DOCUMENT RESUME ED 359 073 SE 053 691 …DOCUMENT RESUME ED 359 073 SE 053 691 AUTHOR Cornu,...

DOCUMENT RESUME

ED 359 073 SE 053 691

AUTHOR Cornu, Bernard, Ed.; Ralston, Anthony, Ed.TITLE The Influence of Computers and Informatics on

Mathematics and Its Teaching. Science and TechnologyEducation Series, 44.

INSTITUTION United Nations Educational, Scientific, and CulturalOrganization, Paris (France). Div. of Science,Technical and Environmental Education.

REPORT NO ED-92/WS/17PUB DATE Oct 92NOTE 140p.PUB TYPE Collected Works Serials (022)

EDRS PRICE MF01/PC06 Plus Postage.DESCRIPTORS Algorithms; Calculators; Calculus; Computer Assisted

Instruction; Computer Science Education; *ComputerUses in Education; *Educational Technology;Elementary Secondary Education; Foreign Countries;Inservice Teacher Education; Mathematics Curriculum;Mathematics Education; *Mathematics Instruction;Microcomputers; Preservice Teacher Education

IDENTIFIERS Computer Algebra; Discrete Mathematics; GraphingUtilities; Representations (Mathematics)

ABSTRACT

In 1985 the International Commission on MathematicalInstruction (ICMI) published the first edition of a book of studieson the topic of the influence of computers on mathematics and theteaching of mathematics. This document is an updated version of thatbook and includes five articles from the 1985 ICMI conference atStrasbourg, France; reports by the leaders of three workshops held atthat meeting; and four new articles on topics related to mathematicsinstruction using technology. The articles are: (1) "Overview" (H.Burkhardt and R. Fraser); (2)"The Effect of Computers on Mathematics"(R. F. Churchhouse); (3) "The Impact of Computers and ComputerScience on the Mathematics Curriculum" (A. Ralston); (4) "Computersas an Aid to Teaching and Learning Mathematics" (B. Cornu); (5)"Living with a New Mathematical Species" (L. A. Steen); (6) "What areAlgorithms? What is Algorithmics?" (S. B. Maurer); (7) "On theMathematical Basis of Computer Science" (J. Sterr); (8) "The Effectof Computers on the School Mathematics Curriculum (K.-D. Graf, R.Fraser, L. Klingen, J. Stewart, and B. Winkelmann); (9) "AFundamental Course in Higher Mathematics Incorporating Discrete andContinuous Themes" (S. B. Seidman and M. D. Rice); (10) TeacherEducation and Training" (B. Cornu); (11) "The Impact of SymbolicMathematical Systems on Mathematics Education" (B. R. Hodgson and E.R. Muller); (12) "Calculus Teaching and the Computer. On theInterplay of Discrete Numerical Methods and Calculus in the Educationof Users of Mathematics" (M. Mascarello and B. Winkelmann); and (13)"Graphic Insight into Mathematical Concepts" (D. Tall and B. West). Alist of 62 annotated references and an index are included. (MDH)

Science and technology education, 44

The influence of computersand informatics on

mathematics and its teaching

Education SectorED-92AVS/17

Edited byProfessor Bernard Cornu

Professor Anthony Ralston

UNESCO

c^s

t)

ParisOctober 1992

The designations employed and the presentation ofthe material in this document do not imply theexpression of any opinion whatsoever on the part ofUNESCO concerning the legal status of any country,territory, city or area or of its authorities, orconcerning the delimitation of its frontiers or boundaries.

Published in 1992 by the United Nations Educational,Scientific and Cultural Organization,7 Place de Fontenoy,75352 Paris 07 SP

Printed by UNESCO

© UNESCO 1992Printed in France

Preface

The three-way interaction between mathemat-ics, computers and mathematics education is be-coming stronger each year. How schools and univer-sities should respond is still an open question. Thisdocument has been prepared to contribute to thedebate. The following quotation from the Overviewchapter states succinctly why this debate is so im-portant:

We are facing a situation in which childrenare taught to do mathematics in ways thatare very largely outmoded, with at least 80%of curriculum time wasted on trying, more orless successfully, to develop fluency in skillsof now limited value.

The International Commission on Mathemati-cal Instruction (ICMI) undertook a study, "The In-fluence of Computers and Informatics on Mathe-matics and Its Teaching", which included a con-ference in Strasbourg, France in 1985, in whichUNESCO co-operated. The outcome of the Studywas a book published by Cambridge UniversityPress, bearing the conference title. With thequick pace of change of computers, mathemat-ics and its teaching, the book's contents have be-come outdated. The development of this newdocument is explained in the Editors' Foreword.

The reader will notice that the authors of thisdocument are all from Europe and North Amer-

ica. One might conclude that school uses of com-puters are only known in those two regions. Thisis certainly not true. But more abundant finan-cial resources have permitted a greater penetrationof computers in schools and universities in Europeand North America than elsewhere. For the situ-ation in 'the rest of the world', see the reference,"An International Perspective", by Jacobsen in theAnnotated References.

Unesco wishes to express its appreciation to theeditors, Professors Anthony Ralston and BernardCornu, to the authors for their contributions, toProfessor Ralston for preparing the final manuscriptand to Cambridge University Press for giving itspermission for UNESCO to include in this docu-ment some updated contributions from the originalpublication.

The views expressed in this report are those ofthe editors or the individual authors and not neces-sarily those of UNESCO.

We welcome comments on the contents of thisdocument, which should be sent to: Mathemat-ics Education Programme Specialist (Science andEnvironmental Education Section) or Mathemat-ics and Computing Programme Specialist (BasicSciences Division), UNESCO, Place de Fontenoy,75700 Paris, France.

UNESCO, Paris

Editors' Foreword to Second Edition

In 1985 the International Commission on Math-ematical Instruction (ICMI) chose The Influence ofComputers and Informatics on Mathematics and ItsTeaching as the topic of the first of a series of stud-ies on topics of current interest within mathematicseducation. ICMI could not have chosen a more aptand important topic. In the seven years since thepublication of the first edition of this book, the im-portance of calculators and computers has grownrapidly and is now the single most impc-Aant factorin creating change in all aspects of mathematics ed-ucation. Thus, ICMI with the cooperation of UN-ESCO asked us to edit an updated version of theoriginal book which would retain the strengths ofthat volume but would also bring the topics in it upto date and, as well, incorporate topics which werenot adequately discussed in the first edition but arenow of major importance (e.g. symbolic mathemat-ical systems, algorithms and algorithmics).

The conference at Strasbourg in 1985 whose pro-ceedings were incorporated in the first edition wasorganized by a Program Committee consisting of R.F. Churchhouse (Cardiff), B. Cornu (Grenoble), A.P. Ershov (Novosibirsk), A. G. Howson (Southamp-ton), J.-P. Kahane (Orsay), J. II. van Lint (Eind-hoven), F. Pluvinage (Strasbourg), A. Ralston (Buf-falo) and M. Yamaguti (Kyoto). The proceedingswere edited by A. G. Howson and J.-P. Kahane andwere published by Cambridge University Press.

iii

For this edition the report of the Strasbourgmeeting itself has been brought up to date by theleaders of the three workshops held at that meetingand five of the articles in the first edition have beenupdated for this edition. In addition, the editorshave solicited four new articles written just for thisedition. The result, we hope, is a volume which willbe as well received as was the first edition and whichwill be useful to mathematics educators throughoutthe world.

