Harper and Row, 1987.

[2] Case, B.A. "Mathematics placement at Florida StateUniversity." PT Newsletter 7:2 (Spring 1984) 1, 6-8.

[3] Cederberg, J. and Harvey, J.G. "Questionnaire results."PT Newsletter 9:1 (Spring 1987) 1-2.College Entrance Examination Board. Academic Prepa-ration for College: What Students Need to Know and BeAble To Do. New York, 1983.College Entrance Board. Academic Preparation in Math-ematics: Teaching for 71.ansition from High School toCollege. New York, 1985.College Entrance Examination Board. ComputerizedPlacemCnt Tests. New York, 1985.Commission on Mathematics. Program for College Pre-paratory Mathe-natics. York: College Entrance Ex-amination Board, 1959.

[8]

[9]

Committee on Placement Examinations (COPE). COPEMeeting Notes (1984-1987) (unpublished).

Demana, F. "News from the campuses: The Ohio StateUniversity." EMPATH, L1 (Sept. 1986) 4.

[10] Demana, F. and Waits, B.K. "Is three years enough?"Mathematics Teacher (to appear).Douglas, R.G. (ed.) Toward a Lean and Lively Calculus.(MAA Notes Number 6) Washington, DC: MathematicalAssociation of America, 1987.

[12] Harvey, J.G. "Placement test issues in calculator-basedmathematics examinations." Paper presented at TheCollege Board and The Mathematical Association ofAmerica Symposium on Calculators in the StandardizedTesting of Mathematics. New York, 1986.

[13] Manf::ed, E. "Using the calculus readiness test at theU.S. Coast Guard Academy." PT Newsletter, 7:2 (Spring1984) 1-3.

[14] Mathematical Association of America. PT Newsletter(volumes 1-9). Washington, DC, 1978-1987.

[15] National Council of Teachers of Mathematics. An Agendafor Action: Recommendations for School Mathematics ofthe 1980s. Reston, VA, 1980.

[16] Ottinger, C.A. 1984-85 Fact Book on Higher Education.New York: American Council on Education, 1984.

[17] Stein, S.K. "What's all the fuss about?" In R.G. Douglas(ed.), Toward a Lean and Lively Calculus (MAA NotesNumber 6). Washington, DC: Mathematical Associationof America, 1987.

[18] Thompson, S.D. College Admissions: New RequirementsBy the State Universities. Reston, VA: National Associ-ation of Secondary School Principals, 1982.

[11]

JOHN G. HARVEY is Professor of Mathematics and ofCurriculum and Instruction at the University of Wiscon-sin at Madison. He is currently Director of the Calculator -Based Placement Test Project of the Mathematical Associ-ation of America, and a member of the MAA Committee onPlacement Examinations, and of the joint MAA-AMATYCTask Force on Remediation in College Mathematics. He isa former editor for Mathematics Education of the AmericanMathematical Monthly.

Calculus from an Administrative PerspectiveRichard S. Millman

WRIGHT STATE UNIVERSITY

Unfortunately, what administrators hear are com-plaints. To ba sure there are complaints both enlight-ened and unenlightened about all subjects. My favoritecomes from the 1820 letter of Farkas Bolyai to JanosBolyai [3]:

You should detest it ... it can deprive you of all yourleisure, your health, your rest, and the whole happi-ness of your life. This abysmal darkness might perhapsdevour a thousand towering Newtons, it will never belight on earth ....

This criticism is from a man who studied the inde-pendence of the parallel postulate and was warning hisson not to pursue it. Fortunately, like many children,Janos didn't listen and eventually showed that the par-allel postulate is independent of the others by finding anon-Euclidean geometry.

Scholars all agree that calculus is a fascinating sub-ject which has had a profound effect on science in par-ticular and human intellectual development in general.Even the skeptic, Bishop Berkeley [1], said that calcu-lus "... is the general key by help whereof the modernmathematicians unlock the secrets of qeometry, andconsequently of Nature."

Ever since the inception of calculus, there has beenand will always be criticism of it from learned individ-uals and those of lesser Ltellectual skills. Certainly thecalculus curriculum is not "devouring thousand tower-ing Newtons"our students are not quite at that level(at least at public universities). On the other hand,many valid discussion points are derived from both theinformed and uninformed questions which are addressedto mathematicians.

Calculus shows its many faces in different ways, de-pending upon the angle of the viewer. As an admin-istrator, as a ckpartment chair, and as a professor ofmathematics, I've seen the subject from different per-spectives. While mathematicians are well aware of thelatter two viewpoints, it is useful to understand the con-cerns that a dean has with calculus.

No matter what your vantage point, what is ulti-mately important, of course, is whether the studentswho finish the course understand and retain the mate-rial well enough to use this knowledge in their futurework. There are, however, many issues which compli-cate this primary objective. In my present position asa Dean and from discussions with administrative col-

leagues, I have come to appreciate the myriad problemswhich surround calculus, none of which have to do withthe chain rule, ei ;4.

An administrative view of calculus should, I believe,focus on four areas i^ this order of priority:

quality of curriculum;accountabilicy of the course;attitude of the department;effectivenes3/coat of the program.

Many of the items on this list would apply far morebroadly than just to calculus.

Quality of CurriculumFirst and foremost is the quality of material pre-

sented in the course. I don't mean that we should reactto occasional lapses in a professor's response to a max-imum/minimum question, but rather that we need toaddress the problem, when it arises, of long-term pas-sage of misinformation. Fortunately, this is the aspectof calculus that is both the most easily monitored and,in the unusual event that there is a problem, the mosteasily corrected. Mathematics departments do this effi-ciently and well.

A second point concerning quality is the idea of "re-alism" in calculus. We have all heard at AMS meetingsof Professor X who decides that differential forms arethe "only real way" to present iLtegration at the fresh-man and sophomore level. While my prejudice as adifferential geometer is in favor of tensor notatio t andmanipulations, the junior or senior levels would be farbetter for such vigorous pursuits (and I have my doubtseven there!).

We need to refrain from teaching "big C" calculusthat is, calculus for engineers, physical scientists andmathematicianswhen the audience is business stu-dents or prospective biologists. This is the notion ofrealism. Can we expect biology majors or business stu-dents to work hard on subjects that they will never useand for which they will not have a real appreciation?

Fortunately we mathematicians have, over the lasttwenty years, realized the error of our ways and splitcalculus into a plethora of different sequences to addressthese matters. The lesson is that we must listen to theadvice of our customers.

152

142 IssuEs: INSTITUTIONAL CONCERNS

Although the students may think they are the cus7tomers, it is actually the university who is the consumerof the service aspect of any department. It is the uni-versity faculty who decide whether calculus, physics, orchemistry, will be required of various majors. Universityfaculty members in all disciplines are the ones who needto be listened to carefully about what they would likein a mathematics course.

It is not enough for us asmathematicians to know that thecourse is a good onewe must be ableto show it to others.

To be sure, outsiders may not know the mathemat-ics that we do and cannot dictate to us what is mostvaluable. For example, there may be items they wouldlike excluded which are actually prerequisites for othersubjects that they prefer to include. The key point,however, is that they do have a good feeling for whatis necessary. They may be myopic, but some myopia isnot necessarily bad.

As we design curriculum, we should beware of thebeast that loves to pack courses with "favorite topics."(I must confess to feeding the beast on occasion: I can'tseem to get through the entire undergraduate calculussequence without emphasizing the notion of torsion andcurvature of plane curves. Mea culpa.)

During my naive youth, I tried to persuade a groupof mathematicians that conic sections could be learnedeasily by anybody who wanted them in later life andwere not needed in college calculus. By dropping theweek devoted to ellipses, parabolas, etc., the calculuscurriculum would loosen up a bit. That turned out tobe someone else's favorite topic and I was summarily cutoff at the knees. Not only do we pack in the numberof hours, but because our students have trouble witha vast amount of the material covered, we are forcingmany of them into a five-year degree program.

Changes in ziudent preparation constitute anotherimportant item. No one can deny that the infusion ofthe "new math" fifteen years ago had a negative effecton the competence of mathematics students as they en-tered college. Whether this should have happened ornot, whether it is the fault of the high school teacher,the math educator or mathematician is no longer rele-vant. In the short run, we must deal realistically withentering students as they present themselves.

We must understand what these individuals know,what they don't know, and what they are prepared todo. As administrators, it is important that we realize

the various departments are well aware of the chang-:ng caliber and preparation of the students for studyingtheir disciplines. In a nutshell, do these students havea chance to pass or are they forced to fail? Because thelong run problem is so important to calculus, a deanwould applaud efforts of an academic department (es-pecially mathematics) to work with middle schools andhigh schools, as well as with the College of Educationto obtain better prepared college students.

AccountabilityIt is not enough for us as mathematicians to know

that the course is a good onewe must be able to showit to others. While we may resent this intrusion onour expertise there is no longer a choice, especially forpublic institutions. In addition to the administration ofthe university- (even from the non-academic side, suchas Vice Presidents for Student Affairs, University LegalCounsel, etc.) there are also legislators who are quiteconcerned with the value of an education. The standardquestions that are asked are often naivesometimes tothe point of ignorancebut that is not the issue. Thesequestions must be answered and deserve to be answered.

One way to respond to the outside pressures on lowlevel mathematics courses (calculus, in particular) is toask if the department is willing to consider mastery lev-els, fundamental learning levels, or exit examinationsfor any of the sequences. For calculus, this could meana modular approach; that is, students will go to a cer-tain level (say the chain rule) and must pass an examon that level before proceeding to the next one. Inaddition, an increasingly popular notion is the "valueaddedness" for all courses of study. Can we really showthat students who have fit...shed the calculus sequencewith a C know it in enough detail that they will beable to handle subsequent physics, chemistry, or busi-ness courses?

Our courses must be demonstrably ofthe highest quality.

I'm not suggesting that we all need to move towardthese modes of education; I'm only asking whether a de-partment is willing to consider it every once in a while.I hope, but am not convinced, that administrators rec-ognize the danger of thinking of all innovative changesas good ones. Innovation really means that somethingis quite new, not necessarily better. On the other hand,we do need to be willing to consider alternate means ofpresenting calculus, even if we are ultimately to rejectthem. Our courses must be demonstrably of the highest

.153

MILLMAN: ADMINISTRATIVE PERSPECTIVE 143

quality. ("To be thus is nothing; But to be safely thus."Macbeth III, i 48).

When I'm aiLed, as dean, to defend a particularinstructional decision, it is very useful to be able topoint to measures of the quality of instruction. Doesthe mathematics department quantify in some fashionand adequately reward teaching at all levels or is theremerely lip service paid? Is there a mechanism in thedepartment to ensure that new instructors are teachingat an appropriately high level? Are pedagogical articlesand thoughts rewarded? Of course, the reward for anarticle in a mathematics education journal must dependonits quality just as that for a research article. Just do-ing something shouldn't be enougheven deans aren'tthat gross. However, it is important that faculty real-ize that good pedagogy in all of its forms is consideredmeritorious.

AttitudesThe next major point is that of the attitude of the

calculus instructors. Students really appreciate an ex-cited, dedicated teacher the classroom. If the atti-tude is "Calculus is a chore and I'm here just to get mytime in so that I can look at my advanced courses anddo my research," the students will compla'm bitterly,with good reason.

While we rarely learn something new about the sub-ject of calculus when teaching it, we do discover quite abit about how students think and learn calculus. Thiscan be fascinating and is a partial reward for goodteaching. We dc a disservice to both our students andour subject when we regard calculus as a chore andthen compound it by communicating that attitude toour students.

Unfortunately, there are many mathematicians who,while they enjoy the discipline tremendously, do notconvey the excitement they feel to their students. Onecan present an exceptionally clear and well thoughtout lecture on trigonometric substitution without be-ing enthusiastic. Students will learn from such a lec-ture. However, it is so much better to have presentedthe ideas with a certain panache, so that the studentscan get an idea of why someone might become a math-ematician (as well as learn the specific calculus skills).Certainly the instructors in all disciplines who are themost popular with students are those who are the mostenthusiastic about their teaching. Enthusiasm is con-tagious. While it won't substitute for content, we needmore excitement in the classroom.

Unfortunately, these are litigious times in higher ed-ucation. Thus we must not only treat all students uni-formly, but must do so across all sections and show that

the treatment is indeed homogeneous. This responsibil-ity usually falls on the department chair, rather thanany individual instructor, but it is one that needs tobe monitored extremely carefully because of the heavyusage of adjuncts, part-time people and new faculty. [2]Most departments have a common syllabus which is afine first start at a uniform treatment. Many have com-mon final exams and a few insist that all exams in acourse be the same across all sections. While the latteris difficult to administer, the former is not and is anexcellent way to demonstrate that students are treatedequitably.

I've learned to loathe any conversation which startswith "I was always bad in math, but ...." The usualcomments describe how hard mathematics is and howit's not surprising, therefore, that little Johnny or Janieis having trouble with it. A useful response is to quotethe international study of high school students [4] whichdeals not only with accomplishments, but attitudes. Ii;is clear that mothers' attitudes in Japan ("Work harderand it will come to you") are quite different than themothers' attitudes in the United States ("It's just toohard for you"). Pointing out the careful research thathas been done through this study helps to enlightenpeople from outside of the physical sciences. They e. nbegin to understand what a "spiral curriculum" is :=1(1what its manifold drawbacks are. (One doesn't have Lodo differential geometry to have a manifold drawbar' .)

Effectiveness vs. CostA final point to be made concerns cost analysis of

the effectiveness of the calculus curriculum. This is notmeant to be an equation which describes the cost perleft-handed student credit hour, bur rather emphasizesthat there are real financial implications which comenot only from class sizes but from grading structuresand repeat/incomplete policies. Most departments donot have enough resources to offer the number of sec-tions that we would like and so need to teach to fullclassrooms. Students who must drop or repeat a classor take an incomplete are occupying chairs that othersneed. They use up an instructor's time and effort andare detrimental to class morale in the long run.

Some drop out is certainly unavoidable. On the otherhand, it severely impacts the faculty work, load, both interms of efficiency and in forcil g the students into cur-ricula which are becoming five-year programs insteadof four. The Accreditation Board of Engineering andTechnology (ABET) is extremely concerned that theengineering curriculum is becoming standardized at fiveyears 1..ther than four, even though we all think of it asa four-year course of study.

154

144 ISSUES: INSTITUTIONAL CONCERNS

The double-hump camel curve for grades happens inmathematics, chemistry and physics, but as dean I onlyhear complaints about the mathematics aspect. Whyis this? Does it happen at your university? If so, why?Answers to such questions would be useful not only tomathematicians, but to anyone who has experienced un-enlightened inquiry from legislators and others.

Students who must drop or repeat aclass ...are occupying chairs thatothers need. They use up aninstructor's time and effort ... .

Changing class size is one way to economize. Whatis the difference between teaching classes of 30, 40, 60or 200? I have taught calculus in sections of 40 and insections of 80 and quite frankly I see no difference ineffort except when grading papers. I do recognize thatvisits outside the classroom to one's office can take upan enormous amount of time and many of my colleagueshave complained about such things. But when teach-ing large classes, I didn't notice any more office visitsthan usual. (Does this mean that students' visits duringoffice hours are not a linear function of students?)

I am not advocating large lecture sections (in fact,classroom structures at some universities would pre-vent this), but rather that people look carefully at theseideas. It is important not to conventional wisdom,but rather to be able to point to v-arious comparativestudies that have been done in the past. While somehave been done in general, there are none I know of thatare specific to college mathematics.

Cor. elusionIt is increasingly clear that we need to resolve issues

through a more organized study of them than has beendone in the past. As resources get tighter, and as leg-islators look more and more carefully at what we aredoing, we will no longer have the luxury of saying 'It'sclear to anybody who knows mathematics that this pro-cedure is right." This translates into an obligation todevote some of our scholarly energy towards mathemat-ics ei,,,cation at the collegiate level.

I am delighted that the National Science Foundationis prepared to infuse money into such efforts. The chal-lenge to us as mathematicians is to decide what specificprojects and comparative studies need to be done; to setup unbiased studies to explore the issues; and to reachsome conclusions. The challenge to us as college admin-istrators is to find the resources to implement conclu-sions of these studies in a fair and equitable manner.

The task cannot be left to our colleagues in the Collegeof Education or to government bureaucrats.

A knowledge of the MAA's committee structure isvery useful in this regard. I'm alighted that thereare subcommittees that deal with service to engineers(chair, Donald Bushaw), preparation of college teachers(chair, Guido Weiss), role of part-time faculty and grad-uate teaching assistants (chair, Bettye Anne Case), andmany others. These committees provide valuable factsto quiet some of the criticism that we hear. This allowsmathematicians to cite specific sources when respond-ing to complaints. We will thus appear to be listeningcarefully, caring and responding in good faith with atremendous spirit of cooperation. I can't overempha-size this point.

As an administrator I say there's a real world out;lime. As a mathematician I say that boundary condi-tions exist. Whichever way one puts it, the implicationsare that we must work within a certain context, listento people, and present well thought-out responses toothers' concerns, whether those concerns be frivolous,unenlightened or substantive.

Can you hear the shape of calculusfrom its complaints?

In 1966 Mark Kac won a Chauvenet prize for hisbeautifully written paper, "Can You Hear The ShapeOf A Drum?" This article asked whether you could tellwha;-, the shape of a domain in Euclidean space was justby knowing the eigenvalues of its Laplacian. Althoughthat problem is unresolved in 3-space, the analogousone in our context is clearly false.

Can you hear the shape of calculus from its com-plaints? No, absolutely not. What you can hear, how-ever, are the concerns of people who have spent timeand money in an effort to get a good education. Theyhave a right to ask us questions about our calculus cur-riculum, even if those questions are naive. It is cru-cial that we not only provide our students with a highquality calculus experience, but that we answer theirquestions in a responsive, thoughtful manner.

One of the early (1734) critics of calculus, BishopBerkeley [1], asked: "And what are these fluxions? Thevelocities of evanescent increments. And what are thesesame evanescent increments? They are neither finitequantities, nor quantities infinitely small, nor yet noth-ing. May we not call them the ghosts of departed quan-tities? ...." To not answer questions carefully is to runthe risk that we become ghosts of depleted qualities.

155

LYONS: INNOVATION IN TEXTBOOKS 145

[1]

ReferencesBerkeley, Bishop, "A discussion addressed to an infi-del mathematician," contained in David Eugene SmithSourcebook in Mathematics.

[2] Case, Bettye Anne, et al, "Teaching Assistants and Part-time Instructors: A Challenge." Mathematical Associa-tion of America, 1987.Struik, D., A Concise History of Mathematics. DoverPublications, 1948.

[4] Travers, Kenneth, et al. The Underachieving Curricu-lum: Assessing U.S. School Mathematics From An Inter-national Perspective. Cnampaign, IL: Stipes PublishingCompany, 1987.

[3]

Innovation in Calculus TextbooksJeremiah J. Lyons

W. H. FREEMAN AND CO.

How can publishers participate in and support inno-vation within the calculus course? I want to examinethe current state of affairs within textbook publishing,giving special attention to the calculus. By demonstrat-ing the assumptions and the current standards which webring to bear on our editorial actions, we will be betterable to assess the options open to usinstructors andpublishersas we consider ways of providing textbooksfor emerging, innovative courses.

Clearly, innovative textbooks will not be publishedin a vacuum, in the hope that courses might be coaxedinto existence by their appearance. With rare excep-tions, textbooks are adopted by instructors for existingcourses; instructors do not adapt course outlines to ac-commodate idiosyncratic textbooks.

What is both refreshing and challenging about thesetimes in publishing is tryinG to assess the direction ofchange. Publishing is a profession in which informa-tion is constantly being gathered and evaluated. Edi-tors work within a strange time system: decisions aboutpublishing must be made today, and the outcome of ourchoices or decisions will not become visible for severalyears. Authors, too, are partners in this process of in-ference making. Correct decisions will result in the pub-lication of the new generation of authoritative mathe-matics textbooks. The increased national concern forthe state of mathematics education at all levels, andthe creation of boards and councils, are very welcomesigns of the seriousness of these discussions.

RICHARD S. MILLMAN is Dean of the College of Sci-ence and Mathematics at Wright State University in Day-ton, Ohio. Previously he sered as Program Director for Ge-ometric Analysis in the Division of Mathematical Sciencesat the National Science Foundation. Millman has been amember of the Council of the American Mathematical So-ciety and Associate Editor of the American MathematicalMonthly. He is currently Chairman of the MAA Committeeon Consultants, and a member of the JPBM Committee onPreparation of College Mathematics Teachers.

Pressure for ConformityThroughout my twenty years in publishing, the orga-

nization and texture of calculus texts have changed verylittle. And yet, every sales representative and math-ematics editor will tell you no mathematician is everpleased with the calculus textbook currently in use. Ev-ery instructor has a pet topic or two which is not donecorrectly, or a particular notation convention which im-mediately signals the worthiness of a given book. Foryears we have heard dissatisfaction about the standardbooks, and yet there has been no radical change in thebooks we are publishing. And, it goes without saying,there has been no serious departure from the conven-tional text in adoption patterns. This stasis is demon-strated vividly by the fact that many of the better-selling texts are in third, fourth, and fifth editions.

Quantitativenot qualitativeexpansion has beenthe striking change in the past twenty years: there aremany more calculus texts, and each text is approxi-mately 1100 pages in length, or greater. When therewere fewer books, instructors had some familiarity withthe texts and committees had the time to give care-ful consideration to the few new texts or new editionswhich were published each year. A text had an iden-tity, a set of distinguishing features that were knownto instructors. And these characteristics came from thebackgrouLd and the interests of the author. Calculustexts were relatively easy to rank by a few simple mea-sures: theoretical or applied; rigorous or less rigorous(there were no "short" or easy calculus texts); books

1.5 o


intended for students in physics and engineering; hon-ors calculus, and so forth.

Beginning in the early 1970s, enrollments grew andmore publiahers entered mathematics. The number ofbooks published each year increased. With more andmore calculus texts sing published, instructors wereforced to become more critical of the new entries. Afterall, not even editors spend evenings and weekends read-ing through introductory textbooks. There was and istremendous selective pressure (to borrow a trope fromPeter Benz) to converge toward the mean, as estab-lished by the best-selling books. The successful books,after all, passed the scrutiny of the curriculum gatekeep-ers. Therefore, these texts must be filling the perceivedtextbook needs of the course. It follows that for a newtextbook to gain a profitable measure of adoptions, thenew text must come close to the best sellers, and offerjust a bit more of one feature or another.

"The major revolution in calculus textsin the last decade has been theintroduction of a second color."

One thing leads to another. IF publishers are dissuad-ing their authors from innovation, because innovativebooks do not sell, what aspects of a textbook can beimproved upon? Sherman Stein [1, p. 169] character-ized this dilemma from a calculus author's perspective:

It seems that a calculus author has the freedom tomake only two decisions: Where to put analytic geom-etry and whether the title should be "Calculus withAnalytic Geometry" or "Calculus and Analytic Ge 3111-,etry." Thus the major revolution in calculus texts inthe last decade has been the introduction of a secondcolor.

Competition without InnovationThere are a few other arenas in which authors and

publishers sought ways to distinguish their books whileleaving the core calculus content and outline intact.These are the very features which, in the end, conspireto inhibit innovation and change. Let's examine thefeatures which are found described in the promotionalmaterial announcing a new calculus textbook or revi-sion.

The number of exercises has been increased signifi-cantly, and they now number six to eight thousand. Theexercises are placed following each major section withina chapter, and a review set is found at the end of eachchapter. Does anyone actually count the number of ex-ercises, including the lettered A, B, C... , subparts of

questions? Yes, they do. Is there an optimum num-ber of exercises for a text? No, in this dimension oftextbook making, the guiding rule seems to be that onecannot have too many exercises.

We can give you more book, for moremoney. Not different books: morebook.

The same can be said for the number of worked-outexamples. The range of applications has been widenedconsiderably, to include examples drawn from biologyand economics, for instance. These additional exam-ples are not replacing existing ones from engineeringor physics. Usually, there are simply more exampies(and step-by-step solutions) added to each succeedingedition.

Publishers are asking for acceptance for our textsfrom a calculus textbook marketplace which is becom-ing increasingly crowded, and in which modifying thetraditional content of the calculus is out of bounds forus. And yet the impetus to publish a new text whichis a commercial success in this lucrative auarket contin-ues. What other aspects of a book can be revised andimproved upon, if the mathematical content must bestandardized in relation to existing books and courseoutlines? As I mentioned above, we can give you morebook, for more money. Not different books: more book.

There have been dramatic improvements in the num-ber and quality of the graphics in calculus texts. Almostall are done in two colors now, and it won't be too farinto the future before even more colors will be used.The use of two colors and high quality airbrush tech-niques have resulted in figures of exceptionally goodquality. This trend towards more and better illustra-tions will continue. Computer- generated curves are be-coming rentine, and color graphs of functions will soonbe incorporated into our textbooks.

The other area in which publishers can modify theexisting textbook model, without cutting into content,is by incorporating pedagogical devices. More heads,use of margins for key terms, learning objectives andsummaries, lists of applied examplcsall these elementsare an essential part of the textbook presectation.

I have described some dimensions of the textbook-making process in order to demonstrate several points.In an increasingly crowded and standardized publish-ing marketplace, publishers seek to outdo each otherand capture your attention and your adoption in wayswhich ensure the preservati m of content, in an accept-able order, and at the same time create some special

LYONS: INNOVATION IN TEXTBOOKS 147

identity for each book. It is no accident that each ofthe features cited here contributes to a longer, morecostly book. Herein is our dilemma.

Ronald Douglas [1, p. 13] stated the dilemma ofpublishers in these words: "Everyone is dissatisfied withthe current crop of calculus textbooks. Yet, if an authorwrites and manages to get published a textbook whichis a little different, most colleges and universities willrefuse to use it. How can we break out of this dilemma?"

Economic ConstraintsThe current models for calculus publishing have been

in place for twenty years. Each year brings new booksand more revisions. In order to break out of this quanti-tative growth pattern, two crucial factors must be con-sidered. First, publishers must be confident that newdirections in calculus teaching and course organization,though many of these courses will be experimental indesign, will be in place for awhile. Second, no pub-lisher is going to sponsor the publication of an innova-tive textbook in calculus that demands an investmentcomparable to that of a mainstream text. Instructorsseeking textbooks for their alternative calculus coursesmust be willing to accept a few tradeoffs. The reasonsfor this should be obvious.

The investment in a mainstream, three semester cal-culus textbook published in two colors will easily sur-pass $350,000 before the first copy is off the press. Inaddition, the unit cost f-sr each copy (paper, printing,and binding) will be $6:O.` average. Given these costsfor entry into a market, no wonder there is enormouspressure to position a textbook close to the center ofthe market.

There are several means by which publishers canrespond to the coming evolutionary changes in calcu-lus. Our strategy for undertaking innovative publish-ing projects can be stated very directly: publishers willsponsor innovative projects intended for modest seg-ments of an introductory market, but only at an in-vestment level commensurate with the expected rate ofreturn on our investment.

Publishers and authors now haveexpedient and economical modes ofproduction by computer available.

Publishers and authors now have expedient andeconomical modes of production by computer avail-able. For the first time we are seeing microcomputer-controlled laser printing of text pages which are of sat-

t

isfactory appearance. It is because of this recent de-velopment that the option of publishing innovative al-ternatives to the big textbooks is a reality. The com-puter is having a forceful impact not only on the contentof mathematics courses, but also on the text materialsmade available for the teaching of these same courses.

Stephen Maurer was direct in his assessment of pub-lishers and our willingness to respond to change in thecurriculum: "It's no use telling publishers to changetheir ways. They are hemmed in by market forces. Wemust show them that a new type of book will attract amarket before we can expect them to help." [1, p. 81]Publishers are always responsive to numbers, so that isvery sound advice.

Making Innovation AffordableIf we agree that substantial investment is not possible

for small segments of the calculus market, what custom-ary features are instructors willing to give up in orderfor a publisher to keep the costs of publishing down?Let's look at the potential for changing textbook re-quirements, along with alternative and more affordablemodes of book production.

First, we should not be thinking of new textbooks ofthe same bulk as existing ones. Eleven hundred pagesis too much, even of a good thing. In addition to be-ing very selective about what topics to include or not(precalculus topics, for instance), an obvious area to re-alize reductions in page length is to sharply reduce thespace given to exercises. Does anyone need eight thou-sand exercises? Most departments have exercises on fileand these can easily be put into an exercise bank on amicrocomputer. Exercises can be generated as needed.This would reduce the publishing costs directly relatedto length.

Next, are we willing to dispense with the pleasingbut ultimately excessive use of color in our new genera-tion textbooks? The preparation of fine line graphics isvery expensive. And the additional cost of separationsand two- and four-color printing drives the investmentin books up, in several ways. Not only are two-colorbooks more expensive for artwork and film, but alsothe cost of two-color printing is prohibitive unless wedo print runs of 7500, 10,000, or more copies. In orderto justify printing in those minimum quantities, we arerapidly departing from the realm of "innovative publish-ing." Can we live with one-colm printing, and use theflexibility of type styles and sizes currently available inmany scientific word processing software packages in-stead of a second color? Imaginative use of differenttype styles and sizes can provide the same visual dis-tinctions and schemes as two-color type.

-8


The next obvious area to look for ways to econo-mize is in artwork. The average investment in a com-plex calculus illustrations program is $75,000, for a firstedition. Substantial savings can be realized if authorscan provide illustrations which are of reproducible qual-ity, thus alleviating the need for costly rendering. Ihave seen excellent computer-generated figures in manymanuscripts. It seems wasteful for publishers to be re-drawing from perfectly acceptable camera-ready art. Infact, experienced textbook authors will appreciate theprecision and control they have over the preparation offigures which they generate by computer.

Eleven hundred pages is too much,even of a good thing.

If we can agree that our innovative texts will not beas big as their more orthodox relatives (a virtue in anyevent), if we can be satisfied with one-color printing,and if we rely wherever possible on computer-generatedillustrations, then we are beginning to describe an af-fordable publishing venture.

Gratuitous AncillariesWhat we have lot considered are the additional ma-

terials which are part of the standard textbook "pack-age." You have no doubt noticed the size of the cartonswhich arrive, sometimes unbidden, from publishers. Itis no longer sufficient to publish only a textbook. So-called ancillaries provide another means for each newor existing textbook to distinguish itself. The devel-opment, production, and distribution of ancillary ma-terials for an introductory mathematics textbook canrepresent additional costs approaching $100,000.

The ancillary or supplementary package consists ofa complete and a partial solutions manual; a stu-dent study guide; a computerized testing program; acomputer-based tutorial program; and overhead trans-parencies. Most of these items are given free to po-tential adopters of a text, although the student studyguides and, in some instances, the student solutionsguides are sold to students. The free supplements arepublished at the publisher's expense, usually providedfor in the marketing budget. This is another stimulat-ing set of reasons to ensure the standard outline andcontent of the book which stands at the center of thebook package: the text has a family to support.

If we can agree to dis,_ nse with some ancillary items,we will be relieved of ue constraint in our budget.

No one admits to really using the supplements, any-way. I sometimes have the feeling that publishers un-dertake the production and distribution of supplementsand other pedagogical aids, in some course areas, on thebasis of matching other publishers and their packages.

Presumably, new calculus textbooks will have quitedistinctive features in terms of mathematical content,,organization, and approach to the subject. Therefore,in functional but unadorned book form, instructors willbe able to see what's been done, and either accept orreject on those bases. In this more specialized and moremodest segment of the calculus market, textbook deci-sions will not depend on such incidentals as the avail-ability of supplements.

Viable MarketsRee:1.ing the physical requirements of textbooks and

their attendant supplements will remove some of the fi-nancial disincentives facing publishers, as we considernew publications. The wonders and portability of desk-top publishing provide us with alternative compositionand sources for accurate line illustrations, tables, andgraphs. These economies, too, will encourage publish-ers to enter emerging course areas.

The question of markets still confronts us, though.While the investment facing publishers might be moremodest, and therefore more attractive than undertak-ing large textbooks, our vision is national in scope.In established publishing companies it costs a certainamount of overhead just to turn on the publishing ma-chine. Contributions to operating costs and other ex-pense requirements preclude our undertaking the text-book equivalent of vanity publishing. No publisher iswilling to publish for a single course being offered inone college. That kind of limited enterprise does not re-quire the editorial, production, and marketing resourcesof even a modest-sized college publisher.

The stability and predictability of standard coursesare two of their assets, in the eyes of publishers. If wepublish a differential equations text from an engineer-ing college in the northeast, we can be confident thecourse looks the same in other colleges. One publisher,in partnership with an author and several manuscriptreviewers, can enter a national, coherent market.

The more cooperation and sharing of ideas that ex-ists among instructors at different colleges, the moreattractive such innovative courses will be for publish-ers. While we all decry the monolithic nature of intro-ductcry textbooks, publication for fragments of small,isolated experimental markets is not a viable alterna-tive. If cooperation among mathematicians helps pub-lishers identify several, rather than many, fragmentary

15D

LAYTON: HIGH SCHOOL PERSPECTIVE 149

proposed future directions in calculus instruction, wewill be responsive.

At the outset of this paper I spoke of publishing as adata-collecting and data-evaluation business. Publish-ers are relieved, frankly, to see that concerted effortsare at work and national commissions are in place tosupport changes in the calculus courses. Too often ourrelations with a discipline are limited to one-on-one ven-tures with authors, and we have little sense or spirit ofcooperation with a discipline.

Some publishers are entrepreneurial risk-takers.New, interesting departures from existing textbookmodels do find their way into print, eventually. But thatis a very slow and unpredictable process. Having accessto the proposed "high-level" information base will beof tremendous help to publishers (and authors) in eval-uating the extent of a potential market for a calculusproject.

With access to such data, and a willingness to useefficient and economical methods of production, inno-vative textbooks will be available. The publication of

Perspective from High SchoolsKatherine P. LaytonBE. ERLY HILLS HIGH SCHOOL

The population of high school students who are af-fected by calculus can be broken into three groups:mathematics students who take AP calculus, those whoare enrolled in a non-AP calculus course of some kind,and those in the precalculus classes such as second yearalgebra, mathematical analysis, or trigonometry whowill take calculus at college.

If a school has an AP calculus program, the studentsenrolled will be the very best mathematics students.These students make up the main pool for future math-ematics majors. Many are already turned onto the ex-citement and beauty of mathematics. This joy of math-ematics must be carefully cultivated. These studentsmust be nurtured both at the high school and collegelevels. Special consideration should be given to themwhen they enter college.

At the college level, mathematics departments needto seek out these students and give them special coun-seling to get them properly placed in mathematics intheir freshman year. Remember, these students havecome from small matheniatics classes where they havebeen nourished and have had much opportunity to in-

innovative alternatives to traditional calculus textbookswilt certainly help the diffusion process for alternativeapproaches to the course. With the right data in hand,,and a moderate investment requirement, publishers willbe your enthusiastic partners in fostering change in cal-culus instruction, and publishing.

Reference

[1] Douglas, Ronald G. (ed.) Toward A Lean and LivelyCalculus. MAA Notes, No. 6. Washington, D.C.: Math-ematical Association of America, 1986.

_.

JEREMIAH J. LYONS is Senior Editor at W. H. Freemanand Company and Scientific American Books. Previously heheld editorial positions at Addison-Wesley and PWS Pub-lishers. He was a member of the Committee on CorporateMembers of the Mathematical Association of America, andis a former chairman of the Faculty Relations Committee ofthe Association of American Publishers.

teract with their teacher and with each other. Usu-ally their high school teachers are among the strongestteachers in the mathematics department.

If a school has an AP calculusprogram, the students enrolled will bethe very best mathematics students.These students make up the main poolfor future mathematics majors.

Students enrolled in a non-AP calculus course alsoneed special care and careful counseling which leads tocorrect college placement in mathematics. They haveexperienced some calculus and may believe they know itall; some may become complacent, cut class, and end upbeing unsuccessful. These students need to be remindedthat attending class is important and that homeworkmust be done; they too need the involvement and con-cern of their professors.

160


I would like to see non-AP calculus courses at thehigh school eliminated and students enrolled in othermathematics courses such as prt. bability, statistics, dis-crete mathematics, or elementary functions includingan introduction to limits.

Preparation for CalculusIf the calculus course is to change, then so must the

precalculus courses. (By precalcuLs courses, I mean al-gebra, geometry, advanced algebra, trigonometry, andmathematical analysis.) Currently, much of the con-tent in these courses is geared to preparing studentsfor calculus. A large portion of the content is devotedto gaining skill in numerical and algebraic operations.Students learn much of their mathematics by memo-rization and may not have had much experience withbeing expected to understand concepts. Instructionaltime is spent on factoring, simplifying rational expres-sions, graphing, simplifying radicals, solving equations,and writing two column proofs.

How much skill is really needed in these areas in lightof the new calculators and computer software? Studentsstill need some proficiency in numerical and algebraicskills and they need some expertise hi symbol pushing.In order to make the best use of a calculator, studentsneed computational facility with paper and pencil, andthe ability to do mental arithmetic.

Yet to use a software package effectively, studentsneed strong conceptual understandings of the subjectmatter and then computers can take care of the me-chanical details. The question arising here involves oheof instructional psychology: how much expertise does astudent need in performing mechanical procedures inorder to understand concepts and to be an effectiveproblem solver?

Understanding ConceptsIf the desired products are students who can think,

who have an understanding of concepts, who have de-veloped logical maturity, and who have the ability toabstract, infer, and translate between mathematics andreal world problems, then this type of training mustbegin early in their mathematical experience.

More time in high school needs to be spent on under-standing concepts, developing logical reasoning, guidingstudents' thoughts, and helping them develop thinkingskills. Students need numerical and graphical experi-ences to help them develop intuitive background. Stu-dents need guidance and experience with the use of bothcalculators and computers. These tools should be an in-tegral part of the overall mathematics program. Issues

of reasonableness of answers, appropriate use of tech-nology, and round-off errors must be considered.

In changing the precalculus program, one needs tobe careful not to overload the courses. Selected contentmust be eliminated if new material is added or if teach-ers are to teach more for concepts and understanding.

I believe students need considerable facility with ba-sic algebraic skills, geometric facts, and numbers factsbut the time and drill spent in these areas can be re-duced, as can the current content. Both Usiskin andFey have made suggestions for content in the precalcu-lus area ([2], [3]).

A New Precalculus CourseThe current calculus course is too full of techniques

which students often memorize without understandingthe fundamental concepts. Changes in calculus will re-quire and depend upon changes in the precalculus pro-gram.

New courses must be constructed that stress devel-opment of concepts as opposed to purely mechanicalunderstanding. Students and teachers will spend moretime setting up problems, analyzing and interpreting re-sults, and creating and solving realistic problems. Stu-dents will have more experiences with estimation, algo-rithms, iterative methods, recursion ideas, experimentalmathematics, and data analysis (how to get data, whatdo these data mean, and how to transform, compare,and contrast the data).

Much more experience with functions will be in-cluded. Students will consider those given by a formula,generated by a computer, and arising from data. Therewill be more emphasis on analysis of graphs. Experiencewith a deductive system will be included.

While in groups students will have the opportunityto read about mathematical ideas new to them andwork out problems using these ideas. They need toconstruct examples illustrating concepts and find coun-terexamples. There should be opportunities to solvenon-standard problems and problems which are multi-step; these problems must push students beyond theblind use of formulas.

Goals of CalculusIn the publication, "Toward A Lean and Lively Cal-

culus" [1], there is a statement of goals giving competen-cies with which students should leave first-year calculus.These include the ability to give a coherent mathemati-cal argument and the ability to be able not only to giveanswers but also to justify them. In addition, calculus

161

LAYTON: HIGH SCHOOL PERSPECTIVE 1S1

should teach students .,ow to apply mathematics in dif-ferent contexts, to abstract and generalize, to analyzequantitatively and qualitatively. Students should learnto read mathematics on their own. In calculus theymust also learn mechanical skills, both by hand and bymachine.

As for things to know, students must understand thefundamental concepts of calculus: change and 0,asis,behavior at an instant and behavior in the average, andapproximation and error. Students must also know thevocabulary of calculus used to describe these concepts,and they should feel comfortable with that vocabularywhen it is used in other disciplines.

Students must understand thefundamental concepts of calculus:change and stasis, behavior at aninstant and behavior in the average,and approximation and error.

At the Sloan Conference where these competencieswere thought through, the content workshop developeda suggested calculus syllabi for the first two semestersof calculus [1]. This is a course I would like to seeimplemented. However, if a new calculus course is im-plemented without changes in precalculus courses, thenthe ca :ulus program must place additional emphasison functions, approximation methods, recursion ideas,data analysis, interpretation of graphs, use of calcula-tors, developing number sense to recognize incorrect an-swers, checking answers for reasonableness, and dealingwith the question of round-off error. Helping studentsto build a conceptual understanding will be extremelyimportant.

StudentsMany students are used to getting through math-

ematics classes by memorizing recipes for doing prob.lams. These techniques seem easier to them th..n havingto reason through a problem using mathematical con-cepts, i.e., having to think their way through a prob-lem. They find mathematics difficult and look for aneasy way out; memorization appears to work.

Students often are more computer literate than manyof their teachers. Many will have computers at homeand will have used some of the mathematics softwarepackages. New high-powered calculators will be in thehands of many students. Even now students are ques-tioning the value of learning certain arithmetic oper-ations since they know their calculators can do them

more quickly and accurately than they can. Calculatorscapable of displaying graphs were in the classrooms lastfall and, of course, some students questioned the timespent on curve sketching. All this calculator and com-puter power must be used to make time for a moreuseful, exciting, and relevant mathematics curriculumfor the students.

ParentsParents also will need to be educated. Many par-

ents were students during the "new math" era and mayhave been burned. Others will say "that's not whatwe learned." Parents have expectations about the skillstheir children should know. The mathematical commu-n4 must convince them that it is all right for studentsto use technology wisely and with discretion.

TestingThere ztre two areas to consider here: teacher-made

tests and the various kinds of standardized tests.Testing influences both student and teacher behav-

ior in the classroom. If students know they will onlybe tested on techniques, they will listen (or day-dream)during concept building experiences but will not worryabout them. When they study for tests, some willonly memorize procedures and techniques. Currentlyteacher-made tests test mostly techniques or skills.These are easier to write, easier to grade, and are thetypes most teachers experienced when they were stu-dents.

Students should be tested on the understanding ofconcepts, should be able to explain ideas, and shouldbe expected to write about mathematics. The difficultyof questions should be graded from quite easy to chal-lenging. Since many teachers use or model their testsafter tests that come with textbooks, publishers mustalso be educated. Calculator use should be encouragedin testing: tests must include questions that make gooduse of the calculator.

Tests that test mostly techniques orskills .. are easier to write, easier tograde, and are the types most teachersexperienced when they were students.

Teachers need to try various forms of testing: studentportfolios containing work completed, open-ended ex-ams, take-home exams, open-book exams, group exams,

1.6


oral exams, and standard questions in nonstandard set-tings, i.e., giving a graphical version of the derivative ofa function and asking for the graph of the function.

Standardized tests, as the saying goes, tend to drivethe curriculum. For the curriculum to change, thesetests must change too: AP Calculus, Scholastic apti-tude Test, College Entrance Examination Board Math-ematics Achievement Test, National Assessment of Ed-ucational, Progress, American College Test, and stateand local competency tests. When will a ective calcu-lator questions be included on these important nationaltests?

MaterialsThe proposed new courses suggest heavy use of cal-

culators and at least classroom demonstration of com-puter experit.nces. The latter necessitates the availabil-ity of well-thought through and simple-to-use software.Teachers want to know quickly how to use _a piece ofsoftware and to know if it is dependable. If somethinggoes wrong, instructions for what to do must be clearand efficient. Guidelines for use of the software with agiven topic are also needed. Calculator exercises andsuggestions for use in discussing a concept and explor-ing mathematical ideas are needed.

When will effective calculator questionsbe included on ...important nationaltests?

I would like to see both computers and calculatorsbecome an integral part of the curriculum. This, ofcourse, implies that appropriate assignments are avail-able which allow students to use these tools to exploremathematical ideas. These materials need to be veryuser-friendly so even the inexperier.ced student can usethem with ease. We are concerned here not with teach-ing how to program the computer but with using thecomputer to teach ..nd explore mathematics.

Textbooks integrating the technology and stressingthe concepts approach are necessary. In many statesthe schnols must furnish each student with a book freeof charge. In planning a new calculus program and sug-gesting changes in the precalculus program, one mustremember that high schools cannot change textbooksvery often. Most schools are on a four to eight-yearcycle.

Equipmentteach student will need a calculator. I would like to

ice a scientific one with graphing capability used in the

precalculus classes. Calculus students would need onewith a "solve" key and au "integrate" key. Computersneed to be available for both classroom demonstrationand for student use. In addition, a large screen monitoris necessary so a class of 30-40 students can see class-room demonstrations of materials on the computer. Forthe most effective use of a classroom tool, a computerand large screen monitor should be in each 'hematicsclassroom.

I cannot leave the discussion of materials and equip-ment without stressing the shortage of funds. Mathe-matics departments in many schools cannot afford thenecessary technology, software, and textbooks. Fund-ing from industry, business, and state or federal gov-ernments is necessary.

TeachersTeachers are the key factor in a successful mathemat-

ics pub ;n. There are many competent high schoolmathematics teachers doing a fine job with 150 youngpeople 180 days a year. Teachers need to be convincedthat a change is necessary and then must be given thetime, tools, resources, and training to do the job. I havebeen told by many mathematics supervisors that highschool teachers are the hardest to change and are leastresponsive to in-service education. They are reluctantto change what they believe has worked.

Many current teachers were educated in the late '50'sand '60's, and if my experience is typical, most have apure mathematics background. This early training doesinfluence what content they feel is important. Theseteachers have been teaching for eighteen to twenty-eightyears. They are committed, experienced, capable, butsome are very tired!

Teacher TrainingTeacher training will be a key to success for a new cal-

culus program and precalculus curriculum. In-service,summer programs, or college courses available duringthe school year are several ways this training could bepackaged. California has had considerable success train-ing teachers during the summer and then using theseexperienced teachers to conduct workshops, speak atconferences, and give in-service programs. Much train-ing will be needed, not only for the calculus teachers,but also for the teachers of the precalculus classes of al-gebra, geometry, advanced algebra, and trigonometry.

A good training program requires several features.Teachers must be given released time or be compensatedfor attending; materials and registration fees should notcome out of their pockets. Courses should be presented

163

LAYTON: HIGH SCHOOL PERSPECTIVE 153

by outstanding teachers familiar with the high schoolenvironment who are modelling what should happen intne high school classroom.

Teachers need continual support such as monthlymeetings to share experiences and to discuss whatworks, what does not work, and what went wrong. Theyshould have the opportunity to observe each other.Care needs to be given as to when and where this train-ing is provided A late afternoon or evening workshop isnot the best experience since teachers are tiredarter teaching five classes during the day and often mustprepare for the next day.

Teacher training will be a key i;osuccess for a new calculus program andprecalculus curriculum.

A vital issue is the teachers' feelings of confidenceand comfort. Th.'s issue must be addressed. If classesbecor ,e more open-ended and more exploratory in na-ture, teachers will feel less in control and will not alwaysbe an authority on the topic. There are teachers whofear something different or new. They need to be will-ing to learn with their students and must be given thenecessary tools and confidence to do so.

Calculators and computers must become integralp. 's of calculus and precalculus courses; this is not truein many schools at this time. Often it is not because theequipment is not available, but because teachers are un-trained and thus uncomfortable using the computer orcalculator. Another critical issue is one of time: teach-ers rarely have time to find ways to implement use ofthis technology in class.

There are teachers who fear somethingdifferent or new.

It is easier to teach mechanical material. It is muchharder to get students to "think" and understand con-cepts. It is also easier to write and grade exams test-ing basic skills. One aspect of "easier" is the issue oftime: time in class, time to plan lessons, and time towrite tests. It takes less time in all these areas to dealwith skills and mechanics instead of concepts. Teacherswill need guidelines and models for writing new typesof test questions, and for conducting different types ofclassroom experiences which help in concept building.

In planning the new curriculum, one muet also beaware of t: quality of people entering the teachingprofession, since many have weal, mathematical back-grounds that need to be strengthened.

Teaching

We want students who can think, reason, apply con-cepts, express themselves with clarity, and use tech-nology effectively. To accomplish these goals, studentsneed to be given of to talk, write, and thinkabout mathematics.

Students should be expected to read and write us-ing the mathematics vcrabulary of the course. Otherexpec.ations should include complete and coherent an-swers which arc well thought out, well developed, andwell written on both tests and homework, and well ex-pressed in class discussions. Students sho';ld spend timeat the board explaining their work, and there should beopportunities for group work.

As mathematics teachers we need to take responsi-bility for all aspects of students' mathematical develop-ment. We must not ignore algebraic errc,rs when stu-dents are writing up solutions to mathematical prob-lems or explaining their reasoning in solving a problem.We should not accept sloppy work; we must encouragestudents and work with them to improve their work

Problems need to be assigned that are thought pro-voking, not just skill oriented, and that require detailedanswers. These must be graded carefully. In all course;teachers should use a variety of techniques such as con-crete materials, chalk and talk, group work, computerdemonstrations, films/videos, games, and discussion tohelp students learn and understand mathematics. Inthis environment students will put though.1 into words,help each other refine answers, and explore ideas.

References

Douglas, Ronald B. (ed.) Toward A Lean and Lively Cal-culus. The Mathematical A.Jsociation of America, Wash-ington, DC, 1986.Fey, James T. (ed) ) Computing and Mathematics: TheImpact On Secondary School Curricula. National Councilof Teachers of Mathematics, Reston, VA, 1984.Usiskin, Zalmnn. The University of Chicago SchoolMathematics Project for Average Students in Cade-, 1-12.

KATHERINE P. LAYTON teaches at Beverly Hills HighSchool in Beverly Hills, California. She is a member of theMathematical Sc.' -ces rducation Board and of the Com-mittee on the -11-.,..1.ical Education of Teachers of theMathematical Ass t of America. In .1983 she servedon the :rational Board Commission on rrecollegeEducation in . v, Science, and Technology.

1641

154 IssuRs: INSTITUTIONAL CONCERNS

A Two-Year College PerspectiveJohn Bradburn

ELGIN COMMUNITY COLLEGE

I start from the premise that it is intuitively obviousfrom the most casual observation that both the teach-ing and learning of calculus is in serious disarray. A fur-ther premise is that if anything of significance, howeverslight, is to be done in the attempt to rectify the situa-tion, then funded pilot projects are a necessary step inthe process.

"If you don't know where you aregoing, any road will get you there."

This paper comments on some of the questions whichneed to be addressed in the design of these projects, of-fers suggestions of ways that two-year college facultycan effectively help in designing and carrying out theprojects, and makes recommendations on disseminat-ing the results of the projects in a manner which wouldbe useable by the greater mathematics ,minunity, es-pecially by two-year college faculty.

As a first step, I recommend a complete perusal ofthe report [1] of the 1986 Tulane workshop on calculusby anyone interested in joining the trek toward a leanand lively calculus. Although this report covers manyof the questions and ideas that must be considered, Ishall repeat some of them in this paper as a means offurther emphasizing them.

My instructional design professor used to say, "Ifyr: don't know where you are going, any road willget you there." Where is calculus going? Where doescalculus fit in the development of mathematical ideas?Where does calculus fit in the development of students'mathematical maturity? What are the proper servicefunctions of calculus? These questions are partially an-swered in the papers in [1], but I do not feel that com-plete enough answers are given there to move to theinstructional design stage for calculus.

The Role of ProofThe central issue that must be addressed in any re-

form of calculus concerns the nature of proof. Whereis the development of the idea of the nature of proofdeliberately cmphasized in the sequence of mathemat-ics courses from beginning algebra through calculus?When I took plane geometry as a high school sophomore

and solid geometry as a high school junior, everyoneknew that you do proofs in those two courses. It now ap-pears that someone looked at the titles of those coursesand decided that since "geometry" but not "proof" isin the titles, then proof is not an appropriate topic in ageometry course.

One of the problems we deal with in attempting toteach calculus today is that proof is not a part of moststudents' mathematical backgrounds. Proof is no longera large part of the geometry course and the algebracourses deal not with proof but with manipulative skills.The debate over the appropriate level of rigor in teach-ing calculus needs to be preceded by a discussion con-cerning whether calculus now represents students' firstintroduction to mathematical proofs. Certainly someproofs are still done in calculus, either using intuitivepremises or very formally constructed premises.

I believe that proof is the very essenceof mathematics.

I believe that proof is the very essence of mathemat-ics. Many students end their formal mathematics train-ing with calculus or shortly thereafter and the nature ofproof should be a part of that training. As a two-yearcollege teacher, I regularly teach 'all the courses in thesequence from basic algebra through calculus. In termsof the textbooks available, with the possible exceptionof analytic geometry, I do not see the idea of develop-ing the nature of proof deliberately being given a highpriority.

Since most two-year colleges teach that whole se-qumce, such institutions need answers to the questionof where the nature of proof is deliberately and con-sciously taught in the sequence. Those answers may becontained in decisions on the content and style of thecalculus or in statements about the prerequisite skillsand knowledge of entering calculus students. Howeverthe answers are given, whether traditional or newly for-mulated, we do need them.