Of course, the nature of computer and calcula-tor technology is that it changes so rapidly - as doits implications for mathematics education - thata third edition will no doubt be needed in severalyears. But the current volume gives a fair pictureof th, impact of computers and calculators on math-ematics education in 1992. It should, therefore, pro-vide a valuable resource for mathematics educatorswho wish to learn about this impact or who wish toincorporate the technology into their mathematicsteaching or their teaching of prospective mathemat-ics teachers.

The editors wish to thank UNESCO for its sup-port for this project, particularly Angelo Marzolloand Edward Jacobsen.

This book was produced using TeX at the StateUniversity of New York at Buffalo.

Bernard CornuAnthony Ralston

June 1992

CONTENTS

H. Burkhardt and R. Fraser: Overview 1

An Update of the 1985 Strasbourg conference: 11

R. F. Churchhouse: The Effect of Computers on Mathematics 12

A. Ralston: The Impact of Computers and Computer Science onthe Mathematics Curriculum 19

B. Cornu: Computers as an Aid to Teaching and LearningMathematics 25

L. A. Steen: Living with a New Mathematical Species 33

S. B. Maurer: What are Algorithms? What Is Algorithmics? 39

J. Stern: On the Mathematical Basis of Computer Science 51

K.-D. Graf, R. Fraser, L. Klirigen, J. Stewart and B. Winkelmann:The Effect of Computers on the School Mathematics Curriculum 57

S. B. Seidman and M. D. Rice: A Fundamental Course in Higher MathematicsIncorporating Discrete and Continuous Themes 80

B. Cornu: Teacher Education and Training 87

B. R. Hodgson and E. R. Muller: The Impact of Symbolic MathematicalSystems on Mathematics Education 93

M. Mascarello and B. Winkelmann: Calculus Teaching and the Computer.On the Interplay of Discrete Numerical Methods and Calculusin the Education of Users of Mathematics 108

D. Tall and B. West: Graphic Insight into Mathematical Concepts 117

Annotated References 124

Index

V

131

AN OVERVIEW

Hugh BurkhardtUniversity of Nottingham, U.K.

Rosemary FraserUniversity of Nottingham, U.K.

The challenge

Where are we going? Where do we want to go?Why? How do we know? How may we find outmore? How do we get it to happen?

As far as the influence of computers and infor-

matics on mathematics and the mathematics cur-riculum is concerned, these are the central questionsthat this volume, like its predecessor, will address.

We shall also be concerned with progress in theseven years since the original Strasbourg meeting.Which aspects have moved quickly and substantiallytowards reasonably firm conclusions? On which ar-

eas is the situation now little different from then?What needs to be done about it?

A mismatch of timescales is one of the centralchallenges of this field which does not normally oc-

cur in the processes of change, either within mathe-matics itself or in the development of new curricula.

The pace of change in the technology is much fasterthan has -,Pr been achieved for school curricula;typical timescales for significant changes to occurare roughly as follows:

computer technology a few yearsmathematics research 10 20 years

school curricula 5 20 years

Thus we should be aware that when we design newcurricula to use the power of new technology, weshall continually be behind the times. This movingtarget problem is well recognised but needs to beaddressed at a strategic level in planning change. Ifthe new curriculum elements are to be robust andwidely useful, the curriculum designer cannot as-sume a specific level of technological provision andsophistication in schools both will vary widelyfrom time to time and from place to place.

This is important. If each student has a 'micro',curriculum possibilities open up which are not therewith one micro per class; even these possibilities de-

pend on the sophistication of the micro one line

of display, a few lines, many lines, graphics, access

to data each step is significant. Equally, it is al-

ready clear that even quite low levels of computerprovision and sophistication still have enormous ed-

ucational potential. Is 'technical restraint' a virtue,

or does it impede progress?

1

The overall pictureIt may be useful to begin with an overview of

the present situation in three separate domains of

activity:A Doing Mathematics - this is the domain of math-

ematical activity; in every sphere, from everyday

uses to research, it has been revolutionised bytechnology.

B Understanding of the Learning and Teaching ofMathematics this domain is concerned withthe processes of learning and teaching con-cepts, skills and strategies in mathematics andits applications; it is clear that technology hasprofound implications here, both through thechanges in doing mathematics and as a potentialaid to learning and teaching, but these phenom-

ena are not yet well-understood.C Mathematics Curricula and Teacher Training

both the first two domains have implications forcurricula, including both materials and teachersupport; the development of new curricula thatreflect the changed learning objectives and usetechnology effectively in their realisation is a ma-

jor task.The pattern of change so far is summarised in

Table 1 on the next page.We shall now discuss each of the three domains

in more detail.

Changes in 'doing mathematics'

In Domain A we now have a situation in whichthe changes in the way mathematics is done, at ev-

ery level from the shopkeeper to the research mathe-matician and engineer, are moving purposefully for-

ward with the advances in the technology, and withthe methods for its utilisation that informatics helpsto develop. Obviously there is some time lag but,at least in comparison with the exploitation withinmathematical education, there is no serious imped-

iment to change. The reasons are fairly clearthose involved have a clear incentive to use the

new methods, which give them more power withless painand at relatively modest cost that is more than

made up for in increased effectiveness.

2 Influence of Computers and Informatics on Mathematics and Its Teaching

A Doing B Learning C SystemMathematics and Teaching Change

Where togo? Why?

lots of ideas+ well triedexperience

How dowe know?

lots of ideas -l-a little exper-ience growingin patches

many examples many effect-work well ive examples

How do wefind outmore?

it will happenbecause par-ticipants haveincentives

How do weget it tohappen?

it. will happenbecause par-ticipants haveincentives

systematicsmall-scalestudies inrealistic cir-cumstances

performancemeasures

more supportfor system-atic r d

Table 1

very littlesign; even cal-culators notintegrated

observation

enrichmentpackages

whole ex-perimentalcurricula

study dynam-ics of change

model ex-perimentsin realis-tic circum-stances

This progress is set out in some detail in thepresent volume, not only in the introductory chap-ters, but in the contributions of Steen and Stern,and to a considerable extent in the other articles.We shall therefore review it briefly and in generalterms the other chapters bring these generalitiesto life.

The changes in mathematics pervade the sub-ject. The methods of first recourse in many areas

are now numerical and, particularly, graphical, al-lowing an experimental approach with much less ef-fort than in the past. Within mathematics mostareas have been affected: discrete mathematics andcombinatorics, number theory, algebraic and differ-ential equations, finite groups, fractals and chaos, aswell as all aspects of data analysis - these have allbeen profoundly changed; the papers in this volumedescribe some examples. The role of algorithmicsis now central (see the chapter by Maurer). Butequally clear is the effect on other aspects of puremathematics - even the definitions of elegance andthe status of proof.

Realistic applications to practical situationshave suddenly become more accessible as thedrudgery associated with realistic numbers andmore realistic models is cut away. No longer is thefocus on extracting the maximum from the few ana-lytic models that are tractable by traditional meth-ods.

Indeed, the role of such models in the future isan important issue for both mathematics and thecurriculum. It seems likely they will continue to becentrally important, not as methods of solving prob-lems, but as vivid illustrations of important effects.A closed algebraic expression displays, for those whocan read it, the dependence on all the variables inthe model something which, if there are many vari-ables, can be very difficult to communicate graph-ically or numerically. (For example, the expressionfor the response of a damped harmonic oscillator toa sinusoidal driving force depends on five variables

understanding the phenomenon of resonance fromnumerical solutions alone is not easy).

Several of the chapters that follow are focussedon the mathematical issues. Churchhouse's reviewranges over the various fields and aspects of mathe-matical activity, looking at the effects of technologyon the way mathematics is being done. Stern ismainly concerned with the impact of computer sci-ence rather than technology on the way mathemat-ics is done while Steen introduces the perspectiveof the computer as a 'new mathematical species'.Maurer addresses what is, perhaps, the central areaof change the dominant place of algorithmic think-ing and its implications.

Finally, a word of warning on the student asmathematician. Largely because of the imitativenature of the current curriculum, it is easy to geta quite false picture of students' capabilities. A ma-ture mathematician has command of a range of con-cepts and techniques (or knows where and how toget such command) and uses them autonomously toexpress and manipulate ideas and relationships to

get answers and understanding. There is clear evi-dence that, on such criteria, students' autonomousperformance is several years at least behind theirperformance on imitative exercises. The calculatoris a useful resource because teenage students canalready use arithmetic for a range of purposes; incontrast it has been shown, for example, that evenvery bright 17 year-old students may not use algebraat all as an autonomous mode of expression, thoughthey have had 5 years of success in manipulatingit (Treilibs et al, 1981); so, for example, the bene-fits of a machine that will manipulate in a languagethey do not speak fluently are elusive, and maybeillusory.

The overall effect of these changes is well-summarised by Mascarello and Winkelmann, in away that clarifies the challenge to designers of cur-riculum:

"In total, there can be observed a specificshift in the spectrum of abilities, from pre-cise algorithmic abilities to more complex in-terpretations, so to speak from calculationto meaning, which in a certain sense is a re-versal of the historical evolution. In this pro-cess the mathematics to be mastered tends tobecome intellectually more challenging, huttechnically simpler."

Changes in mathematics education

As to the other Domains, B and C, nearly all thechapters that follow make suggestions for new cur-riculum elements based on these new methods of do-ing mathematics; readers will find many of these ar-guments stimulating, and even persuasive. Changesarc surely needed and these suggestions seem bettergrounded than most.

Nonetheless, it must be recognised that such sug-gestions are fundamentally speculative at the level oflarge-scale implementation by which we mean thatconverting them into a well-developed and testedcurriculum for the typical teacher and the typicalstudent is still a major challenge. This is the taskof Domains B and C. We can have no reliable ideahow far any suggestions we put forward will provefeasible in any, let, alone every, educational system.Even if they are implemented reasonably faithfully,the full curriculum reality of what occurs will con-tain many surprising side effects; more likely, thetranslation from an idea to a small scale pilot exper-iment with exceptional teachers and facilities, andthen to large scale reality will involve critical dis-tortions of the aims of the exercise which may even,in the end, call into question its curriculum value.

Overview 3

Thus rigour and vigilance are neen,:c1 in this devel-opment process.

In case there are any who believe that we exag-gerate the dangers, let me draw attention to a fewfamous examples of intended innovations in mathe-matical education which turned into something en-tirely different:

The splendid Bourbaki enterprise waslaunched (believe it or not (Weil, 1979)) toestablish a firmer foundation for mathemat-ical education; few now see that as amongthe positive contributions it has made, whilemany are concerned at the effects of overem-phasis on formalism that has arisen from thisapproach in school mathematical education.

Smalltalk was originally devised by the Xe-rox Learning Research Group largely to pro-duce a medium, the Dynabook, that wouldbe 'as natural to a child as pencil and pa-per' (Goldberg, 1978); what has emerged isperhaps the most sophisticated graphics ori-entated data management system so faran important achievement, but a very dif-ferent thing. (The Learning Research Groupwas renamed the Software Concepts Group.)Smalltalk has not, at any rate, done anyharm to the school curriculum, and amongits offspring, the Macintosh microcomputer,may yet contribute notably in a quite differ-ent way.

Our final example must be the reform move-ment in mathematical education of 30 yearsago 'new math', 'modern mathematics' andso on. Comparison of the initial aims agreedat conferences, the pilot schemes in a few ex-ceptional schools, and the classroom realityof today shows the contrasts vividly. For ex-ample, in England the applications of math-ematics occupied a central place in the origi-nal design; in most of the major courses thatemerged applications were mentioned onlyto illustrate techniques with no serious at-tention to the practical situations involved.Equally, new mathematical concepts were in-troduced but often with none of the payoffthat motivated their inclusion because theserious examples originally envisaged provedtoo difficult for most students, and were re-placed with trivial ones. The second wave ofreform over the last decade has been rathermore successful in remedying some of thesedefects, but has left others untouched.

These are cautionary examples to bear in mind

A_ 0


when looking at new possibilities; it is not easy to re-alise the potential vitality of a field of mathematicsin large-scale curriculum implementation. (One caneasily imagine a vivid 'commercial' for the impor-tance and excitement of a 'new' field like calculus;compare it with the reality of a typical introductoryAmerican college text.) That does not mean thatvivid, effective implementation cannot be achieved

but success requires high-quality 'engineering'and some poetry as well as the facts. (We have ar-gued that, if the English language curriculum werelike most mathematics curricula, the readings wouldbe drawn entirely from the telephone directory.)

On the learning and teaching of mathematics

Progress in Domain B continues at a steady pacebut, we would suggest, far too slowly to provide asound comprehensive underpinning for the new cur-ricula that we need now. Some of this progress isdescribed in later chapters in this volume. Thesedevelopments and the associated research representdeeper insights into the way technology can affectand enhance learning and teaching, together withsome elements of curriculum that can be and havebeen used successfully in classrooms. They haverarely been tested on a large scale and thus repre-sent only firm steps along the road towards the newcurricula (Domain C). Let us begin by looking atsome general effects in technology-related change.

In looking at curriculum reform, the first thingto note is the scale of it perhaps 80% of currentschool classroom time is devoted to seeking fluencyin a range of pencil-and-paper technical skills, all ofwhich are now best done on computers of one kindor another. This we call The Big Hole.