Establishing PrioritiesQuestions about the role of proof as well as other

questions raised in [1] need to be answered before de-sign of instructional content and style for the various

BRADBURN: TWO-YEAR COLLEGES 155

projects can begin. Information needs to be gatheredfrom various sources, especially about the service func-tions of calculus. When the questions are answered,the expected outcomes of calculus instruction need tobe clearly prioritizedthere are techniques for doingthatin order to make the difficult decisions aboutwhat topics to include or not include, and which ideasto give primary emphasis and which to give lesser em-phasis.

Another reason for having a clearly prioritized list ofexpected outcomes is that it fac.litates the process ofevaluating the success of the project. The prioritizedlist is also a necessary part of the information to bedisseminated in follow-up workshops in order for oth-ers to judge the effectiveness of the instruction and todetermine if they wish to try the instructional packagedeveloped In that particular pilot project.

I have beer teaching calculus for twenty years and Ino longer have favorite topics to teach. I have fun daysin teaching calculus, but I cannot count on any particu-lar topic being fun to teach on any given day. Therefore,I would not argue for inclusion of any topic because Ienjoy that topic, but would rather point to a prioritizedlisting of what calculus is supposed to accomplish.

Certain topics may be included for completeness, ifcompleteness is high enough on the list of outcomes tooutrank competing topics which meet other outcomes.Proof, as I have argued, should have high priority. Sincemost instructors rely heavily o.i calculus proofs based onintuitive premises, the course should also include someconvincing examples of cases where intuition has led toincorrect conclusions.

Student CharacteristicsAnother question which needs to be answered before

the actual design of the instruction can proceed con-cerns the prerequisite mathematical skills and knowl-edge on which the instruction is to be built. Althoughstudents in calculus have widely differing mathemati-cal backgrounds, student's actual mathematical skillsand knowledge, study skills, and learning skills fit intoa fairly narrow interval.

For examp:e, a couple of years ago I was really dis-appointed in my students' learning abilities and discov-ered that my colleagues in other fields agreed that thisgroup of students was the worst Ever. We assumed thatthe four year schools had dipped lower into the studentpool to keep heir enrollment figures up, leaving lessable students for the two-year colleges. Yet our regis-trar assured me that on paper this was the best group ofstudents we had ever had. Realism rather than wishful

thinking in stating the prerequisite skills and knowledgeleads to a more effective instructional design.

I may appear to be pointing an accusing finger in thepreceding paragraph, but a two-y. ar college mathemat-ics teacher usually cannot afford such a luxury. We haveenough students in our calculus classes who have taken,at our own schools, the sequence of courses leading upto calculus who do no better than the other studentsto keep us from pointing fingers. My earlier suggestionthat two-year college mathematics teachers can use an-swers to the qc.esions of where the nature of proof isdeliberately taught in the sequence of courses also ap-plies to most questions about the development of basicmathematical ideas.

The characteristic of two-year college students thatis most often mentioned is that they tend to be olderthan other lower division undnzraduates. Age is botha blessing and a hinderance: these students tend to bemore serious about school, but have more :esponsibil-ities and demands on their time. Consequently, themajority of them are part-time.

Most two-year colleges are commuter schools. Thesetwo characteristicspart-time and commuterare thevariables that most affect instructional planning. Two-year college students necd to plan schedules in advance(babysitting, jobs, transpori,2tion, etc.) and need afairly fixed schedule and fixed time commitment forschoolwork for the entire semester. A large indepen-dent project assigned on short notice shows a lack ofinstructor's awareness and planning.

Style and ContentPeter Renz' paper, "Style versus Content: Forces

Shaping the Evolution of Textbooks" [1, pp. 85-104speaks specifically to the important issue of style of in-struction. Other papers in the same volume speak tothis issue by discussing sample questions and examplesused in teaching calculus. The order of topics, for ex-ample, has a great effect on the instructional outcomesand is an important part of instructional design. I haveseen useable textbooks issued in a new edition with thechanges consisting primarily of a reordering of topics.Many times the new edition is not a workable text eventhough the style of presentation is the same.

The papers in [1] which deal with the utilizationof computers and symbolic manipulation packages forteaching and learning calculus point out many possibil-ities for change in calculus instruction. However, caremust be taken so that the computer does not take overthe role of performing Mathemagic, that is, "snowing"the students, which the teacher can too easily do al-ready.

1 Pe

156 ISSUES: T1STITUTIONAL CONCERNS

Computers can show many things quickly, but stu-dents still need time to think. They need guidance inhow to look at computer output just as they need guid-ance in reading mathematics texts. The prioritized listof outcomes, the choice of topics selected, and the avail-ability of hardware and software are the major 41ter-mining factors in designing instruction which uses com-puters in a meaningful way. The first two factors arefar more important than the third, since hardware andsoftware continually change.

Computers can show many thingsquickly, but students still :wed time tothink.

One needs to exercise caution when deletinv topicsfrom a course. =Most topics originally had valid rea-sons for being included, although those reasons may nolonger be apparent from the treatment of the topic oreven in the topic itself. One must be aware of the rea-sons the topic was included in the first place and makeconscious decisions about those reasons when deletinga topic. I give two examplesone from trigonometryand one from calculusto clarify this point.

In trigonometry one can be asked to give the valueof sin (75°) or sin(5w/12). It is a very simple matter toget a value by punching a few buttons or, as in my day,by looking it up in a table. However, one purpose ofthat type of problem is to give students an opportunityto practice with various formulas:

(a) sin(571112) = sin(w/6 r/4)(b) sin(5w/12) = sin(2r/3 r/4)(c) sin(5r/12) = sin(1/2)[5 rr/6].In deleting this type of problem from trigonometry,

one is in effect saying that adequate practice is providedelsewhere or that practicing with these formulas to be-come better acquainted with them is not worthwhile.

Techniques of integration is a frequently mentionedcandidate for deletion from calculus. In terms of its im-portance to calculus, I agree. :Iowever, I have told mystudents for years that the real purpose of that chapteris to sharpen their skills in algebra and trigonometry. Ifthat chapter is removed, we are saying implijtly eitherthat other topics provide adequate sharpening or thatsturents do not need to have their skills in algebra andtrigonometry sharpened.

TeachingIf one is looking fo experienced calculus teachers to

help in the design and implementation of p*lot projects,

I would suggest a heavy dose of two-year college math-ematics teachers. Two-year college faculty are first andforemost teachers. A typical regular teaching load is15-17 hours per semester, often with an overload classor pars-time contract on top of that.

A large number of two-year college mathematics in-structors regularly teach calculus, are conscious of whatit takes to be an effective teacher, and are willing to helpimprove the outcomes of teaching calculus. In designingand implementing different instructional strategies forcalculus, the knowledge and experience these teachersbring to the task is very useful.

The feedback that experienced teachers can give onwhat is working and what is not working is invaluablein revising the material for the second and subsequentclasses using the material. Sometimes students sendmixed signals. At that point, decisions about the in-structional process need to be based on teaching expe-rience rather than on the mixed signals.

Two-year college faculty are first andforemost teachers.

Since teaching is such a large part of the professionallives of two-year college instructors, we want teachingto go well. We stand ready to help

to improve calculus instruction;to select topics and teaching strategies for calculus;to provide feedback concerning new instructionalpackages;to disseminate information about successful pilotprojects in a useable form.The designers of calculus projects need to be very

careful in the way that success is defined. Assumingthat each pilot project has a prioritized list of expectedoutcomes, such a list for a particular project would beused in evaluating the success of that project. I stronglyrecommend that statements about the success rates ofstudents not be included in the definition. The successratzs of generally ill-prepared studer,..s who do not putin the necessary study time are not going to changedramatically in the short-term, so project success (orfailure) should not be tied to so insensitive an indicator.

DisseminationIf calculus is to be reformed, then information about

successful projects needs to be widely disseminated ina useable form. Not every school has to jump on thecalculus bandwagon at tile first opportunity. There is asmall list of particular schools that will strongly influ-ence any change that is proposed. If an overwhelming

ANDERSON: LARGE UNIVERSITIES 157

majority of the schools on the list accept the change,then the change will become general and widespread. Irecommend that leaders of this reform movement writedown their own minimal lists of schools and persistentlywork on those schools to be involved as much as possi-ble in each step of the development and implementationprocess.

My hope is that many two-year colleges will be ac-tively involved in the coming changes. Whether a two-year college is involved in the initial phases or not, it willchange its calculus content when the schools to whichmost of its calculus students transfer have changed,since calculus is clearly in the transfer track.

Second, as part of promoting general acceptance of achange in calculus and also as a consequence of a generalacceptance of that change, workshops need to be held toexplain the change in calculus and to explain new waysto think about calculus and new ways to teach calculus.A final written report for each pilot project is useful inspreading the word about the projects. Talks and paneldiscussions about the projects at regional and nationalmeetings are helpful in selling the ideas generated bythe projects. However, I do not feel that those types ofefforts are enough.

With the heavy teaching load that two-year collegefaculty carry, we tend not to be as well read as wewould like. For us, relying primarily on printed ma-terial to spread the word is not enough. I recommendthat each funded project contain, as part of the design

of the project, a workshop component to be used in dis-seminating information about the project and about thecontinuing results of the project. The workshop com-ponent needs to be funded in such a way that the work-shop can be given several times in various locations. Ifurther recommend that the work-hops be given at na-tional, regional, affiliate, and section meetings of AM-Al irC and MAA. Cooperation of these organizationsshould be readily available.

I look forward to the improvement of calculus in-struction over the coming years and to a return to en-joyment in my own teaching of calculus.

Reference

[1] Douglas, Ronald G. (ed.). Toward A Lean and LivelyCalculus. MAA Notes, No. 6. Mathematical Associationof America, Washington, D.C.

JOHN BRADBURN teaches mathematics at Elgin Com-munity College in Elgin, Illinois. He has served as a memberof the Illinois Board of Higher Education Committee on theStudy of Undergraduate Education, as Governor of the Illi-nois Section of the Mathematical Association of America,and as Chairman of the MAA Committee on Two-Year Col-leges. In 1982 Bradburn received the first national "Out-standing Faculty Member" award from the Association ofCommunity College Trustees.

Calculus in a Large University EnvironmentRichard D. Anderson

LOUISIANA STATE UNIVERSITY

There is no single large university environment. Thelarge university category includes the Harvards, theMITs, the Berkeleys, and the Michigans, all with highlyselect student bodies, as well as state universities of theSouth and Midwest with open admission policies. Theroles of beginning -alculus necessarily differ, from thoseof a course by-pasbed by many students in select univer-sities to a course for which only a small percent of en-tering freshmen are eligible as in universities with openadmission policies.

But in their lack of initiative for significant educa-tional reform, the universities of the country are re-markably similar. Except for the addition of some new(optional) topics and the deletion of a few (generally

harder) technical topics, the mainstream calculus booksof today look remarkably like those of fifty years ago.And yet, in that time frame, the technical and scientificworld has changed radically.

There has been one important development in calcu-lus reform in universities over the past 15 or 20 years,namely the growth of non-mainstream courses, e.g.,business or life science calculus. The 1975 and 1985 Un-dergraduate Surveys show a growth in the percentageof non-mainstream calculus from 22.5% of all universitycalculus enrollments in 1975 to 31% in 1985. The 1970Survey did not, list such "soft" or terminal calculus atallpresumably because it was not yet sufficiently wellrecognized for the committee then in charge to have


listed it separately.One can conjecture that, in universities, the growth

in enrollments in calculus courses (or sequences) notleading to upper division mathematics played a rote inreducing the pressure for reform in mainstream calculussince it was easier to believe that the engineering andphysical science students really did need the emphasesand drillwork of traditional calculus.

In their lack of initiative for significanteducational reform, the universities ofthe country are remarkably similar.

Large universities share certain characteristics whichaffect their propensities for change in the calculus cur-riculum.

They are big, with multiple sections taught by a widevariety of faculty.Calculus is the core of the mathematics service coursesequences for students of engineering, physical sci-ences, computer sciences, business, life sciences, andmathematics itself.The user faculties on campus are sensitive to anychanges which run counter to their ideas of what stu-dents should know. Since they don't have much oc-casion to consider details of service course curricula,they are inclined to be conservative about calculuswhat was good for them is good for their students.Because they have diverse individual criteria as towhat is important, any change will step on manytoes. They are more willing to add new thrusts,e.g., linear algebra, to their students' programs thanto make hard decisions about dropping old thrusts.They almost all find calculus useful as a "screen" inwinnowing their students.

Role of UniversitiesIt is important that the large university environment

be a vital part of any efforts to change the nature ofthe teaching of calculus. The universities of our countryare the focus of a very major part of basic research inscience, in engineering, and certainly in mathematics.

They should also be an important origin of educa-tional innovation. However, the reward system in uni-versities is such that faculty time and effort spent onresearch totally dominate time and effort spent on edu-cational reform. And th s educational reform has suf-fered. Very few faculty have either interest in educetional reform or time for such work.

There is now much eviden, e that important elementsof the research community az,: paying attention to basiceducational needs. We must all work to see to it thatthe initiatives now begun are actively carried throughat the university level as well as at the college level.

Science and EngineeringWith the help of the engineering community, ever

present in the university environment and highly cog-nizant in their own educational programs of the roles oftechnology, we in mathematics have an opportunity toally ourselves to forces of change. It is odd indeed that,by all accounts, upper division education in engineer-ing has changed radically in the past third of a century,whereas the basic science and mathematics courses forengineers in the lower division have changed very little.

From discussions with a number of people in vari-ous disciplines at various universities, it seems likely tome that the engineering community is readier to acceptchanges in calculus induced by the age of technologythan is the physical science community. While manyphysical science faculty do employ computers in theirresearch, the nature of research in the physical sciencesas well as in mathematics requires individuals to getaway by themselves in order to think.

Upper division education inengineering has chang-d radically inthe past third of a century, whereasthe basic ...mathematics courses forengineers in the lower division havechanged very little.

Much of that process in mathematics, at least, in-volves conjecturing, drawing pictures, and trial-and-error methods and generally these require paper andpencil activities. What we seek in calculus for the nextcentury is a balance between the old and the new: pa-per and pencil activities to assist in understanding andin problem solving but not in routine computations oralgorithmic processes better done by computers and cal-culators.

The coming reformation of calculus in large universi-ties i3 complicated by the fact that there are mar./ localfingers in the pie. Engineers, computer scientists, physi-cal scientists, life scientists, and business administrationpeopit -a., well as mathematicians all have their own spe-cial and differing needs for calculus level courses. Cal-culus is the dominant introductory mathematics course

1e3


for students in the first two years in these various dis-ciplines.

Technology has brought great changes in the waymathematics is used in the work place, away from pa-per and pencil procedures toward the use of calcula-tors and computers. Symbolic manipulation and com-puter graphics are not only changing the way engineersand others use mathematics in the work place, butthey clearly will force major changes in university levelcalculusstudents' first step on the access route to thework place.

The rote algorithmic paper and pencilprocedures which have been ...largelyunchanged in calculus over the past 50years, at least, are irrelevant in termsof even current work place use.

The rote algorithmic paper and pencil procedureswhich have been developed- over the past century-and-a-half and have been largely unchanged in calculus overthe past 50 years, at least, are irrelevant in terms ofeven current work place use. That does not say thatthe concepts and ideas of calculus are irrelevant, butonly that we must design our educational practices incalculus to conform to students' future needs, both inthe post-calculus learning environment and in the fu-tmre work place.

Calculus in UniversitiesThe 1985-86 Survey on Undergraduate Programs in

the Mathematical and Computer Sciences El] revealsmu-.1 background statistical data on calculus in univer-sities. Although the "university" category used there isfrom Department of Education lists not identical withAMS Group I, II, and III Institutions, percentage fig-ures for enrollments, faculty and teaching phenomenaare close to those applicable for any reasonable defini-tion of "large universities." The data is for calculustaught in mathematics departments in the fall term of1985 and include enrollments in the first, second, orthird terms in the engineering and physical science cal-culus, and in the business and life sciences calculus.

In universities 40% of all undergraduate mathemat-ics enrollments are in ralculus, with a 27% to 13% splitbetween engineering-physical science calculus to busi-ness and life-science calculus. 72% of ail mathematicsenrollments at the calculus I. l or above a, ; in calculusitself; 81% of all such enrollments are in calculus or its

natural successors, i.e., differential equations, advancedcalculus, and advanced mathematics for engineers.

Thus calculus is overwhelmingly the dominant math-ematics course taken by undergraduate students in uni-versities. Total enrollments in calculus as well as totalundergraduate mathematics enrollments in universitieswere essmtially unchanged from 1980 to 1985.

In university calculus courses, approximately 38% ofall students are taught in sections of under 40, 20% insections of size 40 to 80, 12% in lectures of more than80 without recitation sections, and 29% in lectures ofmore than 80 with recitation sections. (The remaining1% are taught in self-paced or other format.) Only 5%of calculus sections in universities have any requiredcomputer use.

The average age of the full time university math-ematics faculty is 44 with 65% over age 40 (up from45% in 1975). 90% of all full time mathematics fac-ulty in universities have doctorates, 63% of whom aretenured. Presumably almost all of the non-doctoratefull time faculty would teach only courses below thecalculus level.

In universities ... 72% of allmathematics enrollments at thecalculus level or above are in calculusitself.

No specific data is available from the Survey on thepercentage of calculus sections taught by teaching assis-tants, but anecdotal information suggests that whereasteaching assistants teach about one-fifth of all sepa-rate university sections in mathematics, the majorityof these sections are at. a level below calculus. In manyuniversities, only select advanced graduate students areassigned calculus-level courses. However, there are afew (selective) major state universities with very lit-tle course load below the calculus level; at such places,teaching assistants teach much of the introductory cal-culus sections.

What Do Students Learn?Student learning procedures consist primarily of

working textbook problemsusually several or manyof the same sortfollowing model procedures given inthe textbook or by a teacher. Thus students learn vari-ous paper-and-pencil algor: ,ims for producing answersto special types of problems. They customarily read thetext only to find procedures for working such problems.

170


Almost no student below the A-level (and few A stu-dents) can cite, much less accurately state, and even lessprove, any of the theorems related to elementary calcu-lus. Student dependence on memorized procedures toproduce answers follows a sirn:lar pattern of learningin pre-calculus mathematics. In calculus, however, itrequires much wider and more readily recalled back-ground information, primarily from algebra, trigonom-etry, and elementary functions.

Almost no students below the A-levelcan cite, much less accurately state,and even less prove, any of thetheorems related to elementarycalculus.

The intellectual achievement for most students inlearning calculus is, nevertheless, considerable. Theyhave had to learn about new concepts much morerapidly than in earlier courses, and they have had todevelop command of a wider and more diverse "bagof tricks." They have learned to solve problems from avariety of geometrical and physical applications. Unfor-tunately, many of the tricks in the calculus bag appearirrelevant in an age where the computer and the calcu-lator are rapidly replacing paper and pencil as the toolsof the trade.

Roles of CalculusIn considering any significant change, much less a

reformation of calculus, it is important for all to keep inmind the various rolel traditionally played by calculusin our universities. We should not, unwittingly, reformcalculus without taking into account the many usefulby-products of the study of calculus. Here are someroles to keep in mind.

Calculus embodies a unity and beauty as one ofthe great and useful intellectual achievements ofmankind. However, many current courses pay onlylip service to these ac,,ects of calculus. In reformingcalculus, we should overtly seek to acquaint all stu-dents with the unity, beauty, and power of calculus.Calculus represents modelling of a mathematical sys-tem with a richly diverse set of applications; physi-cal, geometric, biological, and managerial. Calculus,together with the real number system oa which it isbased is, along with Euclidean geometry, an ultimatemathematical model.Calculus as currently taugnt has been the course inwhich most engineering and physical science students

1 7 ;

really learn algebra and trigonometry: algebraic top-ics are used and reviewed until they come togetheras necessary background for a new and more pow-erful subject. In the age of symbolic manipulationand computer graphics, we must seek to identify andstrengthen those aspects of calculus as well as thoseaspects of algebra and trigonometry which will beimportant in the work place of the future.With its many graphical representations, calculushas been a rich source for the (further) developmentof students' geometric intuition. Along with geomet-ric aspects of linear algebra, introductory multivari-ate calculus offers vital exposure to three and higherdimensional geometry.

Calculus as currently taught has beenthe course in which most engineeringand physical science students reallylearn algebra and trigonometry.

Calculus has been the mathematical and intellectualscreen by means of which students in engineering, inthe sciences, and in mathematics are judged as readyfor more advanced work. It is "the universal prereq-uisite." Both the rapid assimilation of new conceptsand the control of a broad framework of backgroundinformation required in successful study of the calcu-lus are student experiences involving characteristicswhich are manifestly important for further study.

Forces Against ReformInertia. One should never underestimate the inertia

and resistance to change in a large system. The edu-cational system is naturally conservative, with all uni-versities structured along traditional departmental andcollege lines. Significant changes affecting introductorycourses like calculus which are prerequisite to almost allcourses in both mathematics and user disciplines musthave at least grudging approval of most or all of thedepartments and colleges concerned.

For example, even interchanging the order of intro-duction of topics has great implications for normallyconcurrent physics courses. Thus any major changesin emphasis in calculus must be carefully planned andcoordinated with user departments.

Tradition. Virtually all current mathematicians andusers of mathematics have grown up with a calculuscourse largely unchanged in both content and presenta-tion over at least the past half-century. There have been


cycles of relative emphasis on proof and of varying or-ders of presentation of topics but, by and large, studentlearning has been almost totally involved with repetitiveuse of paper-and-pencil procesz,es for producing answersto special types of problems. Based on earlier mathe-mati,:s experiences, students, mathematics faculty, anduser faculty expect standard skills and judge success orfailure by a student's ability or inability to work specialtypes of problems. In my experience, when questioningusers or mathematicians about the content of calculus,there is strong instinctive reaction to judge as importantthose topics the faculty were taught and were successfulin learning (or, for mathematics faculty, in teaching).

Qualifications and Attitudes. The bulk of studentstaking calculus in uni-'ersities are taking it as a means ofgaining access to other subjects. Their primary concernis to get through the course with the least diversion fromtheir other interests.

Virtually all current mathematiciansand users of mathematics have grownup with a calculus course largelyunchanged in both content andpresentation over at least the pasthalf-century.

Furthermore, most have achieved what they haveby working lots of problems by memorized paper-and-pencil algorithmic procedures following textbook orteacher examples. Their natural inclination is to wantmore of the same. They don't want to be shown morethan one way to do z. problem, they don't want to knowwhy but only how, and few of them have been exposedto problems about which they need to think hard orlong.

Many have not learned how to read the to :t ex-cept to follow the step-by-step illustrative examples.Many have difficulty even reading the word problems.A change to conceptual calculus with a downplaying offormal paper-a Id-pencil procedures will run into stu-dent resistance and must be phased in with companionchanges at the school level.

Textbooks. Texts are written and published to besold. They represent the publishers' views of what theteaching community wants. Thus current texts covera wide variety of topics (for faculty choice) and havepages of standard, similar problems, following illustra-tive examples.

Texts are frequently adopted via committee recom-mendation, with committees looking for "teachable"books, those similar to ones liked in the past and thoseconsistent with somewhat traditional values. New textswith radical changes only rarely get adopted, thus au-thors don't write them and publishers don't publishthem.

There is uncertainty among us and among many ofour colleagues about what topics are important in cal-culus for tomorrow and also uncertainty about the needfor gradual continuing change after an initial "reforma-tion." We do not yet understand how completely orrapidly symbolic manipulation and computer graphicswill effect either societal use of calculus or the teachingof calculus nor, at n even more fundamental level, dowe understand which asoects of calculus will turn out tobe most important for users in the evolving and foreverchanging age of technology.

Reference

(1] Albers, Donald J.; Anderson, Richard D.; Loftsgaarden,Don 0. (eds.) Undergraduate Programs in the Mathe-matical and Computer Sciences: The 1985-1986 Survey.MAA Notes No. 7. Washington, D.C.: MathematicalAssociation of America, 1987.

RICHARD D. ANDERSON is Professor of Mathemat-ics, Emeritus, at Louisiana State University. He is PastPresident of the Mathematical Association of America, andserved as Executive Director of the 1985-86 MAA-CBMSSurvey of undergraduate programs in the mathematical andcomputer sciences. Currently he chairs a subcommittee ofthe Committee on the Undergraduate Program in Math-ematics (CUPM) charged with making recommendationsconcerning the first two years of college mathematics.

A Calculus Curriculum for the NinetiesDavid Lovelock and Alan C. Newell

UNIVERSITY OF ARIZONA

There is little disagreement with the premise thatthe mathematical fluency of our educated population isnear an all time low. More serious than the lack of gen-eral literacy is the fact that America is not producing ageneration of students who will become the mathemat-ical scientists and innovators of the future. Our bestbrains are being siphoned off into business, medicine,and other careers before the mathematical science com-munity has even had an opportunity to persuade themthat understanding nature, social behavior, economics,not to mention the vast charges in technology from aquantitative point of view, is a worthwhile pursuit.

There are many reasons for this. First, it might beargued that, except for rare periods, America has neverencouraged its bright, young people into science. Immi-grants have always done the job for us. Second, the peerpressure of materir success suggests that tough, ruggedLee Iacocca is a Later role model than the nerd whoteachers mathematics on the television at six o'clock ona Monday morning. Young people are not exposed torole models of a sufficiently heroic type in the mathe-matical sciences.

America is not producing a generationof students who will become themathematical scientists and innovatorsof the future.

T' .ird, and perhaps the only area in which we mighttake action so as to have immediate impact, is the earlyexperience of students with mathematical thinking andideas. Presently, the preparation of our young peoplein English and mathematics, the two subjects whichshould surely dominate a high school education, is pa-thetic.

Changes in AttitudeTherefore the first suggestions which we offer address

less the details of the curriculum in calculus and moreLie attitudes that we must adopt:

There must be a concerted effort, working with localhigh schools and high school teachers, to a_e that stu-dents are better prepared in algebra, geometry and

trigonometry and, most importantly, that they areaware of the self-discipline required to learn mathe-matics. Real learning requires work.We must stress that mathematical knowledge is cu-mulative and that scientific knowledge has a verti-cal structure. All tests and examinations should re-flect this fundamental premise. Regular homeworkshould be a priority. Homework drill is essential.Tests should be cumulative.

The preparation of our young people inEnglish and mathematics ...is pathetic.

Let us now agree once and for all that not all studentshave equal abilities and introduce some stratification(honors seceonz, sections which coordinate with el-ementary physics or business courses, etc.) into theorganization of classes.Above all, we must put our best teachers, the mostcaring and inspiring, in the first year courses.A program that emphasizes mathematical thinkingand word problems will require more instructor timeper student. Consequently, it will be necessary torestructure the incentive and reward system for uni-versity professors. Research is vital, no one questionsthat. But there is presently far too much recognitiongiven in American universities for mediocre resea*chand far too little for encellence in teaching. This atti-tude has not only led to an imbalance in distributionof personal time and effort, but also has skewed theinternal distribution of faculty among departmentswithin a university.

Changer, in CurriculumWe continue with a similar list of fundamental prin-

ciples directed at the calculus curriculum itself:We must understand that the calculus sequence isthe basic set of courses in mathematical thinking towhich a student is exposed. Althougi: the sequencemust be responsive to the university color unity asa whole and must prepare students to handle themathematics used in the courses of other disciplines,

LOVELOCK & NEWELL: CALCULUS FOR THE NINETIES 163

the teaching of mathematics at this level should nothave a purely service connotation.It is important that students be exposed to logicalthought processes, problem solving and other areas ofmathematics such as probability and number theorywhich stress these activities. The actual materialcovered is not as important as the confidence inducedby gaining understanding through mathematics andsimple logical arguments.

Above all, we must put our bestteachers, the most caring and inspiring,in the first year courses.

We must use to the fullest ext int the products ofmodern technology. Homework can be monitoredand corrected and help can be given on computers.The prerequisites of a given course can be introducedon VCR tapes which can be used on home televi-sions, or on discs which can be used in most standardpersonal computers. Much of the material which astudent must do alone (lots of examples, homework,etc.), all the repetition which is so necessary in learn-ing mathematics, can be done better by the studentwith technological aids.Many students are computer literate (they can oper-ate the computer, play games, etc.), but very few canwrite a good program or use packages effectively. Allsumming operations (series, integrals, etc.) shouldbe done numerically. There is no harm in intro-ducing students to exact methods; it increases theirliteracy and fluency; but it should be stressed thatthere is no difference in principle between calculat-ing f0 sin2 ax clz and f01 V17-f--;7sin2 ax dz. Each isa number and the latter number is no less good thanthe former just because it cannot be exactly calcu-lated but must be approximated.

Mathematics education cannot beegalitarian.

The curriculum should stress word problems, the artof judicious approximation, and the importance ofconverting conservation laws and problems in op-timization into mathematical language. After all,practically all laws used in engineering and scienceare derived from looking at balances of mass, concen-tration, momentum, heat, energy. Moreover, many

of the challenges in modern business involve choosingoptimal configurations of the coordinates to achievecertain desirable ends.

Before writing down an outline of a curriculum, it isimportant to return to the idea that mathematics edu-cation cannot be egalitarian. We suggest therefore thatthe best students be separated into classes which coveran expanded curriculum in the traditional three courses.However, the extra depth and amount of material intro-duced should be recognized and so we recommend thateach of these three courses receive four hours credit.

In order to coordinate the curriculum between thebrighter and the less qualified student, we suggest thatthe vast majority take calculus as a sequence of eitherthree or four three-credit hour courses. The first optionwould cover less material but still emphasize mathe-matical thinking, problem solving, approximation tech-niques, and the importance of computers. The fourcourse option would allow the less qualified student tohave an equal preparation should he or she decide tocontinue in a course of studies where all the calculusmaterial is required.

In addition, it should be possible for a student ma-joring in engineering or the physical sciences (or anyother interested student for that matter) to choose acalculus track which coordinates calculus topics withrelevant material in the parallel science sequence (e.g.,motion problems, Newton's laws, calculation of chem-ical compositions). After all, most major universitiesrun over twenty sections of each calculus course eachsemester. Diversity and choice should have their placein the education system.

Real learning requires work.

We want to .tress again, however, that except forvital prerequisite material, the exact topics introducedare not crucial. We see the role of the national mathe-matics leadership in suggesting a curriculum as limitedto preparing a readable framework about which indi-vidual instructors and teachers can build and which anaverage student can understand. What is important isthat we structure the curriculum so as to nurture thestudents' abilities to develop clear thinking, do wordproblems and be willing to use modern technology withconfidence. Former students who have some work ex-perience are far more apt to say that it is their abilityto think and learn rather than their precise knowledgeof a given topic which is important to them in later life.

1.7 `:

164 ISSUES: MATHEMATICAL SCIENCES

The Honors CurriculumFor honors work we do not suggest any radical

changes from present practice. However, we do recom-mend inclusion of a wider range of topics in the calculussequence. In particular, the mathematical backgroundof almost all students studying the quantitative sciencesshould include

writing programs and using computer packages effec-tively,

basic ideas in probability and statistics,the solution of linear equations, matrices, linear in-equalities,

game theory and the choice of strategies,a litt::. number theory.The last member of this list is included, not because

it has immediate application but because many of theproblems and ideas can be understood without a greatdeal of background, results can be numerically checked,and many tools required in constructing logical proofs(e.g., induction) can be readily introduced.

What is important is that we structurethe curriculum so as to nurture thestudents' abilities to develop clearthinking, do word problems and bewilling to use modern technology withconfidence.

We now introduce what we call the Honors Cur-riculum consisting of three substantial courses, eachwith 4 hours credit. Specific comments will follow eachsemester's outline; general comments will be given atthe end.

1ST SEMESTER: Number systems, functions, .emits,continuity, differentiation, finite differences, indefiniteintegrals, special functions.

Comments: Little change from the present curriculumfor the first 1 and 1/3 semesters. Introduces a littlenumber theory. At least one-sixth of the teacher's timeshould be spent on the mathematization of word prob-lems. Emphasize pictures and curve sketching.

2ND SEMESTER: A mathematical laboratory emphasiz-ing computer programming and use of packages. Defi-nite integration by Riemann sums. Introduction of nu-merical methods. Techniques of exact integration. Im-proper integrals. Series, Taylor series, maximum and

minimum problems. Solution of systems of linear equa-tions. Matrices, determinants. Linear inequalities andprogramming.

Comments: Introduction to the idea and implementa-tion of approximations. Techniques of exact integrationare exhibited only once; the student using exercises anddrills, is responsible for mastering this material out ofclass. Numerical verification of all results should be en-couraged. A microcomputer laboratory is essential andthe student should expect to spend at least three home-work hours per week working on the computer. If timepermits, a little game theory and choice of strategiesmight be introduced here. A knowledge of solutions oflinear equations, matrices nd determinants is impor-tant for the differential equations course, which is oftentaken in the third semester.

3RD SEMESTER: An introduction to probability and el-ementary concepts in statistics (mean, variance, leastsquares fitting). Analytic geometry. Coordinate sys-tems and the differences between Euclidean and othermanifolds. Tangent limts, arc length. Vectors and thedescription of curves surfaces. Partial derivatives, Tay-lor series, maxima and minima in higher dimensions.Multiple integrals. Line and surface integrals. The no-tions of circulation and iiux. The theorems of Gauss,Green, and Stokes.

Comments: Except for the inclusion of probability, thisprogram is not a great deal different from the presentcurriculum. We suggest, however, that proofs be re-placed by a demand that the student understand theresults sufficiently well to carry out nontrivial compu-tations of circulation, heat flux, etc.

The Regular CurriculumThe regular curriculum, which would consist of four

three-credit hour courses, would omit special functions.aid number theory fror semester one, would includespecial functions in semester two but exclude solutionsof linear equations. Semester three would be new andinclude much of the discrete mathematics, probability,statistics, systems of linear equations, matrices, linearprogramming, game theory and strategies and a littlenumber theory. Semester four, which would be requiredof students following cer' ain of the engineering and sci-ence majors, would be the same as the honors sectionwithout probability and statistics.

Logical SequenceOne of the primary aims of a mathematical educa-

tion is to teach students to think logically and correctly

175


about the problems at hand. The problem need not bemathematical in origin or nature, but the solution willrequire logic. It is therefore important that this logicalattack be seen in action. For example, in the standardcalculus course we should not be teaching limits, thendifferentiability, then continuityas do so many texts(and hence courses). We should, instead, be stressingthe logic involved. For example:

Here is the definition of a limit, but who wants to"dwell in Hell"? Here are a few useful results involvinglimits which can be obtained from that definition. Hereare a few good theorems which also can be proved fromthat definition. Here is how we can combine these re-sults and these th orems to produce more results, with-out going back to the initial definition. However, if newfunctions appear (and they do) we will then nzed re-sults associated with them (presumably derived fromthe original definition of limit) to use these theorems.

OK, so some functions have limits and some don't.Let's concentrate on those that do. What other prop-erties might these functions have? Ah, continuity. Nowrepeat tne above paragraph with "limit" replaced by"continuity." Are there any important functions thatare not continuous?

OK, so some functions are continuous and some arenot. Let's concentrate on those that are. What otherproperties might these functions have" Ah, differen-tiability. Now repeat the earlier paragraph with "limit"replaced by "differentiability." Are there any importantfunctions that are not differentiable?

One of the primary aims of amathematical education is to teachstudents io think logically andcorrectly.

Those "theorems" that are missing from the knowntheorems should also be mentioned (no formula for lim-its of composite functions until we have continuity, noresult for the integral of products or quotients).

Checking AnswersAn important aim of mathematics is to solve prob-

lems we haven't seen before. However, it is equally im-portant to develop the expertise to decide whether theresults we get are right. We should be emphasizinghabits that promote this. For example:

While it is true that a curve can be sketched cor-rectly using only a few results, every available re-sult and theorem should be used to ensure that the

curve "hangs together." Asymptotes, first and sec-ond derivative texts, x and y intercepts, symmetry,regions where the function is positive and negative,while frequently giving overlapping information, alsolead to inconsistencies if an error has been made. Butmore importantly, they give the student confidencethat the result is correct without looking at the backof the book.Check constructed formulae by using special cases.For example, in calculating volumes via integrals interms of a cross-sectional area, the formula for thearea, A(x), must be constructed. Before integrating,check A(0) and A at any other value where the resultis known independently of the formula.Habitually check indefinite integrals by differentia-tion.Does the answer make sense? The age-old exampleof this (from absolute extrema) has someone row-ing from an island and then walking along a straightshoreline to a destination in minimum time. Faultymathematics suggests that thc, person should rowpast the destination and walk back.What does the answer tell us? The problem (againfrom absolute extrema) of finding the dimensions ofa cylindrical can, holding 1 liter, with minimum sur-face area should be interpreted in terms of the shapeof the resulting cylinder. (Who cares that the radiusis approximately 5.4 cms?) Much more important isthe conclusion that the cross section of the cylinderis a square, and even more important is the crestion"Why don't manufacturers make them that v.ay?"

Mathematical ExperimentationThat mathematics is an experimental subject needs

to be emphasized more. Contrary to classroom demon-strations, most real problems are not done correctly, oreven the right way, the first time. We should spendtime explaining our intuition as well as our knowledge.This means doing things that don't work! For ex-ample, some time should be spent trying to integratesec(x), and then when it is finally done (by writingsec(x) = cos(x)/(1 sin2 x)) we should point out howmost texts do it (by a trick substitution) presumablybased on kr...nving the answer. We shouldn't be en-couraging the idea that mathematics is just a bunch oftricks.

Related to this is the fact that mathematicians arenot above thinking like physicists when necessary (allfunctions are infinitely differentiable anI have conver-gent Taylor series). One of the main reasons for doingthis is that, as a result, we might be able to guess the

1 76

166 ISSUES: MATHEMATICAL SCIENCE

answer. Knowing a tentative answer frequently helps inproving it. (To a student, knowing what the answer ismeans looking in the back of the book!)

As an example, consider the follow;ng problem: Findall functions that satisfy the conditions of the meanvalue theorem, for which c is always the mid-point, i.e.,find all functions f(x) whose derivative at (a -I- b)/2 isalways MO f(a))/(ba) for all a, b. If we could guessthe answer, we might then define g(x) = f(x)guess,and show that g(x) = 0. So let's assume that the answerhas a Taylor series expannion and substitute it in thecondition on f to see what we get. Lo and behold, aquadratic! So that would now be our guess, which thenwon't depend on the Taylor expansion assumption.

We need to get the student away from the notionthat mathematics is just a collection of formulae thatneed to be memorized. This can be done in at leasttwo ways, by giving interpretations and applications ofthe formulas, and by stressing that memory ch.es not agood mathematician make.

For example: The interpretation of the mean valuetheorem in terms of average velocity, or the formula for1 -1.4 + 9 + bein, used by spies to count how manycannonballs were in a pile, by knowing how many layersthe (square) pyramid has. We should stress the useof tables of integrals, and tables of infinite series, andpoint out, that faced with the problem of Integratingpowers of sin(z) and cos(x) we look it up! We shouldbe stressing, in techniques of integration, not how to dothe integrals which appear in a table of integrals, buthow to get from the integral at hand to the integral inthe table.

We nee,, to stress that "little" things make a bigdifference in mathematics, and attention must be paidto them. The results involving the existence of abso-lute extrema on closed intervals, as opposed to openintervals, is one example. Just as dramatic, and animportant topic which is seldom covered, is sketchingone-parameter families of curves, such as x2 + C/X2,and discussing the three cases for c (positive, negative,zero). This also stresses that a mathematical problemis frequently solved by looking at different cases.

A good place to demonstrate the different case con-cept is in the previously mentioned extreme value prob-lem involving rowing and walking. That is actually aninfinite domain problem which can be split into threecases, land between the closest point to the island andthe destination, land past the destination, and land be-fore the closest point. Ali three cases should be ana-lyzed, not just one as is often the case.

Common SenseWe must emphasize how nature, symmetry, and com-

mon sense can frequently help in understanding a prob-lem or its solution. For example, knowing that the cir-cle is the shape which has a maximum area for a fixedperimeter helps make sense of the solution to the prob-lem of cutting a piece of string into the perimeters ofa circle and a square, the sum of whose areas is to bea maximum. After solving the problem involving theshortest distance from a point to a curve, we should ex-plain that common sense says that the answer could beobtained by considering a small circle centered at thepoint. Then imagine increasing its ,adius until it justtouches the curve. Intuition suggests that -r.t this point,the tangent to the curve should be at right angles to theradius. Is it? Snell's laws should also be mentioned inthis context (why does nature behave the way it does?)to build up some intuition in this area.

Infinite SeriesThe standard tree. lent of infinite series leaves much

to be desired. If the course follows the usual pattern,the good student comes away with the ability to decidewhethe. a series converges or diverges, with the ideathat divergent series are diseased and all convergent se-ries are healthy, but no feeling for what the illness is.They leave with no real idea why convergence series arecategorized as absolutely or conditionally convergent,and no understanding why we care about convergenceor divergence. The main justification for emphasizingthe various tests for convergence seems to be that theymake good exam questions.

The good student comes away ... withthe idea that divergent series arediseased and all convergent series arehealthy, but no feeling for whet theillness is.

We must de-emphasize these tests. Who cares? Wemust stress why a knowledge of convergence or diver-gence is important. We must stress why the distinc-tion between conditional convergence and absolute con-vergence is important. To introduce conditional con-vergence without mentioning Riemann's rearrangementtheorem seems pointless. We should point out that di-ve.gent series are not useless, by doing the "brick prob-lem" (given bricks of the same size, is it possible to placeone brick on top of another, in a vertical plane, in such

.1 7:7


a way that no part of the top brick is above the start-ing brick?). iut overall we must give examplls on theuse of Taylor series. The previously mentioned exam lînvolving the mean value theorem is one sw.thtion. Evaluating definite integrals is anot ny" y = 0 is a third.

Changing Attitudes

It should be repeatedly explained to studeztts thatthe answer alone is not good enough. Whal, is re-quired is the correct logic leading to the solution.There are plenty of examples where the answer isright, but the technique is flawed. Don't check theanswer by looking at the back of the book: find someother way of convincing yourself that you are rightor wrong.

Some material should be covered in class oven thoughit does not lend itself to testing by examination. Forexample, the proof that sin(x)/x goes to l as x goesto 0 is the entire justiT.zation for using ndians incalculus, bht it is not usually important smough toexamine.

Some material should not be covered in class eventhough it does lend itself to testing by examinatiorCovering every single technique of integration gen-erates nice exam questions and kills time, but is itjustified?Like it or not, computers are here (and we believe tostay), so let's learn how to use them effectively.

New MaterialThere are a number of topics, absent from a standard

calculus sequence, to which students should be exposed.

Mathematical induction. Somehow students neverseem to get this in any course. Apart from the useful-ness of the technique, it also puts the ability to create aproof within the grasp and expertise of most students.It also stresses the experimental aspect of mathemat-ics, assuming the actual conjecture is not given, justthe problem (i.e., "what is 1+ 4 + 9 ... ," as opposed to"prove that 1 + 4 + 9 ... = ...").

Approximations. That some integrals, such as erf(x),have no closed form representation should be discussedwithin the context of the approximation of integrals.Various arc length integrals should also be done numer-ically. With the easy access to computers, standardproblems that lead to uncontrived, non-standard alge-bra should be done as a matter of routine.

For example, finding the shortest distance from (0,1)to sin(x) (as opposed to the textbook classic, x2), leadsto solving the equation xcos(x)-1-sin(x) cos(x) = 0. Inhe past we have avoided such problems like the plague,s though they didn't exist. We should no longer shrinkfrom such tasks.

Problems with no solution. Most students are un-der the impression that all problems have solutiorl.("I have a problem, what are you going to do zwoutit?") Non-closed-form integrals come as a shock. Pos-ing problems which have no solutions is important. Forexample, find the area of a four-sided figure given thefour side-lengths. Trying to decide what informationis relevant in solving a problem is also a skill that re-quires fostering. The standard is to eve e..iactly theright amo,Int of information to solve the problem, whichis fine when learning a technique.

However, we should also give problems where thereis overlapping relev-mt information (in addition to giv-ing the rower/walker's velocity, mention that he canwalk so far in so many seconds), too little relevant ..n-formation (no means of calculating the rower/walker'svelocity from the data supplied), or irrelevant informa-tbn (the ior.z.r/walkez has red hair, has recently signeda recall petition, has a girl friend at the destination,etc.).

Problems using many techniques. Calculus has be-come very compartmentalized. Problems using diversetechniques should be tackled. For example, after sur-face area of solids of revolution, find the shape of theco;.e of fixed volume and minimum surface area. Sincemost students (and faculty!) don't know the formulafor the surface area of a cone, this problem is usuallyavoided earlier when extrema are covered. Now, withthe benefit of integration, this formula can i,e derived,and the extrema problem re-attempted.

It should be repeatedly explained tostudents that the answer alone is notgood enough. What is required is thecorrect logic leading to the solution.

After arc lengths has been completed, one can do thefollowing problem (based on Greenspan and Benney'sCalculus). A plane is flying horizontally 5 miles fromthe end of the runway, and is 1 mile'high. It plans toland horizontally following a cubic equa.ion. (Beforeand after, it follows a horizontal stiaight line.) Whatis the maximum slope of the curve it follows? How far

1 7


does it fly? If its horizontal speed is 150 mph, what isits maximum vertical acceleration?

Finite difference calculus. Many students never hearof finite difference calculus, functional equations cr re-currence relations, let alone see them. They bring to-gether a numt ;r of different ideas anu techniques. Somestudents believe that f(x + h) = 1 (z) + 1 (h). Why notsolve it? Show that In(x) is essentially a consequenceof f(xy) = f(x) + f(y).

Euler's formula. By introducing cis(z) in the contextof a little complex arithmetic, at the end of the firstsemester (after exponentials) many of the techniques ofintegration can be bypassed, and many of the trigono-metric identities can safely be forgotten!

Non-mathematical techniques. We should alwayspoint out other ways of doing things. For example:

a. A gardener has a kidnev.shaped lawn to sod.He needs to know the area to order thc correctamount of sod. (Solution: Cut out kidney-shapedpiece of card to scale. Cut out what 1 square yardis. Weigh both on a chemical balance!)

b. Someone wants the volume of some strange shapedcontainer. (Solution: Fill it with liquid, measureamount of liquid used.)

c. Derive Snell's law of reflection. (Solution: Re-flect source about mirror. Join eye to source viaa straight line, reflect line back.)

Use of ComputersThis is not a negotiable itemit is essential. Com-

puter labs must be accessible to mathematics students

and faculty, at all levels. These labs should have pack-ages emphasizing demonstrations, drill and practice, ex-perimentation, numerics, and graphics, in addition toprogram writing. Software such as MACSYMA, Eu-reka, and Gnuplot should be readily available.

With the increase in the number of such labs aroundthe country, it is inevitable, and highly desirable, thatspecialized educational computer packages will continueto be developed by faculty, released to mathematics stu-dents, and placed in the public domain. We recommendthat minimum cost distribution of such software be en-couraged. Furthermore, a central library of such soft-ware should be established and vigorously maintained.Access to it should be via modern.

D. _- .) LOVELOCK is Professor of Mathematics rt theUniversity of Arizona. A relativist by training and repu-tation but t. computer addict by compulsion, Lovelock hastaught at Bristol University, England and the University ofWaterloo, Canada. In 1986 Lovelock received two awardsfrom the University of Arizona for distinguished teachingand high scholarly standards.

ALAN NEWELL is presently Head of the MathematicsDepartment at the University of Arizona and Director ofthe Arizona Center for Mathematkal Sciences, a Center ofExcellence supported by the Air Force Office of ScientificResearch under the University Research Initiative Program.He also serves as Chairman of the Advisory Committee onthe Mathematical Sciences for the National Science Foun-dation.

Computers and Calculus: The Second Stage

R. Creighton Buck

UNIVERSITY OF WISCONSIN

Computers luxe finally reached the mathematicsclassrooms. Recent joint AMS-MAA summer meetingshave highlighted talks and workshops dealing with var-ious aspects of educational software. Leading calculustexts offer floppy disks with programs intended to addvisual or numerical reinforcement to topics treated inthe textbook. The computer magazine Byte featuresfour universities identified as having wade significant ef-forts to compt.cerize their instruction in many subjects,

including mathematics. It is clear that there is stronginterest among college and university mathematicians,yet this interest is restrained by caution, inadequatetechnical experience, funding problems, and sometimesa degree of skepticism.

For many students, the first course in calculus hasbeen a very difficult transition from the algebra andgeometry courses they have taken previously, both incontent and style. The routine algebra courses usu-

17 D

BUCK: COMPUTERS AND CALCULUS 169

ally focus on formal manipulation of symbols accordingto specified rules, accompanied by a collection of algo-rithms which solve relatcd classes of problems.

The geometry course may begin with an extensivenomenclature for plane and solid figures and their com-ponent parts, move to a collection of algorithms for cer-tain geometric constructions, and then to a relativelyshort-list of stattn_.nts that describe various relation-ships among components parts making up geometricfigures. (Sample: "If two sides of a triangle are equalthen ....") Some of these may be given formal proofs,based on a list of accepted axicms. Additional topicsoften covered are right triangle trigonometry and a briefintroduction to coordinate geometry with emphasis onlines and circles.

While some of the assigned problems in either coursewill be interesting and challenging, most are likely tc fitinto templates given in the text. Only a few will be seenby students as having direct connectio with realisticapplications.

Complexity of CalculusIn contrast to these courses, calculus deals with a

vastly more complex collection of skills and concepts:1. Scalar and vector functions of one or more vari-

ables; differentiation, integration and their prop-erties; definition and properties of the standardtranscendental functions.

2. Mathematical models for physical concepts suchas velocity, acceleration, center of mass, momentsof inertia, work, pressure, gradient, gravitationand planetary motion, ... (and this in spite of thefact that only one o'Af, of six high school graduateshas taken a course in physics!).

3. A mathematical and computational approach togeometric concepts such as area, volume, lengthof a curve, area of a surface, curvature.

4. Infinite series, used both as a tool in numericalcomputation and as a way to define and work withspecific functions, or to construct new functions.

5. Polar and spherical coordinates, advanced two andthrce dimensional analytic geometry, conics andquadric surfaces, vector analysis and its applica-tions.

6. Assignments that require the student to build amathematical model for a concrete realistic sit-uation involving time-dependent components orother variables, and then use the model to answerspecific questions about its behavior or properties;this may involve solving a differential equation oroptimizing a related quantity, or perhaps carry ng

out other mathematical procedures suggested bythe model itself.

I' is important to keep in mind that, in addition totraditional mathematic knowledge, a growing numberof students now arrive at college with various levels ofexperience with computers programming, some ofwhich may have involved isolated mathematical con-cepts such as factoring, sorting, and various graphingtechniques. This adds another nonhomogeneous bound-ary condition for the college instructor who is trying todesign a suitable modern calculus course for the incom-ing students.

Current Uses of ComputersOf the calculus software I have examined, many pack-

ages are tied implicitly to familiar texts even if not as-sociated with its author or publisher. In use, manywill be run in the classroom with a suitable projectionapparatus and screen as a demonstration that replacesor supplements the usual chalk and blackboard treat-ment of a concept or technique. The use of color andpartial animation may give a much more professionaltouch that the teacher could not have provided.

Some other programs are intended to be used by stu-dents in a tutorial mode as a review or drill to test aspecific technique (e.g., formal differentiation), usuallywith immediate feedback on success or failure. Thesemay also replace a portion of routine homework if thereis an adequate bookkeeping system for student IDs andperformance records.

Other programs will permit a student to enter a func-tion and thea ask for its graph or its zeros, or even adisplay of its successive derivative!: given as explicit for-mulas. Programs are also available that permit a stu-dent to ask for the formal solution to any second orderconstant coefficient homogeneous ordinary differentialequation, or the numerical solution of any first orderinitial value problem. Similar programs now exist in alimited form an hand held calculators, and more sophis-ticated calculators are lust over the hocizon.

Basic QuestionsThis situation obuieusly poses several crucial ques-

tions to those responsible for planning the content andadministration of undergraduate mathematics courses.When should a student be allowed or encouraged toturn to the omputer for assistance or insight? Whenshould the computer be used in the classroom?

Here is one obvious ground rule: Don t do it ',h acomputer if you can do it better with gestures, pencil

1 SO


and paper, chalk and blackboard, or any other tradi-tional way. However, the word "better" must be inter-preted sorrectly. We know that an uninterested studentwill seldom learn; thus, one must reach a balance be-tween the role of compu 3 as interactive informationsources, and as theatrical attractions.

Don't do it with a computer if you cando it better with gestures, pencil andpaper, chalk and blackboa-d, or anyother traditional way.

However, a third question seems to be more central,and much harder to answer: How does the existence andwide availability of computers modify the objectives ofthe calculus course?

A satisfactory answer to these questions requires thecollective work of individual mathematicians, with sup-port and recognition from their departments. Thesemathematicians must be willing to spend time in athoughtful analysis of the mathematical content of cal-culus and then carry out experiments to test whetheraccess to computers results in more effective instructionand better educated students.

Asking the Right QuestionsThe starting point in this project may be to pose

the right questions. Perhaps sev.:ral samples will clarifythis.