Secondly, the swing towards teaching mathemat-ics that is "intellectually more challenging, but tech-nically simpler" takes both teachers and curriculumdesigners into areas outside the basis of their ex-perience thus such curriculum design should beessentially a research-based exercise, if it is to workwell. It relates not only to content but to learningand teaching style. Everywhere the curriculum isstill based on student imitation (e.g. HMI, 1977),dominated by:

teacher explanation +illustrative examples +

imitative exercises.This can lead to rapid apparent student progress,but much research evidence (see Bell et al, 1983)shows that the skills acquired are not reliably re-tained by most students, nor are they transferable

particularly to non-routine problems in the world

outside the classroom. Fig. 1, for example, showspre-, post- and delayed-test results of individual stu-dents from comparable groups taught by two differ-ent methods. The first 'positive only' method (a) istraditional explanation and reinforcement by prac-tice; the second 'conflict' method (b) is based onstudents' discussion and 'debugging' of errors gen-erated by them from their own misconceptions. Thegreatly improved long-term learning is stable acrossdifferent topics (Bell and Basford 1989).

Score

PRE

Score

POST

(a)

DEL

'1PRE POET DEL

tb)

Figure 1

To achieve the flexible competence of under-standing that the world requires, the pattern ofclassroom activities has to be widened to includesome which give more autonomy, more initiative tothe students. It is encouraging that the microcom-puter shown great promise in supporting suchactivities.

Thirdly, change is often threatening. Technologyappeals to reduce this threat, partly because it pro-duces an obviously new situation and thus cannotimply criticism of the teachers' existing modes of op-eration. This more than compensates for the extrabarrier of learning to use the equipment providedit is reliable. Further, recent work on the teachingof non-routine problem solving of a wide variety ofkinds shows the importance of the strategic skills ofcomprehension, modelling, interpretation and eval-uation just the skills that are brought to the frontof our attention by the computer.

In summary, the two great springs for change inmathematical education in the past and the nextdecade are technology and autonomy. Fortunately,they can help each other, though there is much tobe learnt as to how best it might be done.

However, conversely, we believe that it may heimportant not to discount too easily the value of tra-ditional skills, remembering that the current genera-tion of innovators have the 'tree anal' background,as well as newly acquired skills with computers. Al-most certainly, much of what was learnt is uselessbut we need to check for losses as well as gains ina curriculum change. Mental facility with numbers,graphs and expressions has always been an asset.What is its status now?

Exploratory investigation as a key element inthe curriculum has been a major objective in En-glish mathematical education for at least 30 years

the Association of Teachers of Mathematics wasfounded largely to promote it; in the USA, we knowthat it hal: been a focus since Polya (1945) and offi-cially central at least for a decade, since the NCTMYearbook on problem solving. However, despitestrenuous efforts it has not become a regular partof the curriculum anywhere except in a tiny minor-ity (less that 1 percent) of classrooms. We have a lotof evidence and some understanding of how difficultsuch activities are for the typical teacher to handlein the classroom; appropriate support must be de-veloped. Everyone rightly emphasises the curricu-lum opportunities for exploration, for 'experimentalmathematics', that the computer provides; however,the development of such an investigative element inthe curriculum will succeed only if it confronts thedifficulties such activities present for teachers.

Overview 5

Equally, the challenge to explore must be at alevel matched to the student - if the aim is to 'dis-cover' in an hour or so some important mathemat-ical achievement that took a genius half-a-lifetimeto create, the exploration will have to be so closelyguided as to be essentially fake; on the other hand,interesting, though less global, problems which stu-dents can tackle autonomously on their own re-sources, do exist at every level. For example, pro-gramming projects, at school and university, haveshown some of the possibilities, and the difficultiesfor the teacher. A creative and systematic programof detailed empirical development will be essential ifexploration is not to degenerate in most classroomsinto that closely guided 'discovery learning', whichis really an alternative style of explanation. Thecomputer can, of course, help.

ILLUSTRATIONS OFAPPLICATIONS

MATHEMATICALAND OTHER SKILLS

Figure 2

The emphasis on problem solving encourages ap-plications of mathematics even some with realdata. It is important to note two different kindsof application, the illustrative and the situational(see Fig. 2, from Swan 1990). In illustrative appli-cations the focus is really on the particular mathe-matical topic; th- applications are there in supportto help conceptual understanding through concreteillustration, to show how mathematics can be ap-plied, and to provide practice. In realistic, practi-cal situations from outside mathematics the posi-tion is quite different in principle, any or all of the

6 Influence of Computers and Informatics on

mathematics you know could help you to tackle theproblem, along with other knowledge and strategicskills. Both these kinds of application are needed

but the student must know which 'game' is beingplayed, because the best tactics are quite different.For illustrative applications the aim is to show howmuch mathematics you know; for situations, it isto provide the most powerful understanding of thepractical problem.

Finally, assessment in arriving at a curriculum,assessment can be very helpful in clarifying cur-riculum definitions, particularly by example. Whatrange of types of tasks do we want our students tobe able to do? It can be argued persuasively that atask-defined curriculum has great advantages over a`scope and sequence' approach, though the two arecomplementary one being synthetic and the otheran-iytic.

The chapters that follow range widely over thisfield, complementing analysis with the essentialvivid exemplification which we have had to excludefrom this overview. Ralston's review takes a lookwithout prior assumptions ('zero-based') at whatthe curriculum should contain. The criteria includevalue to the student and, related, the way mathe-matics is done nowadays together with lessons fromthe psychology of learning. Corm's review covers awide range of roles for the computer in enhancingteaching and learning Computer-assisted Learn-ing is as diverse as Paper-assisted Learning. Theremaining chapters that follow present a kaleido-scope of key aspects of computer use in learning andteaching. Tall and West explore and illustrate thevisual aspects of learning and the contributions thatcomputer graphics can make. Hodgson and Mullerlook at the other major shift enabled by the tech-nology automated symbolic manipulation. Graf,Fraser, Klingen, Stewart and Winkelmann take abroader look at the various modes of use of tech-nology in school and the potential of each in theelementary school; both computers and calculatorsare usiscussed. Steen takes a similar approach tocollege mathematics where the issues range fromthe place of discrete mathematics to computer lit-eracy for all students. Seidman and Rice, and Mas-carello and Winkelmann look in a most stimulatingand practical way at a related central problem theintegration of discrete and continuous mathematicswithin a college course.

All of these show what has been achieved in Do-main B in realising, on a pilot scale, somethingof the enormous potential of computers and infor-matics in enhancing the learning and teaching ofmathematics. They also set targets for future de-

Mathematics and Its Teaching

velopment.

New mathematics curricula

The lack of progress in Domain C is the majormismatch between intentions and outcomes over thelast seven years. It is notable that even the use ofsimple calculator .-tas not been fully integrated intothe curriculum in any country in a way that realisestheir known potential for enhancing mathematicalperformance (even on traditional skills!).