Some students who have difficulty with 'Lalculus seemto be deficient in spatial intuition. Is it possible tohelp th 3e students with special graphics programs?Or is this a case where the use of tactile wire orplaster models is better?At present "curve tracing" is a standard topic in cal-culus, ostensibly for the purpose of producing a rea-sonably accurate graph of a specific function or equa-tion, but also to reinforce the techniques of differen-tiation. Now that a computer (or even a I and-heldcalculator) will display the graph immediately, is thisstill important? What do we really mean when werJay "skill in curve tracing is an essential componentof calculus?"

(f.:'ne possible answer: We want students, based ontheir mathematical knowledge and computer exper-imentation, to be able to look at an equation of theform y = f(x), where the formula fix f contains oneor more parameters, be able to say: "The graph ofthis sort o: function has one of the following shapes,depending on the values of the parameters a, b, ....")

When a student has access to programs that do for-mal differentiation, should we continue to stress rou-tine differentiation exercises? (After all, we now al-low students to use calculators to assist with theirarithmetic on homework and tests!) C' early, a stu-dent ought to know how and when to use the formu-las for differentiating the sun., product, or reciprocalof functions, and understand the use of the "chainr .le" for composite functions, since these supply es-..ential skills and insight in 'nany other mathematicalareas. But at what point in this learning process dowe let the student replace paper and pencil manip-ulation by a canned production-style differentiationmachine?It is important that the student be able to describethe geometrical meaning of a statement such as:

!WPM > 0 for all x between -4 and 4.

Is ability to answer this helped if the students havesf_en appropriate classroom demonstrations and car-ried out experimentation i their own?Integration presents a different problem, since wemust deal both with the definite integral, and withanti-differentiation. Accurate numerical integrationprograms obviously should be available. (Studentsalready have access to these on some hand calcula-tors.) However, substitution (change of variable) ineither indefinite or definite integrals remains a use-ful and very important technique in many areas ofanalysis, and so must be retained. When should thisapproach to the evaluation of a definite integral berequired, and when is a numerical integration accept-able?Simple max; min problems provide some of the moreinteresting elementary applications of calculus. How-ever, programs can quickly do a search to locate theabsolute maximum and minimum of a function ofe-..e variable on an interval. Shall such programs bepermitted? Can good problems be selected whichrequire students to locate local extrema?Problems of this type in more than one variable re-quire finding critical points, this in turn often leads tothe solution of complicated non-linear systems. Dowe provide students with the sophisticated black boxNewton-type programs ;that are appropriate here? Ifso, can we make an honest attempt to explain theirnature?What about advanced analytic geometry? Certainly,lines and planes in space must be covered in order towork with normals, tangent planes, and properties"i curves in space. But what about the usual treat-ment of general conics and quadric surfaces? Is this

181

BUCK: COMPUTERS AND CALCULUS 171

something that is now best replaced by a well chs-sen picture gallery or labeled samples, while all therest is left to be covered in a linear algebra course asillustrations of quadratic forms and eigenvalue com-putation?Is there no value for today's calculus student in work-ing through the proofs of the optical properties of theparabola and ellipse, or any of the other classical ge-ometric theorems?In the study of infinite series, which computer pro-grams should be demonstrated in class and whichmade available t) students? For example, shouldthey have access to a program that takes an explicitfunction as input, and supplies a stipulated numberof coefficients of its Mac Laurin series?Calculus courses usually include some introductionto differential equation. In the past, this has oftenbeen a compendium of special techniques, tied to spe-cific classes of equations. Should all this be replacedby a discussion of initial valued problems, both or-dinary and systems, followed by ex;,erimen.s with ablack-box differential equations solver? Would thisrob the student of useful basic mathematical experi-ences and skills? Would such a student be less ableto deal with certain applied problems that do notimmediately fit the standard patterns?

A New SyllabusI hope that it is clear that there are more questions

to be asked abcrt. the :actors that should be examined,ag one moves toward computerization of the calculur se-quence. The end product might be E. radically differenttopic secp.-nce for the calculus course.

At what point ...do we let the studentreplace paper and pencil manipulationby a canned production-styledifferentiation mchine?

Here is an outline for the first four days in one suchcourse. It's purpose is provocative rathei than descrip-tive. (Since it was devised during a conoc ,rip, it cer-tainly .has not been tested in class, r, or researched ex-austively!)

The course starts by discussing familiar functions:polynom'lls and rational functions. These are identifiedas specific algorithms, accepting a numerical input anddelivering a numerical output. Several different formatsor conventions are suggested for describing a function.For example, one might write PO = 602+ 0 2 as well

as P(var) = 6* (var^2) + var 2. The natural domainof a function is the set of admissible inputs. Sequencesare presented as functions on the integers; here it isappropriate to include sequences defined recursively.

Each student receives a disk containing the programsPLOT, TABLE, and ZEROS ,xnd an assortment of oth-ers. Some are demonstrated in class, and the simpleprocedures for use and function input are explained.The assigned problems require students to use the com-puter programs to answer questions about specific func-tions, and then to carry it experiments on their own.

Among the demonstration functions is a polynomialof degree 5 with one parameter. As different valuesfor the parameter are chosen, the effects are seen in thegraph and in the nature and location of the roots; as thenumber of visible zeros change, the existence of complexroots is mentioned, and related to the graph.

Further exploration of functions; exp,rimentationwith combinations of functions and discussion of thearchitecture of TABLE, FLOT, and ZEROS. Introduc-tion of the functions sin and cos, defined by their series.Brief discussion of how series are "summed" and thesources of error. Intuitive explanation of the conceptof convergence of seri,:s. Discussion of time and posi-tion functions that model motion on a line ("dog on aroad").

The assigned problems deal with the new types offunctions; use of PLOT and ZEROS to solve equationsse -h as 3 + 2t t2 sin(4t + 1).

The ZOOM command is used to examine a polyno-mial plot in neighborhoods of a point on the graph.Conci.usion: "In a microscope, curves look like straightlines." Counterexamples: (t2 .1) sin(1/(t2 .1)). Con-clusion: "Some functions are smooth everywhere andsome are not."

Recall "slope of a line;" definition of: "local slopeof a curve." Conclusion: "Smooth functions have localslopes at each point on their graph." Return to func-tions that model motion, and interpret slope as velocity.

Demonstrate program SLOPE, a program that givesthe approximate local slopes of smooth functions. Dis-cussion of nature of the algorithm, as compared with"the exact slope;" formulation of several definitions of

DERIVATIVE introduced as an algorithm that takesa smooth function f as an input and deliver:; a function

= D(f). Sample calculations for linear an i quadratic'"-actions; graphs of f and f' are compared.

Looking to the FutureI am sire this is enough of a sample. My choice of

topics was not quite off-the-cuff. The six objectives I

1 82

172

listed earlier for the calculls course are some of thosethat have been traditionalij cited by textbook authors,publishers, and curriculum &signers. However, as wemove toward the next century, I believe that other ob-jectives should be mentioned. In the last 40 years, therate of change in all sciences has accelerated, and thelevel of mathematical sophistication used in these sci-ences has increc.y..1 dramatically.

In the last 40 years, the rate of eNangein all sciences has accelerated, and thelevel of mathematical sophisticationused in these sciences has increaseddramatically.

The calculus sequence seems to e.Ter an ideal oppor-tunity for us to give students their first exposure tocertain simple tools and unifying viewpoints of mod-ern analysis that will make their future path smoother,whatever their ultimate interests. I would hope thatcomputers can be used intelligently in calculus instruc-tion in such a way that room could be found for suchtopics.

My first candidates are function spaces and linearoperators. As students' gain experience with specificfunctions, and are more at home with the use of for-mulas using symbols that represent arbitrary functionsin some specific class, they will find that the coi:cept"function" has begun to acquire the same concretenessheld by "number."

The moment when a student finds it natural to thinkof an arbitrary cortinuous function as a point-like ob-ject having a specific location with respect to other sim-ilar point-like objects represents a major insight! (Is

ISSUES: MATHEMATICA!, SCIENCES

it perhaps the mathematical equivalent of a "rite ofpassage," with attendant maturity implicat;ons?) Suchstudents then find it easy to move on to the idea of ageometric linear space whose "points" are functions.

As students gain experience ... usingsymbols that represent arbitraryfunctions ... they will find that theconcept "function" has begun toacquire the same concreteness held by"number."

Linear operators are even easier to bring in. In myoutline, the -.alculus instructor has already introducedthe concept of a linear operator by showing that "dif-ferentiation" is merely another type of function thataccepts a suitable numerical function as its input andthen delivers a numerical function as its output.

I own ;...n elementary calculus text from 1831 thatuses fluxions and fluents. I am willing to predict thatcar current elementary texts may look equally strangeand out-of-date to an undergraduate in 2020.

R. CREIGHTON BUCK is Hilldale Professor of Mathe-matics at the University of Wisconsin at Madiam. He hasserved as Vice President of both the Mathematical Associa-tion of America and of the American Mathematical Society.h.. is a former Chairman of the Committee on the Under-graduate Program in Mathematics (CUPM), and is a formerDirector of the Mathematics Research Center t the Uni-versity of Wisconsin. Buck has served on many governmentadvisor.? committees on science and education.

Present Problems and Future ProspectsGail S. Young

NATIONAL SCIENCE FOUNDATION

In 1935, the year I took calculus, the content andspirit of the course were es-entially the same as now.But there were major differences in the environment.

Then a college education was still primarily for anelitea social elite, not an intellectual elite. The stu-dents knew they were exp. -ted to graduate, that there

was a stigma for failure.

Some mathematics was a usual requirement for grad-uation, as "mind-training," but calculus -- -a sophomorecoursewas taken almost exclusively by students inthe physical sciences, engineering, and mathematics, asmall part of the student body, a homogeneous group.

I83

YOUNG: PROBLEMS AND PROSPECTS 173

Almost all were concurrently taking physics. One of ourproblems is that this homogeneity as disappeared.

Since it was a sophomore course, the preparation ofthe calculus student was controlled by the department.When polar coordinates came up, the teacher knew pre-cisely what the students had been taught before. Thatpreparation was usually a year course. covering collegealgebra, trigonometry, and plane and solid analytic ge-ometry. With the placing of calculus in the freshmanyear, starting in the '60s, control over the preparationpassed to the secondary schools.

In the '60s that was not bad. Whatever one thinksof its content, the New Math had changed the entire at-mosphere of secondary school mathematics. In fact, inthe first CBMS Survey, 1965, 75% of reporting depart-ments said it was the New Math that made the changeto freshman calculus possible.

That enthusiasm, that p ash for change, has longended, stopped by the crushing problems of our schools.We are left with-the heterogeneous preparations, but instudents who no longer hake the same enthusiasm andconfidence.

Calculus in ContextSince before Sputnik few students took calculus, for

staffing purposes it could be considered to be essentiallyan upper-division course. In the first few years of teach-ing, a Ph.D. rarely taught a course above calculus, andoften did not even teach calculus. Only three times inmy first five years did I teach a course above calculus,though I got pretty good at "Mathematics for HomeEconomics."

Now 31% of our undergraduate mathematics stu-dent take calculus, and hardly anyone regards it asa privilege to teach it, nor as a compliment to one'sability.

Calculus is our most important course,and the future of our subject...depends on improving it.

These changr9 are all for the worse and it is hare tosee what can be dor.e about some of them. Calculusis our most important course, and the future of oursubject as a separ:te discipline depends on improvingit.

otudent,) no longer enter calculus with enthusiasm.One reason, I think, is that many students, particularlythe better ones, have had experience with the computer,

and expect that their college and university mathemat-ics courses will use computers heavily. However, the1985 CBMS Survey showed that only 7% of the coursesused computers at all. From anecdotal evidence, muchof that usage seems to be as a large calculator, for suchtopics as Simpson's Rule. I do not believe the main rea-son for bringing the computer into calculus is to makethe students nappier, but that would be a desirable re-sult.

I am one of the people who believe that the computerwill revolutionize our subject as greatly as did Arabicnumerals, the invention of algebra, and the invention ofcalculus itself. All these were democratizing discoveries;problems solvable only as re..earch and by au elite sud-denly became routine. The computer will do the samefor our mathematics, and calculus is the place to begin.

Our calculus course (as well as differential equations)comes from the British curriculum of the last part of thelast century with the British emphasis on hard problemsand little theory. It is a course not in the spirit ofcontemporary mathematics, and needs change for thatreason alone. What is left of the British tradition now isan emphasi on working many problems, finding, say, 25centroids, without really understanding what a cent' aidis. That in itself is reason to change.

Using ComputersTo my mind, the most important change will be

the introduction of symbol manipulation (SM). Perhapsthis should come before calculus. The prescnt calculusis really a collection of algorithms with a little theoryand a few applications. It is not easy for a student tosee that. After learning the common substitutions forchanging variables in indefinite integrals, I thought thatgiven some function to integrate, my task was to in-vent some substitution t.c make it integrable. That thewhole topic could be covered by one algorithm neverentered my head. I had no concept of an algorithm.But practically every method in a first calculus courseis an algorithm, never clearly stated, nevci explicit.

Once a topic is -educed to an algorithm, we are inthe realm of the computer, and the probl0:n becomesone of optimal programming. Once progn -inked, thereis no need to repeat it. Except for its poJaible effecton moral character, I see no point in mastering handcomputation of square roots, with calculators availableat $5.95 that will do it with one key punch. Nor do Isee any point in finding the partial fraction expansionof a rational function if a computer can do it.

The intellectual merits of the above discussion, how-ever, are not relevant. The fact is that we can now do all


the manipulation in calculus on the computer cheaplyenough for classrooni use. I mention MAPLE and mu-MATH as examples. One can also consider the HP-28C, a hand-held calculator that can do algebra, andthat is clearly a precursor of more powerful SM calc ila-tors. These developments make the introduction to SMinevitable, if not by us, by our customer departments.

The computer will revolutionize oursubject as greatly as did Arabicnumerals, the invention of algebra, andthe invention of calculus itself.

Are there problems in introducing SM? Yes, ofcourse. Here is one I can't handle. In research papersusing SM, one sees a hand calculation, then Sk, thenmore hand calculation, then more SM, . How dopeople learn the necessary hand techniques? Is that areal problem? I've seen indications that students learnhand techniques better with SM.

At a different level, how do you get the necessarytraining for the faculty? What equipment do,you need,and where does the money come from? Do you havethe necessary additional space? It will probably not bepossible to use as many graduate students and part-time faculty as one can in the ?resent course. How doyou handle that? Where are texts? Etc.

Besides these local problems, there are ones that :rillrequire national effort, to carry out necessary experi-mentation. For example, how much hand computationwill still be needed? Such questions seem to me to re-quire research at a number of schools.

Numbers and Graph.;But SM is not the only use of the computer in cal-

culus. One could say that the objective of most real-world problems in calculus is to give a foreman somedirections. For that, one needs numbers. In most ofmy career, we have not been able to give calculus stu-dents numerical problems of the sort they might actu-ally meet, because of the time required. One gave a fewsimple problems with .,he Trapezoidal Rule or Simpson'sRule, or approximated the sum of a Taylor's Series atsome point, and Oat was it.

We can do better.

We can do better. In addition, there are otherchoices. I suspect, for example, that to evaluate a def-inite integral with the computer, SM is not the way to

go. One should instead immediately turn to numericalintegration, and the question of indefinite integrationbecomes wildly irrelevant for that purpose.

Computer graphics must have a place. For the °tilertwo uses, I know that it is rather easy to learn the nec-essary techniques; I don't know about graphics. Butthe ignorance of geometrical techniques by our studentscould be largely remedied by graphics. For example, onecould sketch a volume determined by several equations.Is it too much to ask for all three techniques?

In my opinion computer methods will let us cut inhalf the time required to teach ottr present course, thatis, the current set of algorithms done by the computerand not by the student, and the same amount of theoryand applications. If so, what do we do with the extratime? One choice is to use it for topics that do not nowusually get into the first two years. This possibility wasdiscussed extensively in the report of a conference atWilliams College [1]; Kemeny's talk is particularly rel-evant. I believe that for many types of students (socialscience majors, for exar wle), this would be best. Butfor students going into fields where analysis is muchused, why not use the time to teach a better under-standing of calculus and how to apply it?

Teaching ApplicationsLet me discuss applications first. Consider the stan-

dard topic of force exerted by water on the face of adam. When I took calculus, the necessary physics hadbeen taught all of us earlier in the year (in Physics),and we all had done simple problems. We had only tolearn how tc set up sums by appropriate partitions thatwould converge on the one hand to the total force onthe dam, and on the other hand to the definite integralof a function.

We had a good basis for an intuitive understandingof the first, and there were results in the text for thesecond. There were many exercises devoted to findingthe force on improbable dam faces, which dams, how-ever, all had the common property that the resultantintegral coi'ld be calculated by indefinite integration.

The ..aly major change I see from then to now is thatmost current students have no knowledge of fluid pres-sure, and must use not physical intuition, but faith asjustification. In such topics we are not teaching the ap-plication of mathematics, and our good students knowthat.

Our applications should be taught as modelling.We are dealing with continuous models; fluids, gases,electrical currents are continuous functions in calcu-lus. Reasons for the use of continuous models insteadof discrete ones should be given, and some examples

1 8 5

YOUNG: PROBLEMS AND PROSPECTS 175

where discrete models are better should be given. Thereshould be enough explanation of the underlying physics,economics, whatever, s that the student can under-stand the basis for the model. The fact that the modelis not reality should be explained over and over. Shouldthere be some purely computer treatments, perhaps asimulation?

Computer methods will let us cut inhalf the time required to teach ourpresent courses. ... Why not use thetime to teach a bat( r understanding ofcalculus and how to apply it?

How much of such treatments can be done is a sub-ject for experiment, but a few good models will beat ourpresent array of poorly understood iiiiiblems apparentlyintended mostly for civil engineers.

Teaching TheoryWe do a poor job of theory. I am not here calling

for rigor; I have long been convinced that epsilon-deltaproofs won't work with students who have no great tal-ent or interest in mathematics per se. (There is the pos-sibility of approaching epsilontics from numerical anal-ysis, as error and control of error. That might work.)What is possible, I believe, is the sort of clear explana-tion found, e.g., in Courant's Calculus, at tinu..3 beingclose to a rigorous proof.

It is difficult to persuade ..lost students that a proofthat the limit of a sum is the sum of th- limits is neces-sary or interesting. On the other hand, the 'proof" thata function continuous and monotone on a closed intervalis integrable mere is perceived differently, as a clarifi-cation of what is going on in integration. At pMsent,time pressures force most teachers to rush through sucha topic, to be understood only by the best students.

With a course of this nature and with freedom fromthe necessity of manipulative skills, more students willgo on to take higher courses, more students will be ableto actually use calculus in their major fields, and moregraduates will be mathematically competent.

Reference

[1] Ralston, Anthony and Young, Gail S. (eds.) The Fu-ture of College Mathematics. New York: Springer-Verlag,1983.

GAIL S. YOUNG is Program Directs. for Mathematicsin the Directorate of Science and Engineering Education atthe National Science Foundation. Young has served as Pro-fessor of Mathematics at the University of Michigan, TulaneUniversity, Rochester University, Case Western Reserve, andthe University of Wyoming. In 1969-70 he served as Pres-ident of the Mathematical Association of America and in1984 he chaired the Mathematics Section of the AmericanAssociation for the Advancement of Science. Young is arecent recipient of the Distinguished Service Award of theMathematical Association of Amc7ica.

z0

r.r r'

s r-"PiPtkri C 1-PAN:1- R 2_51D otvs 'ES cl:

r--. rt

r-___. 1 tn.\ p nw-e_ ccd c LAL i L-. -y-ut Cti oi-1.1 r-

. r.-111.S L41 e-,

. C--:

R e. tAi 0-v-A 11,e- i (-1. S-ty-t.A. c.+0-y, r.-. -E. 1, v i -{-e. ct) EA-10 1 a i..frk \i-x_ ? ro -f-e- cg 0Y-' s C-

cod cu, lAi\s . 0,-. Use O..

i c - Le_c

--ET-be. _e-±.

.

. . . . .c..12.7..e_ st-1,..,-A.Q.An71--s y e. a c.fi ti-,.;,_ --i-e_fm--- 60 G k

' S , 1 k c- La_cs e_s IA) tii,-, h-loiA. vai\ 1 LAsi-ri-t.0-.r-rs e'R

:

et%, ct )214 cp cy i-e-a.c.-1,-.. i ck,t- t?,.e:..4

---S" CIJA-n [ 'L), er.1 cA

4

... ,

e. LI) TA; 2_ .e. -,f7),-.

,

.RCa-tn. epa_SS ckir-cAlutS

A. yl cet- 1 0 I-. -o,j Vt. e-LA) le_i-t-e.y-- ay- CCAthLiDCL sc

tl..6ta-- SL.core4 i 4.e..a....< to-W. e_ il,e_ .6

co... ct, 1 c.A.A.t LA-S 14-0 1 40111. R, LA-) 0 VI t7------167-

iRS

Final Examinations for Calculus I

The following thirteen examinations represent a crosssection of final examinations given in 1986-87 for stu-dents enrolled in the first calculus course. The exami-nations come front 'arge universities and small colleges,from Ph. D. granting institutions and two year colleges.

Brief descriptive information preceding each exam-ination gives a profile of the i'stitution and circum-stances of the examination. Unless otherwise noted, nobooks or notes are allowed during these examinations;calculator usage is explicitly noted where the informa-tion toes provided. Except for information on problemweights and grading protocol, thz texts of the examina-tions are reproduced here exactly as they werz presentedto the students.

INSTITUTION: A public Canadian university :11z nearly

20,000 students that in 1987 awarded 15 bachelor's de-grees, 4 master's degrees, and 6 Ph.D. degrees in math-ematics.

EXAM: A two-hour exam (calculators not allowed) forall students (except engineers) taking first term calculus.Of the 2500 students who enrolled in the course, 50%passed, including 15% who received grades of A or B.

1. a. Use the definition of continuity to sl w that thegiven function is discontinuous at x = 1.

2x 1 if x < 1f (x) = 0 if x = 1

x' if x > 1

1

9 1]b. Evaluate lim

x 1 x2

c. Evaluate lim

2. a. Use the limit definition of derivative to find f1(0)if f (x) = xl(x2 +1).

b. Find dx [sin (cos(eliz))1.

c. Find d x x

dx A/1.7-P]3. a. Find the equation of the line tangent to the curve

2(x3 + Y2)3/ 2 = 27xy at the point (2, 1).

b. If g(-1) = 1, 0-1) = 2 and t(1) = 1, evaluatedx [f (g(-4x))) at x = 2.

4. A spot light located on the ground shines on thewall 12m. away. If a man 2m. tall, walks from thespotlight toward the wall at a speed of 96m./min.,how fast is his shadow on the wall decreasing whenhe is 4m. from the wall?

5. Given f(x) = x/(x 2)2.

a. Findi. the domain of f

ii. the asymptotes of fiii. intervals where f is increasing and decreasing

iv. relative extrema of f .

b. Further, find the intervals where f is concave upand concave down and the points of inflection.And sketch an accurate graph of f.

6. If 1200 cm2 of material is available to make a boxwith a square base, rectangular sides and no top,find the volume of the largest possible box.

7. a. Find f (x3/Vx2 + 1) dx

b. Evaluate fi4(1//i) cos dx

c. Show that f11 (1.7(1 + ax) dx = 1, where a is apositive constant.

INSTITUTION: A st(..e university in the Midwest with20,000 undergraduate and 6 000 graduate students thatin 1987 awarded 45 bachelor's degrees, 14 master's de-grees, and 7 doctor's degrees in mathematics.

EXAM: A two-hour exam (calculators allowed) nn el-ementary calculus for business students. Of the 1440students who enrolled in the course, approximately 74%passed, inrtding 40% who received grades of A or B.

1. Find each of the following limits which exist.x2 + 2x 8 . nn X2 + 2x 8

a.lim-4 X2 - 6x + 8

b. zIz

3x2 4x + 7c. lim d. lim

16

z-400 5x2 + 9x 11 x--.00

16e. lim

x-..-oo 1 +

IRD

+ 4e-31

180 EXAMINATIONS

2. Find the derivative of each of the following functions.

a. f(x)=

b. f(x) = (3x + 2) 79 -1./0

C. f(x) = 3e (-4z2+2z+3)

d. f(x) = -21n (2x3 - 3x +

e. f(x)=:x +-

f. f(x) e(-4z2+2z+3) (4x - 1)

3. Let f (x) = x4 - 8x3 + 216.

a. Determine those intervals of the x-axis on whichf(x) is increasing and those intervals of the x -axison which f(x) is decreasing.

b. Determine where f(x) has a relative maximumvalue and where f(x) has a relative minimumvalue.

c. Determine those intervals of the x-axis on whichthe curve y = (x) is concave upward and thoseintervals of the x-axis on which the curve y = f(x)

is concave downward.

d. Find all inflection points on the curve y = f(x).

e. Sketch the curve y = f(x) carefully and neatly.

f. Determine where f(x) has an absolute maximumvalue t:,11 the closed interval [2,81.

4. Find each of the following.

a. f 3x9 dx

c. f 16x c(-4z2+1) dxe-4x

c. f

r -x2 + 4xb. j x3 - 6x2

ax

d. f 4 [1n(x2)]3 z dx

dx f. fo41/16-+ 5x dx3e-4z

5. Set up the trapezoidal rule ready for calculating anapproximation to f;' 3e- 41`nTiffx dx using 8 subinter-,als.

6. Determine which of the following improper integralsconverges, and find the value of each convergent one.

a. f'°° dxvs+42: b. x2 c-z3 dx

7. Determine where the graph of each of the followingfunctions has a relative maximum point, a relativeminimum point, a saddle point.

a. f (x, y) = 3x2 - 4xy + 5y2 - 14x + 24y - I 7

b. f(x, y) = x2 - 4xy + 27/4 + 11

8. Use the method of Lagrange multipliers to find themaximum value of f(x, y) = -32x2 + 3xy - 2y2 + 45subject to the constraint g(x,y) = 16x2 + y2 = 32.

INSTITUTION: A southern urban two-year college with5,000 students.

EXAM: A two-hour exam (calculators allowed) for thefirst term of calculus for science and business students.Of the 28 students who enrolled, in the course, 64%passed, including 40% who received grades of A or B.

1. Answer either a or b but not both:

a. If f(x) 2x2 - + 1, find fi(vr.-312) by usingthe definition of thz derivative.

b. By means of an appropriate Riemann Sum, findf_31(2x2 - 3x + 1) dx.

2. Answer a or b but not both:

a. Find the area of the region bounded by x = y2and x + y = 2.

b. Find the volume of the solid of revolution gen-erated when the region bounded by x = y2 aadx + y -= 2 is revolved about the line y = 1.

3. Do either a or b but not both:a. Use Newton's Method to estimate starting

with a1 = 2 and finding a3.

Estimate (.94)3 - 1/(.94)3 by using differentials.

4. Answer either a or b below, but not both:

a. Sketch the graph of y = x4/4 - (4x3)/3 + 2x2 - 1by finding the local max and local min; point(s) ofinflection; intervals where the graph is increasing;decreasing; concave upward; concave downward.

b. Estimate fi5 x4 dx using the Trapezoidal Rule withn = 4.

5. Do either part a or b, but not both:

a. A drinking glass is in the shape of a truncated conewith a base radius of 3cm, a top radius of 5cm, andan altitude of 10cm. A beverage is poured into theglass at a constant rate of 48cm3/sec. Find therate at which the level of the beverage is risingin the glass when it is at a depth of 5cm. [Note:Volume of a cone = (1/3)7r(radius)2(altitude).1

190

1 5 1

1 3 1

10

FINAL EXAMS FOR CALCULUS I 181

b. Two straight roads intersect at right angles, onerunning north and south, while the other runs eastand west. Bill jogs north through the intersectionat a steady rate of 6 miles per hour. Sue pedals herbicycle west at a steady rate of 12 miles per hourand goes through the intersection one half hourafter Bill. Find the minimum distance betweenBill and Sue.

6. For each of the following functions, determine bothf'(x) (you need not simplify f ' (x)) and f f (x) dx.a. f (x) = 21x6 + 15x2 - 3x + el'

b. f(x) = 3x (ex7) + x0 ---Ti

c. f(x) 1 1

-s/1 - x2 4 + x2

d. f(x) = ein(z4 +2x2+1)

e. f (x) = (sin x)(ec°")

INSTITUTION: A midwestern liberal arts college with1750 students that in 1987 awarded 24 bachelor's de-grees in mathematics.

EXAM: A two-hour exam (calculators allowed) from onesection of first-term calculus taken all students whoseintended course of study requires calculus. Of the 160students who originally enrolled in the course, approxi-mately 88% passed, including 50% who received gradesof A or B. SO students were in the section that took thisexam.

1. Differentiation.

a. Define f'(x).

b. On the pictures below, indicate Ay and dy.

Y Y = f(x Y y = f(x)

x x + Ax x x + Asc. According to Newton's law of heat transfer, the

rate at which the temperature of a body changesis proportional to the difference between the tem-perature of the body (call the temperature of thebody u) and the temperature T of the surround-ing medium. Express this law as a mathematicalequation.

.1. Given f(2) = e, f'(2) = -1, g(2) = 4, g'(2) = 3,r(x) =if(x))g(z). Find r'(2).

e. Find IX if y/x + xly = 4.

2. Integration.

a. Consider the function f(t) = lit defined on theinterval [1, 2]. Let P be the partition of [1, 2]into n equal subintervals, and in each subintervalExi-1, xd, choose f., = zi-i. Then the sum Su isdefined to be

Sn = f(to)(xi xo) + f(ti)(x2 xi) +...+.qtn-i)(x. xn-i).

Find the two sums, S2 and S3.

b. If you had the use of the tables in your text or acalculator, you should be able to find the numberto which the sums in part a above will converge(that is, the number which will be approached) asn increases. What number would you look up (orenter into your calculator)?

c. Let G(x) = f; Nlic--ildti. Which is correct?

G(2) = -G(--2) or G(2) = G(-2)ii. What is G(0)?

iii. What is G'(2)?

iv. What is G(4)?

d. Find the indicated an::Lrivatives.i. f(t +1)Alidt ii. f Vi/(t3r. + 1) dt

3. Graphs and derivatives. For each of the followingfunctions, find f(x). Give the answer, then find thecritical points, if any, _.... uraw a graph of y = f(x).

a. f(x) = 3x4 - 8x3 + 6x2 - 5

b. f(x) = 4x/(x2 + 2)

c. f(x) =1n x/x

d. f(x) = x2x-x2

e. f(x) = r.1-x

4. The following two que3tions both refer to the func-tion f(x) = In xix gr:plied in part c of question 3above.

a. Write the equation . r the line tangent to the graphat x = Vi.

b. Find the area under the graph from x = Nie to2x = e .

182 EXAMINATIONS

INSTITUTIr N: A western land-grant university with15,000 students that awarded 29 bachelor's, 9 master's,and 3 doctoral degrees in mathematics in 1987.

10. Suppose that the equation y = (2x) /(1 + 27') definesy as a differentiable function of x. Ilse implicit dif-ferentiation to determine "i/.

EXAM: A three-hour exam (calculators not allowed) 11.

given to all students in a first term calculus for en-gineering, science, and mathematics students. Of the435 students who enrolled in the course, 46% withdrewbefore- the final exam, 6% failed the course, and 24%received grades of A or B.

1. Find the equation of the line tangent to the graph ofy = f(x) at the point (0, 4) where Az) = f n x

4 +3x.

2. Determine all relative extreme values of y .A5 20z.(Justify your results.)

3. Find the derivatives of the following functions:

a. f (z) = (x2 1) /(x2 + 1)b. f (z) = (1/z sin 7:)2/3

c. f(z) = e2:8d, f(z) = (ln z)1"2:e. f(x) = arctan ((x + 1) /(x 1))

4. Evaluate the followir g limit in two distinct ways:

8z3 1lim

2z 1

5. Sketch the graph of Az) = x3- 6x2 +12x -4, showingall principal features (such as asymptotes, extrema,and inflecton points).

6. Approximate 1/17 by using differentials.

7. A crate open at the top has vertical sides, a squarebottom, and a volume of 4 cubic meters. If the cratehas the least possible surface area, find its dimen-sions.

8. Compute the following:

a. f_ti EL 2 z3 I dm

b. f z2/(1 z3)clz

c. f1 (t2 (1/t2))2 dt

d. f z.V dz

e. 1/N/16 -2r/z

9. Evaluate lim (sin 3x) /x forz.zoa. zo = 7/3C. Z0 = 0

b. zo = 7r/6

A board 5 feet long slides down a vertical wall. Atthe instant the bottom end is 4 feet from the base ofthe wall, the top end is moving Bowl the wall at therate )f 2 feet per second. At that moment, how fastis the bottom sliding along the horizontal ground?

12. Assume that the half-life of radon gas is 4 days. De-termine a formula which gives the time required for10% of a given quantity of this gas to become harm-less.

13. Evaluate the area under the curve y = on theinterval g, 1].

14. Find a formula for the inverse of f (z) =+ 5

z 4

15.

16.

1:x'2

Sketch the graph of f(x) = ln(2 z:c2). [Note thatx must be such that 2 x z2 >

Determine whether the foltowing statements are trueor false;

a. The function f(z) = 1/z is its own inverse.z2 + 5z + 6

z + 2b. Defining f(-2) = 1 makes f(x) =

continuous at x =

c. lim + z) =1

d. 47 tan z = In I cos x110 = (1n 1 In 1) = 0.

e. If lim f(..e) =x4a+lim Az), the '(a') is continuous

at the point x = a.

INSTITUTION: A highly selective private northeasternuniversity wilt:. 4,500 undergraduate and 4,500 graduate students. In 1987, 68 students received bachelor'sdegrees in mathematics, 2 received master's, and 11 re-ceived mathematics Ph.D. degrees.

EXAM: A three-hour exam for all students enrolled infirst term calculus. Course enrollment rose from 230 to275 during the term, and 94% of the enrolled studentspassed; 62% received grades of A or B.

1. If Az) = z2 cos; then fi (7r/2) =(A) 7 (B) -72/4 (C) 72/4(D) 7r (E) 7 72/4


2. For what value of the constant c will the tangentline to the curve y = x4 + cx2 cx + 1 at the point(x, y) = (1, 2) intersect the x-axis at x = 2?(A) 1 (B) 3 (C) 6 (D) 2 (E) 15

3. Which of the following gives the best approximationfor f (1.01) if f (x) = 2/x7 + 3x4?

(A) 5.00 (B) 5.02 (C) 5.01(D) 4.98 (E) 4.99

4. The equation of the tangent line to the curve x2 +y2 + 3xy3 + 4x3y = 1 at the point (1, 0) is

(A) y = (1/2)x + 1/2 (B) y = (1/2)x 1/2(C) y = x 1 (D) y = 2x 2

(E) y = 2x + 2

5. If f (x) = cos r (x2 - (1/2)x), then f'(1) =(A) 3/2 (B) (2/3)r (C) (3/2)r(D) 3/2 (E) (3/2)7r

6. The curve y = 4x3 3x2 + 2x 1 has a point ofinflection at x =(A) 1 (B) 1/2 (C) 1/3 (D) 1/4 (E) 1/5

7. The curve y = x2/3 + 12/x2 has a(A) relative minimum at x = 6(B) relative minimum at x =(C) relative maximum at x = 6(D) relative maximum at x = vi(E) point of inflection at x = 0

8. If a solid right circular cylinder of volume V is madein such a way as to minimize its surface area thenthe ratio of its height to its base radius is(A) r (B) 1/2 (C) 0 (D) 1 (E) 2

9. A ladder of length 15 ft. is leaning against the sideof a building.' The foot of the ladder is sliding alongthe ground away from the wall at 1/2 ft./sec. Howfast is the top of the ladder falling when the foot ofthe ladder is 9 ft. from the wall?

(A) 3/2 ft./sec. (B) 3/4 ft./sec.(C) 3/8 ft./sec. (D) 1/3 ft./sec.(E) 1/2 ft./sec.

10. Consider the following two statements:P. If a function is continuous then it is differentiable.Q. If a function is differentiable then it is continuous.

(A) P and Q are bOth true.(B) P and Q are both false.(C) P is true and Q is false.(D) P is false and Q is true.

11. For which values of the constants a and b will thefunction f (x) defined below be differentiable?

x < 1f axf(x) = x + 1 x> 1

(A) a=b=1 (B) a = 1, b = 2(C) a = 1, b = 1 (D)a=2,b=0(E) a = 3, b = 1

12. The total area enclosed between the curves y = xand y = x3 is

(A) 1 (B) 1/2 (C) 1/3 (D) 1/4 (E) 1/5

13. The area bounded by the curve y = rsin(rx) and thex-axis between two consecutive points of intersectionis

(A) 1 (B) 2 (C) 3 (D) r (E) 7112

14. If the acceleration of a particle moving in a straightline is given by a = 10t2 and the particle has a veloc-ity of 2 ft./sec. at t = 0, then the distance traveledby the particle between t = 0 and t = 3 sec. is

(A) 277r ft. (B) 147/2 ft. (C) 135/2 ft.(D) 41 ft. (E) 83 ft.

15. If f(x) = fox sine t cosh t dt then f' (7r/4) is

(A) 1/2 (B) 1/4 (C) 1/8 (D) 1/16 (E) 1/32

16. folri2 sin 2x dx =

(A) 0 (B) 1/2 (C) 1 (D) 2 (E) 7r

17. The region bounded above by the curve y = x x2

and below by the x-axis is rotated about the x-axis.The volume generated is

(A) 7116 (B) 7r/30 (C) 7r/12 (D) 7r/15 (E) 7r/3

18. The region in problem 17 is rotated about the y-axis.The volume generated is

(A) r/6 (B) 7r/30 (C) 7r/12 (D) r/15 (E) 1-/3

19. The length of the curve y = (2/3)x312 between x = 1and x = 2 is

(A) 1 (B) 1

(C) 30 (D) 1/2(E) (2/3)(30 20)

20. The average value of the function f(x) = x3 betweenx = 0 and x = 2 is

(A) 1/4 (B) 1/3 (C) 1/2 .(D) 1 (E) 2

Q 7'. t.)

184 EXAMINATIONS

21. The following table lists the known values of a certainfunction f

x 1 2

f(x) 0 1.1 1.4 1.2 1.5 1.6 1.1I

If the trapezoid ru e is used to approx mate f f (x) dxthe answer obtained is

(A) 7.9 (B) 7.5 (C) 6.8(D) 7.35 (E) 7.05

22. If Simpsons's rule is used to estimate g f (x) dx forthe function given in problem 21, the answer ob-tained is

(A) 7.9 (B) 7.5 (C) 6.8(D) 7.35 (E) 7.05

23. If f (x) = e11 n2z, then f' (r /2) =

(A) 2 (B) 1 (C) 0 (D) -1 (E) -224. If f (x) = In (sin-1(x/2)) then f1(1) =

(A) 1/7r (B) (2IS)/v (C) (7r1i)/4(D) 7r.fi (E) 7r/2

25. If lim (ex + 1)/(x4+ In x) = A and limIn x

= Bz-co x3 + 1then

(A) A = 0, B = oo(C) A = oo, B = 0(E) A = 1, B = 1

26. aim ( x In x x2) /(x2 + 2) =

(A) 0 (B) oo (C) 1 (D) 2 (E) 1/2

27. The temperature, T, of a body satisfies the equa-tion dT dt = kT, t in seconds. If T(0) = 30° andT(5) = 5°, then k =(A) -1(C) -(1/5) In 6(E) -(1/7) In 5

3

28. Jo =

(A) 1/3 (B) 2/3 (C) 4/3 (D) 8/3 (E) 16/3

29.

Iv/4sine x cos3 x dx =

(A) (44/81 (B) (bii)/7 (C) (34/17(E) 71i/120

(B) A = 0, B = 0(D) A = oo, B = oo

(B) - In 2(D) -(1/3) In 3

(D) gibq

30.

I2

xez Lix =

(A) 1 (B) 0 (C) e (D) e2 (E) 1/e

31. (2x + 3)/(x3 - 2x2 - 3x) dx =

(A) 2 In 3 + In 2 (B) (1/4) In 3 - 2In 2(C) (1/2) In 3 + (1/2)1n 2 (D) 21n 3 - (1/3) In 2(E) In 3 - In 2

32. The integral

1.

dx

fo xP

(A) diverges for p > 1, converges for p < 1

(B) diverges for p> 1, converges for p < 1

(C) converges for p > 1, diverges for p > I

(D) converges for p > 1, diverges for p > 1

(E) diverges for all p

INSTITUTION: A southwestern public university with12,000 students that in 1987 awarded 24 bachelor's and5 master's degrees in. mathematics.

EXAM: A three-hour exam (calculators not a!,:owed)given to 18 students in one section of first term calculusfor engineering, mathematics, and science majers. Ofthe 269 students who enrolled in the course, 42% com-pleted the course with a passing grade; 60% of those whopassed received grades of A or B.

An object shot upward has height x = -5t2 + 30t mafter t seconds. Compute its velocity after 1.5 sec.,its maximum height, and the speed with which itstrikes the ground

2. Two cars leave an intersection P. After 60 sec., thecar traveling north has speed 50 ft /sec. and distance2000 ft. from P, and the car traveling west has speed75 ft /sec. and distance 2500 ft. from P. At thatinstant, how fast are the cars separating from eachother?

3. Find the maximum and minimum off (x) = xiV7- x2on the interval [-I, :;.

4. Find the coordinates of all local maxima and minimaof f(x) = x3/(1 + x4).

5. Find the point on the graph of the equation y = Nrinearest to the point (1, 0).

6. An athletic field of 400-meter perimeter consists ofa rectangle with a semi circle at each end. Find thedimensions of the field so that the area of the rect-angular portion is the largest possible.


7. Differentiate:a. f(x) = e5z(x2 3x + 6)

b. g(x) = 1/(1+ e')8. In a certain calculus course, the number of students

dropping out each class day was proportional to thenumber still enrolled. If 2000 started out and 10%dropped after 12 classes, estimate the number leftafter 36 classes.

9. Differentiate:a. f (x) = ln(x2 + x)b. g(x) = ln(sec x + tan x)

10. Use the logarithm function to differentiate:a. y = (2 + sin x)z b. y = xliz

11. Evaluate arccos(1 /2) arccos(I/2).12. Differentiate f (x) = x arcsin x + arccos(2x + 1).

13. Differentiate f (x) = x cosh x sinh x.

14. Finda. lim ((1 x)/(ez e))

b. lim (x 7r/2) tan xz--.7r/2

15. Assume the population of a certain city grows at arate proportional to the population itself. If the pop-ulation was 100,000 in 1940 and 150,000 in 1980, pre-dict what it will be in year 2000.

16. Evaluate:a. fi3 (x2 4x + 4) dx b. foz12(sin x cos x) dx

17. Find

a. y = fo5inz Vidt b. y = fox sec t dt

18. Compute the area under the graph f(x) = x + sin xover the interval [0, 7r/2].

19. Show that 2 < 4111(1+ sin2 x) dx < 4. [HINT: If

f (x) < g(x) on [a, b] then fa f (x) dx < fa g(x)

INSTITUTION: A midwestern community college withapproximately 8500 full time students.

EXAM: A two-hour exam (calculators allowed) given to60% of the students who completed a first term calcu-lus course for students specializing in business or liberalarts. Of the 142 students who enrolled in the course,30% withdrew, 1% failed, and 37% received grades of Aor B.

1. A manufacturer has been selling lamps at a price of$15 per lamp. At this price, customers have pur-chased 2,000 lamps per month. Management has de-cided to raise the price to p dollars in an attempt toimprove profit. For each $1 increase in price, it isexpected that sales will fall by 100 lamps per month.

If the manufacturer produces the lamps at a cost of$8 each, then express the monthly profit for the saleof lamps as a function of price p. Then maximize theprofit by finding the price p at which greatest profitoccurs.

2. Find each derivative.

a. f'(2) if f(x)= Vx3 .. 4x

b. g(x) = 3 x6 + 3x2

c. h(x) = xez

d. j(x) = ln(x2 + 4x)

e. n(x) = (x + I)/(x

3. A fine restaurant has purchased several cases of wine.For a while, the value of the wine increases; but even-tually, it passes its prime and decreases in value. In tyears, the value of a case of the wine will be changingat the rate of 62 12t dollars per year. If the an-nual storage costs remain fixed at $2 per case, thenwhen should the restaurant sell the wine in order tomaximize profit? [HINT: Profit depends upon value"minus" storage costs.]

4. Use implicit differentiation to find cl if x2y + 2y2 =30.

5. It is estimated that t days from now a citrus farmer'scrop will be increasing at the rate of 0.3t2 + 0.6t + 1bushels per day. By how much will the value of thecrop increase during the next 6 days if the marketprice remains fixed at $7 per bushel?

6. Evaluate these integrals:

a. fo4(6x2 + 315) dx

b. f 2e2x+1 dx

c. f(3x2)/(z3 + 1) dx

d. f(x2 1)(x3 + 3x + 1)5 dx

e. f xez dx [Use parts.]

f. fr 3/x2 dx

.15)5

186 EXAMINATIONS

7. At a certain factory, the output is Q = 120K1/2L1/3units, where K is the capital investment in $1000units and L is the size of the labor force measuredin worker-hours. Currently, capital' investment is$400,000 and labor force is 1000 worker-hours. Usethe total differential of Q to estimate the changein output which results if capital investment is increased by $2000 and labor is increased by 4 worker-hours.

8. A sociologist claims that the population of a certaincountry is growing at the rate of 2% per year. If thepresent population (1987) of the country is 200 mil-lion people, then what will the population be in theyear 2000? [HINT: Solve the separable differentialequation (.!ii = .02P.]

9. Sketch carefully the graphs of f(x) = 2x X2 andg(x) = 2x 4. Then find the area of the regionenclosed by the two graphs.

10. Determine th .axima and minima for the functionf(x, y) x2 + + y2 3x.

INSTITUTION: A western public university with 23,000undergraduate and 10,000 graduate students that in1987 awarded 350 bachelor's degrees, 19 master's de-grees, and 13 Ph.D. degrees in mathematics.

EXAM: A three-hour exam (calculators not allowed)given to 71 students in one section of a first term calcu-lus course for students intending to major in the life sci-ences and economics. Of the 914 students who enrolledin the course, 79% passed the course, 44% of whom re-ceived grades of A or B.

1. Differentiate the following:

a. A sec (e31b. Ary4.

c. A arcsin (e)

d. 71--ix In(arctan x)

2. Find the area of the 3-sided region in the first quad-rant enclosed by the straight lines y = x and y = x/8and the curve y = 8/x2. Draw a picture.

3. Consider the region enclosed between the curves y =2x2 and y = x2 + 1 as shown:

a. Suppose the region is rotated about the axis x = 5.Write down a definite integral which would givethe volume. Do not evaluate this integral.

b. Same as part (a), except let the axis be the liney = 7.

4. A spring, whose natural length is 10 ft., requires 40ft-lbs. of work to be stretched from a length of 13 ft.to a length of 17 ft. How much work would it requireto stretch it from a length of 15 ft. to a length of 21ft.?

5. An object is removed from an oven and left to sit in a70° room. After 2/3's of an hour it is 170°, and after2 hours it is 120°. Assuming that Newton's law ofcooling applies, how hot was the object when it wasremoved from the oven? (Note: You may leave youranswer in any reasonable form; the actual numericalanswer will contain a Ar.)

6. Find the following indefinite integrals, using substi-tution if necessary:

a. f(e3x)/(1 + e3x)5 dxb. f Vi/(A5 1) dx

7. Using integration by parts twice, find f z2 sin x dx.(Recall you can check your answer by differentia-tion.)

8. Solve the following differential equation, making useof an integrating factor. Then evaluate your constantusing the initial condition.

dy

dx= 3 y/(2x + 1); y(4) = 5

(Assume 2x + 1 > 0 always.)

9. At time t = 0 a tank contains 13 gallons of water inwhich 2 pounds of salt are dissolved. Water contain-ing 6 pounds of salt per "allon is added to the tankat the the rate of 8 gallons per minute. The solu-tions mix instantly and the mixture is drained at therate of 3 gallons per minute. Write down a differen-tial equation which relates the amount of salt s (inpounds) to the time t. Do not solve this equation.


10. Using Theorem 19.3, find the solutions of the follow-ing two differential equations, use the initial data toevaluate your constants, and identify the appropriategraphs.

a. y" + 6y' + 9y = 0; y(0) = 0 and yi(0) = 2.

b. y" + 5y' + 4y = 0; y(0) = 2 and yi(0) = 5.(A) (B)

(C) (D)

(E) (F)

____

Exam: A three-hour exam (calculators not allowed)administered in 1985 to 35,000 high school studentsby the advance placement (AP) program. This is the"AB" exam for students who have studied the prescribedcurriculum, 35% of those who took the exam receivedgrades of 4 or 5, the highest two grades available.

1. .112 Z-3 dx =

(A) 7/8(D) 3/8

(B) 3/4(E) 15/16

(C) 15/64

2. If f(x) = (2x + 1)4, then the 4th derivative of f(x)at x = 0 is

(A) 0 (B) 24 (C) 48 (D) 240 (E) 384

3. If y = 3/(4 + x2), then

(A) 64(4 + x2)2(C) 64(4 + x2)2(E) 3/2x

4. If 2 = cos(2x), then y =(A) i cos(2x) + C(C) i sin(2x) + C(E) sin(2x) + C

dx

(B) 3x/(4 + x2)2(D) 3/(4 + x2)2

(B) cos2(2x) + C(D) i sin2(2x) + C

5. lim (4n2)/(n2 + 10,000n) isoo

(A) 0 (B) 1/2,500 (C) 1(D) 4 (E) nonexistent

6. If f (x) = x, then f'(5) =(A) 0 (B) 1/5 (C) 1 (D) 5 (E) 25/2

1'. Which of the following is equal to In 4?(A) ln 3 + In 1 (B) In 8/ In 2(C) f ct dt (D) fi4 In xdx(E) fi4 i dt

8. The slope of the line tangent to the graph of y =ln(x/2) at x = 4 is(A) 1/8 (B) 1/4 (C) 1/2 (D) 1 (E) 4

9. If f'1 e-x2 dx = k, then f01 e-x2 dx =(A) 2k (B) (C) k/2(D) k/2 (E) 2k

10. If y = 10(x2-1), then =

(A) (In 10)10(x2-1) (B) (2x)10(x2-1)(C) (x2 1)10(x2-2) (D) 2x(ln 10)10(x2-1)(E) x2(In 10)10(x2-1)

11. The position of a particle moving along a straight lineat any time t is given by s(t) = t2 4t + 4. What isthe acceleration of the particle when t = 4?(A) 0 (B) 2 (C) 4 (D) 8 (E) 12

12. If f (g(x)) = ln(x2 + 4), f (x) = ln(x2), and g(x) > 0for all real x, then g(z) =(A) iRrz5-471 (B) 1/(z2 4- 4)(C) Vi2 7-71 (D) x2 + 4(E) x +2

13. If Z2 + zy 4- y3 = 0, then, in terms of x and y, i =(A) (2x + y)/(x + 3y2)(B) + 3y2)/(2x + y)(C) 24(1 + 3y2)(D) 2x/(x + 3y2)(E) (2x + y)/(x + 3y2 1)

14. The velocity of a particle moving on a line at time tis v = 3t1/2 + 5t3i2 meters per second. How manymeters did the particle travel from t = 0 to t = 4?(A) 32 (B) 40 (C) 64 (D) 80 (E) 184

15. The domain of the function defined by f (x) =ln(x2 4) is the set of all real numbers x suchthat

(A) Ix' < 2 (B) Ix' < 2 (C) Ix' > 2(D) lx1 > 2 (E) x is a real number

q.1. 6

188 EXAMINATIONS

16. The function defined by f(x) = x3 3x2 for all realnumbers x has a relative maximum at x =

(A) 2 (B) 0 (C) .1 (D) 2 (E) 4

17. fol xez dx =

(A) 1 2e (B) 1 (C) 1 2e-1(D) 1 (E) 2e 1

18. If y = cos2 x sin2 x, then y' =

(A) 1(C) 2 sin(2x)(E) 2(cos x sin x)

(B) 0(D) 2(cos x + sin x)

19. If f (xi) + f (x2) = f + x2) for all real numbers xland x2, which of the following could define f?

(A) f(x) = x + 1 (B) f(x) = 2x(C) f(x) = 1/x (D) f(x) = ex(E) f(x) =

20. If y = Arctan (cos x), then d =

(A) sin x/(1 + cos2 x) (B) (Aresec(cos x))2sin(C) (Arcsec(cos x))2 (D) 1/((Arccos x)2 + 1)(E) 1/(1 + cos2 x)

21. If the domain of the function f given by f(x) =1/(1 x2) is fx : Ix' > 1 }, what is the range off?(A) {x : oo < < 1}(B) {x : oo < < 0}

(C) : oo < < 1}

(D) : 1 < < cm}

(E) fx : 0 < < ool

22. f2(x2 1)/(x + 1) dx =

(A) 1/2 (B) 1 (C) 2 (D) 5/2 (E) In 3

23. -IL. (1/x3 1/x + x2) at x = is

(A) 6 (B) 4 (C) 0 (D) 2 (E) 6

24. If f22(x7 + k) dx = 16, then k =

(A) 12 (B) 4 (C) 0 (D) 4 (E) 12

25. If f(x) = ex, which of the following is equal to f'(e)?

(A) ano(ez+h)/h (B) lim (ez+h ez)/hh--00

(C) iiirrio(ee+h e)/h (D) ilihno(ez+4 1)/h(E) iihno(ee+h eevh

26. The graph of y2 = X2 + 9 is symmetric with respectto which of the following?

I. The x-axisII. The y-axis

III. The origin

(A) I only (B) II only(C) III only (D) I and II only(E) I, II, and III

27. fo3 Ix 11 dx =

(A) 0 (B) 3/2 (C) 2 (D) 5/2 (E) 6

28. If the position of a particle on the x-axis at time tis 5t2, then the average velocity of the particle for0 < t < 3 is

(A) 45 (B) 30 (C) 15 (D) 10 (E) 5

29. Which of the following functions are continuous forall real numbers x?

I. x2/3

II. =III. y = tan x

(A) None (B) I only (C) II only(D) I and II (E) I and III

30. f tan(2x) dx =

(A) 21n I cos(2x)I + C (B) 121 In I cos(2x) I C

(C) z In I eos(2x)I + C (D) 21n I cos(2x)I C

(E) z sec(2x) tan(2x) +C

31. The volume of a (..one of radius r and height h isgiven by V = 3irr2h. If the radius and the heightboth increase at a constant rate of 1/2 centimeterper second, at what rate, in cubic centimeters persecond, is the volume increasing when the height is9 centimeters and the radius is 6 centimeters?