The reasons are less clear than is sometimesthought by those who ascribe it simply to teacher in-ertia and/or parental opposition. Certainly, teach-ers and other educators do not have the direct in-centives that the use of the technology providesfor those doing mathematics. It does not so obvi-ously promise to increase their power as profession-als, or to make their lives easier or more reward-ing. Rather, it makes obsolete a large part of thestandard professional work of mathematics teachersand threatens them with, at least, a need for newskills, both mathematical and pedagogical. Equallyparents and others tend to believe that their owneducation remains valid after all, look what it hasdone for them!

However, when one compares the support offeredto teachers to make these changes with that whichthey routinely receive simply to sustain the currentcurriculum, the contrast is stark. The textbooksand other materials are not comparable, or ofteneven available. Retraining is sparse, as is coher-ent explanation of what is being attempted, andwhy. The temptation to blame the lack of change onteachers is not only misguided but fruitless theyare who they are. It is up to those who seek changeto find, and to deliver, an effective and appropriatemixture of pressure and support.

We have evidence that teachers actually welcomechange, provided they are confident that the paceand level of support is such that they can cope withit without undue effort. As with many profoundcurriculum changes, systems have so far failed toprovide any basis for such confidence.

In their chapter Cornu and Balacheff look atthe problems of the new pedagogy and how it maybe communicated to teachers in training, a keyarea in Domain C. We already have evidence (seeBurkhardt, 1984, 1985) that the potential of the mi-crocomputer for helping teachers to enhance studentlearning presents a tremendous opportunity for cur-riculum enhancement. The effects on the dynamicsof the classroom can be profound, but they are of-ten subtle; for this reason there is a great deal stillto do before we have even a broad understanding of

(-

what can happen in the various modes of computeruse in education.

We shall illustrate the sort of thing that maybe expected by describing one application that hasbeen developed and studied in some detail, andwhich has proved particularly rich the use bythe teacher of a single micro in the classroom, pro-grammed to be a 'teaching assistant'. We do so forvarious reasons: It is less familiar to most people;it brings out some general points about the over-whelming importance of the people, teacher andpupils, and of the dynamics of their interaction;and it is particularly relevant to schools as we knowthem because it seeks to enhance the performanceof a teacher working with a group of children in theclassroom in the normal way. It also only requiresone microcomputer per class rather than one perchild.

This mode of use, set out by one of us (Fraser,1981), has been shown to have remarkable effects inleading typical teachers in a quite unforced and nat-ural way to broaden their teaching style to includethe 'open' elements that are essential for teachingproblem solving (Fraser et al, 1983, 1988). Sincethis is a crucial aim that reformers have been tryingto achieve for at least thirty years with little or noeffect, this is a valuable result. It is worth explainingbriefly why these effects come about. First, the mi-cro is viewed by the students as an independent 'per-sonality'. It takes over for a time a substantial partof the teacher's normal 'load' of explaining, manag-ing, and task setting. These are key roles played byevery mathematics teacher. The micro takes themover in such a way that the teacher is led into lessdirective roles, including crucial discussion with thechildren on how they are tackling the problem, pro-viding guidance only of a general strategic kindcounselling if you like.

These principles have been incorporated intoa range of teaching materials that enable typicalteachers to sustain in their classrooms, without ex-ceptional effort, these learning activities of a moreopen kind. This book is full of other examples ofcurriculum components that have been shown towork well in typical circumstances, or can be devel-oped to do so. It is also important to recognise thatthere will be disappointments or at least frustra-tions in the development process. In the last fewyears there has been further progress to report inthe thorough, and imaginative, development of sub-stantial curriculum elements. They illustrate whatcan be done. The Journeys in Mathematics project(EDC, 1991), funded by the National Science Foun-dation, has developed a series of modular units that

Overview 7

exploit the potential of computer support for the el-ementary school mathematics classroom in a varietyof powerful ways. Similarly, The Power Series (UC-SMP/Shell Centre, 1992) offers an effective elementin teacher development, through the support thatthe 'single micro classroom' can provide in explor-ing new, more open ways of working. There are nowmany other 'enrichment materials' using the com-puter to support learning, particularly those less-routine activities that many teachers find difficultto handle.

Full technology-integrated curricula, with mate-rials to support them, are hardly available yet. Ifthere is an exception, it is a few new courses inhigher education, such as that described by Hodg-son and Muller. There are early signs of movesto develop materials to support complete curri' ia;some of the latest round of NSF-supported projects,for example Seeing and Thinking Mathematically atEDC and a parallel project at TERC, have a strongemphasis on technology.

It is interesting that all three examples quotedabove (and many others) use the computer as a 'cat-alyst for learning' (Fraser, 1989) rather than as a`tool' for doing mathematics or a 'tutorial system'.

No one doubts that the computer as a tool is acentral element in the curriculum but the develop-ment of curricula to realise this is slow. Of course,the level of computer provision needed to make itmore than a passing experience is still beyond mostschools. It is interesting and ironic to remember thepioneering work of the Computer Assisted Mathe-matics Program (Johnson et al, 1966-68), in whichstudents learned mathematics in a Basic program-ming environment; Kieren (1974) showed that farmore students got through to fluency in algebra inthis way a result that has been confirmed but notyet implemented anywhere. As we have noted, eventhe simple calculator is far from fully integrated intocurricula.

Computer-based tutorial systems have contin-ued to emerge, and to become more sophisticated,sometimes embodying elements of artificial intelli-gence of an 'expert system' kind. Apart from pro-gramming itself, perhaps the first big idea for usingcomputers in mathematical education was in teach-ing technical skills, particularly arithmetic. The ap-proach followed the behaviourist teaching-machinemodel. To provide effective teaching in this way hasproved a much harder problem than was expected.We believe that it is still far from solution. It seemsthat the computer-tutor can be effective in teachingfacts and straightforward techniques to people whohave little difficulty with them; so, of course, are


other methods. However, despite great efforts bysome talented people, it has not so far proved possi-ble to write programs which are successful in diag-nosing and remediating students' conceptual errorsunderlying technical skills that they find difficult. Itremains true that tutorial systems have not begunto tackle the main defects of the traditional (andlargely current) mathematics curriculum, still con-centrating on automating those learning activities(largely drill-based) that are both over-representedand, on their own, ineffective.

The next steps

It seems from the above that. Domains A andB are making steady progress and that Domain Cpresents the greatest difficulties thus it seems thatfurther work on large-scale implementation shouldbecome a priority over the next decade. There aresome small signs of movement in this direction invarious countries. However, the difficulty of achiev-ing large scale change of any kind is often under-rated, or at least neglected. It clearly needs empiri-cal study of the dynamics of change in the educationsystem as a whole, with all the factors this bringsin. We already know far more about the benefitsthat could flow from the use of technology (evenwithin current financial constraints) than is realisedin practice. Without attention to Domain C, thismismatch will simply get worse.