(A) .1.71- (B) l0a (C) 24ir (D) 54ir (E) 108ir

32. fow/3sin(3x)dx =

(A) 2 (B) 2/3 (C) 0 (D) 2/3 (E) 2

33. The graph of the deriva-tive of f is shown inthe figure at the right.Which of the followingcould be the graph of f?

4


(A)

(C)

(E)

(B)

(D)

34. The area of the region in the first quadrants that isenclosed by the graphs of y = X3 + 8 and y = x + 8is

(A) 1/4 (B) 1/2 (C) 3/4 (D) 1 (E) 65/4

35. The figure at the rightshows the graph of a sinefunction for one completeperiod. Which of the fol-lowing is an equation forthe graph?

(A) y = 2 sin (ix)(C) y = 2 sin(2x)(E) y = sin(2x)

36. If f is a continuous function defined for all real num-bers x and if the maximum value of f(x) is 5 andthe minimum value of f(x) is 7, then which of thefollowing must be true?

I. The maximum value of f (14 Is 5.II. The maximum value of 1,f(x)i is 7.

III. The minimum value of f(Ixl) is 0.

(B) y = sin(7m)(D) y = 2 sin(rx)

(A) I only (B) II only(C) I and II only (D) II and III only(E) I, II, and III

37. lim(x cscx) is

(A) oo (B) 1 (C) 0 (D) 1 (E) oo

38. Let f and g have continuous first and second deriva-tives everywhere. If f(x) < g(x) for all real x, whichof the following must be true?

I. P(x) < g'(x) for all real xII. f"(x) < g"(x) for all real x

III. foi ; (x) dx < foi g(x) dx

(A) None (B) I only(C) III only (D) I and II only(E) I, II, and III

39. If f(x) = In x/x for x > 0, which of the following istrue?

(A) f is increasing for all x grea ,er than 0.

(B) f is increasing for all x greater than 1.

(C) f is decreasing for all x between 0 and 1.

(D) f is decreasing for all x between 1 and e.

(E) f is decreasing for all x greater than e.

40. Let f be a continuous function on the closed interval[0,2). If 2 < f(x) < 4, then the greatest possiblevalue of f: f (x) dx is

(A) 0 (B) 2 (C) 4

41. If lim f (x) = L, where L isthe following must be true?

(A) f' (a) exists.

(B) f (x) is continuous at x = a.

(C) f(x) is defined at 73= a.

(D) f(a) = L.

(E) None of the above

42. it- sr, t2 dx =

(A) x/v 1 + x2 (B) ViT-T7 5

(C) i1 - 7 (D) x/N/1.-Fg 1/Nig(E) 1/(2N/1 7E x2)-1/(2N/g)

43. An equation of the line tangent to y = x3 + 3x + 2at its point of inflection is

(A) y = 6x 6 (B) y = 3x + 1(C) y = 2x + 10 (D) y = 3x 1

(E) y = 4x + 1

44. The average value of f(x) = x2Nix371- 1 on the closedinterval [0, 2) is

(A) 26/9 (B) 13/3 (C) 26/3 (D) 13 (E) 26

45. The region enclosed by the graph of y = x2, the linex = 2, and the x-axis is revolved about the y-axis.The volume of the solid generated is

(A) 87r (B) (C) V7r (D) 47r (E)

(D) 8 (E) 16

a real number, which of

190 EXAMINATIONS

FREE-RESPONSE QUESTIONS:

1. Let f be the function given by f(z) = !'÷-::-;'.

a. Find the domain of f.b. Write an equation for each vertical and each hor-

izontal asymptote for the graph of f.c. Find fi(x).d. Write an equation for the line tangent to the graph

of f at the point (0,1(0)).

2. A particle moves along t:.- z-axis with accelerationgiven by a(t) = cost for t ..>_ 0. At t = 0 the velocityv(t) of the particle is 2 and the position x(t) is 5.a. Write an expression for the velocity v(t) of the

particle.

b. Write an expression for the position z(t).c. For what values of t is the particle moving to the

right? Justify your answer.d. Find the total distance traveled by the particle

from t = 0 to t = r/2.

3. Let R be the region enclosed by the graphs of y =e', y = ex, and x =1n 4.a. Find the area of R by setting up and evaluating a

definite integral.

b. Set up, but do not integrate, an integral expres-sion in terms of a single variable for the volumegenerated when the region R is revolved about thex -axis.

Set up, but do not integrate, an integral expres-sion in terms of a single variable for the volumegenerated when the region R is revolved about they -axis.

4. Let f(x) = 14=2 and g(x) = k2sin(ix)/(2k) fork > 0.a. Find the average value of f on [1,4].b. For what value of k will the average value of g on

[0, k] be equal to the average value of f on [1, 4]?

5. The balloon shown at theright is in the shape of a cy-linder with hemisphericalends of the same radius asthat of the cylinder. Theballoon is being inflated atthe-rate of 2617r cubic cen-timeters per minute. Atthe instant the radius ofthe cylinder is 3 centime-ters, the volume of the bal-loon is 1447r

c.

cubic ce ntimeters and the radius of the cylinder isincreasing at the rate of 2 centimeters per minute.(The volume of a cylinder with radius r and heighth is rr2h, and the volume of a sphere with radius ris (4/47rr3.)a. At this instant, what is the height of the cylinder?

b. At this instant, how fast is the height of the cy-linder increasing?

6. The figure below shows the graph of f', the derivativeof a function f. The domain of the function f is theset of all x such that - -3 < x < 3.

.

[NOTE: This is the graph of the deli, if4ve of f, notthe graph of f.]a. For what values of x., 3 < z < 3, does f have a

relative maximum? A relative minimum? Justifyyour answer.

b. For what values of x is the graph of f concave up?Justify your an:;wer.

c. Use the information found in parts a and b andthe fact that f (-3) = 0 to sketch a possible graphof f.

INSTITUTION: A major midwestern research universitywith approximately 40,000 students. In 1987, 75 stu-dents received bachelor's degrees, 30 received master'sdegrees, and 11 received Ph.D. degrees in mathematics.

EXAM: A two-hour exam (calculators not allowed) onfirst term calculus for science and engineering students.Of the 1650 students who enrolled in the course, approx-imately 85% passed, including 40% who received gradesof A or B.

1. For each of the following find 2..

a. y = (cos(5x))3/2

z3

b. y -,/2T:--i- 1

c. y = z tan z

200


X4 4d. y =

4-

4

e. - + y2= 2x3 + 1

y

2. Find the following limits. If the limit is a finite num-ber, write that number. If there is no finite limit,write +00, -00, or "none."

x - 1a. limx-i- X2 2x + 1

b. lim7x5 - 3x2 + 1

x-+ +w x6 + 1

c. limsin(x - 3)

Z-43 x - 3d. lim j":(3t2 - 1) dtx2

3. For the function y = X3 6x2 + 9x - 3 answer thefollowing questions.

a. Give the coordinates of all relative minima andrelative maxima (if any). Justify your answer.

b. Give the coordinates of all inflection points (ifany). Justify your answer.

c. For what interval(s) is the function concave up?For what intervals(s) is it concave down?

d. Sketch the graph using the information from a-c.

4. Let f(x) = 3x2 - 5x + 1.a. State carefully what properties off guarantee that

the Mean Value Theorem applies to f on the in-terval [1, 2].

b. This theorem asserts the existence of a certainnumber c. Find this c for f on [1,2].

5. A flood light is positioned on the ground shining ona white wall 30 feet away. A woman 5 feet tall startsat the wall and walks towards the light at a rate of 4ft./sec. How fast is the height of her shadow on thewall increasing when she is 10 feet from the light?

8. Suppose that f(x) is continuous for all real values ofx and symmetric about the y-axis, F is an antideriva-tive of f, and F has values F(0) = 3, F(1) = 5, andF(2) = 9. Find

a. f0 f(x) dx

b. f21 f (x) dx

9. Evaluate the following definite integrals.

a. f°1 x(x2 + 1)5 dx

34 1b. dx8

10. Let f(x) be the function with graph as shown. An-swer the following questions from the graph. Youshould not try to find an equation for f (x).

11.

a. Find fo4 f

b. For what number a, 0 < a < 4, does foa f(x) dxhave the smallest value?

Find the area enclosed by the two curves y = 2x2 + 1and y = X2 + 5.

12. Suppose that gf (t) dt = x +gt f (t) dt. Give anexplicit algebraic expression (not involving integrals)for f(x).

13. For each part set up do not evaluate the integral(s)which give(s) the result. Both parts relate to thegiven graph of y = cos x.

6. Find the following indefinite integrals. Do not sim-plify your answers.

3x4 + 20a. dx

V

1-7b. f -il 3 - dt

1/2

c. f sin(2x + 1) dx 0 2r/6 r/3 x/2

d. f(4x4 2x + 1) dx a. Find the volume of the solid formed by rotating

7. Find the solution to the differential equation fl = the shaded region about the x-axis.

(.3+1)y2 subject to the initial condition y = 1 when b. Find the volume of the solid formed by rotatingx = 2. the shaded region about the y-axis.

201

192 EXAMINATIONS

14. An object starts from rest and moves with a velocityin m/sec given by v = t2 2t, where t is time inseconds.

a. Find the displacement of the object between thetimes t = 0 and t = 3.

b. Find the fetal distance travelled by the object be-tween the times t = 0 and t = 3.

c. What is the average acceleration between timest = 0 and t = 3?

15. For what value(s) of x does the slope of the tangentline to the curve y = x3 + 3x2 + 1 have the largestvalue? Justify your answer.

16. Set up the computation to find an approximate valuefor the integral fi3(I/ .c) dx, using the trapezoidal rulewith n = 4 subintervals, but do not do the final arith-metic.

17. Find the length of the curve y = (i) x312 from (0,0)to (3,215).

INSTITUTION: A private western university with about4,000 undergraduates that in 1987 awarded 50 bache-lor's degrees in mathematics.

EXAM: A three-hour exam (calculators allowed) on firetem calculus for a mixture of engineering, business, andscience students. Of the 277 students who enrolled inthe course, 41 were in the particular section that wasgiven this exam. Approximately 32% of the students inthe course received grades of A or B.

1. Find formulas (in split-rule form) for the functiongraphed below.

2

1f (x)

-1 0 1 2

2. Find g-X for each of the following:

a. Y =

b. y = sin2(3t), = cos(3t)

c. tan(1 + xy) = 86.

3. Suppose the position of a moving object at time t(seconds) is given by s = (t2 + 1r feet. Find eachof the following:

a. The average rate of change of s between t = 0 andt = 1.

b. The instantaneous velocity at t = 1.

c. The acceleration at t = 1.

4. Evaluate each of the following limits:lim

1x1_7'06'5*

c. aim

5. Suppose y = (x) =a. State the equations of any horizontal or vertical

asymptotes to the graph of f (x).

b. Find 2- (the derivative of the inverse function)and evaluate it at x = 3.

c. Sketch the L, aph of f(x)

d. Find -alz f(f(x))

6. Sketch the graph of y = 3x4 + 16x3 + 24x2 2, aftercompleting tables for the sign of y' and y". Labelturning points. Also, find the values of a and b suchthat the curve is concave down for a < x < b.

a.

b.

7. A printer is to use a page that has a total area of 96in2. Margins are to be 1 in. at both sides and 11 in.at the top and bottom of the page. Find the (outer)dimensions of the page so that the area of the actualprinted matter is a maximum.

8. A variable line through the point (1,2) intersects thex-axis at the point A(x, 0) and the y-axis at the pointB(0, y), where x and y are positive. How fast isthe area of triangle AOB changing at the instantwhen x = 5, if x increases at a constant rate of 16units/sec.?

9. Let f (x) = 07=2. Find a point x = c betweena = 1 and b = 2 at which the slope of the tangentline equals the slope or the line connecting (1,1(1))and (2, f(2)). Also, what is the name of the theoremthat guarantees the existence of the point x = c?

10. a. Find all functions y(x) such that = x2 +

b f VIL:i

C. f x3(2 3x4)7dx


INSTITUTION: A large midwestern state university withapproximately 25,000 students. In 1987, 50 studentsreceived bachelor's degrees, 6 received master's degrees,and 3 received Ph.D. degrees in mathematic.).

EXAM: A two-hour exam (cfaculators not allowed) onelementary calculus for business students. Students areallowed one 5/8 "crib card." Of the 640 students whoenrolled in the course, approximately 60% passed, in-cluding 27% who received grades of A or B.

1. Find the equation of the tangent line to the curvey = 4z3 3z + 30 at the point where z = 2.

2. Evaluate the following limits:x3 + 4z 21

a. urn b.pins

c. urnz-00

z 33z4

1 + 5z4

(z z2

3. a. If y = ln (z4 + z2) + eX3, find lat.

b. If F(x) = (z + 1)4(x + 2)3, find Fi(x).1

'c. If y =

z3

+find y'(1).x+ 2

d. If z2 + zy + y3 = 3, use implicit differentiation tofind :III in terms of z and y, Evaluate a at thepoint (1,1).

4. Find the absolute maximum and the absolute mini-mum of F(x) = z2 4z+ 1 in the interval 1 < z < 4.

5. a. If y = f, find the differential dy.b. Use differentials to estimate ffii.

6. Sketch the graph of y = z3-6x2 and use it to identifya. z-intercepts

b. critical pointsc. inflection points

d. intervals in which y is decreasing

e. intervals in which y is concave up

7. Evaluate the following integrals:z2 ers dx

x dxb. f z2 + 1

c. f z e dx (by parts)

d. fi (3x- + ;15-) dxe. f(2z-1 + 3ex) dx

8. Sketch the curves y = x3 and y = 2z and computethe area between them.

9. The Pullman company knows that the cost of makingz spittoons is C(x) = 50 + z + 16r.

a. Find the member z of spittoons that must be madein order to minimize the cost.

b. What is the minimum cost?

10. Let F (r , y) = 8z2 + 6y2 8z + zy + 10. Findba. FV1, 2) b. Fy(1, 2)

c. Fxx(1,2) d. Fzy(1, 2)

11.z3 y2

3 2a. Find the critical points of F(x, y) = + zY-

b. Determine any relative maximum, relative mini-mum, or saddle points.

12. Use thy: method of Lagrange multipliers to find theminimum of F(x, y) = x3 + y2 subject to the con-straint 4z + 2y = 10.

203

Final Examinations for Calculus II

The following twelve examinations represent a crosssection of final examinations given in 1986-87 for stu-dents enrolled in 0 e second calculus course. The exami-nations come from large universities and small colleges,from Ph. D. granting institutions and two year colleges.

Brief descriptive information preceding each exam-ination gives a profile of the institution and circum-stances of the examination. Unless otherwise :soled, nobooks or notes are allowed during these examinations;calculator wage is explicitly noted where the informa-tion wa.7 provided. Except for information on problemweights and grading protocol, the texts of the examina-tions are reproduced here exactly as they were presentedto the students.

--

INSTITUTION: A southeastern land-grant universitywith 12,500 students that in 1987 awarded 65 bachelor'sdegrees, 26 master's degrees, and 5 doctor's degrees inmathematics.

EXAM: A final exam (calculators allowed) from onesection of second terra calculus for engineering stu-dents. Of the 390 students who originally enrolled in thecourse, 72% passed, including 38% who received gradesofA or B. 32 :;',udents were in the section that took thisexam.

1. Find the area of the region bounded by the curvesy = 6 X2 and y = x.

2. Consicilr the region R bounded by the x. -axis, thecurve y = 1/x, and the lines x r: 1, x = 4. Find thevolume, V, of the solid obtained when R isa. revolved azoun the x-axis.b. revolved around the y-axis.

3. Find the length of the curve y = x3/2 from (0, 0) to

(51A.4. In 1977 the world's population was 4.3 billion persons

and growing at a rate of 2.12% per year. If thiscontinues,

'. how long will it take for the world's population todouble?

b. what will the population in 2000 AD?

5. Find y':a. y = In x2/3

c. y = erre. y = xm

g. y = csc

i. y = x? sin-1 x

b. y = ln(sec x)

d. y = e'""3if. y = 2z7+1

h. y = log2 ez2

j. y = tan-1(7rx)

6. Antiderivatives

u. f yele dy

b. f(3/et)V1 + et dtc. f sect z/(1 tan z) dz

d. f ve7 dxc. f tan(e/z)/x2 dx

f. f 5;1 dx

rj i2

h. f x-3 In x dx

i. f cos3(2z)clz

1dxf (2 + x2)3/2

k. f dx

7. Determine whether the following series are absolutelyconvergent, conditionally convergent, or divergent:show your work.

00 k2a. E ( )k

k =0

c.k=1

oo kcoskb. 2.3(-1) ,

k=1

8. Write the Taylor series expansion for c = 0 and

a. ez b. sin x

c. cos x

9. Find all x's for which the following power series con-verge.

(....1)kzk

2

204

a.k= 1

k

00

b. Ektz 4>k3k

k =0

FINAL EXAMS FOR CALCULUS II 195

INSTITUTION: A midwestern church-related college with400 undergraduates that in 1987 awarded 5 bachelor'sdegrees in mathematics.

EXAM: A two-hour exam (calculators allowed) givento students in a second term calculus course. Of the 7students who enrolled in the course, 6 passed, 3 of whomreceived a grade of A or B.

1. a. Sketch (carefully label) the region bounded by thetwo curves, y = X2 and y = x + 2.

b. Find the volume of the solid generated by revolv-ing the above region about the x-axis.

2. a. Determine the eigenvalues of A = ( 2 22 5)b. Determine the eigenvectors for A.c. Carefully and thoroughly discuss the graph of the

equation 2x2 4xy + 5y2 = 6.

3. a. Carefully sketch (with labels) the graphs of r =1 + cos 8, r = 2 cos O.

b. Identify by name these curves.c. Determine the points of intersection of these

cur ":s.

d. Determine the area inside r = 2 cos B that isoutside r = 1 + cos O.

4. Determine the following:

a. f (xex)/(1 + x2) dxb. f(2x2 + 3)/(x(x 1)2) dx

C. 11111 (X + 1)inxx-40+

d. fc°3 1/(x(ln 2)2) dx

5. Determine the most general form of the solutionthe system of equations

1 4 2 1

1 1 1 x3 4

8. a. Write the equations of the tangent(s) to the curvex(t) = i2 2t + 1, y(t) = t4 4t2 + 4, at the point(1, 4).

b. Is there any point at which the curve has a verticaltangent? Explain.

INSTITUTION: A public Canadian university with 14,000students that in 1987 awarded 52 bachelor's degrees, 5master's degrees, and 2 Ph.D. degrees in mathematics.

EXAM: A three-hour exam (calculators allowed) for allstudents in a year-long course for social science stu-dents. Of the 820 students who enrolled in the course,69% passed, including 32% who received grades of A orB.

1. INVENTORY CONTROL PROBLEM: The success ofdeclin de ]'empire americain" inside Canada has

given hope to the Memphremagog Music companylocated in Saint-Benoit du-Lac, Quebec and they arenow optimistic that Canadians might even start buy-ing Canadian music. They have predicted that theycan sell 20,000 records in the next year and thus a de-cision must be made as to the number of productionruns. Since each run involves a production set-upcost of $200 they do not want too many runs. Onthe other hand there are storage costs involved withkeeping records on hand so they do not want the pro-duction runs to be too big. The total storage cost forthe year is $.50 times the average number of records

of on hand during the year.

a. Let x be the number of records produced in onerun. Assume that the records are sold at a con-stant rate in the peiiod from the beginning ofone run until the beginning of the next run andthat a run is sold ont just as the next one starts.Draw a diagram to indicate the production pro-cess throughout the year.

b. What will be the average number of records onhand from the beginning of one run until the be-ginning of the next? Why is this number also theaverage number on hand during the year?

c. Find the number of runs that the MemphremagogMusic company should have so as to minimize itstotal set up and storage costs.

d. Show that the value obtained does indeed corre-spond to minimum cost.

6. a. Determine the equation of the plane through thepoints P(5, 2, 1), Q(2, 4, 2), R(1, 1, 4).

b. Determine a unit vector that is perpendicular tothe above plane.

7. Assume M2x3 is a vector space. Which of the fol-lowing are subspaces of Mzxs?

a. W ={ (d 0 o) I where b = a + c}

b. U = {(ad 0 0c)I where b d = 1}

205

196 EXAMINATIONS

e. [Optional cultural question.] What other famousmusical organization, aside from the Memphrem-agog Music company, is lccated in Saint-Benoitdu-Lac, Quebec?

2. The Camrose (Alberta) Souvenir corporation spe-cializes in the manufacture of models of oil drillingrigs. Production involves a fixed cost of $5000 formarket research and tooling up, but experience hasshown that the marginal cost is constz at at $2 peroil rig. All items produced are sold. The demandequation for the rigs is x = 10000 - 1000p.a. Find the cost function.b. Sketch the cost and revenue functions on the same

graph using the production level as the indepen-dent variable (abscissa).

c. What range of production levels should the com-pany analyze?

d. For what levels of production does the companymake a profit?

3. The economists at the Bank of New Brunswick havediscovered a new semi-cyclic law governing mort-gage rates; if t is time, in years, measured fromApril 12, 1964 then the interest rate r is given byr(t) = sin(ln[x3 + ln(sin x3)]). On the other hand theNew Brunswick board of realtors has noticed thathousing starts varies with the mortgage rate accord-ing to the formula N(r) = 4.591. (10)3 e-(5r2+2.71).

At what rate was the number of housing starts chang-ing seven years after the April 12, 1964 date? Do notsimplify your answer; at certain points in your calcu-lation you may simply indicate which quantities youwould evaluate numerically.

4. Because of a drop in demand for salmon the ministerof fisheries has decided to reduce the allowable catchfor British Columbia fishermen. The demand hasbeen dropping according to an exponential function.The records for the month of January 1986 were lost,but the demand for March and May were 400 and 100metric tonnes respectively.

a. Find a numerical relationship between the rate atwhich the demand is dropping and the demand atany point in time.

b. Tell the minister what the demand was in January1986. Simplify your answer as much as possible.

c. Tell the minister when the demand for salmon willbe zero.

5 In order to predict how many branch plants it shouldset up in nQrt}'ern Saskatchewan to produce its uni-que brand of junk food, the Aunt Sarah corporation

has made a population study of the Beuval region. In1970 the population was 25. It was found that start-ing in 1970 people entered the region (this includedbirths and immigration) at the rate of 20 + 200t peo-ple per year where t is years measured from January1, 1970. With the passage of time people also leavethe region (due to deaths and emigration) and it wasfound that if there are p people in the region at agiven time, then t time units later peit of thesepeople will still be there.a. Out of the 25 people in the Beuval region in 1970

how many will be there in 2000?b. Approximately how many- people entered the re-

gion in the first two months of 1972?c. Of the people who entered the region in the first

two months of 1972 how many will still be therein the year 2000?

d. Find an expression for the population of the Beu-val region in the year 2000. Do not evaluate thisexpression.

6. The CCCC (Canadian Calligraphy and Clacker Com-pany) produces two kinds of typewriters, manual andelectric. It sells the manual typewriters for $100 andthe electric typewriters for $300 each. The com-pany has determined that the weekly cost of pro-ducing x manual and y electric typewriters is givenby C(x, y) = 2000 + 50x + x2 + 2y2. Assuming thatevery typewriter produced is sold, find the numberof manual and electric typewriters that the companyshould produce so as to maximize profits. Show thatthe value obtained does indeed correspond to maxi-mum profit.

7. Different items that a consumer can purchase havedifferent "utility" values, i.e., how much the item is"worth." If the consumer considers two items at atime, then they will have a joint utility value. Sup-pose that a consumer has $600 to spend on two com-modities, the first of which costs $20 per unit and thesecond $30 per unit. Further let the joint utility of xunits of the first commodity and y units of the sec-ond commodity be given by U(x, y) = 10x6y.4. Usethe method of Lagrange multipliers to determine howmany units of each commodity the consumer shouldbuy in order to maximize utility. [No Le: a joint util-ity function which is of the form U(x, y) = C xaybis called a Cobb-Douglas utility function.]

8. Evaluate:

20;

J23 X3

a. 2 7x4 + 5 dx

c. f x in x dx

b. f cos(4x + 2) dx


9. Use an appropriate 2" Taylor Polynomial to esti-mate 171.7. Do not simplify the arithmetical quanti-ties that you obtain.

INSTITUTION: A mid-Atlantic suburban community col-lege with 3500 students that awarded 750 associate de-grees in 1987.

EXAM: A two and one-half hour exam (calculators al-lowed) for the second term of calculus for engineer-ing, science, and mathematics students. Students areallowed to use a handbook of tables, together with aninstructor-supplied set of basic formulas. Of the 16 stu-dents who enrolled in the course, 93% passed, including63% who received grades of A or B.

1. Find the area under the curve r(z) = 1,/T n.. fromz = 1 to z = 8. Sketch the region.

2. Find the area between the curves f (z) = andandg(x) = x. Sketch the region.

3. Find the volume of the solid of revolution generatedby revolving the region bounded by y = 1/x, thez-axis, x = 1, and x = 6 about the z-axis. Sketch.

4. Find all points of intersection of the curves r = 22 cos 0 and r = 2 cos 0; sketch the system.

5. Using the curves in problem 4, find the area betweenthe curves.

6. Find an equation of the curve where d2y1dz2 = 18z-8 if the curve passes throuet (1, 1) and the slope ofthe tangent line is 9 at (1, 1).

7. Find four derivatives:a. y = 1n3 5z b. y = ln(5x)3c. f (t) = i2eSe d. 7.(0) = 2 cos(202)

e. x2y = tan-1(x/y)

8. Evaluate five integrals (without using tables):6X2

a. fi(x3 + 2)4

dx

b. fo4zez dzc. f 32e-1 dt

-I-d. f Z3 ± dx

cos 30e. fsin4 30

dO

f. f sin4 2z dx

9. Simplify:

a. lim (1 + 1)n-3n--+c n b. lim (1 + krn--.-Foo

10. Find the angle of rotation for the conic Nigx2-1-3xy =Aid /2.

11. For the conic X2 1 4y + 8y + 6z = 3, determine thecoordinates (h, k) for the origin in the x'y' plane.

12. Identify each conic and sketch:a. X2 6z + 12y = 3 b. y2 = 4x2 + 64

13. Find each limit, if it exists:a. lim (1 cos 9)/92 b. lim x2/ex

0-.0c. lim cot x/ In x d. lim (sec z tan z)

x--0+ x.7r/2+

14. Write the Taylor polynomial for n = 4 at c = 1 forf(x) =

15. Determine whether each of the following convergesor diverges:

dx dua. r

0 1 b. JFcc-°° u2 + 4u + 5-

c. dzJc* z

INSTITUTION: A publicly funded Canadian universitywith approximately 30,000 students that in 1987 awarded125 bachelor's degrees, 15 master's degrees, and 6 Ph.D.degrees in mathematics.

EXAM: A three-hour exam (calculators allowed) given.to all students completing a year-long calculus coursefor students of finar. :e, commerce, business, and eco-nomics. Students are allowed one 81 x 11 "aid sheet"in the student's own. handwriting. Of the 750 studentswho enrolled in the course, 69% passed, including 19%who received grades of A or B.

1. You have $1500 to invest for 120 days. Which of thefollowing investment strategies is better, and by howmuch?

(A) 8% annual interest compounded daily.(B) 8.25% simple annual interest.

2. Calculate f22(x+ 1) dx.

3. If y = f(z), find f (z + yy') dz.

4. If y3 = z 1,'find limo(y 1)/z.

20

198 EXAMINATIONS

5. Let g(z,y) = zyf(z, y), where f(2,1) = 1, -2(2,1)= 1, §ty-(2, 1) = 1. Find the value of 94(2,1).

[6. Compute1 1/2

1/3 1/4r

7. Find a saddle point and a local minimum point for(x, Y) = e + y3 3zy.

8. Find (graphically or otherwise) the maximum valueof 6z + 4y, subject to 5z + 4y < 100, 2z + y < 30,

+ 4y < 108, > 0, y > 0.

9. Find the volume obtained by rotating y = X2 aboutthe y-axis, 0 < < 2.

10. If en = 10, find 2..

11. Evaluate file-Ix' dz using Simpson's rule with fourintervals.

12. Evaluate:

a. f z31n(5z) dx

b. f (z2 + 3) / (z2 + 2x + 1) dz

c. f e cosh 2z dz

13. Given f (z) =

a. Graph Az) for 2 < x < 4, showing all localextrema and inflection points.

b. Find the absolute maximum and absolute mini-mum of Az) for 2 < < 4.

c. Evaluate f0 f (z)

15. A rectangular 3-story building is to be construct-ed of material costing $100 per square metre forfloors, $200/m2 for roofing, side and back walls, and$400/m2 for the front wall. If $144,000 is availablefor materials, then, neglecting all other costs, findthe dimensions of the building that will enclose thegreatest volume.

INSTITUTION: A vi-uthwestern public university with45,000 students that in 1987 awarded 36 bachelor's de-grees, 10 master's degrees, and 1 Ph.D. in mathematics.

EXAM: A three-hour, closed book final exam (calcula-tors allowed) from one section of second term calcu-lus for engineering and science students. Students areallowed to use pre-approved outline notes during theexam. Of the 1150 students who originally enrolledin the course, 77% passed, including 35% who receivedgrades of A or B-4 % A's, 31% B's. 113 students werein the section that took this particutar exam.

1. Using limit theorems (like "the limit of a sum is thesum of the limits"), prove that

2nlim sin n = 0.

3n2 + 1

2. State (with reasons) whether the infinite series

CO

E(-1)k/((k + 2) ln k)k=2

14. A factory assembles sedan cars with 4 wheels, 1 en- is absolutely convegent, conditionally convergent, orgine and 4 seats, trucks with 6 wheels, 1 engine and divergent.2 seats, and sports cars with 4 wheels, 1 engine and2 seats.

a.

b.

c.

d.

How many wheels, engines and seats are requiredto assemble x sedans, y trucks and z sports cars?

How many sedans, trucks and sports cars respec-tively can be assembled using exactly 218 wheels,47 engines and 134 seats?

If sedans sell for $10,000, trucks for $12,000 andsports cars for $9,000, find the total value of out-put in part b.

If an extra pair of wheels, an extra engine, andan extra pair of seats become available, can theoutput value in part c be increased? Is so, howmuch?

20

3. Evaluate the improper integral Lc° (z) dz if f (z) =z3e-x . (Note that f' appears in the integral).

4. Find the interval of convergence of

co

E(z)k/(2k In k)k=3

5. Let SZ denote the banana-shaped region betweenthe graph of y = z2 and that of y = z3. Findfnf.zy2 dx dy.

6. Write down (but do not evaluate) a formula usingonly a single integral of a function of one variable forthe area of the region inside the graph of r = 3 cos 0but outside the graph of r = cos 0, where 0 < 0 <

r.,


7. For each limit below, find the limit if its exists and 4.call the limit non-existent otherwise:

a. lim(1 -2h)1/h b. lim n2/"h-40 n-koo

C. lim sin n/70/3n-4--oo

8. Find the Taylor series expansion in powers of x - 2 5.

for cos 3x.

9. Find the Taylor series in powers of x for an an-tiderivative of x'lln(1 + x).

10. Show that the point (x, y) = (97r2,0) is in the inter-section 0 :-= 7r.

11. You're suppose to design a fully enclosed rectangu-lar box (having all six of top, bottom, front, back,left, and right) whose cost is exactly $60, where thematerial from which the six faces are made costs $2per square foot. Find the dimensions 1, w, and li ofsuch a box having the largest possible volume of allsuch boxes.

12. Find lim ln(2/(1 + x))/(x - 1)

13. Find lim (57F 1)1/xz-.00

INSTITUTION: A midwestern liberal arts college with2200 students that in 1987 awarded 3 bachelor's degreesin mathematics.

EXAM: Two two-hour exams (calculators not allowed)from two different sections of second term calculus takenprimarily by prospective mathematics, computer sci-ence, economics, and science majors. Of the 75 stu-dents who originally enrolled in the course, approxi-mately 83% passed, including 55% who received gradesof A or B. 24 students were in the section that tookthe first of these exams, 18 in the section that took thesecond exam.

1. Find the area bounded by the graphs of y2 = 1 - xand 2y = x + 2. (Use a horizontal element of area.)

2. Find y' ifa. y = sin-1(5x - 1)b. y = ln(3x4 + 5)3/2

c. xy2 = ex - elld. y = (2x + 3) sin3(x2)

3. Find:a. f(2x)/( 9 -V------.x4) dx b. f xln x dx

Graph y = ez/x and identify the value of x thatcorresponds to a vertical asymptote, the value of ythat corresponds to a horizontal asymptote, and thevalue of (x,y) at which a minimum occurs. [Note:y' = (ex(x - 1))/x2; y" = (ex((x - 1)2 + 1))/x3]

f(9x + 27)/(x4 + 9x2)dx

6. a. lim (1 - (3/x))2zx-400

b. lim (x In x)/(x2 + 1)Z-400

7. Argue the case for convergence or divergence. Findthe value if convergent:

a. .1;7 : dx xdx

8. Find the Taylor polynomial for f(x) = In x if a = 1and n = 4. Also give R4.

9. The region in the first quadrant bounded by thegraphs of y = 1/VE, x = 1, x = 4 and y = 0 isrevolved about the x-axis. Find the volume of thesolid of revolution.

00

10. a. Explain why E 3 :- L r. i is divergent.n=1

b. Find' the exact sum of the geometric series

1.

2-+1n=2

c. If an = (2n. + 1)/(4n. + 3) then lim an =?n.-..co

Evaluate the following limits:

a. lim (ex + 3x)1/xz.o+

x 4b. lim

z-4-3 ((x2 + 2x - 3) (x + 3)/

2. Differentiate the following functions. Derivativesneed not be simplified.

a. 111/73 b.

c. (cos x)x+1 d.

xln x

sin3 e-2x

3. Evaluate the followhig integrals:

a. f 1/(x + x3) dxc. f X e-4r dx

209

b. f 71--Pix dxd. f ow 11!/T+-1 dx

EXAMINATIONS

4. Set up integrals that could be evaluated (but do notevaluate) to find the volume of the solid generatedby revolving the region bounded by the graphs ofy = x3, x = 2, and y = O.

a. about the x-axis;

b. about the y-axis;

c. about the line x = 3.

5. An above-ground swimming pool has the shape of aright circular cylinder of diameter 12 ft and height5 ft. If the depth of the water in the pool is 4 ft., findthe work required to empty the pool by pumping thewater out over the top.

6. Find the local extrenia for f (x) = xlnx, x > 0.Discuss concavity, find the points of inflection, andsketch the graph of f.

7. The rate at which sugar decomposes in water is pro-portional to the amount that remains undecomposed.Suppose that 10 lb. of sugar is placed in a containerof water at 1:00 P.M., and one-half is dissolved at4:00 P.M. How much of the 10 lb. will be dissolvedat 8:00 P.M?

8. A person on level ground, Zkm from a point at whicha balloon was released, observes its vertical ascent.If the balloon is rising at a constant rate of 2 m/sec.,find the rate at which the angle of elevation of theobserver's line of sight is changing at the instant theballoon is 100 m. above the level of the observer'seyes. [NOTE: 1 km= 1000m.]

9. Find a second degree Taylor polynomial for f(x) =sin(x) around a = ir/4. Use this polynomial to es-timate sin(46°), but do not simplify. Find an upperbound on the error of your estimate.

INSTITUTION: A highly selective northeastern privateuniversity with 8,500 undergraduates that in 1987 award-ed 28 bachelor's degrees, 3 master's degrees, and 10 doc-tor's degrees in mathematics.

EXAM: A three-hour final exam (calculators not al-lowed) for all students who take second term calculus.Of approximately 160 students who originally enrolledin the course, 92% passed, including 47% who receivedgrades of A or B.

1 j34(2x)/(x2 323 + 2) dx =

(A) ln 3 - ln 2 (B) ln(3/4)(C) ln 2 - ln 3 (D) 61n2 21n 3(E) Does not converge.

2. fr xe-3z2 dx =(A) 1/6 (B) -2/9 (C) 0(D) 1 (E) Does not converge.

3. Which integral gives thevolume generated whenthe area under the arc ofy = sin x, 0 <x < ir isrotated about the vertical line x = 2/r?(A) Er x sin2 x dx(C) .17 2rx sin x dx(E) f0 2iral sin x dx

2

4.J dx1

(A) 1 (B) (15 - 1)/2 (C) ir/6

(D) 9/26 (E)

5. The area of the region R shaded below can be rep-resented by which of the following? (R is the regionoutside the circle r = 3 cos° but inside r = 2 - cos 0and above the x-axis.)

(B)IF

2rx sin2 x dx(D) 2r(2r -x) sin x dx

(A)

i1112 (2 cos 0)2 r (2 - cos 0)2r/3 2 Jr/2 2

(B)-1 (2 - cos 0)d0 - I1112 1(3 cos 0)d0

r/3 2 r/3 2

(C)1. (2 - coo 0)2 de - I 1113 1(3 COS 0)2 de

r/3 2 0 2

(D)

Ix

1(2 - cos 0)2 de - f2 1(3 cos 0)2 d0rp 2 ,,./3 2

(E)"J

x1 (2 cos 0)2 d0 - I 1(3 cos 0)2 d0

,,./3 2 ./3 2

6. fe 6 .13-iL12- dx is closest to:

(A) .302 (C) .588(D) .602

(B) .282(E) .988


7. The first few terms of the Taylor series forx/(1 + x) about x = 0 are:(A) x + x2 + X3 + X4 f X5

(B) X X2 + X3 X4 + X5

(C) X X2 + 2X3 3X4 + 4X5

(D) X2 + 2IX3 31x4 + 4!x5

(E) x 2!x2 + 3!x3 4!x4 51x5

8. Let x be a positive but small quantity. Use the firstfew terms of the Taylor series for the functions x,sin x, and ex 1 to determine which of the followinginequalities is correct:

(A) x < sin x ez 1

(B) x < x < ez 1

(C) ez 1 < sin x x

(D) ez 1 < x < sin x

(E) sin x ez 1 < x

9. A medication has a half life of 6 hours. A persontakes a first dose of the medication now, the seconddose in six hours, and continues to take the medicineevery six hours. If the dose is 10 grams, the amountof medication in the blood immediately after the n-thdose will be:

(A) 20(1 (1)n) grams(B) 20(10(x)n-1) grams(C) 20(1 (Dn.") grams(D) 20 grams

(E) 20(f )n grams

10. If !if! = cos x/ev and y(0) = 0, then y =(A) 0 (B) In 2 (C) 1 (D) 1/2 (E) 1

11. A boat is driven forward by its engine, which exertsa constant force F. It encounters water resistancewhich exerts a force proportional to its velocity. Letv = 1(1) be the velocity of the boat at time t, m bethe boat's mass, and k be a positive number. Whichof the following equations describes the situation?

(A) dv /dt = k(F kv) (B) dv Idt = m kv(C) dvldt = kv) (D) dv/dt = krkv(E) dv/dt = k(Ft kv)

12. Find the solution to y" + 6y = 0 which satis-fies the following conditions: i) as t + co, y 0;

ii) V(0) = 6.

(A) y = e-t (2 sin 3x + 3 cos 2x)

(B) y = 6e2t + 2e-3t

f(x) = (C) y = 2e-3t(D) y = 3e-2t(E) y = e-t(3 sin 2x + 2 cos 3x)

13. Which of the following is the graph of the solution2yd dy

to dt2y = 0 and with the initial condition

y(0) = 0, V(0) > 0?(A) (B)

(C)

(E)

14. e(12)i =(A) VF(D) 1 + i

(D)

(B) i(E) cos (ii.)

(C) 1

15. To approximate the value of In 5, use the fact that .In 5 = fi5 e dt and estimate the integral by dividingthe interval of integration into four equal pieces andusing the trapezoidal rule. Which of the followingsums will you get?

(A) [In 2 + In 3 + ln 4 + In 5]

(B) z [1 + 2 (z) + 2 (a) + 2 (1) +

(C) 2[1 +2 +3 +4 +5J( D ) z [1 + 2 (a) + 2 (a) + 2 (q) + sl

(E) El +1+ +1116. a. Find the second degree Taylor polynomial (i.e.,

up to and including the quadratic term) for thefunction f(x)= VI about the point x = 16.

b. What approximation does part (a) give for VIT?(You may leave your answers a sum.)

c. Estimate the error in the above approximation.You may use the fact that 3.8 < 15 < 3.9. Youmust justify all steps. You may leave powers of3.8 or 3.9 unsimplified in your answers.

n

202 EXAMINATIONS

17. A part manager plans to put 100 goldfish in a man-made pond in her park. The birthrate (goldfish perweek) of goldfish is proportional to the populationof goldfish, with proportionality constant 10. Giventhe size and the ecological conditions of the pond, themanager calculates that the death rate (goldfish perweek) is proportional to the interactions between thefish (i.e., is proportional to the square of the goldfishpopulation) with proportionality constant .04.

a. Write a differential equation which describes thissituation. (Let P = f(t) be the number of goldfishin the pond.)

b. Will the number of fish reach an equilibrium? Ifso, what is it?

c. For what value of P is the fish population increas-ing most rapidly? You must give a complete math-ematical justification of your answer.

d. Graph the number of fish versus time.

e. Describe in words what would happen if the parkmanager originally put 350 goldfish into the pond.

18. Barnacles grow on the outside of a cylindrical dock-post. The density of barnacles at a depth d feet belowthe surface of the water is given by 25 (3/4)d bar-nacles per square foot, from the ocean floor to thesurface of the water. It has a radius of 1/3 ft.

a. What Riemann sum approximates the total num-ber of barnacles on the dockpost? Please justifyyour answer carefully.

b. Use your answer to part (a) to find an integralwhich represents the total number of barnacles onthe dockpost.

c. Find the total number of barnacles on the dock-post.

19. A contagious fatal disease is spreading through agrowing population. Let I denote the number of in-fected individuals and let S denote the number ofsusceptible individuals. Assume that in the absenceof the infection, the growth of S would be exponen-tial: S grows at a rate proportional to itself withproportionality constant 2. Assume that if everyonewere infected, the population would die out exponen-tially according to the law I = /0e-3t and hence thedeath rate of the infected population is proportionalto I. Now suppose that the spread of the infection isdirectly proportional to the product of S and I withfactor of proportionality .5.

a. Write a system of differential equations for dS/dtand dI /dt describing the above situation.

21 2

b. Is there any pair of positive values of S and I atwhich both S and I are constant for all time? Ifso, what are these values of S and I?

c. Sketch various trajectories of the system of equa-tions in the first quadrant of the S I plane.(Please draw arrows indicating direction.)

d. What is the average value of S? You need notjustify your answer.

INSTITUTION: A private Midwest university with 7,500undergraduate and 2,000 graduate students that in 1987awarded 33 bachelor's degrees, 6 master's degrees, and6 Ph.D. degrees in mathematics.

EXAM: A two-hour exam (calculators not allowed)given to all students 'in a second term calculus coursefor students enrolled in the colleges of Business andArts and Letters. Of the 263 students who enrolled inthe course, 87% passed the course, including 66% whoreceived grades of A or B.

1. The limit of sequence 1/2,2/3,3/4, 4/5, ... is(A) 5/8 (B) 1/e (C) 1(D) 1(2 (E) diverges

2. In the partial fraction decomposition of (3x2 10z -I-12)/(z 1)(z2 4), the numerator of the term withdenominator z 2 is

(A) 2 (B) 1 (C) 0 (D) 1 (E) 200

3. The sum of the infinite series E (5n 5(1+1)) isn=1

(A) 4/5 (B) oo (C) 0(D) 4/25 (E) 1/5

4. The graph of y = 3 sin (ir/2 most closely resern-bles

(A)

(B)

FINAL EXAMS FOR CALCULUS II

(C)

(D)

(E)

203

9. The sum of the infinte serif (4/5)4 - (4/18 -I-(4/5)12 - (4/5)16 + is

(A) 256/881 (B) 369/125(C) 5/4 (D) 256/369(E) series diverges

10. The improper integral fr 1/x2 dx equals

(A) diverges (B) 1 (C) -1(D) 0 (E) e

11. The function f(x, y) = y3 - x3 - 3y + 12x + 5 pos-sesses the critical points (2, -1), (2, 1), (-2, -1), and(-2,1). The point (2, -1) is(A) a saddle point.

(B) a local maximum.

(C) a local minimum.

(D) an absolute maximum

(E) an absolute minimum.

5. The coefficient of x10 in the Taylor series for sin(2x2)is

(A) 1/20 (B) 4/15 (C) 0(D) -1/6 (E) -1/10

12. If y = In I cos 2x1 then 5-11

(A) -2 tan 2x(C) sin 2x(E) 1/(1cos 2x1)

=(B) 2sec 2x(D) - csc 2x

6. Let z be defines by the implicit equation ln(xy+yz +zx) = 1. Then a is

-(y+z)(A) (B) x z

x + y x + y

(C)x + z

(D)x + y

y + z x + zx + y

(E)y + z

7. If f(x,y) = x sin(x + y), then A- =

(A) sin(x + y) x cos(x + y)

(B) x cos(x + y)

(C) cos(x + y) - x sin(x y)

(D) 2 cos(x + y) - x sin(x +

(E) -x sin(x + y)

8. Let f(x) = f. The 3rd degree Taylor polynomialabout 1 for (x) is

(A) 1 - Nri + 1(.1i)2 - i(Vir(B) 1 -I- (x - 1) - -1(x - 1)2

(C) 1 + 1(x - 1) - i(x - 1)2 + -h(x - 1)3

X2 X3(D) 1 + x + - 71-

(E) 1 + (x - 1) -I- 1(x - 1)2 + 1(x - 1)3

13. d(xln(x + y)) =

(A) (ln(x + y) x/(x + y)) dx + x I (x + y) dy

(B) (1n(x + y) + x/(x + y)) dy + x/(x + y) dx

(C) ln(x + y) dx + xl(x + y) dy

(D) ln(x + y) dy + x/(x + y) dx

(E) x I (x + y) dx + 1 /(x + y) dy

14. The volume generated by revolving about the x-axisthe area bounded by the graphs of y = x2, x = 1,and x = 2 is

(A) 327r/5 (B) 317r/5 (C) 87r/3(D) 77r/3 (E) 47r

15. How many of the following infinite series converge?i) 1 + 1/7 + 1/72 -I- -I- 1/7n -I-

ii) 3/2 - (3/2)2 + (3/2)3 - (3/2)4 +iii) 1-1+1- 1+1-14....iv) 1 -I- 1/2 + 1/3 + 1/4 + + 1/n +v) 1+ 1/2! -I- 1/3! -I- 1/4! -I- -I- 1/n! +

(A) 0 (B) 1 (C) 2 (D) 3 (E) 4

16. f xe-x dx =

(A) xe-x + e- x(C) -xe-x - e'(E) (x2 /2) - e-x

n n4 1.

(B) -xe-x + e'(D) (x2/2)e-x

204 EXAMINATIONS

17. The Taylor series for 1/(1 + t2) dt is(A) 1 + -r- x212 + +(B) x212! x4141 + x5/6! x5/8! +

26. f(7 el"(4")dx =(A) e 1 (B) 1(D) 2 (E) 1/7

(C) eir 1

(C) 1 x2/2 + x4/4 x5/6 + 27. The following is a table of values of the function y =(D) x3/3 + x5/5 x7/7 + f(x)

(E) 1 2x + 3x2 4x3 +

18. The volurrie obtained by revolving about the y-axisthe region bounded b. the graphs of f(x) :::. x2 andf(x) = x3 is(A) 7/6 (B) 7/12 (C) 7/10(D) 7/20 (E) 7/2

19. fott/2(cos x)/(1 + sin x) dx(A) In 1 (B) In 2 (C) 1(D) 2 (E) 7/2

20. limo(csc 2x)/(cot m) =

(A) Does not exist (B) 2(C) 1 (D) 1/2(E) 1

21. f 1/(x In x) dx =(A) In In(C) In x +(E) In x + 1/(ln x)

22. The surface X2 + y2 + z2 2x + 4y 6z 4a sphere with center(A) (-2, 4, 6) (B) (2, 4,6) (C) 1,:-, 3)(D) (1, 2, 3) (E) (4, 8, 12)

23. The solution of the differential equation d = y cos xgiven that y = 8 when x = 0 is(A) y = 8 +sin x (B) y=(C) y = 8e"'"' (D) y = 8 + cc"'(E) y = 8 cos x

24. The area bounded by the graphs of f(x) = X2 andg(x) = x3 is

(A) 1/12 (B) 1/6 (C) 1/4(D) 1/2 (E) 1

25. Let N(t) denote the size of a population of animals attime t. Suppose the growth rate of the population isdirectly proportional to 100 N and the constant ofproportionality is 7. Furthermore, N(0) = 10. ThenN(t) for any time t is given by(A) N(t) = 16e-7t(B) N(t) = 101: 90e-7t

(C) N(t) = 100N(t) = 100/(100 90en)

(E) N(t) = 10 + 90e7t

(B) ln(x In x)(D) 1/(x21n x)

x 0 0.5 1.0 1.5 2

f (x) 3 2 1 2 3

Use the trapezoidal rule (with n = 4) to find theapproximate value of fo2 f (x) dx.(A) 1/8 (B) 11/4 (C) 2(D) 11/2 (E) 4

28. The fifth term of the geometric series whose first termis 2 and whose ratio is 2 is(A) 6 (B) 1/16 (C) 32(D) 0 (E) 8

29. The funbction f (x, = y 1 (x2 + g2 + 9) has the fol-lowing critical points.

(A) (0, 0) only (B) (0, 3) only(C) (0, 3) and (0, 3) (D) (0, 3) only(E) (0,0), (0,3), and (0, 3)

30. If it < a < 37/2 and tan a = 1 then sec a =(A) 2 (B)

10 = 0 is (C) 2/0. (D) 1(E) does not exist

2 I 4

INSTITUTION: A public Canadian university with 14,000students that in 1987 awarded 5/i bachelor's degrees, 5master's degrees, and 2 Ph.D. degrees in mathematics.

EXAM: A three-hour exam (calculators allowed) for allstudents in a year-long course for mathematics, physicsand computer science students. Of the 190 students en-rolled in the course, 33% passed, including 13% whoreceived grades of A or B.

ANSWER QUESTIONS 1-4:

1. Find the following limits, if they exist. Explain yourreasoning.

a. lim sin (i(x 1))1

b. lim (1 + 5/t)tt,o+

c.zlim (sin z + z)/(cos z 3z)

d. lim ((2 1n(2 + h) 21n 2)


2. Determine convergence or divergence of the followingseries. Justify your answers.

1 1 1a. 1 + +V2 V3 V71

bln 3 _ In 4 ln 5 _ ln 6

3 4 5 6

x' 2kc.

k7....o 3k + (-2)k

3. Evaluate these integrals and explain clearly what theimproper integral in part (c) means.

a. 03 z2 z 6 dx

b. f x2 arcsin(3x) dx

c. f at3 + tz

4. a. Find the solution to the differential equation(sec x)il y = 4 satisfying the condition y = 1when x = 0.

b. Find the general solution of the differential equa-tion y" - 4y' + 5y = x.

ATTEMPT ANY THREE QUESTIONS:

5. Consider the function f (x) = (e-x)/(x+1) for x E R,x -1.a. Find the local extreme values of f and the inter-

vals where f is increasing and decreasing.

b. Find the points of inflection of f and the intervalswhere its graph is concave up and concave down.

c. Find all asymptotes.

d. Using this information (and any other informationyou need), sketch the graph of y = f (x).

6. a. Let 0 < K < 1. Sketch the region bounded bythe curves y = x2, y = 1fx, and y = K (only thepart where y > K should be considered). Findthe area of this region as a function of K.

b. Compute the volume of revolution obtained by re-volving the region defined in part (a) about thex-axis.