What, specifically, are we to do about this?This is not the place for a serious discussion ofmethodologies of research and curriculum develop-ment (Burkhardt, Fraser and Ridgway, 1990). Verybriefly, there is no proven successful answer butsome seem to be less susceptible to corruption ofoutcomes than others. We believe that the essenceis an empirical approach find out what actuallyhappens to your draft ideas in practice, in circum-stances sufficiently representative of what you areaiming for, and then revise the materials repeat-edly until they work in the way intended. We havefound (e.g. Shell Centre, 1984) that such an ap-proach, taken for granted in other fields, can com-bine educational ambition with user-friendliness to alevel not achievable with more casual development.Structured classroom observation makes a key con-tribution to this approach, providing much richerfeedback than is often acquired in the developmentof educational materials (Burkhardt, Fraser et al,1982). However, more rigorous comparative evalu-ation of alternative approaches is sorely needed. Afew more comments are made below.

It is important to ask of everyone in the sys-tem, but particularly teachers, "Why should they

change?". It seems (Fullan, 1980) that both pres-sure and support are needed for effective change butproducing a balanced 'well-engineered' package thatworks is still an unsolved problem. One lever forchange that cannot be ignored is the assessment sys-tem; if a system does not recognise, measure and re-ward new curriculum elements then they will not betaken seriously by many WYTIWYG (what youtest is what you get). Pressure is often preferred bypoliticians because it IPss expensive than support,so that an effective balance is destroyed.

The questions we have raised imply a great dealof work, integrating research techniques with cur-riculum development, before we have even a basicunderstanding of the classroom potential that wesee so vividly illustrated in this book. Experiencesuggests that, along the way, we shall find otherpossibilities of at least as much promise.

In order to realise any of these possibilities, theywill need to be systematically developed in detailwith representative samples of teachers and stu-dents, using structured detailed data from the class-room.

Systematic research and development

The slow progress in B and, particularly, Cpartly reflect a general problem in educatithi thatthe level of expenditure on designing and developingsoundly-based changes is remarkably low; in Eng-land and the US, for example, this 'research anddevelopment ratio' is substantially less than 0.1% ofeducational expenditure, whereas in other changingfields such as medicine or modern industry it is typi-cally between 5 and 15%. We believe that this arisesbecause education is still dominated by the 'craft-based' approach, which assumes that experiencedprofessionals have satisfactory methods of handlingeach situation that presents itself that everythingis basically well under control.

This craft-based approach works well when twoconditions are satisfied:

the system is working satisfactorilyand

there is no expectation of major change

Otherwise it involves the extrapolation of reliableexperience beyond its domain of validity always ahazardous process.

In this respect the situation in education israther similar to that in medicine a century ago,or engineering further in the past. There are signsthat the more systematic 'research-based' approach(which now dominates the other two fields) is more

widely recognised as important in education, butthere is still a long way to go. No one would dreamof using a drug, or flying in an aeroplane, that hadonly been developed and tested as sketchily as mostnew (or, indeed, old) curricula. Nonetheless, the sit-uation persists perhaps because educational disas-ters are much less immediately visible.

Some dismiss such arguments on the groundsthat teaching and learning are much more variedand less controllable than engineering or medicalsituations. Though this is true in some respects,systematic observation of typical mathematics class-rooms (HMI, 1979) shows a remarkable uniformityin delivering a curriculum that is inappropriate andserious/ impoverished by current standards. Simi-lar res' are observed in most countries.

The research and development ratio illustrateshow little serious attempt has been made in educa-tion to work systematically to do better. The pro-cesses of systematic development are either absentor sketchy, particularly in the quality of feedbackand the typicality of the development environment.The search for better development methods is al-most non-existent. We believe that we are still es-sentially leaping off cliffs, flapping 'wings' tied toour shoulders but nobody notices the mess.

These things need not be. So far from costingmoney, they represent a potential source of largesaving in the overall operation of the education sys-tem.

These considerations are not, of course, confinedto technology-inspired change. They are, however,particularly acute in that circumstance because ofthe pace of change needed. We are facing a situationin which children are taught to do mathematics inways that are very largely outmoded, with at least80% of curriculum time wasted on trying, more orless successfully, to develop fluency in skills of now-limited value.

In planning a systematic approach, we think it isuseful to distinguish different levels of research anddevelopment in education. Studies at each succes-sive level involve an order of magnitude more stu-dents, and teachers, than at the previous one. Fourlevels are:

L Learning studies of student's learning, thenature of cognitive processes, difficulties andmisconceptions (10° 101 children minimum)

T1 Teaching Possibilities studies of differentkinds of stimuli and their effects on studentlearning (101 - 102 children minimum)

'I'2 Realizable teaching studies on what can ac-tually be achieved with typical teachers underrealistic circumstances (102 - 103 children mini-

Overview 9

mum)C Curriculum change on a large scale studies of

how curriculum change can be effected and whatother school or social factors affect it? (104 - 107children minimum)All these levels are important, the earlier ones

contribute to a fundamental understanding of thelater ones. However, much more serious work hasbeen done at the early L and T1 levels; a morebalanced effort would be productive. The crucialdistinction between T1 and T2 is often not made.At the T1 level the teacher' variables are almost ir-relevant work there simply shows that there is ateacher, usually a member of the development team,who can make these things happen. At the T2 level,curriculum developers face the challenge of show-ing that a wide range of unexceptional teachers, innormal circumstances of support, can also functionin the desired ways. Similarly, the C level bringsin all the variables that relate to the pressures onthe classroom from school and society, which are socritical to the implementation of any change. Thesedistinctions appear to be important in the limitedimpact of technology so far and in doing better inthe future.

Acknowledgements

We have benefited from many conversations without friends in the ITMA Collaboration, particularlyRichard Phillips, Jan Stewart, Jim Ridgway and JonCoupland, and others around th.> world, particularlyDavid Tall and Tony Ralston.

REFERENCES

Bell A.W., Costello J. and Kuchemann D. [1983]:A Review of Research on Mathematical Educa-tion, Part A: Research on Learning and Teach-ing, Windsor: NFER- Nelson.

Bell A.W. and Basford D. [1989]: A conflict and in-vestigation teaching method and an individu-alised learning scheme a comparative exper-iment on the teaching of fractions, Notting-ham: Shell Centre for Mathematical Educa-tion; also Teaching for the Test, Times Educa-tional Supplement, 27 Oct.

Burkhardt II., Fraser R. et al [1982]: Design and De-velopment of Programs as Teaching Material,London: Council for Educational Technology.

Burkhardt H. [1984]: How can micros helpin schools?: research evidence, Nottingham:Shell Centre for Mathematical Education.


Burkhardt H. [1985]: The Microcomputer: Miracleor Menace in Mathematical Education in Pro-ceedings of ICM1.7 (M. Carss, Ed.) Berlin:Birkhauser.

Burkhardt H., Fraser R. and Ridgway J. [1990]:The Dynamics of Curriculum Change in De-velopments in School Mathematics Around theWorld, Vol 2 (I. Wirszup and R Streit, Eds.),Reston, VA: National Council of Teachers ofMathematics.

EDC [1991]: Journeys in Mathematics, Scotts Val-ley, CA: Wings for Learning.

Fraser R. [1981]: How to use Homo Sapiens in aComputer Environment in Proceedings of the1980 ADV Conference, Vienna: ADV; alsoDesign and evaluation of educational softwarefor group presentation in Microcomputers inSecondary Education (J. Howe and P. Ross,Eds.), London: Kogan Page.

Fraser R., Burkhardt H., Coup land J., Phillips R.,Pimm D. and Ridgway J. [1983, 1988]: Learn-ing Activities and Classroom Roles, Notting-ham: Shell Centre for Mathematical Educa-tion and J. Math. Behaviour, 6, 305-338.

Fraser R. [1989]: Introduction to Computers and theTeaching of Mathematics (E. Dubinsky and R.Fraser, Eds.), Nottingham: Shell Centre forMathematical Education.

Fullan M. [1980]: The Meaning of EducationalChange, New York: Teachers College Press.

Goldberg A. [1978]: Trends in Hardware and Soft-ware in Informatics and Mathematics in Sec-ondary Schools: Impacts and Relationships,1977 IFIP Conference (D.C. Johnson and J.D.Tinsley, Eds.), Amsterdam: North Holland.

HMI [1979]: Aspects of Secondary Education inEngland, Report of the HMI Secondary Sur-vey, London: HMSO.

ICMI [1984]: The Influence of Computers and In-formatics on Mathematics and its Teaching,L'Ense:gnement Mathmatigue, 30, 159-172.

Johnson D.C. with Hatfield, L., Walther, J., LaFrenz, D., Katzman, P. and Kieren, T. [1966-68]: Computer Assisted Mathematics Pro-gram, Glenfield, IL: Scott Foresman.

Kieren T [1978]: Informatics and the SecondarySchool Mathematics Curriculum in Informat-ics and Mathematics in Secondary Schools:Impacts and Relationships, 1977 IFIP Confer-ence (D.C. Johnson and J.D. Tinsley, Eds.),Amsterdam: North Holland.

Shell Centre [1984]: Problems with Patterns andNumbers, a Module of the Testing StrategicSkills Programme, Nottingham: Shell Centrefor Mathematical Education.

Swan M. [1990]: Mathematical Modelling for AllAbilities in Proceedings of ICTMA4, Chi-chester, U.K.: Ellis Horwood.

Treilibs V., Burkhardt H. and Low B. [1981]:Formulation Processes in Mathematical Mod-elling, Nottingham: Shell Centre for Mathe-matical Education.

UCSMP/Shell Centre [1992]: The Power Series (sixmodules for the single micro classroom in theelementary school), Chicago: University ofChicago School Mathematics Project and Not-tingham: Shell Centre for Mathematical Edu-cation.

Weil, A. [1979]: History of Mathematics in Pro-ceedings of the 1978 International Congress ofMathematicians, Helsinki: ICM.

1. '

-1 I

An Update of the 1985 Strasbourg Conference

The first edition of this book grew out of a conference in Strasbourg in March 1985. The attendees atthat conference divided themselves into three Working Groups on the subjects of The Effect of Comput-ers on Mathematics, The Impact of Computers and Computer Science on the MathematicsCurriculum and Computers As an Aid to Teaching and Learning Mathematics. The reports ofthese three working groups formed the first three chapters in the previous edition. In this edition the leadersof the three workshops have updated the reports which appeared in the previous edition. These updatedreports appear on the following pages.

11

1 0

Part I

THE EFFECT OF COMPUTERS ON MATHEMATICS

R. F. ChurchhouseUniversity of Wales, Cardiff, UK

1.0 Introduction

Mathematical concepts have always dependedon methods of calculation and methods of writing.Decimal numeration, the writing of symbols, theconstruction of tables of numerical values all pre-ceded modern ideas of real number and of function.Mathematicians calculated integrals, and made useof the integration sign, long before the emergence ofRiemann's or Lebesgue's concepts of the integral. Ina similar manner, one can expect the new methodsof calculation and of writing which computers andinformatics offer to permit the emergence of newmathematical concepts. But, already today, theyare pointing to the value of ideas and methods, oldor new, which do not command a place in contem-porary "traditional" mathematics. And they permitand invite us to take a new look at the most tradi-tional ideas.

Let us consider different ideas of a real number.There is a point on the line R, and this representa-tion can be effective for prompting the understand-ing of addition and multiplication. There is also anaccumulation point of fractions, for example, con-tinued fractions giving the best approximation of areal by rationals. There is also a non- terminatingdecimal expansion. There is also a number writtenin floating-point notation. Experience with even asimple pocket calculator can help validate the lastthree aspects. The algorithm of continued fractions- which is only that of Euclid - is again becoming astandard tool in many parts of mathematics. Corn-plicated operations (exponentiation, summation ofseries, iterations) will, with the computer's aid, be-come easy. Yet even these simplified operations willgive rise to new mathematical problems: for exam-ple, summing terms in two different orders (startingwith the largest or starting from the smallest) willnot always produce the same numerical result (see,e.g., Churchhouse, 1980, 1985).

Again, consider the notion of function. Teach-ing distinguishes between, on the one hand, elemen-tary and special functions - that is, those functionstabulated from the 17th to the 19th century and,on the other, the general concept of function intro-duced by Dirichlet in 1830. Even today, to "solve" adifferential equation is taken to mean reducing thesolution to integrals, and if possible to elementaryfunctions. However, what is involved in functional

12

equations is the effective calculation and the qual-itative study of solutions. The functions in whichone is interested therefore are calculable functionsand no longer only those which are tabulated. Thetheories of approximation and of the superpositionof functions - developed well before computers - arenow validated The field of elementary functionsis extended, through the discretisation of nonlinearproblems. Informatics, too, compels us to take anew look at the notion of a variable, and at the linkbetween symbol and value. This link is stronglyexploited in mathematics (for example, in the sym-bolism of the calculus). In informatics, the necessityof working out, of realizing the values has presentedthis problem in a new way. The symbolism of func-tions is not entirely transferable, and the attributesof a variable are different in languages such as For-tran, Lisp and Prolog.

In the sections that follow we look at some as-pects of how computers and informatics have al-ready affected mathematics and mathematical re-search and present some thoughts on what futureeffects might be seen. We do not claim that oursurvey is comprehensive, especially so in the disci-plines of applicable mathematics, but we hope thatit provides some pointers. In ary event informationtechnology, in the widest sense, is advancing far toofast for any predictions to be of value for a periodof more than a few years.

1.1 New and revived areas of mathematicalresearch

Computers not only provide a new tool in math-ematical research and teaching. They are, at thesame time, themselves the source of new areas ofresearch. Not all of the research stimulated by theavailability of computers is in new branches of math-ematics; some is of ancient lineage, going back to the19th or 18th century, but open now to attack witha weapon not available to Euler, Gauss, Jacobi, Ra-manujan, etc. Who can doubt, though, that thesegiants of the past would have exploited these newpossibilities with enthusiasm had they been avail-able? It is one of the unique features of mathe-matics that it is based upon a body of results thatnever loses its value. Fashions and interests maychange, but the neglected subject of the last cen-

tury, or even of the last millennium, may prove tobe of new interest at any time when conditions areright for its re-emergence. So the corpus expands;nothing ever dies, though it may remain dormantfor centuries. In the age of information technologywe need to emphasize this fact, for it underlies ev-erything that follows.