7. a. Define what is meant by the radius of convergenceof a power series. If the radius of convergence isR, what does this tell you about the convergenceof the power series? Give examples of power serieswith R = 0, 1, and oo (no proofs required).

b. Write down the power series for ex and find itsradius of convergence. Use this to find a powerseries for e-t and give its radius of convergence.

c. Using (b.), and explaining the theorems which youuse, find a power series for F(x) = fo e-t2 dt.

d. Show that 0.09666 < F(0.1) < 0.09668.

8. a. State and prove the Intermediate Value Theorem.

b. Prove that the polynomial p(x) = x3 - 3x + 1 hasat least 2 roots in the interval [0, 2].

9. a. Show that the curve given parametrically byx(t) = e-t cost and y(t) = e-t sin t for 0 < t < oohas finite length, and find this length.

b. An object of mass m falling near the surface of theearth is retarded by air resistance proportional toits velocity so that, according to Newton's secondlaw of motion, mg = mg - kv where v = v(t)is the velocity at time t, and g is the accelerationdue to gravity near the earth's surface. Assumingthe object falls from rest at time t = 0, find thevelocity v(t) for any time t > 0. Show v(t) ap-proaches a limit as t oo. Find an expression forthe distance fallen after time t.

INSTITUTION: A publicly-funded Canadian universitywith approximately 30,000 students that in 1987 awarded125 bachelor's degrees, 15 master's degrees, and 6 Ph.D.degrees in mathematics.

EXAM: A three-hour exam (calculators allowed) givento all stuc,'..lts completing a year-long calculus course forspecialist students in mathematics. Of the 100 studentswho enrolled in the course, 39% passed, including 14%who received grades of A or B.

ANSWER BOTH QUESTIONS 1 AND 2.

1. State and prove one of the following theorems.

a. The Fundamental Theorem of Calculus (bothparts).

b. The Intermediate Value Theorem for ContinuousFunctions.

c. Taylor's Theorem with Remainder for a functionf which is n 1-times differentiable on a neigh-bourhood of a point a. (Give whatever form ofthe Remainder you wish.)

206 EXAMINATIONS

2. The graph of a function f defined on R is given below.4

2

1

-5 -4 -3 -2 -1 0

b. tan x < (44/7r (0 < x < 7r/4)

c. lim In =ncod. 1 - 1/3 + 1/5 - 1/7 + 1/9 - = 7r/4

e. 1/2 - 1/4 + 1/6 - 1/8 + - = (1/2) log 2.

5. Consider the lollowing property (P) of a set S of real2 3 4 5 6 7 8 9 numbers:

Sketch the graphs of the following functions.

a. f; f(t)dt b. f (x)c. 1/[f (x))2 d. 1(2 - x)e. 1 + f(x2) f. f (1(0))

ANSWER ANY FOUR OF QUESTIONS 3-8.

1 if x = 03. a. Let f(x) = {

(sin x)/x otherwise.State whether fi(0) and f"(0) exist, andcalculate these derivatives.

b. Verify that

sint t dt _ sint2t

sin2 x sin2 t+ at

U t2

and deduce that the improper integral

f sin tdt

t

converges.444. Let In = fo tan's x dx. Show that

a. i n = 1/(n + 1) - (n = 0, 1,2, ...)

2(P) x E X =. ES.

a. Prove that, if S is bounded and satisfies (P), thensup S > 1.

b. Determine, with justification, all sets S that sat-isfy (P) and for which sup S = 1.

c. Give ail example, with justification, of a set S forwhich 1) (P) holds, ii) sup S = 2, iii) 2 gt S.

6. a. Find the solution to the equation (1 4- x2)yi + (1 -if so, x)2y = xe-z which satisfies y = 0 when x = 0.

b. Sketch the' graph of the solution obtained inpar t (a).

7. Consider I = fo V sin(x2) dx.

a. By using a series expansion of sin x2, evaluate Ito 4 decimal places,

b. By dividing the interval [0,1) into four parts, ap-proximate the value of I by using either the trape-zoidal rule or Simpson's rule.

8. a. Determine lim x log x.

b. Determine the volume of the solid of revolutionobtained by rotating the region bounded by thecurves y = 0 and y = x log x (0 < x < 1) aboutthe x-axis.

2 0

Final Examinations for Calculus III

The following seven examinations represent a crosssection of final examinations given in 1986-87 for stu-dents enrolled in the third calculus course. The exami-nations come from large universities and small colleges,from Ph. D. granting institutions and two year colleges.

Brief descriptive information preceding each exam-ination gives a profile of the institution and circum-stances of the examination. Unless otherwise noted, nobooks or notes are allowed during these examinations;calculator usage is explicitly noted where the informa-tion was provided. Except for information on problemweights and grading protocol, the texts of the examina-tions are reproduced here exactly as they were presentedto the students.

INSTITUTION: A western, state university with 16,000students that in 1987 awarded 36 bachelor's degrees, 9master's degrees, and 3 Ph.D. degrees in mathematics.

EXAM: A two-hour exam (calculators allowed) givento 26 students in one section of a third term calcu-lus course for scier.:e, mathematics, and computer sci-ence students. Of the 349 students who enrolled in thecourse, 74% passed the course, including 32% who re-ceived grades of A or B.

1. Find the length of the arc of x = t3, y = t2 fromt = 1 to t = 0.

2. Find the. area inside the graph of r = cos 20.

3. Find thelimits of these sequences, if they exist:a. an = (sin n)/nb. an = sin (1;-* -I- 71,-)

c. an = " 11/TrTT1.

d. an = (1 2/n)'"4. Test and decide if the following series are absolutely

convergent, conditionally convergent, or divergent.State the test used.

In nEc° ( -1)n In n2n=2

E00( -1)n

+1 (n!)2

=1 (2n)!

a.

b.

5.

6.

7.

0.c. E (-1)" In n3n=2

4 1Evaluate fo el= if it converges.

Estimate fo.1 In(l+ x)

dx to within .001.

e.,Find the sum E (_1)2k ..

.2k

k=0 (2k)!

8. Find the Taylor series at x = 0 for f (x)

(in powers of x).

9.

10.

ez -I- e-22

Nif-1- x 1 (x/2)Find the limit lim7-.0 x2

Find the third Taylor polynomial P3(x) for f(x) =Er:T. in powers of (x 5).

11. Find the interval of convergence of E(2x 5)t

k=1k2

INSTITUTION: A southwestern public university with12,000 students that in 1987 awarded 24 bachelor's and5 master's degrees in mathematics.

EXAM: A three-hour exam (calculators not allowed)given to 21 students in one section of third term calcu-lus for engineering, mathematics, and science majors.Of the 138 students who enrolled in the course, 50%passed, including 38% who received grades of A or B.

1. Given XV) = (et, 2et, 3et). Determine the arclengthof i(t) for t E [0, 1].

2. Let x(t) = (x(t), y(t)) = (3 cost, 4 sin t); t E (0,4Find d2y/dx2.

3. Determine the maximum absolute value of the cur-vature of y = f (x) = In(x); x > 0.

4. Determine the unit tangent vector and the curvatureK of x(t) = (t cos t, t sin t) and show that K > 0 fort E R.

5. Let x(t) = (cost, sin t, sin 2t). Show that the curva-ture of gt) vanishes at t = 1/2 arccos (± 17/12).

6. Find the unit normal vector of i(t) = (3t, sin t, cost).

2:17

208 EXAMINATIONS

7. Using the chain rule, find '4 if W=exp(x)y2 a;= t, y = 2t, arid z = 4t.

8. Find the equation to the tangent to y + sin(xy) = 1at (0, 1) by evaluating the gradient of the r pectivelevel curve f(x,y) = 0.

9. Compute the directional derivative of f(x,y,z) = 11.y2z2

10. Invert the order o' integration in

1.3 x -1

J1 Je

at (1, 1,2) in the direction of (1,0,1).

10. Find all unit vectors ii for which the directional deriv-ative of f(x, y, z) = x2 ay yz at (-1, 1, 1) in thedirection of i vanishes.

INSTITUTION: A private midwestern liberal arts collegeof 2206 students that in .1987 awarded 17 bachelor's de-grees in mathematics.

EXAM: A three-hour final exam (calculators allowed)from the only section of the third term of a calculuscourse for prospective mathematics and science majors.Of the 20 students who enrolled in the course, 85%passed, it.cluding 33% who received grades of A or B.

1. Find the cosine of the angle between the two linesegments which start at (0,0,0) and end at (2,3, -4)and (5, 2,4), respectively.

2. Let z = x2 2y2, x = 3s + 2t, and y = 3s 2t. Findz, in two ways.

3. Let z = (2x2 3x2)/(xy3). Calculate xzx yzy.

4. Let z = a2 + ay + y2. Find the directional derivativeof z at (3, 1,13) in the direction r+

5. Solve y" p' 2y = e2x given y(0) = ys(0) = 1.

6. Suppose a particle has position F(t) = t2rd- (t +t3/3)Y+ (t to /3)% at time t. Find the particle'svelocity and acceleration, and decompose its acceler-ation into tangential and normal components.

7. Solve xy' = y V;2 ----1-y2.

8. Write two triple integralsin rectangular and cylin-drical coordinatesfor the volume bounded belowby z = x2 + y2 and above by z = 2y, and evaluateone of them.

9. Find the length of the curve x = 3t2, y = 3t t3 for0 < t <2.

f (x, dy dz.

If (x, y) (0, 0), f(x, y) = (4xy)/(4x2 + y2). As(x, y) approaches (0,0) along a line through the ori-gin, f(x, y) approaches 1. What is the slope of theline?

12. Find the y-coordinate of the centroid of the first-quadrant region bounded by y = x3 and y =

13. Sketch the first-octant portion of the graph of x2 +2z2 = 3y2 and find the equatrns of the tangent planeand normal line at (2, 2, 2).

14. Find the minimum distance from (1, 1,4) to thesurface x2 + y2 + z2 + 2z 4y + 4z = 16.

15. Find the equation of the plane passing through(1, 2,3) and (3, 2,3) which is ptipendicular to theplane with equation 3x 2y + 4z = 5.

--INSTITUTION: A private university in the south with6,600 undergraduates that in 1987 awarded 8 bachelor'sdegrees, 3 master's degrees, and 3 Ph.D. degrees.

EXAM: A four-hour final exam (calculators allowed)from the only section of third term calculus for Arts andSciences students. (A separate course serves engineer-ing students.) Of the 20 students who enrolled in thecourse, 50% passed, including 20% Wu- received gradesof A or B.

1. a. Find an equation of the line through (2, 1,5)parallel to the line 1(t) (3t, 2 + t,2 t) where

oo<t<co.b. Find the parametric equation of the line of :in-

tersection of the planes 2x + y + z = 4 and3x y+ z = 3.

c. Find an equation of the plane passing through(1,0,-1), (3,3,2) and (4,5,-1).

d. Find the distance between the parallel planes 2x+ 2z = 4 and 2x y + 2z = 13.

FINAL EXAMS FOR CALCULUS III 209

2. Find (a) the unit tangent, (b) the unit normal, (c) theunit binormal, (d) the curvature, (e) the torsion ateach point of the curve

t37(t) = (t,

1/2-2 t21 ).

Also, if 7 gives the path of a moving particle, whatare the (f) tangential and (g) normal accelerations ofthis particle at time t?

3. Write down parameters equations for the two dimen-sional torus centered at the origin obtained by rotat-ing (x 2)2 + z2 = 1 about the z-axis.

4. a. What is the partial differential equation obtainedfrom the equation

02u 02u 02u5 + 200 + 2 = 00x2 xy 8y2

by substituting x = 2s t, y = s t?b. Suppose that y = g(x, z) satisfies the equation

F(x, y, z) = 0. Find a in terms of the partials ofF.

5. a. An open topped rectangular box is to have a totalsurface area of 300 in2. Find the dimensions whichmaximize the volume.

b. Find the minimum distance between the circlex2 + y2 = 1 and the line 2x + y = 4.

6. a. Find and classify all critical points of the functionf (x ,y) = 2y2 x(x 1)2.

b. Find the maximum and minimum values off(x,y,z) = x y + 2z on the ellipsoid M =

z) x2 + y2 2x2 2 }.

7. Find the total mass of an object with densit6(x,y,z) = z + 3 bounded by X2 + y2 = 4 andz = y2 and the xy-plane.

8. Evaluate f fit cos(y x)/(y+ x) dx dy where R is theregion bounded by the lines x + y = 2, x + y = 4,x = 0, y = 0.

9. Write the parametric equation of the surface givenby the equation z = 4 (x2 + y2) and then find thearea of the part of this surface which lies above thexy-plane.

10. Let F(x,y) = .1:111n(y/t)dt and compute 2: andaFay

11. Compute the line integral of F = ( y, x) around thecardiod r = 1 + sin 0 where 0 < 0 < 27r and find thearea enclosed by this curve.

12. Compute

a. 5 (y dx + x dy)/(x2 + y2), if 7 is a closed curveabout the origin.

b. 5 (x dx + y dy)/(x2 + y2) if 7 is a closed curveabout the c .igin.

13. Let = 0(71711113) where q is a constant and F =xi+yj+zk. Let V be a region surrounding the originand let S be its surface.

a. Compute f fs f fi ds.b. Compute 5 f ds where is a closed curve within

the unit sphere.

c. Explain the different effects of the singularity at(0,0, 0) on the value of the integrals.

14. By first showing that

/11V/112 dx dy dz = fV OV

conclude that the steady state temperatures within aregion V are determined by the surface temperatureswhere the temperatures u satisfy V2u = a + 4 +a2u - 0.

INSTITUTION: A major midwestern public research uni-versity with 27,000 undergraduate and 9000 graduatestudents. In 1987, this university awarded 184 bache-lor's degrees, 32 master's degrees, and 13 doctor's de-grees in mathematics.

EXAM: A three-Lur final exam (calculators allowed)from one section of third term calculus for engineeringand science students. Of the 1200 students who origi-nally enrolled in the course, 87% passed, including 55%who received grades of A or B. 150 students were in thesection that took this particular exam.

1. Locate and classify all the critical points on the sur-face z = x3 + y3 3xy.

{x = 2 t x = 1 + 2s

2. Let Li : y = 1 + 3t and L2 : y = 7 + 4s .

z =2t z = 3 2sFind the following:

a. The point where La int -cts the ,xy plane.b. The point where Li and L2 intersect.c. The equation of the plane containing Li and L.

210 EXAMINATIONS

d. The equation of the line through the point of in-tersection (part b) and perpendicular to the plane(part c).

3. Sketch the region of integration and evaluate

x2Ye ju.x dy.

o v-y- z3

4. Consider the curve with the parametric equationsx = 3t2, y = 2t3.

a. Carefully sketch the graph of the curve.

b. Find the unit tangent vector T and the unit nor-mal vector N at the point (3, 2).

c. Find the equation of the tangent line at the point(3, 2).

d. Find the equation of the normal line at the point(3, 2).

e. Find the curvature at the point (3,-2).

f. Find the equation of the osculating circle at thepoint (3,-2).

5. Let w = f(u) and suppose u = x3 y3. Find anexpression for 7.--1 and abl.

6. The circle (y 1)2 z2 1 (lying in the yz-plane)is revolved around the z-axis.

a. Write the equation of the solid of revolution.

b. Set up but do not evaluate a triple integral incylindrical coordinates to calculate the volume ofthe solid.

7. Find the equation of the line tangent at (1, - 2, 3) tothe curve of intersection of the surface z y2and the plane x y + 2z = 3.

8. Consider the surface x2 + y2 z2 1.

a. Sketch the surface.

b. Use Lagrange multipliers to find the point(s) onthe surface which are closest to (4, 0, 0).

c. Is there a point on the surface farthest away from(4,0,0)? Is so, find it; if not, explain why not.

9. Let w(t) = (t, 3t3-,t2) be the position vector of aparticle moving in three space.

a. F.nd the distance that is travelled by the parti-cle along the curve as it moves from (1,1,1) to(3,18,9).

b. What is the speed of the particle at (1, 3, 1)?

10. Suppose T is the region between the spheres x2 +y2 + z 1 and x2 + y2 + z2 = 5. Evaluate

e_(.2+y2+,2)dV.Li Vx2 + y2 + z2

11. Find the centroid of the region R shown. AssumeP(x, Y) = 1.

-2 -1 I 2

12. Show that the surface area of the region S on thecone z = + y2 which lies about the region R,satisfies the following relationship:

Area of S = VT127-1 (Area of R).

x

Tz

:42:22i

>y

13. Consider the surface Aix + A/g + = 1.a. Find the plane tangent to the surface at (a, b, c)

(where a > 0, b > 0, c > 0).b. Show that the sum of the x-intercept, y-intercept,

and z-intercept of this plane is 1.

INSTITUTION: An eastern liberal arts college for womenwith 2600 students that in 1987 awarded 25 bachelor'sdegrees in mathematics.

EXAM: A two and one-half hour exam (calculatorslowed) from one section of third term calculus for fresh-men and sophomore science and mathematics majors.Of the 70 students originally enrolled in the course,approximately 90% passed, including 67% who receivedgrades of A or B. 30 students were in the section thattook this particular exam.

1. Suppose the temperature at the .point (x, y, z) isgiven by the function T(x, y, z) y2 e2X4-3Z What is

Tz(0,2,1)? What is the significance of this numberin terms of how the temperature changes?

220

FINAL EXAMS FOR CALCULUS III 211

2. Let f(x, y) = x I (x y) and suppose a particle is atthe point (1,0). What direction should the particletravel in order that the function f decrease as rapidlyas possible?

3. What is the equation of the plane tangent to thesurface z = f(x, y) at the point (xo,Y0, f(x0, Yo))?

4. Let f(x, y) =11 + x2 - yl. Does h (0,1) exist? Doesfy(0, 1) exist? Be sure to give reasons for your an-swers.

5. Suppose f (x, y) = ev(x2-1). What does the contourline for f =1 look like?

6. Find all the critical points of xy4 + cos x but do notclassify them.

7. Use the method of Lagrange multipliers to find thepoint (or points) (x, y) on the circle X2-Fy2 = 4 wherethe value of the function x3 + y3 is greater than orequal to its value at any other points on the givencircle.

8. Evaluate f fit xy dA over the region enclosed by thecurves y = Ix and y = f.

9. Below are some values of a differentiable functionf(x, y). The numbers are placed directly over thecorresponding points so, for example, 1(2.5,2) =1.899, 1(2.5,2.5) = 1, 1(4,1) = 4, 1(4,2) = 2.203,etc. Assuming that nothing unexpected happens be-tween the given values, answer the following ques-tions.

a. If u is a unit vector in the northwest direction,which of the following numbers best approximateshz(3, 2)?

(A) (B) -1 (C) 0 (D) 1 (E)

b. Which of the following numbers best approxi-mates h(3, 2)?

(A) 0 (B) 0.1 (C) 0.2 (D) 0.5 (E) 1

3r

1,695 1.797 1.999 k 2.101 2.203 2.:05 2.407

I 3

I

,I.

1

[ i , I2 3 5

INSTITUTION: A midwestern public university with8500 students that in 1987 awarded 30 bachelor's and 5master's degrees in mathematics.

EXAM: A two-hour final exam (calculators allowed) forone section of the third term of a calculus course for en-gineering, physical science, and mathematics students.Of the 106 students who enrolled in the course, 71%passed, including 25% who received grades of A or B.

1. Find a rectangular equation of the curve F(t) =07+- 4 r+ 2ty and sketch the curve in the xy-plane.

2. Find the length of the curve whose parametric equa-tions are x = 3t, y = 2t3/2 where 0 < t < 4.

3. Find the equation for the plane through the pointsP(1,2,3) and Q(2,4, 2) that is parallel to a =(-3, -1, -2).

4. For 4x2 + y2 z2 16,

a. Find and name the traces in the coordinate planesand in other planes as needed;

b. Sketch the surface;

c. Name the surface.

5. Find parametric equations for the tangent line tothe curve x = et, y = jet, z = t2 + 4 at the pointP(1,0,4).

6. Find the critical points and then the extrema off(x, x2y 6y2 3x2.

7. Let f(x, y, z) = x2 -1- 3yz 4xy

a. Find the directional derivative of f (x, y, z) at thepoint P(1,0, -5) in the direction of a. = (2, -3, 1).

b. Find a vector in the direction in which f increasesmost rapidly at 1' and find the rate of change off in that direction.

8. Reverse the order of integration and evaluate the re-sulting integral: jy, y cos(x2) dx dy.

9. Find the volume of the solid in the first octant bound-ed by the sphere p = 2, the coordinate planes andthe cones = 7r/6 and = 7r/3. (Draw appropriatepictures.)

10. Use the change of variables x = 2u, y = 3v toexpress the integral Lin (x2/4 + y2/9) dx dy, whereR = {(x,y)1 (x2/4 + y2/9) < 1 }, as a double inte-gral over a region S in the uv-plane. (Do not evalu-ate.)

.1

A-k:n c ref-vr\fr REr1Sh&-t- CCw le.

\.5

-i--D

I kA r v e. tc cA LAI i S+rtA cti

D cA.1 ctiv.1 g C

4.1 r e, PA.% 0 LA. 1 t". S h% LA_ c-1-01e'S S 0 1-1, ck-t-. C-

C-.1De_rs o 110, I c_014,-(-0,c_i-- Is Tocs 1 loie_ . .

, .

AA ctk e... si.-1 A. e-t- tv r isfe., wtct.1-2-14,xct-i-L'c_ c C. ,C -

,...............

R-sio,. s -{- Lk. a etn-1--,S +- j u- -.k-i -Eua, all_

_ -t-LouLc c\,1A6A. -

N inA.c. re_ a.. tn...< ut) e-r_c 1/\

kook.Tri e_c-tor v olive_oc

E t7"-e_ VA.volv e. 14-\ ck.v. 1-1-e4

&ctAA oy)

cvn

fl 'it-Si-- v-of\,o v v

vscrg_r 40 0(.4* O p ri-r-Le ("y) ct

5

2 "t.

A Special Calculus Survey: Preliminary ReportRichard D. Anderson and Donald' O. Loftsgaarden

LOUISIANA STATE UNIVERSITYUNIVERSITY OF MONTANA

A preliminary report from a special survey of calcu-lus, conducted during October 1987 by the MAA-CBMSSurvey Committee:

The MAA Survey Committee has, over the pastmonth and with support from NSF (under Grant SRS-8511733), conducted a survey among four-year collegesand universities to get much more detailed statisticalinformation on calculus than that available elsewhere.The (CBMS) Surveys conducted every five years tradi-tionally are concerned with the fall term only and lumpall calculus courses in just two categories with no dif-ferentiation by level.

The same survey questionnaire was sent to a strat-ified random sample of about one-sixth of the almost1500 mathematics departments. The preliminary re-sults reported below are projections to the total indi-cated four-year college and university populations. Amuch more detailed report including breakdowns byclass of institution is to be submitted shortly to Focusand the Notices of the AMS.

This survey sought information on mainstream (M)calculus, i.e., freshman-sophomore calculus courses orsequences designed as a basis for eventual student ac-cess to upper-division mathematics courses, and non-mainstream (N-M) calculus, i.e., calculus courses or se-quences (such as most business calculus) not intendedspecifically for student access to upper-division math-ematics courses. The calculus courses considered wereCalculus I, II, III, and IV with Calculus I the beginningsemester or quarter course. Unless otherwise specified,all data here refer to phenomena in the two semestersor three quarters of the acal:cmic year AY 1986-87.

At the time of the preparation of this preliminaryreport, the full data were still being analyzed. It isbelieved that the percentage figures cited here are quiteaccurate, but some of the total enrollment figures maybe off by a few percent (probably low). The over-allresponse rate from the sample was a little over 50%,with a higher response rate from the public universitycategory and a somewhat lower response rate from theprivate college "ategory.

Enrollments and Success RatesFor all of AY 1986-87, there were a total of slightly

more than 300,000 enrollments in mainstream Calcu-lus I and just under 260,000 in non-mainstream Calcu-lus I. Thus 54% of all Calculus I enrollments were inmainstream calculus. There were also 16,000 to 17,000enrollments in each of Calculus I (M) and Calculus I(N-M) in the summer of 1987. Total semester or quar-ter enrollments in all of Calculus I to Calculus IV in AY1986-87 were 975,000.

In AY 1986-87, there were a little more than 140,000students who completed (with a D or better) the finalcourse of the first year of mainstream calculus (Calcu-lus II for semester schools and Calculus III for quarterschools) and there were about 113,000 students who en-rolled in the next term of calculus beyond the first yearof mainstream calculus. About 85% of all enrollmentsin non-mainstream calculus were in Calculus I whereasless than 50% of mainstream calculus enrollments werein Calculus I.

Slightly over 20% of all students in Calculus I (Mor N-M) were enrolled in lecture-recitation section for-mats. Almost three-fourths of all enrollees in CalculusI passed the course with a D or better (with 11-12% ofall enrollees getting a D). Enrollees in lecture-recitationsections had a 3% better passing rate than enrollees insingle-instructor sections.

More than four-fifths of all enrollments in Calculus Iwere in institutions using the semester calendar.

Section SizesThe average of sections in calculus taught on the

semester basis in AY 1986-87 are shown below:

Mainstream Non-MainstreamSection I II III I II

Single Instr. 29 26 26 38 35Lecture 126 108 95 153 115

In mainstream calculus the typical lecture section wassplit into about four recitation sections and in non-mainstream calculus the figure was about five.

Who Teaches Calculus?The table below shows the percentage distribution

by type of faculty for all single-instructor sections inAY 1986-87 for institutions on the semester schedule:

224.

216 READINGS

Faculty Type:Mainstream Non-Mainstream

I II III I II

FT ProfessorFT InstructorPart-timeTeach. Assist.

70% 73% 82% 47% 45%9% 14% 10% r13% 12%6% 4% 3% 13% 20%

15% 9% 5% 25% 23%

For classes taught in the lecture-recitation section for-mat, almost 100% of the recitation sections were han-dled by TA's. Almost 100% of the lecture sections inmainstream calculus were taught by professorial faculty,as contrasted to 80% in non-mainstream calculus.

From this data as well as section size data, it is clearthat, as expected, departments generally give priorityto mainstream calculus.

Two Proposals for CalculusLeonard Gillman

UNIVERSITY OF TEXAS, AUSTIN

A reprint of "From the President's Desk" in theSeptember 1987 issue of Focus, the News lett(' of theMathematical Association of America:

In 1946, in the town of Sullivan, Indiana, a man wasaccused of murdering his estranged wife in what was anopen and shut case. The county put its greenest attor-ney in charge, and no lawyer seemed willing to under-take the defense. Finally, Norval Kirkhan Harris (laterJudge Harris), a well-known local attorney, agreed totake it on. He decided on the line that the wholething was an unfortunate accident, and he played upthe phrase unsparingly. ("You say you were at the gro-cery the morning of this unfortunate accident. Wherewere you the afternoon of this unfortunate accident?")At the final summation, the prosecutor got up and be-gan, "I intend to show that this unfortunate accident..." (The entire courtroom from the judge down burstinto a guffaw, and the defendant got away with wmeasly2-1/2 years for manslaughter.)

I am reminded of the incident every time I encounterthe phrase the crisis in calculus. There is no crisis in cal-culus. Students come into the course unpreparedyes.Textbooks are too bigof course. Emphasis should bemodified to reflect the world of computerscertainly.Crisisno.

225

Characteristics of Calculus Classes

A little more than 60% of all single-instructor main-stream semester calculus sections met four hours perweek whereas about 50% of all mainstream quarter cal-culus sections met five hours per week.

About 3% of all calculus students have some com-puter use required in homework assignments.

About 55% of all calculus students on semesterschedules rarely or never had their homework pickedup and graded. About 30% of calculus students onsemester schedules were not given short quizzes. If theywere, almost all the quizzes were instructor designed.About one-eighth of such calculus students took groupor departmental-designed hour exams and about 50%took group or departmental finals.

Any crisis that may exist is in education, or in so-ciety. Up to World War II, only a minor portion ofcollege-age youth went on to college, and of those, onlya small fraction took calculus. Students typically beganwith a full semester of analytic geometry or a preced-ing semester of trigonometry. Calculus was big stuff; atColumbia University, where I studied, the final require-ment for the mathematics major was a comprehensiveexam in calculus and analytic geometry. In those days,a class of 40 students was considered huge. There waslittle concern with "motivating" students; when a pro-fessor told you something was important, you learnedit.

The ProblemToday, we bellow at 100 poorly prepared students at

a time and "cover" in two terms what used to use upthree. Students are reluctant to ask questions in frontof so many people, and no sensible exchange of ideasis possible anyhow. Faced with student evaluations, weplay to the gallery, giving easy quizzes and grades thatstudents did not earn. (Why should I split a gut tryingto buck the system? If they didn't study, that's theirproblem. I'll go back to my research or my garden orwhatever.)

GILLMAN: CALCULUS PROPOSALS 217

I am unpersuaded by results purporting to show thatlearning is independent of class size. To me, the ex-periments prove once again that the standard tests areinsufficient. We test only a small, easily quantifiablepart of what we hope the student is learning. We leaveout subject matter that cannot be easily adapted to thetest, as well as searching questions that require thought-ful responses and equally thoughtful, time- consuminggrading. And we perforce omit a host of delicate intan-gibles, such as the little ways the instructor respondedto a question or attacked a problem, which can makea lasting impression on students and shape their at-titudes. Certainly I can never forget a discussion inGeorge Adam Pfeiffer's class :A Columbia when oneof the studentsa very bright one, by the waywaswrestling with epsilon and delta. Finally, in desper-ation, he blurted out, "But suppose I choose epsilonlarge?" "Ah," said Pfeiffer, "but you don't choose ep-silon. / choose epsilon."

American students spend hours watching television.That most of what they look at is without merit is theminor crime. The major one is the fact of passive look-ing, encouraging them to sit back and let things cometo them. "Good" television programs are still televisionprograms: Sesame Street and Square One are still pas-sive entertainment. There is no opportunity for viewersto hold up the show while they sit back and reflect, tomull over ideas and express them in their own wordsasthey can when reading a book. The constant, mindlessblare trains people not to listen. Mathematics requiresintense concentration; television encourages nonconcen-tration. I sometimes wonder how many of my studentsare capable of concentrating on one idea, uninterrupted,for ten full minutes.

Today there are vast numbers of families where bothparents work, or in which there is only one parent (whoperforce works). Parents therefore have less time tospend with their kids, stimulating their curiosity, an-swering their questions, reading to them, relaxing withthem, inculcating a love of books. (If they can. AllanBloom, in The Closing of the American Mind, assertsthat though families may eat, play, and travel together,they do not think together.) As a result, students don'tknow anything. They don't know who Grant and Leewere (reported by E.D. Hirsch in Cultural Literacy). Ahistory colleague tells of a student's question: "I keepforgetting. Which side was Hitler on?" I remember acollege algebra class being tripped by a problem becausethey don't know a revolution of the moon takes abouta month. (Can they never have remarked on the simi-larity of the words?) In 1980 I gave a ounting problemthat depended on knowing the number of days in that

year, but the class stumbled because they didn't knowit was a leap year; the day of tli': test was February29th.

Neither can we count on mathematical prerequisites,on the elementary facts. What we really hope for issome true mathematical understanding; but you can'tunderstand ideas without knowing the facts they reston. It is always exciting to me to announce to a cal-culus class that we are about to enter a new realm ofmathematical powerin computing areas, we will jumpfrom parallelograms and trapezoids or other polygons tocurved boundaries. Many students fail to share my ex-citement. The big jump in power I am so excited aboutis to them a confused blur. They have no clear pictureof what they have been able to do thus far. They arenot confident about computing the area within a par-allelogram. As for trapezoids, they are not ever surewhat they are.

Our students live in a world of morale-sappinghypocrisy. (When was the last time you braked on theyellow light?) They see America being run by crooks.(Where is Richard Nixon now that his country needshim?) The glamorizing of senseless violence by themovies and TV pays off in real life: during 1986, thecity of Detroit averaged one child murder per day. Whatour young people see about them are not incentives toscholarship and learning but causes for despair. If allour values are a mess, what's the point of clean living?

Two ProposalsMathematicians cannot single-handedly solve the

problems of society, but we can do better than leaveout related rates. Updating the curriculum is a wor-thy goal but addresses only one of the variables underour control. I suppose it is an improvement to go fromoutmoded methods of teaching ill-prepared students aragged curriculum to outmoded methods of teaching ill-prepared students a spruced-up curriculum, but we canset our sights higher. I have two proposals, both sim-ple, although to put them into effect may require someunglamorous hard work. But I think they would go a

*long way toward setting our classes on a more realisticfooting.

Proposal 1: Let computers handle the drill. In learn-ing a subject, there are two things you have to doabsorb the ideas, and acquire skill in the routines. Theappropriate setting for learning ideas is some thoughtfulgive and take with a teacher. For skill with the routines,you have to have a lot of just plain drill. Today we don'tneed humans to oversee routine drill. That task shouldbe taken over by computers. This requires some trulyfirst-rate programs; but such things are possible.

226.

218 READINGS

Advantages of computer drill are well known, but Iwill mention some anyhow. Students do their practic-ing at times convenient to them. They work at theirown,pace. They not only get feedback but instant feed-back. (In contrast, homework papers are often gradedwithout comment by a teaching assistant and returnedseveral days after being handed in.) Students work inprivacy, with no one scolding or laughing at them orchiding them for being slow. A well-designed program,with thoughtful conditional branching, will offer gaid-ance while at the same time allowing students to pickthe topics they need practice on. The instructor is freedto devote full time to the exchange of ideas. Finally,classes can meet less often, and large classes can be di-vided into smaller ones.

Proposal 2: Enforce the prerequisites. Not only dowe award grades that were not earned, but we do a dis-service at the beginning when we admit students whoare not qualified for the course. These students usuallydo poorly and end up soured on math. Instructors feelobliged to review background material in class, cuttinginto time for the regular syllabus, degrading the charac-ter of the course, and shortchanging the better-prepared

Calculus in Secondary Schools

Text of a letter endorsed by the governing boards ofthe Mathematical Association of America and the Na-tional Council of Teachers of Mathematics concerningcalculus in the secondary schools:

Memo

TO: Secondary School Mathematics Teachers

FROM The Mathematical Association of America

The National Council of Teachers of Math.

DATE: September, 1986

RE: Calculus in the Secondary School

Dear Colleague:

A single variable calculus course is now well estab-lished in the 12th grade at many secondary schools, andthe number of students enrolling is increasing substan-tially each year. In this letter, we would like to discusstwo problems that have emerged.

students.I propose we all be brave and enforce the prerequi-

sites. This is consistent with the MAA-NCTM reso-lution of last fall on calculus in high school. Just re-member to check your plan with your engineering andbusiness colleagues, pointing out that they too will gainfrom the new standardsotherwise, they may put intheir own mathematics courses.

The resolution just referred to lists algebra, trigon-ometry, analytic geometry, complex numbers, and el-ementary functions, studied in depth, as prerequisitesfor the high school calculus course. For the college pre-calculus course, I would say be sure to include a thor-ough treatment of the conic sections (with the byprod-uct of freeing up the calculus course from that hereto-fore obligatory chapter). It's a difficult course to han-dle, because the students know some of the materialand become easily bored; but that's a poor reason forputting them directly into calculus, where the materialis assumed to be known. Do an honest job, make thecourse excitingthere is plenty of exciting materialand entrust it to your conscientious teachers. And don'tinflate the grades.

The first problem concerns the relationship betweenthe calculus course offered in high school and the suc-ceeding calculus courses in college. The Mathemati-cal Association of America (MA A) and the NationalCouncil of Teachers of Mathematics (NCTM) recom-mend that the calculus course offered in the 12th gradeshould be treated as a college-level course. The expec-tation should be that a substantial majority of the stu-dents taking the course will master the material and willnot then repeat the subject upon entrance to college.Too many students now view their 12th grade calculuscourse as an introduction to calculus with the expecta-tion of repeating the material in college. This causesan undesirable attitude on the part of the student bothin secondary school and in college. In secondary schoolall too often a student may feel "I don't have to studythis subject too seriously, because I have already seenmost of the ideas." Such students typically have con-siderable difficulty later on as they proceed further intothe subject matter.

2 2, :7

CALCULATORS IN STANDARDIZED TESTS 219

MAA and NCTM recommend that all students takingcalculus in secondary school who are performing satis-factorily in the course should expect to place out of thecomparable college calculus course. Therefore, to verifyappropriate placement upon entrance to college, stu-dents should either take one of the Advanced Placement(AP) Calculus Examinations of the College Board, ortake a locally - administered college placement examina-tion in calculus. Satisfactory performance on an APexamination carries with it college credit at most uni-versities.

The second problem concerns preparation for thecalculus course. MAA and NCTM recommend thatstudents who enroll in a calculus course in secondaryschool should have demonstrated mastery of algebra, ge-ometry, trigonometry, and coordinate geometry. Thismeans that students should have at least four full yearsof mathematical preparation beginning with the firstcourse in algebra. The advanced topics in algebra,trigonometry, analytic geometry, complex numbers, andelementary functions studied in depth during the fourthyear of preparation are critically important for students'latter courses in mathematics.

It is important to note that at present many well-prepared students take calculus in the 12th grade, placeout of the comparable course in college, and do sellin succeeding college courses. Currently, the two mostcommon methods for preparing students for a college-level calculus course in the 12th grade are to begin thefirst algebra course in the 8th grade or to require stu-dents to take second year algebra and geometry con-currently. Students beginning with algebra in the 9thgrade, who take only one mathematics course each yearin secondary school, should not expect to take calculusin the 12th grade. Instead, they should use the 12thgrade to prepare thetnselves fully for calculus as fresh-men in college.

We offer these recommendations in an attempt tostrengthen the calculus program in secondary schools.They are not meant to discourage the teaching ofcollege-level calculus in the 12th grade to strongly pre-pared students.

LYNN ARTHUR STEENPresidentMathematical Associationof America

Calculators in Standardized Testing of Mathematics

Recommendations from a September .1986 Sympo-sium on Calculators in the Standardized Testing ofMathematics sponsored by the The College Board andthe Mathematical Association of America:

In the ten years that inexpensive hand-held calcula-tors have been available, a great deal of considerationhas been given to their proper role in mathematics in-struction and testing. In September 1986, the CollegeBoard and the Mathematical Association of Americanarranged a Symposium on Calculators in the Standard-ized Testing of Mathematics to focus on a specializedbut essential aspect of this debate.

Thr participants found in this arena of rapid changea variety of old and new issues. Many of the old issuesagain yield the same educational conclusions, ones thatcause everyone to be frustrated with the delays thatsurround implementation of calculator usage.

At the same time, rapid changes in technology andprice are presenting fresh issues. In particular, falling

JOHN A. DOSSEYPresidentNational Council ofTeachers of Mathematics

prices essentially resolved the old equity issue ofstudent a cess to calculators and have introduced in-stead a more important issue of student access to ade-quate preparation for using these devices appropriatelyand well.

Likewise, the earlier complications arising from dif-ferent calculator capabilities that generated test admin-istration policies which gave upper limits to the type ofcalculators permitted have been superseded. The newissue is that test makers now must specify the minimumlevel of sophistication necessary for a calculator to beappropriate for use in taking a given examination. Ex-cess sophistication in a machine can be self-defeating;surplus machine capabilities may detract far more thanthey contribute to a student's performance.

Recommendations1. The symposiuni endorses the recommendations made

by the National Council of Teachers of Mathemat-

228

220 READINGS

ics, the Conference Board of the Mathematical Sci-ences, the Mathematical Sciences Education Board,and the National Science Board that calculators beused throughout mathematics instruction and test-ing.

2. The symposium calls for studies to identify contentareas of mathematics that have gained importance asa result of new technologies. How achievement andability in these areas are measured should be studied,and new testing techniques should be considered.

3. The symposium points to a need for research anddevelopment on:a. item types and formats;b. characteristics of new item types;c. student responses to items that allow the use of a

calculator; andd. instructional materials that require the use of a

calculator.4. The symposium believes that a mathematics achieve-

ment test should be curriculum based and that noquestions should be used that measure only calcula-tor skills or techniques;

5. The symposium recognizes,that choosing whether ornot to use a calculator when addressing a particu-lar test question is itself an important skill. Conse-quently, not all questions on a calculator-based math-ematics achievement test should require the use of acalculator.

6. The symposium gives strong support to the devel-opment of examinations in mathematics that requirethe use of a calculator for some questions. In partic-ular, we support The College Board in its E..udy ofmathematics achievement examinations that requirethe use of calculators, and we commend the Math-ematical Association of America for its intention todevelop a new series of "calculator-based" placementexaminations in mathematics.

7. The symposium recommends that nationally devel-oped mathematics achievement tests requiring theuse of a calculator should provide descriptive ma-terials and sample questions that clearly indicatethe level of calculator skills needed. However, thereshould be no attempt to define an upper limit to thelevel of sophistication that calculators used on suchtests should have. Any calculator capable of perform-ing the operations and functions required to solve theproblems on a particular examination should be al-lowed.

8. The symposium notes that different standardizedtests in mathematics are used to serve different ed-ucational purposes. Therefore, some tests need tobe revised as soon as possible to allow for the use of

2

calculators, whereas others may not need to changevery soon. In every case, however, the integrity of thetest must be maintained in order for its relevance tothe mathematics taught in schools and colleges to besustained.

9. Because of the importance of the SAT in the collegeadmission process, as well as the nature of its math-ematical content, the symposium carefully examinedthe use of calculators on that test. We recommendthat calculators not be used on the SAT at this time.However, this issue should be reconsidered period-ically in light of the status of school mathematicspreparation.

Participants:

College Board Staff Members:JAMES HERBERT, Executive Director, Office of Aca-

demic AffairsGRETCHEN W. RIGOL, Executive Director, Office of

Access ServicesHARLAN P. HANSON, Director, Advanced Placement

ProgramROBERT ORRILL, Associate Director, Office of Aca-

demic Affairs

Educational Testing Staff Members:CHANCEY 0. JONES, Associate Arca DirectorJAMES BRASWELL, Senior Examiner, Test Develop-

mentBEVERLY R. WHITTINGTON, Senior Examiner, Test

Development

The College Board's Matherriatical Scicsmcs Ad-visory Committee:JEREMY KILPATRICK, MSAC Chair. University of

GeorgiaJ.T. SUTCLIFFE, St. Mark's School, TexasCAROL E. GREENES, Boston UniversityTHOMAS W. TUCKER, Colgate UniversityEDWARD SIEGFRIED, Milton Academy, MassachusettsR.O. WELLS, JR., Rice University

The MAA Committee on Placement Examina-tions;JOHN W. KENELLY, COPE Chair: Clemson UniversityJOAN R. HUNDHAUSEN, Colorado School of MinesBILLY E. RHOADES, Indiana UniversityLINDA H. BOYD, DeKalb CollegeJOHN G. HARVEY, University of WisconsinJACK M. ROBERTSON, Washington State UniversityJUDITH CEDERBERG, St. Olaf College

NSF WORKSHOP ON UNDERGRADUATE MATHEMATICS 221

MARY MCCAMMON, Pennsylvania State University

Invited Speakers:JOAN P. LEITZEL, Ohio State UniversityBERT K. WAITS, Ohio State UniversityJAMES W. WILSON, University of Georgia

Invited Participants:GEOFFREY AKST, Chair: AMATYC Education Com-

mittee; CUNY, Manhattan Community CollegeDONALD KREIDER, Treasurer: Mathematical Associa-

tion of America; Dartmouth College

ALFRED B. WILLCOX, Executive Director, Mathemat-ical Association of America

MICHAEL J. CHROBAK, Texas Instruments Corpora-tion

GERALD MARLEY, UC/CSU Project; California StateUniversity, Fullerton

STEVE WILLOUGHBY, NCTM: New York University

SHIRLEY HILL, Chairman: Mathematical Sciences Ed-ucation Board of the National Research Council

TAMMY RICHARDS, Texas Instruments Corporation

NSF Workshop on Undergraduate Mathematics

Text of a report prepared in June 1986 as a sequel tothe National Science Board Task Committee Report onUndergraduate Science, Mathematics, and EngineeringEducation:

The mathematical sciences are both an enabling forceand a critical filter for careers in science and engineer-ing. Without quality education in mathematics wecannot build strong programs in science and engineer-ing. NSF policy for science and engineering educationboth precollegiate and collegiatemust be built on thiscentral fact: mathematics is not just one of the sci-ences, but is the foundation for science and engineer-ing.

In our view the most serious problem facing un-dergraduate mathematics is the quality of teachingand learning for the three million students in allfields who study undergraduate mathematics each term.We are concerned about the professional vitality ofthe faculty, the "currency" of the curriculum, andthe shortage of mathematically-trained students, es-pecially those preparing for careers in mathematicsteaching or research. These problems are all interre-lated and must be addressed collectively and simulta-neously.

The explosion of new applications of mathematicsand the impact of computers require major change inundergraduate mathematics. Moreover, recent trendstowards unification of basic theory must be integratedinto the curriculum. These trends reinforce the ur-gency of an NSF initiative in undergraduate mathe-matics based on the themes of leadership and lever-age elaborated in the report of the NSB Task Commit-

tee on Undergraduate Science and Engineering Educa-tion.

We propose programs in four categories: faculty,student support, curriculum development, and globalprojects. Within each category we list programs in or-der of priority, but overall there should be at least oneprogram supported from each category.

Faculty

A. SUMMER FACULTY SEMINARS. A nation-wide, cen-trally coordinated, continuing series of attractiveseminars linking new mathematics with curricularreform. Seminars should cover a wide spectrum ofissues and levels, from two-year college concerns toexposition of recent research. Some seminars shouldbe taught by industrial mathematicians to facilitatethe transfer of mathematical models into the cur-riculum. To be effective there must be enough sem-inars of sufficient duration to provide a significantrenewal opportunity to at least 50% of the Nation'stwo- and four-year college and university facultiesat least once every five years. (That suggests posi-tions for 10%about 3,000 peopleper summer ina steady state.) Budget Estimate: $344 million peryear.

B. FACULTY-INDUSTRY LINKAGES. A special programof limited duration in which NSF would help ini-tiate nation-wide models for faculty to become ac-quainted with how mathematics is used in indus-try and government. The models could includefaculty summer internships and arrangements for

,23(0

222 READINGS

faculty-student teams to work on real problems. Thegoal of NSF should be to establish a nation-widesystem of industry programs to be sustained in-definitely by support from industry. Budget Es-timate: Start-up funds of $1 million over threeyears.

C. INDIVIDUAL TEACHING GRANTS. Like researchgrants, teaching grants woulit provide support toindividuals with innovative plans tr.% renew specificmathematics courses or create new ones. At least100 grants (say two per year per state) is a minimumnumber to have significant effect.

D. FACULTY FELLOWSHIPS. To supplement sabbaticals,perhaps matched by a teaching fellowship from thehost departments. Proposed level: At least 100 peryear.

StudentsA. NSF UNDERGRADUATE FELLOWSHIPS. A program

of stipends for talented undergraduates, adminis-tered by departments, to provide active mathemati-cal experiences to undergraduates outside class, e.g.,working with precollege st , summer opportu-nities for research apprentk, ,, assistants in highschool summer institutes, indiviuual study, and spe-cial seminars. We are especially concerned about thecontinued low number of women and the almost to-tal absence of minority students among those whopursue careers in the mathematical sciences. Specialefforts to encourage talented women and minoritystudents should be included under this program.In steady state, this program should provide sup-port for approximately the top 25% of undergraduatemathematics majors (about 4,000 students). BudgetEstimate: $4-$5 million per year.

B. STUDENT INTERNSHIPS. To provide undergra;:lateswith opportunities in industry or government tc em-ploy mathematics in a realistic setting. Initial 'lac-ommendation: 10-12 pilot programs.

Curriculum

A. CALCULUS RENEWAL. A multi-year special under-taking, perhaps involving several consc,tia of insti-tutions, to transform both texts and teaching prac-tice in calculusthe major entry point (and impedi-ment) to college mathematics, science, and engineer-ing. This is of immediate priority. Budget Estimate:$500,000 per consortium per year.

B. MODEL PROGRAMS. A continuing series of projectsto identify and develop examples of courses, mathe-matics majors, instructional environments, and cur-ricular experiments that are right now successfullystimulating interest in undergraduate mathematics,backed up by dissemination activities (perhaps linkedwith the Summer Seminars) to stimulate others todevelop their own programs. Budget Estimate: $2million per year.

Global NeedsA.

B.

ASSESSMENT OF COLLEGIATE MATHEMATICS. Amajor study of standards, human resources, carterpatterns, curriculum patterns, and related issuesmust be undertaken to enable all institutions re-sponsible for undergraduate mathematics to under-stand clearly the nature and magnitude of the prob-lems facing collegiate mathematics. This study mustdeal both with introductory courses that representthe primary undergraduate mathematics experiencefor most students, and with the mathematics ma-jor that forms the base for a wide variety of post-baccalaureate careers. Because of preparations al-ready underway for such an assessment, it is veryimportant that this project be supported from theFY 1987 budget. Budget Estimate: $1 million overtwo years.Consider the establishment of INSTITUTES FOR UN-DERGRADUATE EDUCATION IN TIIE MATHEMATICAL

SCIENCES to develop professional and research exper-tise in issues related to undergraduate mathematicseducation.

Current ProgramsThree current NSF programs are also very impor-

tant to the renewal of collegiate mathematicsresearchsupport for mathematics, instructional scientific equip-ment, and graduate fellowships. For different reasons,all three programs are now making inadequate impacton undergraduate mathematics:

As documented in the David Report, the NSF re-search budget for mathematics is out-of-balance withrespect to support for other disciplines, thus insur-ing that many capable young investigators get cutoff early from the frontiers of the discipline. Avail-ability of research support for a larger number ofmathematicians is crucial to regenerating the profes-sional vitality of the faculty which is so important toeducation.

NSF WORKSHOP ON UNDERGRADUATE MATEZMATICS 223

For various reasons, mathenntics departments donot seek resources for such things as computers asvigorously as they should. Consequently, they do notapply for their fair share of support from programssuch as CSIP. Both NSF and the mathematics pro-fession must work to insure that significant resourcesfor computing are available and utilized by collegiatemathematics departments.Because of the decline in interest among Americansfor graduate study in mathematics, too few mathe-matics graduate students receive support from NSFGraduate Fellowships. The community must workto reverse this trend, and NSF must be prepared tosupport increased numbers of graduate students inmathematics.

Other NeedsThese program recommendations do not address all

the problems facing collegiate mathematics. The bur-den of remedial mathematics and the lack of pub-lic understanding that mathematics is pervasive andimportant significantly impede the ability of collegesand universities to maintain high-quality undergradu-ate programs in mathematicsthe former by drainingresources, the latter by limiting resources. These issuesare able to be addressed by other NSF programs, so werecommend that new undergraduate resources not beused in these areas.

Making Programs EffectiveFinally, we make a few recommendations for how

NSF can most effectively develop undergraduate math-ematics programs in a way that resonates with the ex-isting professional structures of the mathematical sci-ences:

1. INTERNAL STRUCTURE. There is a need that fund-ing be specifically directed to support and improveundergraduate mathematics education. Such fund-ing should not divert funds from mathematical re-search nor should such fund: be diverted to math-ematical research or to other educati gal problems.We propose that a well-defined unit staffed by math-ematical scientists knowledgeable about educationalproblems be established to deal with problems inundergraduate mathematics education. The unitshould have a specific budget of sufficient size to re-flect the very significant role undergraduate math-

2.

3.

4.

ematics plays in science and engineering educa-tiou.

Generally, but not exclusively, programs supportedshould have implications national in scope. Projectsshould be undertaken by broadly based consor-tia, networks, professional societies, and other na-tional organizations. Programs calling for pro-posals should be focused in intent but not be sospecific as to rule out innovative and imaginativeapproaches. Proposals addressing significant lo-cal problems should receive significant local fund-ing or contributions in kind. Proposals addressingbroad national concerns should not necessarily be ex-pected to ata,act significant funding other than fromNSF.

Professional societies and other national organiza-tions need to assume a significant responsibility indefining and obtaining consensus as-to national con-cerns, informing their members of NSF programs,disseminating results, and identifying necessary hu-man resources. In light of the low response rate ofmathematicians to NSF educational proposals, werecommend that a consortium of professional soci-eties provide the community with proposal consul-tants. It may be necessary in the short run for NSFto fund such an activity.The results of course and curriculum developmentneed to be widely disseminated. There should beboth short-term and long-term evaluative follow upsof the effectiveness of various programs supportedunder this effort.

Participants:RICHARD D. ANDERSON, Louisiana State UniversityKENNETH COOKE, Pomona College

JOSEPH CROSSWHITE, Ohio State University

RONALD G. DOUGLAS, SUNY at Stony BrookHARVEY KEYNES, Un:versity of Minnesota

ANNELI LAX, Courant Institute, New Mork UniversityDONAJD J. LEWIS, University of Michigan

BERNARD MADISON, University of Arkansas

INGRAM OLKIN, Stanford University

HENRY 0. POLLAK, Bell Communications Research

THOMAS W. TUCKER, Colgate UniversityLYNN ARTHUR STEEN (Chair), St. Olaf College

2 I 2

224 READINGS

Transition from High School to College Calculus

Donald B. Small

COLBY COLLEGE

Report of a CUPM subcommittee concerning thetransition from high school to college calculus, reprintedfrom THE AMERICAN MATHEMATICAL MONTHLY, Oc-tober, 1987:

There is a widespread and growing dissatisfactionwith the performance in college calculus courses of manystudents who had studied calculus in high school. Inresponse to this concern, in the fall of 1983, the Com-mittee on the Undergraduate Program in Mathematics(CUPM) formed a Panel on Calculus Articulation toundertake a three-year study of questions concerningthe transition of students from high school calculus tocollege calculus and submit a report to CUPM detail-ing the problems encountered and proposals for theirsolution.

The seriousness of the issues involved in the Panel'sstudy is underscored by the number of students involvedt.nd their academic ability. During the ten-year period1973 to 1982, the number of students in high schoolcalculus courses grew at a rate exceeding 10% annually.Of the 234,000 students who passed high school cal-culus course in 1982, 148,600 received a grade of B- orhigher [2]. Assuming a continuation of the 10% growthrate and a similar grade distribution there were approx-imately 200,000 high school students in the spring of1985 who received a grade of B- or higher in a calculuscourse. Thus possibly a, third or more of the 500,000college students who began their college calculus pro-gram (in Calculus I, Calculus II, or Calculus III) in thefall of 1985 had already received a grade of B- or higherin a high school calculus course.

The students studying calculus in high school consti-tute a large majority of the more mathematically capa-ble high school students. (In 1982, 55% of high schoolstudents attended schools where calculus was taught[2].) Students who score a 4 or 5 on an AdvancedPlacement (AP) Calculus examination normally do wellin maintaining their accelerated mathematics programduring the transition from high school to college. How-ever, this is a very small percentage of the students whotake calculus in high school. For example, in 1982, ofthe 32,000 students who took an Adv:nced Placementcalculus examination, just over 12,000 received scores of4 or 5, which represents only 6% of all high school stu-dents who took calculus that year. The primary concern

of the Panel was with the transition difficulties associ-ated with the remaining almost 94% of the high schoolcalculus students.

Problem AreasPast studies and the Panel's surveys of high school

teachers, college teachers, and state supervisors suggestthat the major problems associated with the transitionfrom high school calculus to college calculus are:1. High school teacher qualifications and expectations.2. Student qualifications and expectations.3. The effect of repeating a course in college after having

experienced success in a similar high school course.4. College placement.5. Lack of communication between high schools and col-

leges.(Copies of the Panel's Report including the surveys andsummaries of the responses can be obtained from theWashington Office of the MAA.)

These problems were addressed by first consideringaccelerated programs in general, high school calculus(successful, unsuccessful), and the responsibilities of thecolleges.

Accelerated ProgramsAccelerated mathematics programs, usually begin-

ning with algebra in eighth grade, are now well estah--fished and accepted in most school systems. The suc-cess of these programs in attracting the mathematicallycapable students was documented in the 1981-82 test-ing that was done for the "Second International Math-ematics Study." The Summary Report [9] states withreference to a comparison between twelfth grade pre-calculus students and twelfth grade calculus studentsin the United States:

We note furthermore that in every content area (setsand relations, number systems, algebra, geometry, el-ementary functions/calculus, probability and statis-tics, finite mathematics), the end-of-the-year averageachievement of th, orecalculus classes was less (and inmany cases considerably less) than the beginning-of-the-year achievement of the calculus students.

The report continues:It is important to observe that the great majority ofU.S. senior high school students in fourth and fifthyear mathematics classes (that is, those in precalculus

23

SMALL: TRANSITION FROM HIGH SCHOOL 225

classes) had an average performance level that was ator below that of the lower 25% of the countries. Theend-of-year performance of the students in the calculusclasses was at or near the international means for thevarious content areas, with the exception of geometry.Here U.S. performance was below the internationalaverage.

Thus those students in accelerated programs cul-minating in a calculus course perform near the inter-national mean level while their classmates in (non-accelerated) programs culminating in a precalculuscourse perform in the bottom 25% in this internationalsurvey. The poor performance in.geometry by both theprecalculus and calculus students correlates well withthe statistic that 38% of the students were never taughtthe material contained in the g -..metry secti.ut of thetest [9, p. 59]. The test data unuerscores the concernexpressed by many college teachers that more emphasisneeds to be placed on geometry throughout the highschool curriculum. This data does not, however, indi-cate that accelerated programs emphasize geometry lessthan non-accelerated programs.

The success of the accelerated programs in com-pleting the "normal" four year high school mathemat-ics program by the end of the eleventh grade presentsschools with both an opportunity and a challenge for a"fifth" year program. There are two acceptable options:

1. Offer college-level mathematics courses that wouldcontinue the students' accelerated program and thusprovide exemption from one or two semesters of col-lege mathematics.

2. Offer high school mathematics courses that .9uldbroaden and strengthen a student's background andunderstanding of precollege mathematics.Not offering a fifth year course or offering a watered-

down college level course with no expectation of studentsearning advanced placement are not considered to 1.c ac-ceptable options.

A great deal of prestige is associated with offeringcalculus as a fifth year course. Communities often viewthe offering of calculus in their high school as an in-dication of a quality educational program. farents,school board officials, counselors, and school Pdminis-trators often demonstrate a competitive pride in theirschool's offering of calculus. This prestige factor caneasily manifest itself in strong political pressure fcz aschool to offer calculus without sufficient regard to thequalifications of teachers or students.

It is important that this political pressure be resistedand that the choice of a fifth year program be madeby the mathematics faculty of the local school and bemade on the basis of the interest and qualifications of

the mathematics faculty and the quality and number ofaccelerated students. School officials should be encour-aged to develop public awareness programs to extendthe prestige and support that exists for the calculusto acceleration programs in general. This would helpdiffuse the political pressure as well as broaden schoolsupport within the community.

Schools that elect the first option of offering a col-lege level course should follow a standard college coursesyllabus (e.g., the Advanced Placement syllabus for cal-culus). They should use placement test scores alongwith the college records of their graduates as primarymeasures of the validity of their course.

For schools that elect the second option, a variety ofcourses is possible. The following course descriptionsrepresent four possibilities.

ANALYTICAL GEOMETRY. This course could go wellbeyond the material normally included in second yearalgebra and precalculus. It could include Cartesian andvector geometry in two- and three-dimensions with top-ics such as translation and rotation of axes, characteris-tics of general quadratic relations, curve sketching, po-lar coordinates, and lines, planes, and surfaces in three-dimensional space. Such a course would provide spe-cific preparation for calculus and linear algebra, as wellas give considerable additional practice in trigonometryand algebraic manipulations.

PROBABILITY AND STATISTICS. This course couldbe taught at a variety of levels, to be accessible tomost students, or to challenge the strongest ones. Itcould cover counting methods and some topics in dis-crete probability such as expected values, conditionalprobability, and binomial distributions. The statisticsportion of the course could emphasize exploratory dataanalysis including random sampling and sampling dis-tributions, experimental design, measurement theory,measures of central tendency and spread, measures ofassociation, confidence intervals, and significance test-ing. Such an introduction to probability and statisticswould be valuable to all students, and for those whodo not plan to stud: mathematics, engineering, or thephysical sciences, probably more valuable than a calcu-lus course.

DISCRETE MATHEMATICS. This type of course couldinclude introductions to a number of topics that are ei-ther ignored or treated lightly within a standard highschool curriculum, but which would be stimulating andwidely useful for the college-bound high school student.Suggested topics include permutations, combinations,and other counting techniques: mathematical induc-tion; difference equations; some discrete probability; el-ementary number theory and modular arithmetic; vec-

226 READINGS

for and matrix algebra, perhaps with an introductionto linear or dynamic programming; and graph theory.

MATRIX ALGEBRA. This course could include ba-sic arithmetic operations on matrices, techniques forfinding matrix inverses, and solving systems of linearequations and their equivalent matrix equations usingGaussian elimination. In addition, some introduction tolinear programming and dynamic programming couldbe included. This course could also emphasize three-dimensional geometry.

High School CalculusThere are many valid reasons why a fifth year pro-

gram should include a calculus course. Four major rea-sons: (1) calculus is generally recognized as the startingpoint of a college mathematics program, (2) there ex-ists a (nationally accepted) syllabus, (3) the AdvancedPlacement program offers a nation wide mechanism forobtaining advanced placement, and (4) there is a largeprestige factor associated with offering calculus in highschool. Calculus, however, should not be offered un-less there is a strong indication that the course will besuccessful.

Successful Calculus CoursesThe primary characteristics of a successful high

school calculus course are:1. A qualified and motivated instructor with a mathe-

matics degree that included at least one semester of ajunior-senior real analysis course involving a rigoroustreatment of limits, continuity, etc.

2. dministrative support, including provision of addi-onal preparation time for the instructor (e.g., as

recommended by the North Central AccreditationAssociation).

3. A full year program based on the Advanced Place-ment syllabus.

4. A college text should be used (not a watered-downhigh school version).

5. Advanced placement for students (rather than merepreparation for repeating calculus in college) is a ma-jor goal.

6. Course evaluation based primarily on college place-ment and the performance of its graduates in thenext higher level calculus course.

7. Restriction of course enrollment to only qualified andinterested students.

8. The existence of an alternative fifth year course thatstudents may select who are not qualified for or in-terested in continuing in an accelerated program.

The bottom line of what makes a high school calculuscourse successful is no surprise to anyone. A qualifiedteacher with high but realistic expectations, using some-what standard course object:yes, and students who arewilling and able to learn result in a successful transi-tion at any level of our educational process. Problemsappear when any of the above ingredients are missing.

Unsuccessful Calculus Courses

Two types of high school calculus courses have anundesirable impact on students who later take calculusin college.

One type is a one semester or partial year course thatpresents the highlights of calculus, including an intuitivelook at the main concepts and a few applications, andmakes no pretense about being a complete course inthe subject. The motivation for offering a course of thiskind is the misguided idea that it prepares students fora real course in college.

However, such a preview covers only the glory andthus takes the excitement of calculus away from the col-lege course without adequately preparing students forthe hard work and occasional drudgery needed to un-derstand concepts and master technical skills. Profes-sor Sherbert has commented: "It is like showing a tenminute highlights film of a baseball game, including thefinal score, and then forcing the viewer to watch theentire game from the beginningwith a quiz after eachinning."

The second type of course is a year-long, semiserious, but watered-down treatment of calculus tha.does not deal in depth with the concepts, covers noproofs or rigorous derivations, and mostly stresses me-chanics. The lack of both high standards and emphasison understanding dangerously misleads students intothinking they know more than they really do.

In this case, not only is the excitement taken away,but an unfounded feeling of subject mastery is fosteredthat can lead to serious problems in college calculuscourses. Students can receive respectable grades in acourse of this type, yet have only a slight chance of pass-ing an Advanced Placement Examination or a college-administered proficiency examination. Those who placeinto second term calculus in college will find themselvesin heavy competition with better prepared classmates.Those who elect (or are selected) to repeat first termcalculus believe they know more than they do, and themotivation and willingness to learn the subject are lack-ing.

2 qi5

SMALL: TRANSITION FROM HIGH SCHOOL 227

College ProgramsSeveral studies ([1], [3], [5], [6], [7]) have been con-

ducted on the performance in later courses by studentswho have received advanced placement (and possiblycollege credit) by virtue of their scores on AdvancedPlacement Calculus examinations. The studies showthat, overall, students earning a score of 4 or 5 on eitherthe AB or BC Advanced Placement Calculus examina-tion do as well or better in subsequent calculus coursesthan the students who have taken all their calculus incollege. It is therefore strongly recommended that col-leges recognize the validity of the Advanced PlacementCalculus program by the granting of one semester ad-vanced placement with credit in calculus for studentswith a 4 or 5 score on 'le AB examination, and twosemesters of advanced p' .cement with credit in calculusfor students with a 4 or .; score on the BC examination.

The studies reviewed by the Panel do not indicateany clear conclusions concerning performance in sub-sequent calculus courses by students who have scoreda 3 on an Advanced Placement Calculus examination.The treatment of these students is a very importanttransition problem since approximately one-third of allstudents who take an Advanced Placement Calculus ex-amination are in this group and many of them are quitemathematically capable.

It is therefore recommended that these students betreated on a special basis in a manner that is appro-priate for the institution involved. For example, sev-eral colleges offer a student who has earned a 3 on anAdvanced Placement Calculus examination the oppor-tunity to upgrade this score to an "equivalent 4" bydoing sufficiently well on a Department of Mathemat-ics placement examination. Another option is to givesuch students one semester of advanced placement withcredit for Calculus I upon successful completion of Cal-culus II. A third option is to give one semester of ad-vanced placement with credit for Calculus I and providea special section of Calculus II for such students.

Other important transition problems are associ-ated with students who have studied calculus in highschool, but have not attained advanced placement ei-ther through the Advanced Placement Calculus pro-gram or effective college procedures. These studentspose an important and difficult challenge to collegemathematics departments, namely: How should thesestudents be dealt with so that they can benefit fromtheir accelerated high school program and not succumbto the negative and (academically) destructive attitudeproblems that often result when a student repei.ts a.course in which success has already been experienced?There are three major factors to consider with respect

to these students.

1. The lack of uniformity of high school calculus courses.The wide diversity in the backgrounds of the stu-dents necessitates that a large review component beincluded in their first college calculus course to guar-antee the necessary foundation for future courses.

2. The mistaken belief of most of these students thatthey really know the calculus when, in fact, they donot. Thus they fail to, study enough at the begin-ning of the course. When they realize their mistake(if they do), it is often too late. These students of-ten become discouraged and resentful as a result oftheir poor performance in college calculus, and be-lieve that it is the college course that must be atfault.

3. The "Pecking Order" syndrome. The better the stu-dent, the more upsetting are the understandable feel-ings of uncertainty about his or her position relativeto the others in the class. Although this is a commonproblem for all college freshmen, it is compoundedwhen the student appears to be repeating a course inwhich success had been achieved the preceding year.This promotes feelings of anxiety and produces anaccompanying set of excuses if the student does notdo at least as well as in the previous year.

The uncertainty of one's position relative to the restof the class often manifests itself in the student notasking questions or discussing in (or out of) class forfear of appearing dumb. This is in marked contrast tothe highly confident high school senior whose ques-tions and discussions were major components in hisor her learning process.The unpleasant fact is that the majority of students

who have taken calculus in high school and have notclearly earned advanced placement do not fit in eitherthe standard Calculus I or Calculus II course. The stu-dents do not have the level of mastery of Calculus Itopics to be successful if placed in Calculus II and areoften doomed by attitude problems if placed in CalculusI. In modern parlance, this is the rock and hard place.

An additional factor to consider is the negative effectthat a group of students who are repeating most of thecontent of Calculus I has on the rest of the class as wellas on the level of the instructor's presentations.

What is needed are courses designed especially forstudents who have taken calculus in high school andhave not clearly earned advanced placement. Thesecourses need to be designed so that they:

1. Acknowledge and build on the high school experi-ences of the studenb-;

2. Provide necessary review opportunities to ensure an

228 READINGS

acceptable level of understanding of Calculus I top-ics; 10.

3. Are clearly different from high school calculus courses(in order that students do not feel that they are es-sentially just repeating their high school course); 11.

4. Result in an equivalent of one semester advancedplacement.Altering the traditional lecture format or rearrang-

ing and supplementing content seem to be two promis- 12.ing approaches to developing courses that will satisfythe above criteria. For example, Colby College hassuccessfully developed a two semester calculus coursethat fulfills the four conditions. The course integratesmultivariable with single variable calculus, and therebycovers the traditional three semester program in twosemesters [10].

Of course, the introduction of a new course entailsan accompanying modification of college placement pro-grams. However, providing new or alternative coursesshould have the effect of simplifying placement issuesand easing transition difficulties that now exist.

Recommendations1. School administrators should develop public aware-

ness programs with the objective of extending thesupport that exists for fifth year calculus courses toaccelerated programs including all of the fifth yearoptions.

2. A fifth year program should offer a student a choiceof courses (not just calculus).

3. The choice of fifth year options should be made bythe high school mathematics faculty on the basis oftheir interest and qualifications and the quality andnumber of the accelerated students.

4. If a fifth year course is intended as a college levelcourse, then it should be treated as a college levelcourse (text, syllabus, rigor).

5. A fifth year college level course should be taught withthe expectation that successful graduates (B- or bet-ter) would not repeat the course in college.

6. A fifth year program should provide an alternativeoption for the student who is not qualified to continuein an accelerated program.

7. A mathematics degree that includes at least onesemester of a junior-senior real analysis course involv-ing a rigorous treatment of limit, continuity, etc., isstrongly recommended for anyone teaching calculus.

8. A high school calculus course should be a full yearcourse based on the Advanced Placement syllabus.

9. The instructor of a high school calculus course shouldbe provided with additional preparation time for this

13.

14.

15.

course.High school calculus students should take either theAB or BC Advanced Placement calculus examina-tion.The evaluation of a high school calculus courseshould be based primarily on college placement andthe performance of its graduates in the next levelcalculus course.Only interested students who have successfully com-pleted the standard four year college preparatoryprogram in mathematics should be permitted to takea high school calculus course.Colleges should grant credit and advanced placementout of Calculus I for students with a 4 or 5 score onthe AB Advanced Placement calculus examination,and credit and advanced placement out of Calculus IIfor students with a 4 or 5 score on the BC AdvancedPlacement calculus examination. Colleges should de-velop procedures for providing :special treatment forstudents who have earned a score of 3 on an Ad-vanced Placement calculus examination.Colleges should individualize as much as possible theadvising and placement of students who have takencalculus in high school. Placement test scores andpersonal interviews should be used in determiningthe placement of these students.Colleges should develop special courses in calculus forstudents who have bee.i successful in accelerated pro-grams, but have clearly not earned advanced place-ment.

Colleges have an opportunity and responsibility to de-velop and foster communication with high schools. Inparticular:

16. Colleges should establish periodic meetings wherehigh school and college teachers can discuss expek ta-tions, requirements, and student performance.

17. Colleges should coordinate the development of en-richment programs (courses, workshops, institutes)for high school teachers in conjunction with schooldistricts and state mathematics coordinators.

Members of the Panel:GORDON BUSHAW, Central Kitsp.p High School, Sil-

verdale, WashingtonDONALD J. NUTTER, Firestone High Schoch, Akron,

OhioRONALD SCHNACKENBURG Steamboat Springs High

School, ColoradoBARBARA STOTT, Riverdale High School, Jefferson,

Louisiana

TUCKER: CALCULATORS 229

JOHN H. HODGES, University of ColoradoDONALD R. SHERBERT, University of IllinoisDONALD B. SMALL (Chair), Colby College

References

[1] C. Cahow, N. Christensen, J. Gregg, E. Nathans, H. Stro-bel, G. Williams. Undergraduate Faculty Council of Artsand Sciences Committee on Curriculum; Subcommitteeon Adv;kr.ced Placement Report, Trinity College, DukeUniversity, 1979.

[2] C. Dennis Carroll. "High school and beyond tabulation:Mathematics courses taken by 1980 high school sopho-mores who graduated in 1982." National Council of Ed-ucation Statistks, April 1984 (LSB 84-4-3).P.C. Chamberlain, R.C. Pugh, J. Schellhammer. "Doesadvanced placement continue throughout the undergrad-uate years?" College and University, Winter 1968.

[4] "Advanced Placement Course Description, Mathemat-ics." The College Board, 1984.E. Dickey. "A study comparing advanced placement andfirst-year college calculus students on a calculus achieve-ment test." Ed.D. dissertation, University of South Car-

[3]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

olina, 1982.

D.A. Frisbie. "Comparison of course performance of APand non-AP calculus students." Research MemorandumNo. 207, University of Illinois, September 1980.D. Fry. "A comparison of the college performance incalculus-level mathematics courses between regular-prog-ress students and advanced placement students." Ed.D.dissertation, Temple University, 1973.

C. Jones, J. Kenelly, D. Kreider. "The advanced place-ment program in mathematicsUpdate 1975." Mathe-matics Teacher, 1975.

Second International Mathematics Study Summary Re-port for the United States. Champaign, IL: Stipes Pub-lishing, 1985.

D. Small, J. Hosack. Calculus of One and Several Vari-ables: An Integrated Approach. Colby College, 1986.D.H. Sorge, G.H. Wheatley. "Calculus in high schoolAt what cost?" American Mathematical Monthly 84(1977) 644-647.

D.M. Spresser. "Placement of the. first college course."International Journal Mathematics Education, Science,and Technology ,'.O (1979) 593-600.

Calculators with a College Education?

Thomas Tucker

COLGATE UNIVERSITY

Reprint of a lead article that appeared in the January1987 issue of Focus, the Newsletter of the Mathemati-cal Association of America:

The title of this article should sound familiar; it is avariation on Herbert Wilf's "The Disk with the CollegeEducation," which appeared in the Monthly in January,1982. In that article, Wilf sent a "distant early-warningsignal" that powerful mathematical computer environ-ments like muMATH, which were just then becomingavailable on microcomputers, would someday soon ap-pear in pocket computers.

That day may have arrived. The Casio fx7000-G, in-troduced in early 1986, is no bigger than the usual $10hand calculator, but plots functions on its small dot ma-trix screen. In January 1987, Hewlett-Packard releasedits HP-28C; again, a normal-sized hand calculator thatnot only plots curves but also does matrix operations,equation-solving, numerical integration, and, last butnot least, symbolic manipulation!

Neither the Casio ($55-$90) nor the HP-28C (around$180) are cheap by calculator standards, but any stu-dent who has bought a calculus textbook without flinch-ing can afford a Casio and in a couple of years I wouldexpect the textbook to cause more flinching than theHP-28C. The questions these calculators raise for math-ematics educators are the same Wilf asked in 1982 (afterwhich he "beat a hasty retreat"). The answers are notany clearer today.

Here is a more detailed look at the two calculators.The Casio fx7000-G differs from other key-stroke pro-grammable, scientific calculators by having a larger dotmatrix screen that makes graphing possible and thatallows the user to see clearly the expressions being eval-uated. The plotter can graph many functions at once.The window (range of t- ml y-coordinates) can be con-trolled by the user and can be easily magnified to zoomin on a particular poi Lion of the graph. A moving pixeltracing out a given curve can be stopped at any time to

230 READINGS

specify a zooming-in point. The plotter can be used inthis way to find the points of intersection of two curves.

The calculator, however, has no built-in routine tosolve an equation numerically, nor does it have a Simp-son's rule key for numerical integration like some otherscientific calculators. Since every key has three func-tions, the keyboard is cluttered with symbols and ab-breviations:

The HP-28C does a lot more than the Casio, and itis perhaps unfair to compare them at all. It is truly thefirst in a new generation of calculators. Although thinenough to fit easily in a breast pocket, it fold:: open toreveal two keyboards. On the right is a four-line screenand below it an array of 37 buttons that looks vaguelylike a standard calculator keyboard. On the left is a35-button alphabetic keyboardthere is only a single"shift key" so each key has only two functions.

But where are the sin, cos, and exp buttons? Andwhy does the pop row of six buttons on the right key-board have no labels at all? Because the HP-28C ismenu-driven, and the top row contains all-purpose func-tion keys! If you want sin, press the TRIG button,which activates a trigonometry menu at the bottom ofthe screen, and then press the function button directlybelow SIN (press NEXT to see six more functions onthe TRIG menu).

There arc other menus for logs and exponentials,equation-solving, user-defined functions, statistics, plot-ting, matrix operations and editing, binary arithmetic,complex arithmetic, string operations, stack operations,symbolic manipulation, program control (DO UNTIL,etc.), special real arithmetic (modulo, random numbergenerator, etc.), printing (yes, one can buy a thermalprinter with infrared remote control), and, of course, acatalog of all operations.

Like other Hewlett-Packard calculators, the HP-28Cis a stack machine with operators and operands enteredin reverse Polish. Stack entries can be commands, realor complex numbers, lists, strings, matrices, or alge-braic expressions such as '2 * 3 + 5' or 'X * SIN(X)'.Expressions can be evaluated by pressing the EVAL key,which can be used like the "=" key on algebraic calcu-lators by those averse to reverse Polish.

If 'X * SIN(X)' is at the top of the stack and youwant it symbolically differentiated, put the variable ofdifferentiation 'X' on the stack and press the derivativebutton. To compute its degree 5 Taylor polynomialcentered at 0, enter 5 and press TAYLR on the algebraicmenu. To compute its definite integral from 0 to 1 enterthe list 'X', 0, 1 and a tolerance .0001 and press theintegral button.

To plot 'X * SIN(X)', press DRAW on the plotting

menu: Window parameters are controlled by the useras on the Casio fx7000-G; x- and y-coordinates of anypoint on the screen can be found by moving cross-hairsto th,.. desired point and pressing INS.

To find a root of the equation 'X = TAN(X)', put iton the stack followed by an initial guess, or an interval,or an interval and a guess; then press ROOT on theequation-solving menu.

To invert the 2 by 2 matrix with rows [1, 2] and [3, 4],enter [[1, 2] [3, 4]] on the stack and press 1/x. To find itsdeterminant, press DET on the array menu. To solve asystem of equations, put the left-side coefficient matrixon the stack, followed by the right-side vector, and thenpress ÷. To multiply two matrices on the stack, pressx.

The numerical routines are high quality. There are12 digits of accuracy displayed and 16 digits internal.For example, with display set at three digits to the rightof the decimal point, '2A39 EVAL' yields 549755813888;multiply the result by 2 and 1.100 E 12 appears; dividethat by 2 and 549755813888 reappears. The routinesare also fast. A short program written to multiply amatrix times itself 100 times ran in about 2 minutes fora 6 by 6 matrix!

There are some problems. Memory is limited com-pared to a microcomputer: 8 by 8 matrix multiplication:s about all the calculator can handle and a request forthe degree 5 Taylor polynomial for ec(r-i) overflows thesymbolic differentiator (try computing the fifth deriva-tive without regrouping to see whyWilf's micro took 4minutes to get the degree 9 polynomial in his 1982 arti-cle). Although the matrix algebra routines are accurateand fast, the HP-28C has never met a square matrix itcouldn't invert (presumably, matrix entries such as 1 E500 should tell the user something is wrong).

How hard are these calculators to use? Although justclearing the Casio fx7000-G memory can be a challengewithout the manual, students used to a scientific cal-culator should feel comfortable after an hour or two.Hewlett-Packard designs more for engineers than forprecalculus students, and the HP-28C is no exception.It takes ten hours to become proficient enough to beginto realize the potentia! of this calculator, and one couldspend weeks exploring Ile nooks mid cran.iies of themachine. Luckily, documentation isn't too bad. Ba-sically, if a student can learn PASCAL, he or she canlearn to use the HP-28C.

Who will vse the HP-28C? It is not powerful enoughto help a professional mathematician do symbolic ma-nipulation the way MACSYMA helped Neil Sloane.(See his January 1986 Notices article "My Friend MAC-SYMA.") But plenty of calculus students would find

2 3

STEEN: WHO STILL DOES MATH WITH PAPER AND PENCIL? 231

the HP-28C handy when they are asked on their nexttest to differentiate (1 x2)91112, and so would I if Ihad to graph that function. The calculator can't solvea twenty-variable linear program by the simplex algo-rithm, but it could be programmed to do eight or ninevariables, and I can already see using the HP-28C in mylinear algebra class to find eigenvectors and eigenvaluesfor arbitrary 5 by 5 matrices using the power method.

One might cringe at a student integrating 1/(1 + x2)from 0 to 1 by pressing a button and getting .785 (actu-ally, it takes about 20 button-pushes to get the expres-sion on the stack and another 10 to enter the integra-tion parameters and perform the integration, althoughit all takes less than a minute). But what about the arclength of y = x3 from x = 0 to x = 1? Something lost,something gained.

Mathematicians are traditionally wary of technology.Perhaps their qualms are justified. To think of the area

under the curve y = 1 /(1 + x2` from 0 to 1 as .785and not vr/4 is to miss all the beauty of a mysteriousrelationship between circles and triangles and areas andrates of change. Mathematicians are notoriously slowto come to grips with technology.

At the Sloan Conference on Calculus at Tulane lastJanuary, a syllabus was proposed that recommendedthe use of programmable calculators with a Simpson'srule button, although such calculators have existed forfive years. The participants at that conference had noidea that a Casio fx7000-G or an IIP-28C was loom-ing on the horizon. How long will it take to recognizepedagogically the existence of these calculators, whichmany students will already have? Must it be "some-thing gained, somethin; lost?" Cati't it just be "some-thing gained?" Good questions to ask, but like Wilf in1982, it is probably tame to beat a hasty retreat.

Who Still Does Math with Paper and Pencil?Lynn Arthur Steen

ST. OLAF COLLEGE

Reprint of a "Point of View" column that appeared inTHE CHRONV;LE OF HIGHER EDUCATION on October14, 1987:

Mathematics is now so widely used in so many differ-ent fields that it has become the most populatedbutnot the most popularundergraduate subject. Eachterm an army of three million students labors withprimitive tools to master the art of digging and fill-ing intellectua, ditches: instead of using shovels andpick axes, they use paper and pencil to perform mil-lions of repetitive calculations in algebra, calculus, Indstatistics. Mathematics, the queen of the sciences, hasbecome the serf of the curriculum.

People who use mathematics in the workplaceaccountants, engineers, and scientists, for examplerarely use paper and pencil any more, certainly not forsignificant or complex computations. Electronic spread-sheets, numerical-analysis packages, symbolic-algebrasystems, and sophisticated computer graph .cs have become the power tools of mathematics in industry. Evenresearch mathematicians use computers to help themwith exploration, conjecture, and proof. In the col-lege classroom, however, mathematics haswith few

exceptionsremained in the paper-and-pencil era.Academic inertia alone is not a sufficient explana-

tion for this rtate of affairs. Other disciplinesnotablychemistry, economics, and physicshave adapted theirundergraduate curricula to include appropriate use ofcomputers.

In contrast, many mathematicians believe that com-puters are rarely appropriate for mathematics instruc-tion; theirs is a world of mental insight and abstractconstructions, not of mechanical caLulation or concreterepresentation. Most mathematicians, after all, choosemathematics at least in part because it depends onlyon the power of mind rather than on a variety of com-putational contrivances.

All that is going to change in the next two or threeyears, which in education are the equivalent of a twin-ding of an eye. The latest pocket calculators withcomputer-like capabilities can perform at the touch of afew buttons many of the laborious calculations taughtin the first two years of college mathematics. They can,among other things, graph and solve equations, performsymbolic differentiation as well as numerical and SOT.ICsymbolic integration, manipulate matrices, and solve si-multaneous equations. Although such computations do

232 READINGS

not form the heart of the ideal curriculum as it existsin the eye of the mathematician, they do account forthe preponderance of the achieved curriculum that isactually mastered by the typical undergraduate.

Computation has become significant for mathemat-ics because of a major change not just in scale but ofmethods: the transition from numerical mathematics,the province of scientists, to symbolic and visual mathe-matics, the province of mathematicians. Large comput-ers have beer. doing "real" mathematics for years, butcost and relative scarcity kept them out of the class-room. No more. Mathematics-speaking machines areabout to sweep the campuses, embodied both as com-puter disks and as pocket calculators. Beginning thisfall, college students will be able to use such devicesto find the answers to most of the homework they areassigned.

Much as professors like to believe that educationstandards are set by the faculty, the ready availabil-ity of powerful computers will enable students to setnew ground rules for college mathematics. Templateexercise and mimicry mathematicsstaples of today'stextswill vanish under the assault of computers thatspecialize in mimicry. Teachers will be forced to changetheir approach and their assignments. They will nolonger be able to teach as they were taught in the paper -andpencil era.

Change always involves risk as well as benefit. Wehave no precedents for learning in the presence ofmathematics-speaking calculators. No one knows howmuch "patterning" with paper-and-pencil methods isessential to provide a foundation for subsequent ab-stractions. Preliminary research suggests that it maynot be as necessary as many mathematics teacherswould like to believe.

On the other hand, many students tolerate (and sur-vive) mathematics courses only because they can getby with mastery of routine, imitative techniques. Amathematics course not built on the comfortable foun-dation of mindless calculation would almost surely betoo difficult for the student whose sole reason for takingmathematics is that it is required.

Despite such risks, mathematicsand societyhasmuch to gain from the increasing use of pocket comput-ers in college classes:

Undergraduate mathematics will become more likereal mathematics, both in the industrial work placeand in academic research. By using machines to ex-pedite calcuktions, students can experience math-

ematics as it really isas a tentative, exploratorydiscipline in which risks and failures yield clues toticcess. Computers change our perceptions of whatis possible and what is valuable. Even for unsophis-ticated users, computers can rearrange the balancebetween "working" and "thinking" in mathematics.Weakness in algebra skills will no longer preventstudents from pursuing studies that require collegemathematics. Just as spelling checkers have en-abled writers to express ideas without the psycho.logical block of worrying about their spelling, so thenew calculators will enable students weak in algebraor trigonometry to persevere in calculus or statis-tics. Computers could be the democratizer of collegemathematics.Mathematics learning will become more active andhence more effective. By carrying most of the compu-tational burden of mathematics homework, comput-ers will enable students to explore a wider variety ofexamples, to study graphs of a quantity and varietyunavailable with pencil-and-paper methods, to wit-ness the dynamic nature of mathematical processes,and to engage realistic applications using typicalnot oversimplifieddata.Students will be able to explore mathematics on theirown, without constant advice from their instruc-tors. Although computers will not compel studentsto think for themselves, these =chi-es can provklean environment in which student-generated mathe-matical ideas can thrive.Study of mathematics will build long-lasting knowl-edge, not just shwt-lived strategies for calculation.Most students take only one or two 1,,-rms of col-lege mathematics, and quickly forget whdt little theylearned of memorized methods for calculation. Inno-vative instruction using a new symbiosis of machinecalculation and human thinking can shift the balanceof mathematical learning toward understanding, in-sight, and mathematical intuition.Mathematics-capable calculators pose deep questions

for the undergraduate mathematics curriculum. Byshifting much of the computational burden from stu-dents to machines, they leave a vacuum of time andemphasis in the undergraduate curriculum. No one yetknows what, if anything, will replace paper-and-pencilcomputation, or whether advanced mathematics can bebuilt on a c, :nputer-reliant foundation. What can besaid with certainty, however, is that the era of paper-and-pencil mathematics is over.

241

ZORN: COMPUTING IN UNDERGRADUATE MATHEMATICS 233

Computing in Undergraduate MathematicsPaul Dorn

ST. OLAF COLLEGE

An "issues paper" prepared in conjunction with aJune 1987 conference organized by the Associated Col-leges of the Midwest to examine the role of computingin undergraduate mathematics as part of an effort bythe staff of the National Science Foundation to increasethe impact of computing on undergraduate departmentsof mathematics. Views expressed in this paper are thoseof the author, and do not necessarily reflect the viewsof the National Science Foundation.

Modem computing raises unpreceder.,ed opportuni-ties, needs, and issues for undergraduate mathemat-ics. The relation between computing and mathemat-ics is too young, and changing too quickly, to admitdefinitive positions. None are taken here. In math-ematical language, this paper is not an authoritativemonograph but a topical survey with many oven ques-tions.

We do not a4...rt i,hat computing serves every worth-while purpose of undergraduate mathematics. The in-teresting issues are genuine questions: Where in thecurriculum is computing appropriate, and why? Whatdoes computing costin time, money, and distrac-tion fr.= other purposes? If computers handle rou-tine mathematical manipulations, what will studentsdo insteLd? Wilt students' manipulative skills and in-tuition survive? Should we teach things machines doLetter?

So much said, it would bt: dit,i. genuous to deny theviewpoint that m. tivat,..1, this paper, :rd is implicitthroughoutthat we can and should use modern computing more than we have done to imrove mathemat-ical learning and teaching. Alt%ough we will arguethat the computing resource hp., scarcely been tapped,this paper is not simply a plea for computers in theclassroom. Mathematical computing is educationallyvaluable only as it alters and E,' 'JCS curricular goals ofundergraduate mathematics. It follows that curriculargoals should guide teaching uses of computing, not theother way around.

What is Computing?Until recently, computing in undergraduate mathe-

matics usually meant writing or running programs (in

Basic or Pascal) for floating point numerical operations.Much more is now possible: symbolic algebra, sophis-ticated graphics, interactive operating modesall withlittle or no programming required of the user. Comput-ing should be understood broadly, comprising hardware,software, and peripheral equipment.

Other forms of educational technology, such as video-discs, might someday become important teaching tools.They are not addressed here. Covering everythingthat now exists would be difficult; anticipating whatmay exist is impossible. Our scope is comprehen-sive only in the sense that many kinds of educationaltechnology, like computing, amount to new ways ofrepresenting and manipulating mathematical informa-tion.

Mathematics and ComputingComputing drives the modern mathematical revolu-

tion. As Gail Young puts it in [3],

...Me are participating in a revolution in math-ematir:s as profound as the introduction of Arabicnumerals into Europe, or the invention of the cal-culus. Those earlier revolutions had common fea-tures: hard problems became easy, and solvable notonly by an intellectual elite but by a multitude ofpeople without special mathematical talent; prob-lems arose that had not been previously visualized,and their solutions changed the entire level of thefield.

Like Arabic numerals and the calculus, computing isa sharper tool, but it is also more than that. Comput-ers do mop- In help solve old problems. They leadto new pro',L.-..as, new approaches to old problems, andnew notions of what it means to solve problems. Theychange fundamental balances that have defined the dis-cipline of mathematics and how it is pursued: continu-ous and discrete, exact and approximate, abstract andconcrete, theorltical and empirical, contemplative andexperimental. Computers change what we think possi-ble, what w..t think worthwhile, and even what we thinkbeat.tiful

Computing is becoming commonplace (if not ubiqui-tous) in mathematical research, even on classical prob-lems. Without computers, research in many new ar-

242

234 READINGS

eas would stop. Computing figured, more or less fun-damentally, on the way to several recent spectacularadvances, including the four color theorem and theBieberbach conjecture. With numerical, graphical; andsymbol-manipulating abilities, computers check calcu-lations, test conjectures, process large data sets, searchfor structure, and represent mathematical objects innew ways. They make possible entirely new viewpointson problems in mathematical researchviewpoints thatare more empirical than deductive, more experimentalthan theoretical.

Computing has changed how mathematics is used atleast as much as it has changed how mathematics is cre-ated. The changes are broad and deep, touching areasfrom arithmetic to statistics to differential equations.This part of the computer revolution, moreover, is hap-pening in public. Changes at the rarefied research levelmay affect only a few people; changes at the "user"level reach a much broader constituency. Already, un-dergraduates freelyif sometimes naively use sophis-ticated numerical "packages" in science and social sci-ence courses. Seeing computing all around them, stu-dents naturally expect some in mathematics courses,too.

It seems axiomatic (certainly to students!) thatthe profound effects of computing on research and ap-plied mathematics should be reflected in undergrad-uate mathematics education. Honesty to our disci-pline and our own best interests as mathematics teach-ers both dictate so. Honesty requires at least thatwe keep ourselves and our students abreast of impor-tant developments in our field. Self-interest says weshould do more than report what happens outside: weshould avail ourselves of the enormous opportunitiescomputers offer for teaching and learning mathemat-ics.

Despite all this, computing has not yet changedthe daily work of undergraduate mathematics verymuch. The standard freshman calculus course, forexample, sti71 consists largely of paper and pencilperformance of mechanical algorithmsdifferentiation,graph-sketching, antidifferentiation, series expaaz.ier,etc.just what machines do best. Graphical and nu-merical methods are usually treated as side issues. Witha little computing power, they could illuminate impor-tant interplays between discrete and continuous ideas,exact and approximate techniques, geometric and ana-lytic viewpoints.

Statistical as well as anecdotal evidence shows thatmathematics lags behind the other undergraduate sci-ences in teaching uses of computing. In 1985 and 1986,for example, only 32 of approximately 2800 proposals

to the NSF's College Science Instrumentation Programcame from mathematics deparLments. Galling as it isto be elbowed from the trough, this paucity of mathe-matics proposals is only a symptom. Our real prob-lem is not too few proposals, but the opportunitiesmathematicians miss to revitalize teaching and learn-ing.

Problems of the PastReasons for the lag in undergraduate mathematical

computing are easy to guess. The clearest differencebetween mathematics and the physical sciences is inthe roles experiment and observations play in each. Al-though mathematics has an essential (if informal) ex-perimental aspect, especially in research, mathematicsis not a laboratory discipline in the formal, ritualis-tic sense that applies to the other sciences. The valueof "instruments," whether computers or chemicals, tosupport the experimental side of the natural sciencesis taken for granted, but there is no similar consensusabout undergraduate mathematics. This may change,but for now, the idea of mathematical "instrumenta-tion" is still a novelty. Machines to support undergrad-urte mathematical experimentation are just appearing,and we are just learning to use them. Unlike our col-leagues in the natural sciences, we mathematicians mustconvince our departments and college administratorsand sometimes ourselvesnot just that we need partic-ular items of equipment, but that we need equipmentat all.

Computing may reshape college mathematics slowlyalso because computers raise harder, more fundamen-tal pedagogical questions in mathematics than in otherdisciplines. Computers can thoroughly transform ac-tivities in a chemistry laboratory, but they need notchange the basic ideas studied there. By contrast, mod-ern computers handle so much of what we mathemati-cians traditionally teach that we are forced to rethinknot only how we teach, but also what and why. Ironi-cally, computers may have contributed so little in un-dergraduate mathematics just because they can do somuch.

The cost of computer programming, measured intime and distraction, has been another impedimentto mathematical computing. Is the effort of imple-menting, say, a simple Riemann sum program in Bu-sic worth the mathematical insight it offers? Similarquestions might seem to apply in other sciences, butexperimental data generated in natural science labo-ratories is well suited to routine numerical manipula-tions; a few programs go a long way. Mathematical

243


computing, being less circumscribed, is harder to "pack-age," and the programming overhead is correspondinglyhigher.

Prospects for the FutureGiven this history, why should things change? The

simplest reason might be called "manifest destiny."Like it or not, computing is already changing un-dergraduates' views of and experience with mathe-matics, and the rate of change is increasing. Infreshman calculus, symbol-manipulating and graph-sketching programs on handheld calculatins already re-duce a good share of canonical exam questions tobutton-pushing. (See [4] and [5]; note particularly theirmetaphorical titles.) We mathematicians can eitherapplaud or condemn these changes, but we can't ig-nore them. We will either anticipate and use com-puting developments, or we will have to fend themoff.

Good omens for computing in undergraduate math-ematics can also be seen in hardware and softwareimprovements. Mathematical computing is becomingmore powerful, cheaper, and easier to use. For ex-ample, "computer algebra systems" Nacsyina, Maple,Reduce, SMP, and others) are starting to appear onstudent-type machines. These systems do much morethan algebra; they are actually powerful ("awesome," instudentspeak) and flexible mathematics packages thatperform a host of routine operationssymbolic alge-bra, formal differentiation and integration, series expan-sions, graph-sketching, numerical computations, matrixmanipulations, and much more. Because no program-ming is needed (one-line commands handle most opera-tions), all this power costa virtually nothing in distrac-tion. For good or ill, computing is changing the mix of"working" and "thinking" that determine what it is toknow and do mathematics.

The possibilities and problems computing raises forteaching would be important. even if undergraduatemathematics were thriving. On the contrary, toofew students study mathematics; of those who do,too few learn it deeply or well. Freshman calcu.,lus is a squeaky wheel, but general complaints aleheard up and down the curriculum: students can'tfigure, can't estimate, cent read, can't write, can'tsolve problems, can't harxdtc theory and so on. Lack-ing computers didn't augt: all these problems, andhaving computers won't, volve them all. Neverthe-less, the climate for change is favorable (see, e.g.,[2]).

Benefits and OpportunitiesComputing can benefit undergraduate mathematics

teaching in many ways. Understanding the context isimportant: Our goal is not more computers, but bettermathematical learning.

1. To make undergraduate mathematics more like "real"mathematics. Mathematics as it is really used hasmany parts: formal symbol manipulation, numeri-cal calculation, conjectures and experimen.a, "pure"ideas, modeling, and applications. Undergraduatestudents, especially in beginning courses, see mainlythe first ,,ao at the expense of the others. Anothermixture of ingredients might give students a bet-ter sense of context, and help them calculate moreknowledgeably and effectively. By handling routineoperations, computing can free time and attentionfor other things.

2. To illustrate mathematical ideas. Analytic con-cepts such as the derivative have numerical, geo-metric, and dynamical (i.e., time-varying) as wellas analytic meaning. Pursuing graphical, numeri-cal, and dynamical viewpoints is tedious or impossi-ble by hand techniques. Doing so is easy and help-ful with computing, especially if algebraic, graph-ical, and numerical manipulation are all available.Given these, a student might compute differencequotients algebraically, tabulate numerical valuesas a parameter varies, and observe the geometricbehavior of the associated secant lines at variousscales.

3. To help students work examples. Mathematiciansknow the value of concrete examples for under-standing theorems and their consequences. Stu-dents need examples, too, but the points examplesmake are often obscured by computational difficul-ties. With computers, students can work more andbetter examples. In matrix algebra and statistics,large-scale problems become feasible. In calculus,subtle points can be clarified. The fundamentaltheorem, for example, is often misunderstood be-cause students have insufficient experience with the"left side"the integral defined by Riemann sums.II;flth a machine to crunch the numbers, the "leftside" makes numerical and geometric sense. Whenthe area-under-the-graph function can be tabulated,graphed, and geometrically differentiated, for severalintegrands, then the idea of the theorem is hard tomiss.

4. To study, not just perform, algorithms. Algorithmicmethodsfor matrix operations, polynomial factor-

236 READINGS

izati)n, finding gcd's, and other operationsare nowa way of mathematical life. Students continually per-form algorithms, but seldom study them in their ownright. Rudimentary algorithm analysis (e.g., the 0-notation) could be an important and timely appli-cation of elementary calculus. Recursion, iteration,and list processing, viewed as general mathematicaltechniques, also deserve more attention than theyget. Treating these topics efficiently means imple-menting them, or seeing them implemented, on com-puters.

5. To support more varied, realistic, and illuminatingapplications. Limited by hand computations, manyapplications of college mathematics are contrived andtrite. There are too many farm animals, rivets,and exotic fencing schemes. Computing allows bothlarger-scale versions of traditional applications (e.g.,larger matrices, larger data samples) and new ap-plications altogether (e.g., ones requiring numericalmethods.) More flexible, less circumscribed appli-cations not only do more, they also show more ofthe mathematics underlying them. Physical appli-cations, for example, should be part of the histor-ical, mathematical, and intuitive fiber of elemen-tary calculus. Hand techniques restrict the scopeof feasible physical problems to those few that canbe solved in closed form, using elementary func-tions. Other conceptually simple applications, likemany from economics, lead ..,o high-order polyno-mial equations, and so are also taboo. Only sim-ple numerical methods (root finding, numerical inte-gration, etc.) are needed to make such applicationstractable.

6. To exploit and improve geometric intuition. Graphsof all kinds give invaluable insight into mathematicalphenomena. With computer graphics, attention canshaft from the mechanics of obtaining graphsa sub-stantial topic in elementary calculusto how graphsrepresent analytic information. Sophisticated graph-ics (surfaces in three dimensions, families of curves,fractals) practically require computing. They easedifficult learning transitions: from one variable toseveral, from function to family of functions, fromreal domain to complex domain, from pointwise touniform convergence, from step to step in iterativeconstructions.

7. To encourage mathematical experiments. The pol-ished theorem-proof-remark style of mathematicalwriting hides the fact that mathematics is createdactively, by trial, error, and discovery. Students can

learn the same way, if the labor of experimenting isnot too great. With computing, students can dis-cover that square matrices "usually" have full rank,that differentiable functions look straight at smallscale, and that there is pattern to the coefficients ofa binomial expansion.

Mathematical experimentation is good both as ateaching tool and as an active, engaged attitude to-ward mathematics. We mathematicians often tryto inculcate this attitude in students, begging themto "try something." Interestingly, the oppositeprobleman excessively experimental attitude, or"hacking"plagues computer science. Will comput-ers breed mathematical hackers? Would that be abad thing?

8. To facilitate statistical analysis and enrich probabilis-tic intuition. Data analysis in mathematical statis-tics is highly computational. Machines allow largersamples, and thus greater reliability and verisimil-itude. Students see more analysis, and more ofits power, with less distraction. Computing in un-dergraduate statistics is already becoming routine.As methods of data analysis becomes easier, choos-ing methods and interpreting their results becomesharder. Informed statistical analysis requires soundprobabilistic intuition. Probabilistic viewpoints arealso essential, of course, in classical analysis and inmodern physical applications. By simulating randomphenomena, computers illustrate probabilistic view-points concretely. Monte Carlo integration meth-ods, for example, combine ideas from elementary cal-culus and probability, and show relations betweenthem.

9. To teach approximation. The idea of approxima-tion is important throughout mathematics and itsapplications. When students use mathematicsinother courses and in careersthey will certainlyuse numerical methods. Yet students' learningexperience, especially in beginning courses, is al-most entirely based on exact, algebraic methods:explicit functions, closed-form solutions, elemen-tary antiderivatives, and the like. Numerical andgraphical illustrations of approximation ideas arecomputationally expensive, but essential for under-standing. Machine computing makes them possi-ble.

To ase approximation effectively, students need someidea of error analysis. Error estimate formulasare especially intractable for hand computation be-cause they usually involve higher derivatives, upper

245

ZORN: COMPUTING IN UNDERGRADUATE M.:THEMATICS 237

bounds, and other mysterious ingredients. High-level computing (root-finding, symbolic differentia-tion, numerical methods) helps students understanderror estimation without fourrlsring in distractingcalculations.

10. To prepare students to compute effectivelybut skep-ticailyin careers. Applied mathematics studentswho pursue technical professions (engineering, actu-arial science, business, and industry) will use math-ematical computing in many forms. Some kinds ofsoftware (e.g., differential equation solvers, statisti-cal analysis packages) are standard; students shouldsee some of them in advance. Even more importantthan working knowledge of particular programs is asound mathematical understanding of how those--oranyprograms address the problems they purportto solve. Arbitrary choicesestimates, simplifica-tions, stopping rulesare always implicit in appli-cations software. Duly skeptical users must under-stand these choices and how they affect computedresults.

11. To show the mathematical significance of the com-puter revolution. The relation between mathematicsand computer science offers excellent object lessonsin the interconnectedness of knowledge. As a matterof general education, and for practical reasons, stu-dents should learn something about the mathemati-cal foundations of computing, and about the mathe-matical problems computing raises.

12. To make higher-level mathematics accessible to stu-dents. Undergraduates in the natural sciences havealways participated in serious research. Comput-ing offers similar opportunities in mathematics as itstrengthens the concrete, empirical side of mathe-matical research. With powerful graphics, studentsmight investigate the fine structure of Mandelbrotsets, observe evolution of dynamical systems, and ex-plore geodesics on complex surfaces.

Resource RequirementsNo one doubts that educational uses of computing re-

quire hardware and software. It is less well understoodthat resource requirements only begin there. Chemistrydepartments require more than chemicals and equip-ment to support their laboratory courses. In the sameway, more than hardware and software is needed if thebenefits of having hardware and software are to be re-alized.

College mathematics teachers who use computingface common problems. Some problems are local (e.g.,

securing institutional support) and some are national.Many stem from the fact that computing is relativelynew to mathematics. Mathematicians are just begin-ning the resource management tasks our scientific col-leagues have worked at for decades: convincing ad-ministrators, purchasing and maintaining equipment,modifying time-hardened comes, developing curricu-lar rationales, and articulating what we are doing. Un-dergraduate science departments write proposals, carryout supported projects, and administer grants as amatter of habit. In mathematics departments, thesenabits are less ingrained, and "machinery" to sup-port themadministrative help, program and deadlineinformation, accounting proceduresis usually primi-tive.

1. Technical support. Natural science departmentsmaintain a complete apparatus of support servicesfor their laboratory courses: equipment mainte-nance, classroom demonstration equipment, dedi-cated space, and paid student assistants. As %lath-ematics departments develop and use their own ver-sions of "instrumentation," the same support needsarise.

2. Institutional support. Most colleges have comr,.;,1rs,but not necessarily the right ones, or in the Atplaces, for mathematical use. Less tangible forms ofinstitutional support are just as important: teach-ing loads that credit faculty time for developing andstaffing mathematical labor tories, tenure and pro-motion procedures that reward such work, and ad-ministrative support for matching money for grantproposalr

3. Time. Realizing the mathematical benefits of com-puting, whatever they are, costs timeours and ourstudents'. Not all benefits will prove worth having,but for those we judge worthwhile, time s'noule beprovided, and accounted for honestly. Instructionalcomputing, like other new things, often begins witha trail-blazing department zealot, for whom the workis its own reward. Eventually, Jwnership and re-sponsibility should be shared. Unless time is madeavailable; computing will remain the zealot's privateprovince.

4 Courseware. It seems historically inevitable thatcomputing will change mathematics courses andcou-se material.,. If we mar.ematicians are to man-age the process, we neect nardwar- ,,oftware, and"courseware"instructional material (,manuals, ex-ercises, tests, discursive material, and full text-books) that thoughtfully integrates, rather than sim-

246

238 READINGS

ply appends; the computing viewpoint into substan-tial mathematics courses. The necessary hardwareand software exist, or will soon. The laggard, in-evitably, is courseware. Robust courseware is ex-pensive and time-consuming to develop, and it mustfind a precarious balance between being too spe-cific to be portable and being too general to be use-ful.

5. Technical information. Hardware and software cha-nge rapidly. In order to choose equipment wiselyand use it effectively, mathematicians need techni-cal specifications for hardware and software, crit-ical reviews of educational software, reliable pricedata, and (hardest of all) a sense of the future. Be-cause hardware usually outruns software, the naivestrategy (choose software, then hardware to runit) guarantees obsolete hardware. Should our pro-fessional societies marshal the expertise we need?How?

6. Shared experience. Who is doing what? Where?With what equipment? What worked? How wastopic X treated? Undergraduate mathematical com-puting, like any quickly developing field, dependson communication if the wheel is not repeatedlyto be reinvented. Several models exist: The SloanFoundation supports several college projects anda newsletter on teaching uses of computer alge-bra systems, and occasional conferences on the sub-ject. The Maple group at the University of Wa-terloo publishes a users' group newsletter and or-ganizes electronic communications for an interestgroup in teaching uses of computer algebra. TheCollege Mathematics Journal carries a regular col-umn (see[1]) on instructional computing. The MAAand its Sections sponsor minicourses nationally andshort courses regionally. More such efforts areneeded.

Open QuestionsTechnical, financial, institutional, and logistical con-

sideratic ns notwithstanding, the most difficult and in-teresting questions computing raises are mathemati-cal and pedagogical. Although few answers are haz-arded here, most users face some of these questions, atleast implicitly. Asp discipline, we face them all regu-larly.

1. Will computers reduce students' ability to calculateby hand? If so, is that a bad thing? When comput-ers do routine manipulations in mathematics courses,

students must do something else. How will stu-dents who have never mastered routine calculation,or those who enjoy it, fare in such courses? Willseeing the results but not the ,process of calcula-tions help them understand, or further mystify them?If hand computation builds algebraic intuition"symbol sense"will machine computation destroyit?

2. How should analytic and numerical viewpoints bebalanced? To estimate fl 1 1 dx numerically as+7:-1.57 is routine. To see analytically that the answeris is memorable. Both facts are worth know-ing. Will students learn them both in the calcu-lus course of the future? To paraphrase RichardAskey, exact solutions are precious because they arerare. Will students learn this? Will we rememberit?

3. How does computing change what students shouldknow? Traditional courses are full of methods andviewpoints that arose to compensate for the limita-tions of hand computation. Do new computationaltools render these topics obsolete? More generally,should we teach mathematical techniques machinescan do better? Some things, surely, but which, andwhy? Partial fraction decomposition? The square-root algorithm? For topics we keep, will we for-bid computers? What will replace topics we dis-card?

4. Will the mechanics of computing obscure the mathe-matics? We teach mathematics, not computing, andmathematics syllabi are already full. How will we usecomputing to teach mathematics without distractingtechnical excursions? How will we gauge whethercomputing effort is commensurate with the mathe-matical insight it gives? Can computing save teach-ing time? Anecdotal evidence suggests that calculusstudents can use high-level programs without unduedifficulty or distraction. Can pre-calculus studentsdo the same?

5. How will computing affect advanced courses? Math-ematical computing frequently occurs in lower-levelcourses, like calculus, which have other educationalgoals than preparation for advanced courses. Willalumni of such courses be better or worse pre-pared for advanced mathematics? Should advancedcourses change along with introductory courses? Cancomputing improve advanced courses in their ownright?

6. Computing and remediation. Remedial course em-phasize mechanical operations. Will relegating rou-

2 4 7


tine operations to machines reduce the need for reme-diation? Or could deeper, more idea-oriented coursesrequire more remediation for weaker students? In ei-ther case, how can computing help students in re-medial courses? Can such students use advancedcomputing, or do they lack some necessary sophis-tication?

7. CAI vs. tool-driven computing. In some applica-tions, computers act as intelligent tutors, leading stu-dents through carefully prescribed tasks. In others,computers are flexible tools: students decide where.and how to use them. Where is each model use-ful?

8. Equity and access to computing. Se..phisticated com-puting on powerful computer:: is itiil financiallyexpensive and, for most peopic, not always onhand. Calculators, for the price of four books, al-ready handle graphical, numerical, and algebraic(including matrix) calculations. Sophisticated cal-culators might both radically "democratize" high-level computing, and make it as natural as hand-held arithmetic computation. Will they? Shouldthey?

9. Will computing help students lec-n mathematicalideas more deeply, more easily, and more quickly?Conjecturally, yes, but the -onclusion is not foregone.For undergraduate mathematics, this is the bottomline.

References

[1] R. S. Cunningham, David A. Smith. "A Math-ematics Software Database Update." The Col-lege Mathematics Journal, 18:3 (May, 1987) 242-247.

[2] Ronald G. Douglas, Ed. Toward a Lean and LivelyCalculus: Report of the Conference/Workshop ToDevelop Curriculum and Teaching Materials for Cal-cutus at the College Level. Mathematical Associationof America, 1986.

[3] Richard E. Ewing, Kenneth I. Gross, Clyde F. Mar-tin, Eds. The Merging of L iplines: New Direc-tions in Pure, Applied, and c ,mputational Mathe-matics. Springer Verlag, 1987.

[4] Thomas Tucker. "Calculators with a College Ed-ucation?" FOCUS, 7:1 (January-February, 1987)

[5] Herbert E. Wilf. "The Disk with a College Educa-tion." The American Mathematical Monthly, 89:1(Januoxy, 1982) 4-8.

Planning Committee:

LESLIE V. FOSTER, San Jose State UniversitySAMUEL GOLDBERG, Alfred P. Sloan FoundationELIZABETH R. HAYFORD, Assosaated Colleges of the

MidwestRICHARD KAKIGI, California State University, Hay-

ware.

ZAVEN KARIAN, Denison UniversityJOHN KENELLY, Clemson UniversityWARREN PAGE, Ohio State UniversityDAVID SMITH, Duke UniversityLYNN ARTHUR STEEN (Chair), St. Olaf CollegeMELVIN C. TEWS, College of the Holy CrossTHOMAS TUCKER, Colgate UniversityPAUL ZORN, St. Olaf College

National Science Foundation Participants:KENNETH GROSS, Division of Mathematical SciencesDUNCAN E. MCBRIDE, Office of College Science In-

strumentationBASSAM Z. SHAKHASHIRI, Assistant Director for Sci-

ence and Engineering EducationARNOLD A. STRASSDNBURG, Director, Division of

Teacher Preparation and EnhancemmtJOHN THORPE, Deputy Director, Division of Materials

Development, Research, and Informal Science Edu-cation

ROBERT WArsoN, Head, Office of College Science In-strumentation

24

-.:

r 0-0 e-G- p ry E. i--z_ CO RA Y1,\. ii.--N-IS

?Pt RTI L 1 ? Pt.IV 1r

C;''`

51-1 -12-- -k-- CtO V149 *--- iC\A In k. 0/_. cst. l Ls ..0_ck_c okAAAl

.

L...e_s s e,,,,pt, is.._ 5,,,-- ?,Ls-..1.,- --1-4i). or90or.---,. e___,y--.c - i ::: _-.4

.....--._Thf 547 ect..1,rs_a± L10 L4) - -c,..Art -I:- :::,',:'.'":..-,-

D re. a...L2a+ t7,st --ti-:c.,,, L-L, e_ _...., ..

cL>i-te..4-'11A-xe---c. C.0.____c-(c).e..0. A.z,..:: JP--,s1 j.-s.:.:.:.. It....

'Put. 6 l. s Le/r_s in 04/._e -p)-1). .(,,. _1,--

O , 4 atej clic r-e,,( IL%-t+L-c. . ,.

: .

tAi.e._ I", ut s -4 b.e_. -peL-t-i ÷- .. ... .

i tivia-e- a-60 k_÷- GL I( p-\,,0 IP CO e....,t. AD $;-1-i-e-L-E- ': -0..,r,e, -..5.0 .- --..'

. . .

Lk ce__72._c_y, t. e_.' -.1 '.'

,

,

1-1, e.s e 1 - ts -povi s e.s W e e se../ e_c+e_cA 1,. IIIV.w;-L-2A i'.1-(5,- C L 11- S )_____fi,\S cp±,t_e_c_-,4t4

tl...Q._ To 1,---H ce_re.t L 12_i/u

e A (-Eor___'_. ca--i-e_ ct.-eccribe( P(,/)0 rA tr_

25

Calculus for a New Century: Participant List

Washington, D.C., October 28-29, 1987

ADAMS, WILLIAM. National Science Foundation, Wash-ington DC 20550. [Program Director, 1800 G StreetNW] 202-357-3695.

AGER, TRYG. Stanford University, Stanford CA 94305.[Sr. Research Associate, IMSSS-Ventura Hall] 415 -723-4117.

ALBERS, DONALD. Menlo College, Menlo CA 94025.[Chairman, Dept of Math & CS 1000 El Camino Real]415-323-6141.

ALBERTS, GEORGE. National Security Agency, FortMeade MD 20755. [Executive Manager, PDOD] 301-688 -7164.

ALEXANDER, ELAINE. California State University,Sacramento CA 95819. [Professor, Dept of Math & Stet6600 J Street] 916-278-6534.

ALLEN, HARRY. Ohio State University, Columbus OH43210. [Professor, Dept of Math 231 West 18th Avenue]614-292-4975.

ALLEN, PAUL. University of Alabama, Tuscaloosa AL35487. [Dept of Math P.O. Box 1416] 205-348-1966.

ANDERSON, R.D. Louisiana State University, LatonRouge LA 70809. [Professor Emeritus, 2954 FritchieDr.]

ANDERSON, EDWARD. Thomas Jefferson High School,Alexandria VA 2231? [Chair Math Dept, 6560 Brad-dock Road] 703-941-7954.

ANDERSON, NANCY. University of Maryland, CollegePark MD 20742. [Professor, Dept of Psychology] 301-454 -6389.

ANDREWS, GEORGE. Oberlin College, Oberlin OH44074. [Professor, Dept of Mathematics] 216-775-8382.

ANDRE, PETER. United States Naval Academy, An-napolis MD 21402. [Professor, Math Dept] 301-267-3603.

ANTON, HOWARD. Drexel University, Cherry Hill NJ08003. [304 Fries Lane] 609-667-9211.

ARBIC, BERNARD. Lake Superior State College, SaultSte. Marie MI 49783. [Professor, Dept of Mathematics]906-635-2633.

ARCHER, RONALD. Unviersity of Massachusetts,Amherst MA 01003. [Professor, Dept of Chemistry] 413-545 -1521.

ARMSTRONG, JAMES. Educational Testing Service,Princeton NJ 08541. [Senior Examiner] 609-734-1469.

ARMSTRONG, KENNETH. University of Winnipeg, Win-nipeg Man. Canada R3B2E9. [Chair, Dept of Math andStatistics] 204-786-9367.

ARNOLD, DAVID. Phillips Exeter Academy, Exeter NH03833. [Instructor] 603-772-4311.

ARZT, SHOLOM. The Cooper Union, New York NY10003. [Professor, Cooper Square] 212-254-6300.

ASHBY, FLORENCE. Montgomery College, Rockville

MD 20850. [Professor, 109 South Van Buren St.] 301-279 -5202.

AVIOLI, JOHN. Christopher Newport College, NewportNews VA 23606. [50 Shoe Lane] 804-599-7065,

BAILEY, CRAIG. United States Naval Academy, An-napolis MD 21402. [Associate Professor, PMath Dept]301-267-3892.

BAKKE, VERNON. University of Arkansas, FayettevilleAK 72701. [Professor, Dept of Mathematical Sciences]501-575-3351.

BARBUSH, DANIEL. Duquesne University, PittsburghPA 15221. [Assistant Professor, 512 East End Avenue]412-371-1221.

BARRETT, LIDA. Mississippi State University, Missis-sippi State MS 39762. [Dean College of Arts & Sciences,P.O. Drawer AS] 601-325-2644.

BARTH, CRAIG. Brooks-Cole Publishing Co., PacificGrove CA 93950. [Editor, 511 Forest Lodge Rd] 408-373 -0728.

BARTKOVICH, KEVIN. N.C. School of Science *z Math-ematics, Durham NC 27705. [Broad St. & West ClubBlvd.] 919-286-3366.

BASS, HELEN. Southern Connecticut State Univ, NorthHaven CT 06473. [Professor, 59 Garfield Avenue] 203-397 -4220.

BATRA, ROMESH. University of Missouri, Rolla MO[Professor, Dept of Engineering Mechanics] 314 -341-4589.

BAUMGARTNER, EDWIN. LcMoyne College, SyracuseNY 13214. [Associate Professor, Dept of Mathematics]315-445-4372.

BAVELAS, KATHLEEN. Manchester Community College,Wethersfield CT 06109. [Instructor, 46 Westway] 203-647 -6198.

BAXTER, NANCY. Dickinson College, Carlisle PA17013. [Assistant Professor] 717-245-1667.

BEAN, PHILLIP. Mercer University, Macon GA 31207.[Professor, 1406 Coleman Avenue] 912-744-2820.

BECKMANN, CHARLENE. Western Michigan University,Muskegon 10 49441. [Student, 1716 Ruddiman] 616-755 -4300.

BEDELL, CARL. Philadelphia College of Textiles & Sci,Philadelphia PA 19144. [Associate Professor, HenryAve. & Schoolhouse Lane] 215-951-2877.

BEHR, MERLYN. National Science Foundation, Wash-ington DC 20550. [Program Director, 1800 G Street NWRoom 638] 202-357-7048.

BENHARBIT, ABDELALI. Penn State University, YorkPA 17403. [Assistant Professor, 264 Brookwood DrSouth] 717-771-4323.

BENSON, JOHN. Evanston Township High School,

2 'o

244 PARTICIPANTS

Evanston IL 60202. [Teacher, 715 South Blvd.] 312 -328-8019.

BENTON, EILEEN. Xavier University of Louisiana, NewOrleans LA 70125. (Professor, 7325 Palmetto] 504 -486-7411.

RERAN, DAVID. University of Wisconsin, Superior WI54880. [Associate Professor] 715-394-8485.

BESSERMAN, ALBERT. Kishwaukee College, De Kalb IL60115. [Instructor, 924 Sharon Dr] 815-756-5921.

BEST, KATHY. Mount Olive College, Mount Olive NC28365. [Professor, Dept of Mathematics] 919-658-2502.

BLUMENTHAL MARJORY. National Research Council-CSTB, Washington DC 20418. [Study Director, 2101Constitution Ave NW] 202-334-2605.

BOBO, RAY. Georgetown University, Washington DC20057. [Associate Professor, Dept of Mathematics] 202-625 -4381.

BODINE, CHARLES. St. George's School, Newport RI02840. [Teacher] 401-849-4654.

BORDELON, MICHAEL. College of the Mainland, TexasCity TX 77591. [8001 Palmer Hwy] 40-938-1211.

BORK, ALFRED. University of California, Irvine Ca92717. [Professor, ICS Department] 714-856-6945.

BOSSERT, WILLIAM. Harvard University, CambridgeMA 20318. [Professor, Aiken Computational Labora-tory] 617-495-3989.

B.:;;TILIER, PHYLLIS. Michigan Technological Univer-sity, Houghton MI 49931. [Director] 906-487-2068.

BOYCE,'WILLIAM. Rensselear Polytechnic Institute,Troy NY 12180. [Professor, Dept of Mathematical Sci-ences] 518-276-6898.

BRABENEC, ROBERT. Wheaton College, Wheaton IL60187. [1006 North Washington] 312-260-3869.

BRADBURN, JOHN. AMATYC MAA, Elgin IL 60123.[1850 Joseph Ct] 312-741-4730.

BRADEN, LAWRENCE. Iolani School, Honolulu HI96826. [563 Kamoku St] 808-947-2259.

BRADY, STEPHEN. Wichita State University, WichitaKS 67208. [Associate Professor, Dept of Math 1845North Fairmount] 316-689-3160.

BRAGG, ARTHUR. De:aware State College, Dover DE19901. [Chair, Dept of Mathematics] 302-736-5161.

BRITO, DAGOBERT. Rice University, Houston TX77251. [Professor, Department of Economics] 701 -527-4875.

BROADWIN, JUDITH. Jericho High School, Jericho NY11753. [Teacher, 6 Yates Lane] 516-681-4100.

BROWNE, JOSEPH. Onondaga Community College,Syracuse NY 13215. [Associate Professor, Dept of Math-ematics] 315-469-7741.

BROWN, CLAUDETTE. National Research Council-MSEB, Washington DC 20418. [2101 Constitution Av-enue NW]

BROWN, DONALD. St. Albans School, Washington DC20016. [Chairman Dept Math, Mount St. Alban] 202-537 -6576.

BROWN, JACK. University of Arkansas, Fayetteville AR72701. [Student, Box 701 Yocum Hall] 501-575-5569.

BROWN, ROBERT. University of Kansas, Lawrence KS66045. [Director Undergrad Studies, Dept of Mathemat-ics] 913-864-3651.

BROWN, MORTON. University of Michigan, Ann ArborMI 48109. [Professor, Dept of Mathematics] 313 -764-0367.

BRUBAKER, MAR/IN. Messiah College, Grantham PA17027. [Professor] 717-766-2511.

BRUNELL, GLORIA. Western Connecticut State Univer-sity, New Milford CT 06776. [Professor, 8 Howe Road]203-355-1569.

BRUNSTING, JOHN. Hinsdale Central High School,Hinsdale IL 60521. [Dept. Chair, 55th and GrantStreets] 312-887-1340.

BUCCINO, ALPHONSE. University of Georgia, AthensGA 30602. [Dean, College of Education] 404-542-3030.

BUCK, CREIGHTON. University of Wisconsin, MadisonWI 53705. [Professor, 3601 Sunset Drive] 608-233-2592.

BUEKER, R.C. Western Kentucky University, BowlingGreen KY 42101. [Head, Dept of Mathematics] 502 -745-3651.

BURDICK, BRUCE. Bates College, Lewiston ME 04240.[Assistant Professor, P.O. Box 8232] 517-786-6143.

BYHAM, FREDERICK. State Univ College at Fredonia,Fredonia NY 14063. [Associate Professor, Dept of Mathand Computer Science] 716-673-3193.

BYNUM, ELWARD. National Institutes of Health,Bethesda MD 20892. [Director MARC Program, West-wood Bld. Room 9A18] 301-496-7941.

CALAMIA, LOIS. Brookdale Community College, Eat,tBrunswick NJ 08816. [Assistant Professor, 12 SouthDrive] 201-842-1900.

CALLOWAY, JEAN. Kalamazoo College, Kalamazoo MI49007. [Professor, 1200 Ace lemy Street] 616-383-8447.

CAMERON, DWAYNE. Old Rochester Regional SchoolDistrict, Mattapoisett MA 02739. [Coordinator, 135Marion Rd.] 617-758-3745.

CAMERON, DAVID. United States Military Academy,West Point NY 10996. [Head, Dept of Mathematics]914-938-2100.

CANNELL, PAULA. Anne Arundel Community College,Annapolis MD 21401. [1185 River Bay Rd] 301 -260-4584.

CANNON, RAYMOND. Baylor University, Waco TX76798. [Professor, Mathematics Dept] 817-755-3561.

CAPPUCCI, ROGER. Scarsdale High School, ScarsdaleNY 10583. [Teacher] 914-723-550.

CARLSON, CAL. Brainerd Community College, BrainerdNY 56401. (Teacher, College Drive] 218-829-5469.

CARLSON, DONALD. University of Illinois-Urbana, Ur-bana IL 61801. [Professor, 104 South Wright Street}217-333-3846.

CARNES, JERRY. Westminster Schools, Atlanta GA30327. [Chair Math Dept, 1424 West Paces Ferry Road]404-355-8673.

CARNEY, ROSE. Illinois Benedictine College, Lisle IL60532. [Professor, Dept of Math 5700 College Road]312.960 -1500.

21t ,

PARTICIPANT LIST 245

CARR, JAMES. Iona College, New Rochelle NY 10801.[Assistant Professor, PDept of Mathematics] 914 -633-2416.

CASCIO, GRACE. Northeast Louisiana University, Mon-roe LA 71209. [Assistant Professor, Dept of Mathemat-ics] 31C 342-4150.

CASE, BETTYE-ANNE. Florida State University, Talla-hassee FL 32306. [Professor, Department of Mathemat-ics] 904-644-2525.

CASTELLINO, FRANCIS. University of Notre Dame,Notre Dame IN 46556. (Dean, College of Science]

CAVINESS, B.F. National Science Foundation, Washing-ton DC 20550. [Program Director, 1800 G Street NWRoom 304] 202-357-9747.

CHABOT, MAURICE. University of Southern Maine,Portland ME 04103. [Chair, Dept of Math 235 ScienceBldg] 207-780-4247. -

CHANDLER, EDGAR. Paradise Valley Community Col-lege, Phoenix AZ 85032. [Instructor, 18401 North 32ndStreet] 602-493-2803.

CHAR, BRUCE. University of Tennessee, Knoxville TN37996. [Associate Professor, Dept of Computer Sci] 615-974 -4399.

CHILD, DOUGLAS. Rollins College, Winter Park FL32789. [Professor, Dept Math Sciences 100 Holt Ave]305-646-2667.

CHINN, WILLIAM. 539 29th Avenue, San Francisco CA94121. [Emeritus, City College of San Francisco] 415-752 -1657.

CHIPMAN, CURTIS. Oakland University, Rochester MI48309. [Associate Professor] 313-370-3440.

CHRISTIAN, FLOYD. Austin Peay State University,Clarksville TN 37044. [Associate Professor, Dept ofMathematics] 615-648-7821.

CHROBAK, MICHAEL. Texas Instruments Inc., DallasTX 75265. [Product Manager, P.O. Box 655303] 214-997 -2010.

CLARK, ROBERT. Macmillan Publishing Co., New YorkNY 10022. [Editor, 866 3rd Avenue 6th floor] 212 -702-6773.

CLARK, CHARLES. University of Tennessee, KnoxvilleTN 37996. [Professor, Dept of Mathematics] 615 -974-4280.

CLEAVER, CHARLES. The Citadel, Charleston SC29409. [Head, Dept of Math & Computer Science] 803-222 -8069.

CLIFTON, RODNEY. Brown University, Providence RI02912. [Professor, Division of Engineering] 401-863-2855.

COALWELL, RICHARD. Lave Community College, Eu-gene OR 97401. [356 Paradise Court] 503-747-4501.

COCHRAN, ALLAN. University of Arkansas, FayettevileAR 72701. [Professor, Dept of Mathematical SciencesSE] 501-575-3351.

COHEN, MICHAEL. Bay Shore High School, Merrick NY11566. [Teacher, 2036 Dow Avenue] 516-379-5356.

COHEN, SIMON. New Jersey Institute of Technology,Newark NJ 07102. [Dept of Mathematics] 201-596-3491.

COHEN, JOEL. University of Denver, Deliver CO 80208.[Associate Professor, Dept Math & Computer Science]303-871-3292.

COHEN, MICHAEL. University of Maryland, CollegePark MD 20742. [Associate Professor, PSchool of PublicAffairs Morrill Hall] 301-454-7613.

COLLINS, SHEILA. Newman School, New Orleans LA70115. [Chair, Math Dept 1903 Jefferson Ave.] 503 -899-5641.

COLLUM, DEBORAH. Oklahoma Baptist University,Shawnee OK 74801. [Assistant Professor, 500 West Uni-versity] 405-275-2850.

COMPTON, DALE. National Academy of Engineering,Washington DC 20418. [Senior Fellow, 2101 Constitu-tion Avenue NW] 202-334-3639.

CONNELL, CHRIS. Associated Press.CONNORS, EDWARD. University of Massachusetts,

Amherst MA 01003. [Professor, Dept of Mathematics& Statistics] 413-545-0982.

COONCE, HARRY. Mankato State University, MankatoMN 56001. [Professor, Box 41] 507-389-1473.

COPES, LARRY. Augsburg College, Minneapolis MN55075. [Chair, 731 21st Avenue South] 612-330-1064.

CORZATT, CLIFTON. St. Olaf College, Northfield MN55057. [Associate Professor] 507-633-3415.

COVENEY, PETER. Harper & Row Publishers Inc., NewYork NY 10022. [Editor, 10 East 53rd Street] 212 -207-7304.

COX, LAWRENCE. National Research Council-BMS,Washington DC 20418. [Staff Director, 2101 Constitu-tion Avenue NW] 202-334-2421.

COZZENS, MARGARET. Northeastern University,Bostor. MA 02115. [Associate Professor, Dept of Math360 Huntington Ave] 617-437-5640.

CRAWLEY, PATRICIA. Nova High School, Sunrise FL33322. [Dept Head, 2021 N.W. 77th Avenue] 305 -742-6452.

CROOM, FREDERICK. University of the South, SewoneeTN 37375. [Professor, University Station] 615 - 598 - 1..48.

CROWELL, RICHARD. Dartmouth College, Hanover NH03755. [Professor, 16 Rayton Road] 603-646-2421.

CROWELL, SHARON. O'Brien & Associates, AlexandriaVA 22334. [Associate, Carriage House 708 Pendleton St]703-54P-7587.

CUMMINGS, NOP.MA. Arapahoe High School, LittletonCO 80122. [Teacher, 2201 East Dry Creek Rd] 303 -794-2641.

CURTIS, PHILIP. UCLA, Los Angeles CA 90024. [Pro-fessor, Dept of Mathematics] 213-206-6901.

CURTIS, EDWARD. University of Washington, SeattleWA 98195. [Professor, Padelford Hall] 206-543-1945.

CUTLER-ROSS, SHARON. Dekalb College, ClarkstonGA 30021. [Associate Professor, 555 North IndianCreek] 404-299-4163.

CUTLER, ARNOLD. Moundsview High School, NewBrighton MN 55112. [Teacher, 1875 17th Street NW]612-633-4031.

53

246 PARTICIPANTS

DANCE, ROSALIE. District of Columbil Public Schools,Washington DC 20032. [Teacher, Ballow High School3401 4th St. SE] 202-767-7071.

DANFORTH, KATRINE. Corning Community College,Corning NY 14830. [Associate Professor, P.O. Box 252]607-962-4034.

DANFORTH, ERNEST. Corning Community College,Corning NY 14830. [Associate Professor, P.O. Box 252]607-962:9243.

DARAI, ABDOLLAH. Western Illinois University, Ma-comb IL 61455. [Assistant Professor, Dept of Math Mor-gan Hall] 309-298-1370.

DAVIDON, WILLIAM. Haverford College, Haverford PA19041. [Professor, Mathematics Dept] 215-649-0102.

DAVIES, RICHARD. OTG U.S. Congress, Washington .DC 20510. [Analyst, Office of Technology Assessment]202-228-6929.

DAVISON, JACQUE. Anderson College, Anderson SC29621. [Instructor, 316 Boulevard] 803-231-2165.

DAVIS, NANCY. Brunswick Technical College, ShallotteNC 28459. [Instructor, Rt 2 Box 143-2A] 919-754-6900.

DAVIS, RONALD. Northern Virginia Community Col-lege, Alexandria VA 22205. [Professor, 3001 NorthBeauregard St] 703-845-6341.

DAVIS, FREDERIC. United States Naval Academy, An-napolis MD 21402. [Professor, Mathematics Dept] 301-267 -2795.

DAWSON, JOHN. Penn State York, York PA 17403.[Professor, 1031 Edgecomb Avenue] 717-771-4323.

DE-COMARMOND, JEAN-MARC. French Scientific Mis-sion, Washington DC 20007. [Scientific Attache, 4101Reservoir Road NW] 202-944-6230.

DEETER, C. .LES. Texas Christian University, FortWorth TX )129. [Professor, Dept of Math Box 32903]817-921-7335.

DEKEN, JOSEPH. National Science Foundation, Wash-ington DC 20550. [Program Director, 1800 G Street NWRoom 310] 202-357-9569.

DELIYANNIS, PLATON. Illinois Institute of Technology,Chicago IL 60616. [Associate Professor, Dept of Mathe-matics] 312-567-3170.

DELLENS, MICHAEL. Austin Community College,Austin TX 78768. [Instructor, P.O. Box 2285] 512 -495-7256.

DEMANA, FRANKLIN. Ohio State University, Colum-bus OH 43210. [Professor, Dept of Math 231 Wc...t 18thAvenue] 614-292-0462.

DEMETROPOULUS, ANDREW. Montclair State Col-lege, Upper Montclair NJ 07043. [Chah, Dept of Math& Computer Science] 201-893-5146.

DENLINGER, CHARLES. Millersville University,Millersville PA 17551. [Professor, Dept of Math & Com-puter Science] 717-872-4476.

DEVITT, JOHN. University of Saskz tchewan, SaskatoonSask. Canada S7NOWO. [Asociate Professor, Dept ofMathematics College Drive] 306-966-6114.

DIFRANCO, ROLAND. University of the Pacific, Stock-ton CA 95211. [Professor, Mathematics Dept] 209.946-

3026.DICK, THOMAS. Oregon State University, Corvallis OR

97331. [Assistant Professor, Mathematics Dept] 503 -754-4686.

DIENER-WEST, MARIE. Johns Hopkins University,Baltimore MD 21205. [550 North Broadway 9th floor]301-955-8943.

DION, GLORIA. Penn State Ogontz, Abington PA19001. [1600 Woodland Avenue] 215-752-9595.

DIX, LINDA. National Research Council-OSEP, Wash-ington DC 21408. [Project Officer, 2101 Cons.itutionAvenue NW] 202-334-2709.

DJANG, FRED. Choate Rosemary Hall, Wallingford CT06492. [Chairman Math, P.O. Box 788] 203-269-7722.

DODGE, WALTER. New Trier High School, Winnetica. IL60093. [Teacher, 385 Winnetica. Ave] 312-446-7000.

DONALDSON, JOHN. Amer. Society for EngineeringEducation, Washington DC 20036. [Deputy ExecutiveDirector, 11 Dupont Circle Suite 200] 202-293-7080.

DONALDSON, GLORIA. Andalusia High School, Andalu-sia AL 36420. [Chair Math Science Division, P.O. Box151] 205-222-7569.

DORNER, GEORGE. Harper College, Palatine IL 60067.[Dean, Algonquin/Ruselle Roads] 312-397-3000.

DORNER, BRYAN. Pacific Lutheran University, TacomaWA 98447. [Associate Professor, Dept of Mathematics]206-535-8737.

DOSSEY, JOHN. Illinois State University, Eureka IL61530. [Professor, RR #1 Box 33] 309-467-2759.

DOTSETH, GREGORY. University of Northern Iowa,Cedar Falls IA 50613. [Dept of Math & Computer Sci-ence] 319-273-2397.

DOUGLAS, RONALD. SUNY - Stony Brook, StonyBrook NY 11794. [Dean, Physical Sciences & Math] 516-632 -6993.

DREW, JOHN. College of William and Mary, Williams-burg VA 23185. [Associate Professor, Math.Dept] 804-253 -4481.

DUTTON, BRENDA. Spring Hill College, Mobile AL36608. [4000 Dauphin St.] 205-460-2212.

DWYER, WILLIAM. University of Notre Dame, NotreDame IN 46556. [Chair, Dept of Mathematics]

DYER, DAVID. Prince George's College, Largo MD20772. [Associate Professor, Math Dept 301 LargoRoad] 301-322-0461.

DYKES, JOAN. Edison Community College, Ft. MyersFL 33907. [Instructor, 8099 College Parkway SW] 813-489 -9255.

DYMACEK, WAYNE. Washington and Lee University,Lexington VA 24450. [Associate Professor] 703-463-8805.

EARLES, GAIL. St. Cloud State University, St. CloudMN 56301. [Chair, Dept of Math & Statistics] 612 -255-3001.

EARLES, ROBERT. St. Cloud State University, St.Cloud MN 56301. [Professor, Dep, of Math & Statistics]612-255-2186.

EBERT, GARY. University of Delaware, Newark DE19716. [Professor, Math Sci Dept] 302-451-1870.

20L":-


EDISON, LARRY. Pacific Lutheran University, TacomaWA 98447. [Professor] 206-535-8702.

EDLUND, MILTON. Virginia Polytech & State Univ,Blacksburg VA 24061. [Professor, Dept of MechanicalEngineering] 703-951-1957.

EDWARDS, CONSTANCE. IPFW, Ft. Wayne IN 46805.[Associate Professor, Math Dept] 219-481-6229.

EDWARDS, BRUCE. University of Florida, GainesvilleFL 32611. [Associate Chair, Dept of Math 201 WalkerHall] 904-392-0281.

EGERER, GERALD. Sonoma State University, RodnertPark CA 94928. [Professor, Dept of Economics] 707 -664-2626.

EHRET, ROSE-ELEANOR. Holy Name College, OaklandCA 94619. [Professor, 3500 Mountain Blvd] 415 -436-0111.

EIDSWICK, JACK. University of Nebraska, Lincoln NE68588. [Professor, Dept of Mathematics and Statistics]402-472-3731.

EMANUEL, JACK. University of Missouri-Rolla, RollaMO 65401. [Professor, Dept of Civil Engineering] 314-341 -4472.

ERDMAN, CARL. Texas A & M University, College Sta-tion TX 77843. [Associate Dean, 301 Wisenbaker EngrRes Center] 409-845-5220.

ESLINGER, ROBERT. Hendrix College, Conway AR72032. [Associate Professor, Dept of Mathematics] 501-450 -1254.

ESTY, EDWARD. Childrens Television Workshop, Chevye3h a s e MD 20815. [4104 Leland Street] 301-656-7274.

ETTELSEEK, WALLACE. Calif State University, Sacra-mento CA 95819. [Professor, Math Dept 6000 J Street]916-278-6361.

FAIR, WYMAN. University of North Carolina, AshevilleNC 28804. [Professor, Mathematics Dept 1 UniversityHeights] 704-251-6556.

FAN, SEN. University of Minnesota, Morris MN 56267.[Associate Professor, Math Discipline] 612-589-2211.

FARMER, THOMAS. Miami University, Oxford OH45056. [Associate Professor, Dept of Math & Stat] 513-529 -5822.

FASANELLI,-FLORENCE. National Science Foundation,Washington DC 20550. [Associate Program Director,1800 G Street NW Room 635] 202-357-7074.

FERRINI-MUNDY, JOAN. University of New Hampshire,Durham NH 03824. [Associate Professor, Dept of Math-ematics] 603-862-2320.

FERRITOR, DANIEL. University of Arkansas, Fayet-teville AK 72701. [Chancellor, 425 AdministrationBuilding] 501-575-4148.

FIFE, JAMES. University of Richmond, Richmond VA23173. [Professor, Dept of Math/Computer Sci] 804 -289-8083.

FINDLEY-KNIER, HILDA. Univ of the District ofColumbia, Washington DC 20008. [Associate Professor,2704 Woodley Place NW] 202-282-7465.

FINK, JAMES, Butler University, Indianapolis IN 46208.[Head, Der of Mathematical Sciences] 317-283-9722.

FINK, JOHN. Kalamazoo College, Kalamazoo MI 49007.[Associate Professor] 616-383-8447.

FISHER, NEWMAN. San Francisco State University, SanFrancisco CA 94132. [Chairman, Mathematics Dept1600 Holloway Ave] 415-338-2251.

FLANDERS, HARLEY. University of Michigan, Ann Ar-bor MI 48109. [Professor, Dept of Mathematics] 313-761 -4666.

FLEMING, RICHARD. Central Michigan University, Mt.Pleasant MI 48659. [Professor, Dept of Mathematics]517-774-3596.

FLINN, TIMOTHY. Tarleton State University, Stephen-ville TX 76402. [Associate Professor, Dept of Mathe-matics Box T-519] 817-968-9168.

FLOWERS, PEARL. Montgomery County PublicSchools, Rockville MD 20850. [Teacher Specialist, 850Hungerford Dr CESC 251] 301-279-3161.

FLOWERS, JOE. Northeast Missouri State Univ,Kirksville MO 63501. [Professor, Div of Math VioletteHall 287] 816-785-4284.

FOLIO, CATHERINE. Brookdale Community College,Lincroft NJ 07738. [Math Dept Newman Springs Road]201-842-1900.

FRAGA, ROBERT. Ripon College, Ripon WI 54971.[Box 248] 414-748-8129.

FRANCIS, WILLIAM. Michigan Technological Univer-sity, Houghton MI 49931. [Associate Profees:;r, Dept ofMathematical Sciences] 906-487-2146.

FRANKLIN, KATHERINE. Los Angeles Pierce College,Northridge CA 91324. [Associate Professor, 8827 Ju-milla Ave] 818-700-9732.

FRAY, BOB. Furman University, Greenville SC 29613.[Professor, Mathematics Dept] 803-294-2105.

FRIEDBERG, STEPHEN. Illinois State University, Nor-mal IL 61761. [Professor, Dept of Math 119 Doud Drive]309-438-8781.

FRIEL, WILLIAM. University of Dayton, Dayton OH45469. [Assistant Professor, Mathematics Dept] 513 -229-2099.

FRYXELL, JAMES. College of Lake Coui..", GrayslakeIL 60030. [Professor, 19351 W. Nrlshington Street] 312-223 -6601.

FULTON, JOHN. Clemson University, Clemson SC29634. [Professor, Dept of Mathematics] 803-656-3436.

GALOVICH, STEVE. Carleton College, Northfield MN55057. [Professor, Dept of Math & Computer Science]507-663-4362.

GALWAY, ALISON. O'Brien & Associates, AlexandriaVA 22314. [Associate, Carriage House 708 Pendleton St]703-548-7587.

GASS, FREDERICK. Miami University, Oxford OH45056. [Associate Professor, Dept of Math and Statis-tics] 513-529-3422.

GEGGIS, DAVID. PWS-Kent Publishing Co., BostonMA 02116. [Managing Editor, 20 Park Plaza] 617 -542-3377.

GENKINS, ELAINE. Collegiate School, New York NY

. 2 5 5

248 PARTICIPANTS

10024. [Head Math Dept, 241 West 77th St] 212 -873-0677.

GETHNER, ROBERT. Franklin and Marshall College,Lancaster PA 17604. [Professor] 717-291-4051.

GEUTHER, KAREN. University of New Hampshire,Durham NH 03824. [Assistant Professor, Dept of Math-ematics] 603-862-2320.

GIAMBRONE, AL. Sinclair College, Dayton OH 45402.[Profess Or, Math Dept] 513-226-2585.

GILBERT, JOHN. Mississippi State University, MississipiState MS 39762. [Associate Professor, Dept of MathP.O. Drawer MA] 601-325-3414.

GILBERT, WILLIAM. University of Waterloo, Water-loo Ont. Canada N2L3G1. [Professor, Pure Math Dept]519-888-4097.

GILFEATHER, FRANK. University of Nebraska, LincolnNE 68588. [Professor, Department of Mathematics] 402-472 -3731.

GILMER, GLORIA. Math-Tea Connexion Inc., Milwau-kee WI 53205. (President, 2001 West Vliet Street] 414-933 -2322.

GLEASON, ANDREW. Harvard University, Camb!idgeMA 02138. [Professor, Dept of Math 110 LarchwoodDrive] 617-495-4316.

GLEICK, JIM. New York Times.GLENNON, CHARLES. Christ Church Episcopal School,

Greenville SC 29603. [Instructor, P.O. Box 10128] 803-299 -1522.

GLUCHOFF, ALAN. Villanova University, Villanova PA19085. [Assistant Professor, Dept of Math Sciences] 215-645 -7350.

GOBLIRSCH, RICHARD. College of St. Thomas, St.Paul MN 55105. [Professor, 2115 Summit Avenue] 612-647 -5281.

GODSHALL, WARREN. Susquehanna Twp High Shool,Harrisburg PA 17109. [Teacher, 414A Amherst Drive]717-657-5117.

GOLDBERG, SAMUEL. Alfred P. Sloan Foundation, NewYork NY 10111. [630 Fifth Avenue Suite 2550]

GOLMFAG, MORTON. Broome Community College,Binghamton NY 13902. [Professor, P.O. Box 1017] 607-771 -5165.

GOLDBERG, DOROTHY. Kean College of New Jersey,Union NJ 07083. [Chairperson, Dept of Math MorrisAve.] 201-527-2105.

GOLDBERG, DONALD. Occidental College, Los Ange-les CA 90041. [Assistant Professor, Dept of Math 1600Campus Road] 213-259-2524.

GOLDSCHMIDT, DAVID. University of California, Berke-ley CA 94720. [Professor, 970 Evans Hall] 415:642-0422.

GOLDSTEIN, JEROME. 'Mane University, New OrleansLA 70118. [Dept of Mathematics] 504-865-5727.

GOODSON, CAROLE. University of Houston, HoustonTX 77004. [Associate Dean, 4800 Calhoun] 713 -749-1341.

GORDON, SHELDON. Suffolk Community College, EastNorthport NY 11731. [Professor, 61 Cedar Road] 516-451 -4270.

GRAF, CATHY. Thomas Jefferson High School, BurkeVA 22015. [Teacher. 6101 Windward Drive] 703 -354-9300.

GRANDAHL, JUDITH. Western Connecticut State Uni-versity, Danbury CT 06810. [Associate Professor, 181White Street] 203-797-4221.

GRANLUND, VERA. University of Virginia, Char-lottesville VA 22901. [Lecturer, Thornton Hall] 804 -924-1032.

GRANTHAM, STEPHEN. Boise State University, BoiseID 83725. [Assistant Professor, Dept of Mathematics]208-385-3369.

GRAVER, JACK. Syracuse University, Syracuse NY13244. [Professor, Dept of Mathematics] 315-472-5306.

GRAVES, ELTON. Rose-Hulman Institute of Technology,Terre Haute IN 47803. [Associate Professor, Box 123]812-877-1511.

GREEN, EDWARD. Virginia Tech, Blacksburg VA 24061.[Professor, Dept of Mathematics] 703-961-6536.

GROSSMAN, MICHAEL. University of Lowell, LowellMA 01851. [Associate Professor, 185 Florence Road]617-459-6423.

GROSSMAN, STANLEY. University of Montana, Mis-soula MT 59801. [Professor, 333 Daly Avenue] 406 -549-3819.

GUILLOU, LOUIS. Saint Mary's College, Winona MN55987. [Dept of Mathematics and Statistics] 507 -457-1487.

GULATI, Bopp. Southern Connecticut State Univ,Cheshire CT 06410. [Professor, 954 Ott Drive] 203-397-4486.

GULICK, DENNY. University of Iviaryiand, College ParkMD 20742. [Professor, Dept of Mathematics] 301 -454-3303.

GUPTA, MURLI. George Washington University, Wash-ington DC 20052. [Professor, Dept of Mathematics] 202-994 -4857.

GUSEMAN, L.F. Texas A & M University, College Sta-tion TX 77843. [Professor, Dept of Mathematics] 409-845 -3261.

GUSTAFSON, KARL. University of Colorado, BoulderCO 80309. [Professor, Campus Box 426-Mathematics]303-492-7664.

GUYKER, JAMES. SUNY College at Buffalo, BuffaloNY 14222. [Professor, Math Dept 1300 Elmwood Ave.]716-837-8915.

HABER, JOHN. Harper & Row, New York NY 10022.[Editor, 10 East 53rd Street] 212-207-7243.

HAINES, CHARLES. Rochester Institute of Technology,Rochester NY 14623. [Associate Dean, College of Engi-neering] 716-475-2029.

HALLETT, BRUCE. Jones & Bartlett Publishers, Brook-line MA 02146. [Editor, 208 Fuller St] 617-731-4653.

HALL, LEON. University of Missouri, Rolla MO 65401.[Associate Professor, Dept of Math & Statistics] 314-341 -4641.

HAMBLET, CHARLES. Phillips Exeter Academy, ExeterNH 03833. [Instructor] 603-772-4311.


HAMMING, RICHARD. Naval Postgraduate School,Monterey CA 93943. [Code 52Hg] 408-646-2655.

HAMPTON, CHARLES. The College of Wooster, WoosterOH 44691. [Chair, Mathematical Sciences Dept] 216-263 -2486.

HANCOCK, DON. Pepperdine University, Malibu CA90265. [Associate Professor, Math Dept Natural ScienceDivision] 213-456-4241.

HANSON, ROBERT. James Madison University, Har-risonburg VA 22807. [Coordinator, Dept of Math &Computer Science] 703-568-6220.

HARTIG, DONALD. Calif. Polytechnic State University,San Luis Obispo CA 93407. (Professor, Math Dept] 805-756 -2263.

HART, THERESE. National Research Council-MS 2000,Washington DC 20418. [2101 constitution Ave. NW]202-334-3740.

HARVEY, JOHN. University of Wisconsin-Madison,Madison WI 53706. [Professor, Dept Math 480 LincolnDrive] 608-262-3746.

HAUSNER, MELVIN. CIMS/NYU, New York NY 10012.[Professor, 251 Mercer Street] 212-998-3190.

HAYNSWORTH, HUGH. College of Charleston, Charles-ton SC 29424. [Associate Professor, Dept of Mathemat-ics] 803-792-5735.

HEAL, ROBERT. Utah State Unversity, Logan UT84322. [Associate Dept Head, Mathematics Dept] 801-750 -2810.

HECKENBACH, ALAN. Iowa State University, Ames IA50011. [Associate Professor, Dept of Mathematics] 515-294 -8164.

HECKLER, JANE. MAA - JPBM, Washington DCHECKMAN, EDWIN. Central New England College,

Westboro MA 01581. [Professor, 4 Lyman Street] 617-366 -5527.

HEID, KATHLEEN. Pennsylvania State University, Uni-versity Park PA 16802. [Assistant Professor, 171 Cham-bers Building] 814-865-2430.

HELLERSTEIN, SIMON. University of Wisconsin, Madi-son WI 53706. [Professor, Dept of Math Van Vleck Hall]608-263-3302.

HENDERSON, JIM. Colorado College, Colorado SpringsCO 80903. [Assistant Professor, PDept of Mathematics]303-473-2233.

HENSEL, GUSTAV. Catholic University of America,Washington DC 20064. [Assistant Dean, Dept of Mathe-matics] 202-635-5222.

HERR, ALBERT. Drexel University, Philadelphia PA19104. [Associate Professor, Dept Math/Comp Sci 32nd& Chestnut Sts] 215-895-2672.

HILDING, STEPHEN. Gustavus Adolphus College, St.Peter MN 56082. [Professor, Dept of Mathematics] 507-931 -7464.

HILLTON, THOMAS. Educational Testing Service,Princeton NJ

HILL, THOMAS. Lafayette College, Easton PA 18042.[Dept of Mathematics] 215-250-5282.

HILL, SHIRLEY. University of Missouri, Kansas CityMO 64110. [Professor, Dept of Mathematics] 816 -276-2742.

HIMMELBERG, CHARLES. University of Kansas,Lawrence K3 66045. [Chair, Dept of Mathematics] 913-864 -3651.

HINKLE, BARBARA. Seton Hill College, Greensburg PA15601. [Chair, Dept of Math & Computer Science] 412-834 -2200.

HODGSON, BERNARD. Universite Laval, QuebecCanada G1K7P4. [Professor, Dept De Maths Stat. &Actuariar] 418-656-2975.

HOFFER, ALAN. National Science Foundation, Wash-ington DC 20550. [1800 G Street NW]

HOFFMAN, DALE. Bellevue Community College, Belle-vue WA 98005. [Professor, 12121 5.E. 27th Street] 206-747 -8515.

HOFFMAN, KENNETH. Massachusetts Institute of Tech-nology, Washington DC 20036. [Professor, 1529 18thStreet NW] 202-334-3295.

HOFFMAN, ALLAN. National Academy of Sciences,Washington DC 20418. [Executive Director COSEPUP,2101 Constitution Avenue NW]

HOLMAY, KATHLEEN. JPBM Public Information,Washington DC

HORN, P.J. Northern Arizona University, Flagstaff AZ86011. [Assistant Professor, Box 5717] 602-523-6880.

HORN, HENRY. Princeton University, Princeton NJ18544. [Professor, Dept of Biology] 609-452-3000.

HOWAT, KEVIN. Wadsworth Publishing Co., BelmontCA 94002. [Publisher, 10 Davis Drive] 415-595-2350.

HSU, Y1.1-KAo. University of Maine, Bangor ME 04401.[Professor, Room 111 Bangor Hall] 207-581-6138.

HUANG, JANICE. Milligan College, Milligan College TN37682. [Associate Professor, Dept of Mathematics]

HUDSON, ANNE. Armstrong State College, SavannahGA 31413. [Professor, Dept of Math & CS 11935 Aber-corn St] 912-927-5317.

HUGHES-HALLETT, DEBORAH. Harvard University,Cambridge MA 02138. [Senior Preceptor, Dept of Math-ematics] 617-495-5358.

HUGHES, RHONDA. Bryn Mawr College, Bryn MawrPA 19010. [Associate Professor, Dept of Mathematics]215-645-5351.

HUGHES, NORMAN. Valparaiso University, ValparaisoIN 46383. [Associate Professor] 219-464-5195.

HUNDHAUSEN, JOAN. Colorado School of Mines,Golden CO 80401. [Dept of Mathematics] 303-273-3867.

HUNSAKER, WORTHEN. Southern Illinois University,Carbondale IL 62901. [Professor, Dept of Mathematics]618-453-5302.

HUNTER, JOYCE. Webb School of Knoxville, KnoxvilleTN 37923. [Teacher, 9800 Webb School Drive] 615 -693-0011.

HURLEY, SUSAN. Siena College, Loudonville NY 12209.[Science Division] 518-783-2459.

HURLEY, JAMES. University of Connecticut, Storrs CT

250 PAATICIPANTS

06268. [Professor, 196 Auditorium road Rm 111] 203-486 -4143.

HVIDSTEN, MICHAEL. Gustavus Adolphus College, St.Peter MN 56082. [Assistant Professor, Dept of Mathe-matics) 507-931-7480.

INTRILIGATOR, MICHAEL. UCLA, Los Angeles CA90024. [Professor, Dept of Economics] 213-824-0604.

JACKSON, ALLYN. American Mathematical Society,Providence RI 02940. [Staff Writer, P.O. Box 6248] 401-272 -9500.

JACKSON, MICHAEL. Earlham College, Richmond IN47374. (Assistant Professor, PP) 317-983-1620.

JACOB, HENRY. University of Massachusetts, AmherstMA 01003. [Professor, Dept Math/Stat Lederle Re-search Tower] 413-545-0510.

NJANSON, BARBARA. Janson Publications Inc, Provi-dence RI 02903. [President, 222 Richmond Street Sdte105] 401-272-0009.

JAYNE, JOHN. Univ. of Tennessee at Chattanooga,Chattanooga TN 37413. [Professor, Math Dept] 615-755 -4545.

JENKINS, FRANK. John Carroll University, UniversityHeights OH 44138. [Assistant Professor, MathematicsDepartment] 216-397-4682.

JENKINS, JOE. SUNY University at Albany, AlbanyNY 12222. [Chaif Dept of Math, 1400 Washington Ave.]518-422-4602.

JENSEN, WALTER. Central New England College, Dud-ley MA 01570. [Head, 10 Shepherd Avenue) 617 -943-3053.

JEN, HORATIO. W.C.C. College, Youngwood PA 15601.[Professor] 412-9Z5-4184.

JOHNSON, DAVID. Lehigh University, Bethlehem PA18015. [Associate Professor, Mathematics Dept #14)215-758-3730.

JOHNSON, JERRY. Oklahoma State University, Stillwa-ter OK 74078. [Professor, Dept of Mathematics) 405-624 -5793.

JOHNSON, LEE. Virginia Polytech & State University,Blacksburg VA 24061. [Professor, Dept of Mathematics460 McBryde Hall) 703-961-6536.

JOHNSON, ROBERT. Washington and Lee University,Lexington VA 24450. [Professor, Dept of Mathematics]703-463-8801.

JONES, LINDA. National Research Council-MS 2000,Washington DC 20418. [2101 Constitution Ave NW]202-334-3740.

JONES, ELEANOR. Norfolk State University, Norfolk VA23504. [Professor, Dept of Mathematics)

JONES, WILLIAM. Univ of the District of Columbia,Washington DC 20011. [Assistant Professor, Dept ofMat114200 Connecticut Avenue NW] 202-282-3171.

JUNGHANS, HELMER. Montgomery College, Gaithers-burg MD 20878. [Professor, 220 Gold Kettle Drive] 301-926 -4403.

KAHN, ANN. Mathematical Science Education Board,Washington DC 20006. [Consultant, 818 ConnecticutAve NW Suite 325) 202-334-3294.

KALLAHER, MICHAEL. Washington State University,Pullman WA 99164. [Professor, Mathematics Dept] 509-335 -4918.

KANIA, MAUREEN. Earl Swokowski. LTD, West AllisWI 53227. [Executive Assistant, 12124 West Ohio Av-enue] 414-546-3860.

KAPUT, JAMES. Educational Tech Center-HarvardUniv., North Dartmouth MA 02747. [473 Chase Road)617-993-0501.

KARAL, FRANK. NYU - Courant Institute, New YorkNY 10012. [Professor, 251 Mercer Street) 212-998-3162.

KARIAN, ZAVEN. Dension University, Granville OH43023. [Dept of Math Sciences] 614-587-6563.

KASPER, RAPHAEL. National Research Council-CPSMR, Washington DC 20418. [Executive Director,2101 Constitution Avenue NW)

KATZEN, MARTIN. New Jersey Institute of Technology,West Paterson NJ 07424. [Associate Professor, 11 Wash-ington Drive)

KATZ, VICTOR. Univ of the District of Columbia,Washington -DC 20011. [Professor, Dept of Math4200Connecticut Avenue NW] 202-282-7465.

KAY, DAVID. University of North Carolina, AshevilleNC 28804. [Professor, Mathematics Dept .1 UniversityHeights) 704-251-6556.

KEEVE, MICHAEL. Norfolk State University, NorfolkVA 23504. [Instructor, Math & Computer Sci Dept] 804-623 -8820.

KEHOE-MOYNIIIAN, MARY. Cape Cod CommunityCollege, West Barnstable MA 02668. [Professor] 617-362 -2131.

KENELLY, JOHN. Clemson University, Clemson SC29631. [Alumni Professor, 327 Woodland Way] 803 -656-5217.

KEYNES, HARVEY. University of Minnesota, Minneapo-lis MN 55455. [Professor, School of Math 127 VincentHall] 612-625-2861.

KIMES, THOMAS. Austin College, Sherman TX 75090.[Chairman, Dept of Mathematics] 214-892-9101.

KING, ELLEN. Anderson College, Anderson SC 29621.[Instructor, 316 Boulevard] 803-231-2162.

KING, ROBERT. Westmar College, LeMars IA 51031.[Assistant Professor, 115 7th Street SE] 712-546-6117.

KIRKMAN, ELLEN. Wake Forest University, Winston-Salem NC 27109. [Associate Professor, Box 7311Reynolds Station] 919-761-5351.

KLATT, GARY. Unversity of Wisconsin, Whitewater WI53190. [Professor, Math Dept 800 West Main] 414 -472-5162.

KNIGHT, GENEVIEVE. Coppin State College, ColumbiaMD 21045. [Professor, 2500 W. North Ave., Baltimore,21216] 301-333-7853.

KOKOSKA, STEPHEN. Colgate University, Hamilton NY13346. [Assistant Professor, Dept. of Mathematics] 315-824 -1000.

KOLMAN, BERNARD. Drexel University, PhiladelphiaPA 19104. [Professor) 215-895-2683.

25


KOPERA, ROSE. National Research Council-BMS,Washington DC 20418. [2101 Constitution Ave NW)202-334-2421.

KRAMAN, JULIE. National Research Council-MSEB,Washington DC 20006. [818 Connecticut Ave NW Suite325] 202-334-3294.

KRAUS, GERALD. Gannon University, Erie PA 16541.[Chair, Math Dept University Square] 814-871-7595.

KREIDER, DON. Dartmouth College, Sharon VT 05065.[Vice Chairman, Math & CS D-pt RR #1 Box 487]

KUHN, ROBERT. Harvard Uni.ersity, Cambridge MA02138. [Lecturer, Dept of Mathematics] 617-495-1610.

KULM, GERALD. Amer Assoc for Advancement of Sci-ence, Washington DC 20005. [Associate Program Direc-tor, 1333 H Street NW) 202-326-6647.

KULNARONG, GRACE. National Research Council-MSEB, Washington DC 20006. [818 Connecticut AveNW Suite 325] 202-334-3294.

KUNZE, RAY. University of Georgia, Athens GA 30602.[Chair, Dept of Mathematics] 404-542-2583.

LATORRE, DONALD. Clemson University, Clemson SC29631. [Professor, Dept of Mathematical Sciences] 803-656 -3437.

LACEY, H.E. Texas A & M University, College StationTX 77843. [Head, Dept of Mathematics]

LAMBERT, MARCEL. Universite du Quebec a Trois-Rivieres, Trois-Rivieres Que Canada G9A5H7. [DeptHead, Department de Math-Info] 819 -S76 -5126.

LANE, BENNIE. Eastern Kentucky University, Rich-mond KY 40475. [Professor, Wallace 402] 606-622-5942.

LANG, JAMES. Valencia Community College, OrlandoFL 32811. [Professor, 1800 South Kirkman Road] 305-299 -5000.

LAUBACHER, MICHAEL. Holland Hall School, TulsaOK 74137. [Teacher, 5666 East 81st Street] 918 -481-1111.

LAUFER, HENRY. SUNY at Stony Brook, Long IslandNY 11794. rrofessor, Dept if Mathematics] 516 -632-8247.

LAX, PETER. NYU-Courant Institute of Math Science,New York NY 10012. [251 Mercer Street] 212-460-7442.

LECKRONG, GERALD. Brighton Area Schools, BrightonMI 48116. [Teacher, 7878 Brighton Road] 313-229-1400.

LEE, KEN. Missouri Western State College, St. JosephMO 64507. [Professor] 816-271-4284.

LEINBACH, CARL. Gettysburg College, Gettysburg PA17325. [Chair, Computer Science P.O. Box 506] 717 -337-6735.

LEITHOLD, LOUIS. Pepperdine University, Pacific Pal-isades CA 90272. [Professor, 336 Be llino Drive) 213 -454-2500.

LEITZEL, JAMES. Ohio State University, Columbus OH43210. [Associate Professor, Dept of Math 231 West18th Avenue] 614-292-8847.

LEVINE, MAITA. University of Cincinnati, CincinnatiOH 45221. [Professor, Dcpt tf Mathematical Sciences]513-475-6430.

LEVY, BENJAMIN. Lexington High School, LexingtonMA 02173. [Teacher, 215 Waltham Street] 617-862-7500.

LEWIN, JONATHAN. Kennesaw College, Marietta GA30061. [Associate Professor) 404-423-6040.

LEWIS, KATHLEEN. SUNY at Oswego, Oswego NY13126. [Assistant Professor, Dept of Mathematics] 315-341 -3030.

LEWIS, GAUNCE. Syracuse Univ,-"ty, Syracuse NY13126. [Mathematics Dept] 315-3% 0788.

LINLEY, DAVID. Nature.LIPKIN, LEONARD. University of North Florida, Jack-

sonville FL 32216. [Chairman, Dept of Math 4567 St.Johns Bluff Rd] 904-646-2653.

LISSNER, DAVID. Syracuse University, Syracuse NY13210. [Professor, Math Dept] 315-423-2413.

LITWHILER, DANIEL. U.S. Air Force Academy, Col-orado Springs CO 80840. [Head, Dept of Math Sciences]303-472-4470.

LIUKKONEN, JOHN. 'Iltlane University, New OrleansLA 70118. [Associate Professor, Mathematics Dept] 504-865 -5729.

LOCKE, PHIL. University of Maine, Orono ME 04469.[Associate Professor, 236 Neville Hall] 207-581-3924.

LOFQUIST, GEORGE. Eckerd College, St. PetersburgFL 33712. [Math Dept P.O. 12560) 813-864-8434.

LOGAN, DAVID. University of Nebraska, Lincoln NE68588. [Professor, Dept of Mathematics]

LOMEN, DAVID. University of Arizona, Tucson AZ85721. [Professor, Mathematics Dept) 602-621-6892.

LOVELOCK, DAVID. University of Arizona, Meson AZ85721. [Dept of Mathematics] 602-621-6855.

LOWENGRUB, MORTON. Indiana University, Blooming-ton IN 47405. [Professor, Bryan Hall 104] 812-335-6153.

LUCAS, WIGWAM. National Science Foundation, Wash-ington DC 20550. [1800 G Street NW Room 639] 202-357 -7051.

LUCAS, JOHN. University of Wisconsin-Oshkosh,Oshkosh WI 54901. [Professor, Dept of Math Swart Hall206] 414-424-1053.

LUKAWECKI, STANLEY. Clemson University, ClemsonSC 29634. [Professor, Dept of Mathematical Sciences]803-656-3449.

LUNDGREN, RICHARD. University of Colorado at Den-ver, Denver CO 80202. [Chairman, Math Dept 110014th Street] 303-556-8482.

LYKOS, PETER. Illinois Institute of Technology, ChicagoIL 60616. [Consultant, Dept of Chemistry] 312-567-3430.

MADISON, BERNARD. National Research'Council,Washington DC 20418. [Project Director MS 2000, 2101Constitution Ave. NW) 202-334-3740.

MAGGS, WILLIAM. EOS - American GeophysicalUnion, Washington DC 20009.

MAGNO, DOMINIC. Harper College, Palatine IL 60067.[Associate Professor, Algonquin/Ruselle Roads] 312 -397-3000.

MAHONEY, JOHN. Sidwell Friends School, WashingtonDC 20016. [3825 Wisconsin Avenue NW) 202-537-8180.

259

252 PARTICIPANTS

MALONE, J.J. Worcester Polytechnic Institute, Worces-ter MA 01609. [Professor, Dept of Math Sciences 100Institute Rd) 617-793-5599.

MANASTER, ALFRED. University of California-SanDiego, La Jolla CA 92093. [Professor, Dept of Mathe-matics C-012) 619-534-2644.

MANFLZ'IS, ANDRE. National Szience Foundation,Washington DC 20550. [Deputy Director biv. of MathSc., nO6 G Street NW)

MARC DU, MARGARET. Montgomery County Schools,'';iievy Chase MD 20815. [Teacher, 5 Farmington CO

301-656-2789.MARSHALL, JAMES. Western Carolina University, Cul-

lowhee NC 28723. [Assistant Professor, P.O. Box 684)704-227-7245.

MARSHMAN, BEVERLY. University of Waterloo, Water-loo Ontario Canada N2L3G1. [ Assistant Professor, Deptof Applied Mathematics) 519-885-12:1.

MARTINDALE, JOHN. Random House "nc., CambridgeMA 02142. [Editorial Director, P21F; 1st Street) 617 -491-2250.

MARXEN, DONALD. Loras Czilege, Dubuque IA 52001.[Professor) 319-588-7570.

MASTERSON, JOHN. Michigan State University, EastLansing MI 48824. (Professor, Math Dept 211 D WellsHall) 517-353-4656.

MASTROCOLA, WILLIAM. Colgate University, HamiltonNY 13346. [Associate Professor, Dept of Mathematics)315-824-1000.

MATHEWS, JEROLD. Iowa State University, Ames IA50011. [Professor, Dept of Mathematics) 515-294-5865.

MATTUCK, ARTHUR. Massachusetts Institute of Tech-nology, Cambridge MA 02139. [Dept of MathematicsRoom 2-241) 617-253-4345.

MAYCOCK-PAr.KER, ELLEN. Wellesley College, Welles-ley MA 02181. [Assistant Professor, Dept of Mathemat-ics) 617-235-0320.

MAZUR, JOSEPH. Marlboro College, Marlboro VT05344. [Professor) 802-257-4333.

MCARTHUR, JAMES. Bethesda -Chevy Chase HighSchool, Bethesda MD 20814. [Teacher, 4301 East WestHighway) 301-654-5264.

MCBRIDE, RONALD. Indiana University of Pennsylva-nia, Indiana PA 15701. [Professor, Mathematics Dept)412-357-2605.

MCCAMMON, MARY. Penn State University, UniversityPark PA 16802. [Math Dept 328 McAllister Bldg) 814-865 -1984.

MCCARTNEY, PHILIP. Northern Kentucky Univer-sity, Annapolis MD 21401. [Professor, 1886 CrOwnsvilleRoad) 301-224-3139.

Mk;CLANAHAN, GREG. Anderson College, AndersonSC 29621. [Instructor, 316 Boulevard] 803-231-2165.

MCCOLLUM, MARY-ANN. Jefferson County GiftedProgram, Birmingham AL 35226. [Teacher, 1707 Kest-wick Cir.) 205-879-0531.

MCCOY, PETER. United States Naval Academy, An-napolis MD 21402. [Prof -sor, Mathematic Dept) 301-

267-2300.MCCRAY, LAWRENCE. National Research Council-

CPSMR, Washington DC 22050. [Associate ExecutiveDirector, 2101 Constitution Ave. NW) 202-334-3061.

MCDONALD, KIM. Chronicle of Hither Education.MCDONALO, BERNARD. National Science Foundation,

Washington DC 20550. [1800 G StreetNW]MCGEE, IAN. University of Waterloo, Waterloo Ontario

Canada N2L3C41. [Professor, Applied Math Dept) 519-885 -1211.

MCGILL, SUZANNE. University of South Alabama, Mo-bile AL 36688. [Chair, Dept of Mathematics and Statis-tics) 205-460-6264.

MCINTOSH, HUGH. The Scientist.MCKAY, FRED. National Research Council-MS 2000,

Washington DC 20418. [2101 Constitution Avenue NW)MCKEON, KATHLEEN. Connecticut College, New Lon-

don CT 06320. [Box 1561] 203-447-1411.MCLAUGHLIN, Rr,NATE. Univers:ty of Michigan-Flint,

Flint MI 48502. [Professor, Dept of Mathematics] 313-762 -3244.

MCNEIL, PHILLIP. Norfolk State University, NorfolkVA 23504. [Professor, Dept. Math 2401 Corprew Av-enue) 804-623-8820.

MELLEMA, WILBUR. San Jose City College, San JoseCA 95128. (Instructor, Math Dept 2100 Moorpark) 408-298 -2181.

MELMED; AD rifuR. New York University, New YorkNY 10003. [Research Professor, SEHNAP-23 Press BldgWashington Square) 212-998-5228.

MESKIN, STEPHEN. Society of Actuaries, Columbia MD21044. [Actuary, 5626 Vantage Point Road) 202.872-1870.

METT, CAREEN. Radford University, Radford VA24142. [Professor, Dept of Mathematics and Statistics)703-831-5026.

MILCETICH, JOHN. Univ of the District of Columbia,Washington DC 20011. [Professor, Dept of Math4200Connecticut Avenue NW) 202-282-7328.

MILLER, ALICE. Babson College, Babson Park MA02157. [As;istant Professor) 617-239-4476.

MINES, LINDA. National Research Council-MSEB,Washington DC 20006. [R18 Connecticut Ave NW Suit.,325) 202-334-3294.

MISNER, CHARLES. University of Maryland, CollegePark MD 20742. [Professor, Physics Deeartnient) 301-454 -3528.

MITCHELL, GEORGE. IndianaiUniversity of Pennsylva-nia, Indiana PA 15701. [Professor, 120 Concord Street)412-357-2305.

MOCHIZUKI, HORACE. Univ California - Santa Bar-bara, Santa Barbara CA 93106. [Professor, Dept ofMathematics) 805-961-3462.

MOOD: :4If.:11AEL. Washington State University, Pull-man o lee ii4. [Assistant Professor, MathematicsDept) t. 3172.

MOORE, L Z5 2NCE. Duke University, Durham NC

260

PARTICIPANT LIST

27706. [Associate Professor, PDept of Mathematics] 919-684 -2321.

MOORE, JOHN. Univ California - Santa Barbara, SantaBarbara CA 93106. [Professor, Dept of Mathematics]805-961-3688.

MORAWETZ, CATHLEEN. NYU-Courant Institute ofMath Science, New York NY 10012. [Director, 251 Mer-cer Street] 212-460-7100.

MORLEY, LANNY. Northeast Missouri State Univ,Kirk. -tine MO 63501. [Head, Div of Math Violette Hall287] 816-785-4547.

MORREL, BERNARD. IUPUI, Indianapolis IN 46223.[Associate Professor, Dept of Math 1125 East 38thStreet] 317-274-6923.

MORTON, PATRICK. Wellesley College, Wellesley MA02181. [Assistant Professor, Dept of Mathematics] 617-235 -0320.

MOSKOWITZ, HERBERT. Purdue University, WestLafayette IN 47907. [Professor, Krannert GraduateSchool of Management] 317-494-4600.

MOSLEY, EDWARD. Arkansas College, Batesville AR72501. [Professor] 501-793-9813.

MOVASSEGHI, DARIUS. CUNY - Medgar Evers College,Brooklyn NY 11225. [Professor, 1150 Carroll Street]717-735-1900.

MULLER, ERIC. Brock University, St. Catherine On-tario Canada L2S3A1. [Professor] 416-688-5550.

MURPHY, CATHERINE. Purdue University Calumet,Hammond IN 46323. [Head, 1i ept of Mathematical Sci-ences] 219-989-2270.

NAIL, BILLY. Clayton State College, Morrow GA 30260.[Professor, 5900 Lee Street] 404-961-3429.

NARODITSKY, VLADIMIR. San Jose State University,San Jose CA 95192. [Associate Professor, Dept of Math& Computer Science] 408-277-2411.

NEAL, HOMER. University of Michigan, Ann Arbor Nil48109. [Chair, Dept of Physics 1049 Randall Lab] 313-754 -4438.

NELSON, ROGER. Ball State University, Muncie IN47306. [Associate Professor, Dept of Math Sciences] 317-285 -8640.

NELSON, JAMES. University of Minnesota, Duluth MN55812. [Associate Professor, 10 University Avenue] 218-726 -7597.

NEWMAN, ROGERS. Southern University, Baton RoageLA 70813. [Professor, Dept of Mathematics] 504 -771-4500.

NORDAI, FREDERICK. Shippensburg University, Ship-pensburg PA 17257. [Associate Professor, P621 GlennStreet] 717-532-1642.

NORFLEET, SUNNY. St. Petersburg Junior College,Tarpon Springs FL 34689. [Teacher, 1309 Vermont Av-enue] 813938-7049.

NORTHCUTT, ROBERT. Southwest Texas State Uni-versity, San Marcos TX 78666. [Professor, MathematicsDepartment] 512-245-2551.

NOVAK, CAROLYN. Syracuse University, Utica NY13502. [Student, 213 Richardson Avenue] 315-733-4590.

253

NOVIKOFF, ALBERT. New York University, New YorkNY 10012. [Dept of Math 251 Mercer Street] 212 -982-5019.

NOVINGER, PHIL. Florida State University, TallahasseeFL 32306. [Associate Professor, Dept of Mathematics]904-644-1479.

O'BRIEN, RUTH. O'Brien & Associates, Alexandria VA22314. [President, Carriage House 708 Pendleton St.]703-548-7587.

O'DELL, RUTH. County College of Morris, aandolphNJ 07869. [Associate Professor, PRoute 10 and CenterGrove Road] 201-361-5000.

O'DELL, CAROL. Ohio Northern University, Ada OH45810. [Associate Professor, Dept of Math & ComputerScience] 419-772-2354.

O'MEARA, TIMOTHY. University of Notre Dame,Notre Dame IN 46556. [Provost, Administration Build-ing Room 202] 219-239-6631.

O'REILLY, MICHAEL. University of Minnesota, MorrisMN 56267. [Math Discipline] 612-589-2211.

OFFUTT, ELIZABETH. Spriugbrook High School,Bethesda MD 20814. [Teacher, 9304 Elmhirst Dr.] 301-530 -6238.

ORTIZ, CARMEN. Inter American Univ of Puerto Rico,Humacao PR 00661. [Lecturer, P.O. Box 204] 809 -758-8000.

OSER, HANS. SIAM News.OSTEBEE, ARNOLD. St. Olaf College, Northfield MN

55057. [Associate Professor] 507-663-3420.OST, LAURA. Orlando Sentinel, Orlando FLPAGE, WARREN. NYC Technical College SUNY, Brook-

lyn NY 10705. [Professor, 30 Amberson Ave. YounkersNY] C14-965-3893.

PALLAI, DAVID. Addison-Wesley Publishing Co., Read-ing MA 01867. [Senior Editor, Route 128] 617-944-3700.

PALMER, CHESTER. Auburn University-Montgomery,Montgomery AL 36193. [Professor, Dept of Mathemat-ics] 205-271-9317.

PAOLETTI, LESLIE. Choate Rosemary Hall, WallingfordCT 06492. [Teacher, P.O. Box 788] 203-269-7722.

PARTER, SEYMOUR. University of Wisconsin, MadisonWI 53706. [Dept of Mathematics Van Vleck Hall] 608-263 -4217.

PASSOW, ELI. Temple University, Bala Cynwyd PA19004. [Professor, 30 North Highland Avenuej 215 -664-6854.

PATTON, CHARLES. Hewlett-Packard Co., Corvallis OR97330. [Software Engineer, 1000 N.E. Circle Blvd. MS34-L9] 503-757-2000.

PAUGH, NANCY. Woodbridge Township School District,Woodbriige NJ 07095. [Supervisor, P.O. Box 428 SchoolStreet] 201-750-3200.

PEACOCK, MARILYN. Tidewater Community College,Portsmouth VA 23703. [Assistant Professor, State Rte.135] 804-484-2121.

PENNEY, DAVID. University of Georgia, Bogart GA30622. [Associate Professor, 235 West Huntington Road]404-542-2610.

261

254 PARTICIPANTS

PENN, HOWARD. United States Naval Academy, An-napolis MD 21402. [Professor, Mathematics Dept] 301-267 -3892.

PETERSEN, KARL. University of North Carolina,Chapel Hill NC 27514. [Professor, Dept of Mathemat-ics] 919-962-2380.

PETERSON, DORN. James Madison University, Har-risonburg VA 22807. [Physics Dept] 703-568-6487.

PETERSON, BRUCE. Middlebury College, MiddleburyVT 05753. [Professor] 802-388-3711.

PETERSON, IVARS. Science News.PETZINGER, KE'I. College of William and Mary,

Williamsburg VA 23185. [Professor] 804-253-4471.PHUA, MEE-SEE. Univ of the District of Columbia,

Washington DC 20011. [Dept of Math4200 ConnecticutAvenue NW] 202-282-7465.

PICCOLINO, ANTHONY. Dobbs Ferry Public Schools,Yonkers NY 10710. [Math Coordinator, 33 Bonnie BriarRd] 914-793-2645.

PIRTLE, ROBERT. Join Wiley & Sons, New York NY10158. [Editor, 605 3rd Avenue 5th floor] 212-d50-6348.

PLOTTS, RANDOLPH. St. Petersburg Junior College,St. Petersburg FL 33733. [Ins'. actor, 6605 lith AvenueNorth] 813-341-4738.

POIANI, EILEEN. Saint Peter's College, Jersey City NJ07306. [Professor, 2641 Kennedy Boulevard] 201 -333-4400.

POLLAK, HENRY. , Summit NJ 17901. [40 EdgewoodRoad] 201-277-1143.

POLUIHIS, JOHN. St. John Fisher College, RochesterNY 14618. [Professor, 3497 East Avenue] 716-586-4600.

PONZO, PETER. University of Waterloo, Waterloo On-tario Canada N2L3G1. [Professor, Applied Math Dept]519-885-1231.

PORTER, JACK. University of Kansas, Lawrence KS66045. [Professor, Dept of Mathematics] 912-864-4367.

POSTNER, MARIE. St. Thomas Aquinas College,Sparkill NY i0976. [Assistant Professor, PRoute 340]914-359-9500.

POWELL, WAYNE. Oklahoma State University, Stillwa-ter OK 74075. [Associate Professor, Dept of Mathemat-ics] 405-624-5790.

PRESS, FRANK. National Academy of Sciences, Wash-ington DC 20418. [President, 2101 Constitution AvenueNW] 202-334-2100.

PRICE, CHIP. Addison-Wesley Publishing Company,Reading MA 01867. [Editor-in-Chief, Route 128] 617-944 -3700.

PRICE, ROBERT. Addison - Wesley Publishing Company,Reading MA 01867. [Editor-in-Chief, Route 128] 617-944 -3700.

PRICHETT, GORDON. Babson College, Wellesley MA01157. [Vice President, Babson Park] 617-239-4316.

PRIESTLEY, W.M. University of the South, Sewanee TN37:75. [Professor] 615-598-5931.

PROSL, RICHARD. College of William and Mary,Williamsburg VA 23185. [Chair, Dept of Computer Sci-ence] 804-253-4748.

PROTOMASTRO, GERARD. St. Peter's College, Bloom-field NJ 07003. [Professor, 96 Lindbergh Blvd.] 201 -333-4400.

PURZITSKY, NORMAN. York University, DownsviewOntario Canada M3J1P3. [Associate Professor, Dept ofMathematics] 416-736-5250.

QUIGLEY, STEPHEN. Scott Foresman and Co., Glen-vier IL 60025. [Editor, 1900 East Lake Avenue] 312-729 -3000.

QUINE, J.R. Florida State University, Tallahassee FL32306. [Professor, Dept of Mathematics] 904-644-6050.

QUINN, JOSEPH. University of North Carolina, Char-litte NC 28223. [Chairman, Dept of Mathematics] 704-547 -4495.

RADIN, ROBERT. Wentworth Institute of Technology,West Hartford CT 06119. [Professor, 781 FarmingtonAvent,e] 203-233-8106.

RAGER, KEN. Metropolitan State College, Denver CO80204. [Professor, 1006 11th Street] 303-556-3284.

RAJAH, MOHAMMED. Miracosta College, Oceanside CA92056. [Professor, 1 Barnard Drive] 619-757-2121.

RALSTON, ANTHONY. SUNY - Buffalo, Buffalo NY14260. [Professor, Dept of Computer Science 225 BellHall] 716-878-4000.

RAMANATHAN, G.V. University of Illinois-Chicago,Chicago IL 60580. [Professor, Dept of Math StatisticsCS; Box 4348] 312-996-3041.

RAMSEY, THOMAS. University of Hawaii, HonoluluHI 96822. [Associate Professor, PMath Dept 2565 TheMall] 808-948-7951.

RAPHAEL, LOUISE. National Science Foundation,Washington DC 20550. [Program Director, DMS 1800G Street NW] 202-357-7325.

RASMUSSEN, DOUG. Chemeketa Community College,Salem OR 97309. [Instructor, P.O. Box 14007] 503 -399-5246.

RAY, DAVID. Bucknell Unviersity, Lewisburg PA 17837.[Professor, Dept of Mathematics] 717-524-1343.

REDISH, EDWARD. University of Maryland, CollegePark MD 20742. [Professor, Dept of Physics] 301-454-7383.

REED, MICHAEL. Duke University, Durham NC 27706.[Chair, Dept of Mathematics] 919-684-2321.

REED, ELLEN. Tria:ty School at Greenlawn, SouthBend IN 46617. [107 South Greenlawn Avenue] 219 -287-5590.

REICHARD, ROSALIND. Elon College, Elon College NC27244. [Assistant Professor, Dept of Mathematics P.O.Box 2163] 919-584-2285.

RENZ, PTER. Mathematical Association of America,Washington DC 20036. [1529 18th Street NW]

RICE, PETER. University of Georgia, Athens GA ,A302.[Professor, Mathematics Dept] 401-542-2593.

RIESS, RONALD. Virginia Polytech & State Univer-sity, Blacksburg VA 24061. (Dept of Mathematics ,.'60McBryde Hall] 703-961-6536.

RISEBERG, JOYCE. Montgomery College, Rockville '..`::D20850. [Professor, 51 Mannakee Street] 301-279-5203.


ROBERTS, WAYNE. Macalester College, St. Paul MN55113. [Professor, 1500 Grand Avenue] 612-696-6337.

RODGERS, PAMELA. O'Brien & Associates, Alexan-dria VA 22314. [Senior Associate, Carriage House 708Pendleton St.] 703-548-7587.

RODI, STEPHEN. Austin Community College, AustinTX 7872;;. [Chair, Dept of Math & Pnys Sci 2008 Lazy-brook] 512-495-7222.

ROECKLEIN, PATRICIA. Montgomery College, RockvilleMD 20850. [Associate Professor, 51 Mannakee Street]301-279-5199.

ROGERS, LAUREL. University of Col .7ado, ColoradoSprings CO 80933. [Assistant Professor, PDept of MathP.O. Box 7150] 303-593-3311.

ROITBERG, JOSEPH. Hunter College, New York NY10021. [Professor, 695 Park Avenue] 212-772-5300.

ROITBERG. YAEL. New York Institute of Technology,Old Westbury NY 11568. [Associate Professor] 516 -686-7535.

ROLANDO, JOSEFINA. St. Thomas. University, MiamiFL 33054. [Professor, 16400 N.W. 32nd Avenue] 305-625 -6000.

ROLANDO, TOMAS, St. Thomas University, Miami FL33054. [Professor, 16400 N.W. 32nd Avenue] 305 -625-6000.

ROLWING, RAYMOND. University of Cincinnati, Cincin-nati OH 45221. [Professor, Dept of Mathematical Sci-ences] 513-475-6430.

ROSENHOLTZ, IRA. University of Wyoming, LaramieWY 82071. [Dept of Mathematics] 307-766-3192.

ROSENSTEIN, GEORGE. Franklin and Marshall College,Lancaster PA 17604. [Professor, Box 3003] 717-291-4227.

ROSENSTEIN, JOSEPH. Rutgers University, NewBrunswick NJ 08904. [Professor, Dept of Mathematics]201-932-2368.

ROSENTHAL, WILLIAM. Ursinus College, CollegevillePA 19426. [Asristant Professor. Dept of Math and Com-puter Science] 215-489-4111.

ROSEN, LINDA. National Research Council-MSEB,Washington DC 20006. [Project Officer, 818 Connecti-cut Avenue NW Suite 325] 202-334-3294.

ROSS, KENNETH. University of Oregon, Eugene OR97403. [Professor, Dept of Mathematics] 503-686-4721.

ROUSSEAU, T.H. Siena College, Loudoaville N Y. 12180.[Head, Dept of Mathematics] 518-783-2440.

ROXIN, EMILIO. University of Rhode Island, KingstonRI 02881. [Professor] 401-792-2709.

RuBENSTEIN, PATRICIA. Montgomery College,Gaithersburg MD 20879. [Professor, 19038 WhetstoneCircle] 301-948-2737.

Russo, PAULA. Trinity College, Hartford CT 06106.[Assistant Professor, PDept of Mathematics] 203 -527-3151.

RYFF, JOHN. National Science Foundation, WashingtonDC 20550. [Program Director, 1800 G Street NW Room339] 20.4-357-3455.

SACHDEV, SOHINDAR. Elizabeth City State University,

Elizabeth City NC 27909. [Chairman, Dept of Math &Computer Sci Box 951] 919-335-3243.

SADLOWSKY, ROGER. Columbia Heights High School,New Brighton MN 55112. [Teacher, 2393 Pleasant ViewDr.] 612-574-6530.

SAHU, ATMA. University of Maryland, Princess AnneMD 21853. [Assistant Professor] 301-651-220U.

SALAMON, LINDA. Washington University, St. LouisMO 63130. [Dean, College of Arts & Science] 314 -889-5000.

SALZBERG, HELEN. Dhode Island College, ProvidenceRI 02908. [Professor, Dept of Math & Computer Sci]401-456-8038.

SAMPSON, KIRSTEN. JPBM, Washington DCSANDEFUR, JAMES. Georgetown University, Washing-

ton DC 20057. [Professor, Mathematics Dept] 703 -687-6145.

SA''AGOPAN, K.P. Shaw University, Raleigh NC 27606.[Associate Professor, 4303-3 Avent Ferry Road] 919 -755-4877.

SAYRAFIEZADEH, MAHMOUD. Medgar Evers College-CUNY, Brool-lyn NY 11225. [Associate Professor, 1150Carroll St.j 718-735-1897.

SCHEPPERS, JAMES. Fairview High School, Boulder CO80303. [Chair Math Dept, 1515 C;reenbriar Blvd.] 303-499 -7600.

SCHICK-LENK, JUDITH. Ocean County College, TomsRiver NJ 08723. [College Dr] 201-255-0400.

SCHLAIS, HAL. University of Wisconsin Centers,Janesville WI 53534. [2909 Kellogg Avenue] 608 -755-2811.

SCHMEELK, JOHN. Virginia Commonwealth University,Richmond VA 23284. [Associate Professor, 1015 Westrain St] 804-257-1301.

SCHMIDT, HARVEY. Lewis and Clark College, PortlandOR 97219. [Associate Professor, Campus Box 111] 5q3-293-2743.

SCHNEIDER, DAVID. University of Maryland, CollegePark Mi.) 20742. [Associate Professor, Dept of Mathe-matics] 301-454-5002.

SCHREMMER, ALAIN. Community College of Philadel-phia, Philadelphia PA 19130. [Associate Professor, 1700Spring Garden Street] 215-751-8413.

SCHROEDER, BERNIE. Univ. of Wisconsin-Platteville,Platteville WI 53818. [Avsociate Professor] 608 -342-1746.

SCHURRER, AUGUSTA. Univ of Northern Iowa, CedarFalls IA 50614. [Professor. Dept of Math & ComputerScience] 319-273-2432.

SCHUTZMAN, ELIAS. National Science Foundation,Wvslangton DC 20550. [Program Director, 1800 GStreet NW] 202-357-9707.

SEIDLER, ELIZABETH. Mercy High School, BaltimoreMD 21239. [Teacher, 1300 East Northern Parkway] 301-433 -8880.

SEIFERT, CHARLES. University of Central Arkansas,Conway AK 72032. [Chairma , Math & Computer SciDept Main 104] 501-450-3147.

2' -,

256 PA liTICIPANTS

SELDEN, JOHN. Tennessee Technological University,Cookeville TN 38505. [Assistant Professor, Math DeptBox 5054] 615-372-3441.

SELDON, ANNIE. Tennesbee Technological University,Cookeville TN 38505. [Assistant Professor, PPMathDept Box 5054] 615-372-3441.

SELIG, SEYMOUR. National Research Council-BMS,Washington DC 20418. [2101 Constitution Ave NW]202-334-2421.

SESAY, MOHAMED. Univ of the District of Columbia,Silver Spring MD 20910. [Professor, 8750 Georgia Av-enue #118A] 301-565-2623.

SESSA, KATHLEEN. D.C. Heath and Co., LexingtonMA 02173. [Developmental Editor, 125 Spring Street]617-960-1544.

SHARMA, MAN. Clark Col?ege, Atlanta GA 30314. [Pro-fessor, Dept of Math 240 James Brawley Dr] 404 -577-6685

SHARP, JACK. Floyd Junior College, Rome GA 30161.[Associate Professor, P.O. Box 1854] 404-295-6357.

SHIFLETT, RAY. Calif State Polytechnic University,Pomona CA 91768. [Dean, 3801 West Temple Avenue]714-869-3600.

SHOOTER, WILLIAM. Gloucester County College,Sewell NJ 08080. [Coordinator] 609-465-5000.

SIEBER, JAMES. Shippensburg University, ShippensburgPA 17257. [Professor, Dept of Math & Computer Sci]717-532-1405.

SIEGEL, MARTHA. Towson State University, TowsonMD 21204. [Professor, Dept of Mathematics] 301 -321-2980.

SIMPSON, DAVID. Southwest State University, MarshallMN 56258. [Professor] 507-537-6141.

SINGH, PREMJIT. Manhattan College, Riverdale NY10471. [Assistant Professor, PDept of Math & ComputerSci] 212-920-0385.

SKIDMORE, ALEXANDRA. Rollins College, Winte: ParkFL 32789. [Professor] 305-646-2516.

SKITZKI, RAY. Shaker Hr-ghts High School, ShakerHeights OH 44120. [Teacher, 15911 Aldersyde Drive]216-921-1400.

SLACK, STEPHBN. Kenyon College, Gambier OH 43022.[Associate Professor, Mathematics Department] 614 -427-5267.

SLINGER, CAROL. Marian College, Indianapolis IN46222. [Head, Dept of Math 3200 Cold Spring Road]317-929-0281.

SLOUGHTER, DAN. Furman University, Greenville SC29613. ` ssistant Professor, Mathematics Dept] 803 -294-3233.

SLOYAN, STEPHANIE. Georgian Court College, Lake-wood NJ 08701. [Professor, Dept of Mathematics] 201-364 -2200.

SMALL, DON. Colby College, Waterville ME 04601. [As-sociate Professor] 207-872-3255.

SMITH, DAVID. Duke University, Durham NC 27706.[Associate. Professor, Dept of Mathematics] 919 -684-2321.

SMITH, ROBERT. Millersville University, Millersville PA17551. [Professor, Dept of Math & Computer Science]717 - 872 -3780.

SMITH, ROSE-MARIE. Texas Woman's University, Den-ton TX 76204. [Chair, Dept of Math P.O. Box 22865]817-898-2166.

SMITH, RICK. University af Florida, Gainesville FL32611. [Associate Professor, Dept of Mathematics] 904-392 -6168.

SNODGRASS, ALICE. John Burroughs School, WebsterGrove MO 63119. [Teacher, 440 East Jackson Rd] 314-993 -4040.

SOLOMON, JIMMY. Mississippi State University, Mis-sissipi State MS 39762. [Professor, Dept of Math P.O.Drawer MA] 601-325-3414.

SOLOW, ANITA. Grinnell College, Grinnell IA 50112.[Associate Professor, Dept of Mathematics] 515 -269-4207.

SPANAGEL, DAVID. St. John Fisher College, RochesterNY 14618. [Instructor, Dept of Math & CS 3690 EastAve] 716-385-8190.

STAHL, NEIL. Univ Wisconsin Center-Fox Valley,Menasha WI 54952. [Associate Professor, PMidwayRoad] 414-832-2630.

STAKGOLD, IVAR. University of Delaware, Newark DE19716. [Professor, Dept of Math 501 Ewing Hall] 302-451 -2651.

STARR, FREDERICK. Oberlin College, Oberlin OH44074. [President]

STEARNS, WILLIAM. University of Maine, Orono ME04469. [Associate Professor, 228 Neville Hall] 207 -581-3928.

STEEN, LYNN. St. Olaf College, Northfield MN 55057.[Professor, Dept of Mathematics] 307-663-3114.

STEGER, WILLIAM. Essex Community College, Reister-stown MD 21136. [Associate Professor, 12717 Gores MillRd] 301-522-1393.

STEPP, JAMES. University of Houston, Houston TX77004. [Professor, Dept. of Mathematics] 713-149-4827.

STERN, ROBERT. Sunders College Publishing, Philade-phia PA 19105. [Senior Editor, 210 West WashingtonSquare]

STERRETT, ANDREW. Denison University, GranvilleOH 43023. [Professor] 614-587-6484.

STEVENSON, JAMES. Ionic Atlanta Inc, ;-..tlanta GA30309. [CEO, 1347 Spring Street] 404-876-5166.

STEVENS, CHRISTINE. National Science Foundation,Washington DC 20550. [Associate Pr:3ram Director,1800 G Street NW Room 635] 202-357-7074.

STODGHILL, JACK. Dickinson College, Carlisle PA17013. [Associate Professor] 717-245-1743.

STONE, DAVID. Gesgia Southern College, StatesboroGA 30460. [Professor, Dept of Math & CS L. Box 8093]912-681-5390.

STONE, THOMAS. PWS-Kent Publishing Co., BostonMA 01970. [Editor, 20 Park Plaza] 617-542-3377.

STOUT, RICHARD. Gordon College, Wenham MA01984. [Chair, Dept of Mathematics] 617 -927 -2300.

PARTICIPANT LIST 25'

STRALEY, TINA. Kennesaw College, Marietta GA30061. [Chair, Dept of Mathematics] 404-423-6104.

STRANG, GILBERT. Massachusetts Institute of Tech-nology, Cambridge MA 02139. [Professor, Room 2 -240]617-253-4383.

STRONG, ROGER. Livonia Public Schools, Plymouth MI48170. [39651 Mayville] 313-455-1530.

SUMMERHILL, RICHARD. Kansas State University,Manhattan KS 66506. [Associate Professor, Dept ofMathematics] 913-532-6750.

SUNLEY, JUDITH. National Science Foundation, Wash-ington DC 20550. [Director, DMS 1800 G Street NW]202-357-9669.

SURI, MANIL. University of Maryland, Catonsville MD21228. [Assistant Professor, Dept of Mathematics] 301-455 -2311.

SWAIN, STUART. University of Maine-Machias, MachiasME 04654. [Assistant Professor, Science Division] 207-255 -3313.

SWARD, GILBERT. Montgomery College, Chevy ChaseMD 20815. [Professor, 9101 Levelle Dr.] 301-657-3056.

SWARD, MARCIA. National Research Council-MSEB,Washington DC 20006. [Executive Director, 818 Con-necticut Ave NW Suite 325] 202-334-3294.

SWOKOWSKI, EARL. Marquette University, MilwaukeeWI 53227. [12124 West Ohio Avenue] 414-546-3860.

SZOTT, DONNA. South Campus - CCAC, West MifflinPA 15102. [Professor, 1750 Clairton Road] 421-469-6228.

TALMAN, Lows. Metropolitan State College, DenverCO 80204. [Assistant Professor, Dept Math SciencesBox 38] 303-556-8438.

TELES, ELIZABETH. Montgomery College, TakomaPark MD 20912. [Associate Professor, Department ofMathematics] 301-587-4090.

TEMPLE, PATRICIA. Choate Rosemary Hall, Walling-ford CT 06492. [Calculus Head, P.O. Box 788] 203 -269-7722.

THESING, GARY. Lake Superior State College, SaultSte. Marie MI 49783. [Head, Dept of Mathematics] 906-635 -2633.

THOMPSON, MELVIN. Howard University, WashingtonDC 20059. [Director Develop. & Research Admin., Rm.1116 2300 6th St. NW] 202-636-5077.

THOMPSON, DON. Pepperdine University, Malibu CA90265. [Natural Science Division] 213-456-4239.

THOMPSON, THOMAS. Walla Walla College, CollegePlace WA 99324. [Professor, Dept of Mathematics] 509-527 -2161.

THONGYOO, SUTEP. Syracuse University, Syracuse NY13210. [Student, B5 Apt #4 Slocum Heights] 315 -423-2373.

THORNTON, EVELYN. Prairie View A&M University,Prairie View TX 77446. [Professor, Dept of Mathemat-ics] 409-857-4091.

THRASH, JOE University of Southern Mississippi, Hat-tiesburg MS 39406. [Associate Professor, PI,Jx 5045Southern Station] 601-266-4289.

TILLEY, JOHN. .Mississippi State University, MississipiState MS 39762. [Professor, Dept of Math P.O. DrawerMA] 601-325-3414.

TOLBERT, MATTHEW. U. S. Congress, Washington DC[House Committee on Sci Space & Tech]

TOLER, CHARLES. Wilmington Friends School, Wilm-ington DE 19803. [Teacher, 101 School Road] 302 -575-1130.

TOLLE, JON. University of North Carolina, Chapel HillNC 27514. [Professor, Curriculum in Math Phillips Hall]919-962-0198.

TOUBASSI, ELIAS. University of Arizona, Tucson AZ85721. [Professor, Dept of Mathematics] 602-621-2882.

TREFZGER, JIM. McHenry County College, CrystalLake IL 60012. [Professor, Route 14 at Lucas Road] 815-455 -3700.

TREISMAN, URI. University of Calif-Berkeley, BerkeleyCA 94720. [PDP-230B Stephens Hall] 415-642-2115.

TRIVIERI, LAWRENCE. Mohawk Valley CommunityCollege, Utica NY 13501. [Professor, 1101 ShermanDrive] 315-792-5369.

TROYER, ROBERT. Lake Forest College, Lake Forest IL60045. [Professor] 312-234-3100.

TUCKER, THOMAS. Colgate University, Hamilton NY13346. [Professor, Dept of Mathematics] 315-824-1000.

TUCKER, RICHARD. Mary Baldwin College, St' .untonVA 24401. [Assistant Professor] 703-887-7112.

TUFTE, FREDRIC. University of Wisconsin-Platteville,Platteville WI 5C 18. [Associate Professor, Dept Math 1University Plaza] 608-342-1745.

UPSHAW, JANE. University of South Carolina-Beaufort,Beaufort SC 29928. rCl'air, 800 Cartaret Street] 803-524 -7112.

URION, DAVID. Winona State University, Winona MN55987. [Professor, Dept of Mathematics and Statistics]507-457-5379.

VANVELSIR, GARY. Anne Arundel 0- munity Col-lege, Arnold MD 21012. [Professor, 1 Jollege Park-way] 301-260-4565.

VAVRINEK, RONALD. Illinois Math and ScienceAcademy, Aurora IL 60506. [1500 West Sullivan Road]312-801-6000.

VEI,EZ-RODRIGUEZ, ARGELIA. Department of Educa-tion, Washington DC 20202. [Director, Room 3022ROB-3 7th & D Streets SW] 202-732-4396.

VICK, WILLIAM. Broome Community College, Iling-hamton NY 13902. [Professor, P.O. Box 1017] 607 -771-5165.

VIKTORA, STEVEN. Kenwood Academy, Chicago IL60615. [Chair, 5100 South Hyde Park Blvd Apt 3D] 312-947 -0882.

VOBEJDA, BARBARA. Washington Peat.VONESCHEN, ELLIS. Suffolk County Community Col-

lege, Selden NY 11784. [Professor, 533 College Rd] 516-451 -4270.

WAGONER, RONALD. California State University,Fresno CA 93710. [Professor, 617 East Teal Circle] 209 -438 -5512.

265

258 PARTICIPANTS

WAGONER, JENETTE. California State University,Fresno CA

WAITS, BERT. Ohio State University, Columbus OH43201. [Professor, Dept of Math 231 West 18th Avenue]614-292-0694.

WALKER, RUSSELL. Carnegie Mellon University, Pitts-burgh PA 15601. [Senior Lecturer, Dept of Mathematics]412-268-2545.

WALSH, MARY-LU. D.C. Heath & Co., Lexington MA02173. [Editor, 125 Spring Street] 617-860-1144.

WALSH, JOHN. Science Magazine.WANG, AMY. Montgomery College, Vienna VA 22180.

[Assistant Professor, 1834 Batten Hollow Road] 703 -281-0579.

WANTLING, KENNETH. Washington College, Chester-town MD 21620. [Assistant Professor] 301-778-2800.

WARDROP, MARY. Central Michigan University, Mt.Pleasant MI 48859. [Professor. Dept of Mathematics]517-774-3596.

WARD, JAMES. Bowdoin College, Brunswick ME 04011.[Chair, Dept of Mathematics] 207-725-3577.

WASHINGTON, HARRY. Delaware State College, DoverDE 19901. [Associate Professor, Dept of Mathematics]302-736-3584.

WATKINS, SALLIE. American Institute of Physics,Washington DC 20009. [Sr. Education Fellow, 2000Florida Ave NW] 202-232-6688.

WATSON, ROBERT. National Science Foundation,Washington DC 20550. [Acting Head UndergraduateEd, 1800 G Street NW]

WATSON, MARTHA. Western Kentucky University,Bowling Green KY 42101. [Professor, Dept of Mathe-matics] 502-745-6224.

WATT, JEFFREY. Indiana University, Indianapolis IN46206. [Associate Instructor, PP.O. Box 2813] 317 -849-4136.

WELLAND, B013. Northwestern University, Evanston IL60208. [Associate Professor, Lunt Hall] 312-492-5576.

WELLS, DAN. Western Carolina University, CullowheeNC 28723. (Associate Professor, Box 837] 704-586-5797.

WENCER, RONALD. University of Delaware, NewarkDE 19716. [Director, Math Center 032 Purnell Hall]302-451-2140.

WESTERMAN, JOAN. O'Brien & Associates, Alexan-dria VA 22314. [Senior Associate, Carriage House 708Pendleton Street] 793-A8-7587.

WHITAKER, PATRICIA. Elon College, Elon College NC27244. [Assistant Professor, Dept of Mathematics P.O.Box 2163] 919-584-2285.

WHITE, ROBERT. National Academy of Engineering,Washington DC 20418. [President, 2101 ConstitutionAvenue NW] 202-334-3200.

WICK, MARSHALL. Univ of Wisconsin-Eau Claire, EauClaire WI 54702. [Chair, Dept of Mathematics] 715 -836-2768.

WILLARD, EARL. Marietta College, Marietta OH45750. [Dept of Mathematics] 614-374-4811.

WILLCOX, ALFRED. Mathematical Association ofAmerica, Washington DC 20036. [1529 18th StreetNW]

WILSON, JACK. University of Maryland, College ParkMD 20742. [Professor, Dept of Physics] 301-345-4200.

WINGO, WALTER. , Design News, 22207. [WashingtonEditor, 4655 N 24th St. Arlington VA] 703-524-3816.

WOLFSON, PAUL. West Chester University, WestChester PA 19383. [Associate Professor, Dept of Mathe-matical Sciences] 215-436-2452.

WOODS, JOHN. Oklahoma Baptist University, ShawneeOK 74801. [Professor] 405-275-2850.

WRIGHT, DONALD. University of Cincinnati, Cincin-nati OH 45221. [Professor, Dept of Mathematical Sci-ences] 513-475-3461.

YOUNG-DAVIS, PATSY. Lake Mary High School, LakeMary FL 32746. [Chair Math Dept, 655 Longwood-Lakt.Mary Road] 305-323-2110.

YOUNG, GAIL. National Science Foundation, OssiningNY 10562. [Program Director, 53B Van Cortland Ave]

YOUNG, PAUL. University of Washington, Seattle WA98915. [Chair, Computer Science Board] 206-543-1695.

YUHASZ, WAYNE. Random House Publishing Co.,Cambridge MA 02142. [Senior Editor, 215 1st Street]617-491-3008.

ZIEGLER, JANET. UPI.ZORN, ?AUL. Purdue University, West Lafayette IN

47906. [Associate Professor of Mathematics, St. OlafCollege] 317-494-1915.

21;6

CALCULUS AS A PUMP, NOT A FILTER

The challenge that Robert M. White, President of theNational Academy of E .gineering, made to the mathemat-ical community at the close of his keynote address to theCALCULUS FOR A NEW CENTURY symposium was tomake the introductory calculus course into a pump thatfeeds more students into science and engineering, not afilter that cuts down the flow. This challenge provides thesubtitle to this volume and a common purpose for thediverse efforts to reform the teaching of calculus over thenext few years. These proceedings with contributionsfrom over eighty authors, show the full sweep of con-cerns and approaches of all the groups involved in cal-culus reform, including those currently teaching tradi-tional and innovative courses, those whose students oremployees need to use calculus as a tool, and the depart-ment chairs, deans, and others who must mobilize theresources needed for this reform.

CALCULUS FOR A NEW CENTURY is divided intoseven parts. Colloquium, containing the plenary and paneladdresses, Responses, solicited from representatives ofthe concerned constituencies; Reports, gathered from theconference working groups, Issues, containing the back-ground papers for the conference, Examinations, a selec-tion of examination questions from representative schoolsof all types; Readings, position papers from other sourcesthat are important to calculus reform; and Participants, alist of names and addresses of those attending, includedto help facilitate future exchanges of ideas. The Readingswere added so that these proceedings, together with theearlier volume, TOWARD A LEAN AND LIVELY CALCU-LUS (edited by Ronald G. Douglas and published as MAANotes 6). wo ild give a complete picture of the recentefforts to reform the teaching of calculus.

Dr. White along with other speakers stressed tne impor-tance of giving students a deep and practical understand-ing of calculus. This kind of understanding will becomeever more important as calculus moves into a new cen-tury when scientific ani engineering calculations will beincreasingly done by machine. Ronald G. Douglas, Deanof Physical Scie- ices at SUNY Stony Brook, and othershave pointed out that existing hand-held symbolic andgraphic calculators can perform all the routine calcula-tions and handle all the graphing needed to get a B orbetter in the standard course. Moreover, such machinesand computer algebra systems running on microcom-puters can solve realistic problems that would have beenbeyond the capabilities of the very best students in thepast. Many of the op -ortunities that such machines offerare discussed in this v..)lume. Fully realizing these opportu-nities will require nothing less than a rethinking of the rela-tionship between calculation and calculus both in learn-ing the subject and in applying it.

The rethinking of introductory calculus must extend tohow the course is taught and how learning is assessed.Lead speaker Ronald G. Douglas describes calculus asbig, important, and in trouble. Lynn Arthur Steen'saddress, "Calculus Today," provides figures that stronglysupport Douglas's characterization. Pedogogical consid-erations are a strong theme in this volume and in tti,_earlier TOWARD A LEAN AND LIVELY CALCULUS.

This overview of opportunities and needs for new initia-tives in teaching calculus does not provide a tested andproven model course that meets Dr. White's challenge.But there -.1e many ideas here and many experiments nowunderwa.. More importantly, this symposium shows abroad interest in the community in moving ahead with cal-culus reform. Well over six hundred people attended theCALCULUS FOR A NEW CENTURY symposium at theNational Academies of Sciences and Engineering. The mixof mathematicians, scientists, engineers, social scien-tists, college and university administrators, people frombusiness, economics, foundations, and granting agenciesbodes well for these reforms. But it is our intended readerwho must carry this new course into the classroom andprovide the final answers to Dr. White's challenge. Wemay not yet know these, but at least our readers will beon the path toward those answers and, we trust they willfind them.

Peter L. Renz, Associate DirectorThe Mathematical Association of America

The CALCULUS FOR A NEW CENTURY colloquium washeld by the National Research Council in collaborationwith the Mathematical Association of America. The collo-quium was funded by a grant from the Alfred P. SloanFoundation. See the Preface, page ix, for further details.

The image used on the cover of this book and on the openingpages of each of its sw.fen parts shows a periodic minimal surfacediscovered in 1987 by Michael Callahan, David Hoffman, and BillMeeks III at the University of Massachuse.,s. This connutergenerated image was created by James T. Hoffman, 1987.

ISBN 0-88385-058-3

Date post:	20-Apr-2020
Category:	Documents
Upload:	others
View:	1 times
Download:	0 times

DOCUMENT RESUME ED 300 252 AUTHOR Steen, …ED 300 252 AUTHOR TITLE INSTITUTION REPORT NO PUB DATE...

Documents