One of the most famous examples of mathe-matical research being stimulated by the use of acomputer is the soliton (solitary wave) solution ofthe Korteweg-de Vries equation by Zabusky andKruskal (1965), which was initially suggested by nu-merical results. Continuing experimental investiga-tions have indicated the existence of other, related,solutions and theoretical research has provided asubstantial framework for investigating soliton so-lutions of several nonlinear wave equations.

Another example is found in the work of Ya-maguti, which may be summarized briefly by say-ing that he observed continuous, but nowhere-differentiable, functions via numerical experimentson dynamical systems defined iteratively whose so-lutions exhibit very chaotic behaviour. Particularcases produce the Weierstrass function and the Tak-agi function (see also the chapter by Tall and West);the latter may be written

T(X) =

where

E 2ko(k)(x)k=1

f 2x 0 < x < 1/2(4x) 2(1 x) 1/2 < x < 1

and has recently been used in teaching elementaryanalysis. Further research, in collaboration withHata on a family of finite difference schemes led toLebesgue's Singular Function.

Among long-established branches of Pure Math-ematics where computers have had a major impactare Group Theory, Combinatorics and Number The-ory. Many applications of computers in these areashave been published in proceedings of conferences(for example, Churchhouse and Herz (1968), Atkinand Birch (1971), Leech (1970)).

The applications are already too numerous tolist in full or describe in detail but it is clear thatthe search for sporadic groups, the investigation ofBurnside's problem, the study of rational points onelliptic curves, and the search for large primes wouldbe quite impossible without computers. The fac-torisation of large integers is another example; al-though intrinsically it is not an exciting topic it hasrecently assumed considerable importance in rela-tion to cryptography and public-key systems (Beker

Effect of Computers on Mathematics 13

and Piper, 1982). Many of these applications havebenefited considerably from the availability of pro-gram packages specifically designed as an aid for re-searchers in the field; the CAYLEY system for thestudy of finite simple groups is a well-known exam-ple. Another is the development of Symbolic Math-ematical Systems (see Section 1.6 and the chapterby Hodgson and Muller). Such systems relieve re-search workers of a great deal of drudgery. Indeed,they make possible manipulations which just couldnot be done manually in any reasonable time or withany valid hope of an accurate result. Another "old"topic that has taken on a new lease of life is thatof continued fractions, both as providing approxi-mations to real numbers and, in analytical form, innumerical analysis.

The availability of colour graphics displays andpackages has opened up exciting possibilities for re-search not only in geometry, modelling and fluidflow but in less obvious areas such as analysis (seethe chapter by Tall and West). The study of the it-eration of complex-valued functions has been trans-formed recently; the complex nature of Julia setsand their descendants is made beautifully apparentby the use of colour graphics, even through much oftheir mathematical nature remains unknown (see,for example, Section 1.4 below).

It is clear to us that the computer is having, andwill continue to have, a significant impact on thedirections of mathematics research and on the wayin which mathematicians carry out their research.Computers will not only be commonly used to arriveat conjectures but also to assist in finding proofs.In addition, some important questions are raised:(i) How should computers be used to assist math-ematicians in communicating their discoveries andin keeping abreast of the research of others? and(ii) What are likely to be the intellectual and so-cial consequences, so far as mathematics and math-ematicians are concerned, of the widespread interestin, and use of computers?

1.2 Proof

In mathematics a "proof" is, strictly, a chain ofdeductions from the axioms; in practice, of course,a proof is accepted if it makes use of results whichhave themselves been deduced from the axioms, orfrom other results, etc., etc. It would be possible,but exceedingly tedious, to write out a proof of thetheorem that every positive integer is the sum ofthe squares of four integers by starting from the ax-ioms of arithmetic, but few people would regard thisas necessary and would accept various intermediatesteps - an identity of Jacobi, or representation of

r;rw 0


integers by binary quadratic forms - as valid rungson the ladder, since each of these steps is deduciblefrom other results which are deducible from theaxioms.

Computers might be used in mathematicalproofs; they might, ,nitially, suggest what is trueand, equally important, what is not, they might beused for computations which are required in a proof;they might be used - as in the proof of the 4-colourtheorem (Appel and Haken, 1976) - to examine all ofthe finite s' t of cases, on which the truth of the the-orem ulti tely depends; they might even be pro-grammed to find part of the proof by trying manypossible combinations of known axioms, theorems oridentities, though "combinatorial explosion" makessuch an approach infeasible except in very specialcases.

As examples, computers have been used to sug-gest results in group theory, combinatorics, numbertheory, coding theory and to support the truth ofconjectures such as the Riemann Hypothesis. For anearly Purvey article see Churchhouse (1973). Amongnotable theorems which were initially conjecturedon the basis of numerical evidence are the PrimeNumber Theorem (Gauss) and several important re-sults of Ramanujan (1927) including the congruenceproperties of the partition function and of the func-tion r(n). On the other hand Lander and Parkin(1967) and a computer found that

275 + 845 + 1105 + 1335 = 1445

and so disproved a conjecture of Euler that hadstood for nearly 200 years. One very specific re-cent achievement deserves special mention viz: thedisproof by Elkies (1988) of the Euler Conjecture onsums of fourth powers. Euler conjectured that (in-ter alia) no fourth power could be the sum of threefourth powers. Elkies however found that

(2682440)4 + (15365639)4 + (18796760)4

= (20615673)4and went on to prove that not only are there an in-finity of such counterexamples but, when expressedas the representation of 1 as the sum of three ra-tional fourth powers, the solutions are dense in< 0,1 >.

Accuracy and reliability of the computationsshould not be an issue today. Where a result issufficiently important or in doubt it can be checkedby someone else on a different machine; this hasbeen done on several occasions and if the result isconfirmed and, assuming that the underlying math-ematics is correct, the result can he accepted withconsiderable confidence, if not certainty. Computer-assisted proofs need not be any more suspect than

purely human proofs; many false "proofs" - includ-ing some of the 4-colour theorem - have been pub-lished in the past; we do not believe that the com-puter will increase the number of false proofs, quitethe contrary.

It is, of course, accepted that no amount, of nu-merical evidence constitutes a proof of a theoremrelating to an infinite set; the numerical evidencemay be misleading even for a very large set of val-ues of the variables involved. A well-known examplefrom analytic number theory is Littlewood's proof(see Ingham, 1932) that despite all the numericalevidence then, and even now, available

ix dtA In t

(where r(x) indicates the number of primes lessthan or equal to x) not only eventually changes sign,but does so infinitely often.

A criticism of computer-assisted proofs - such asthat of the 4-colour theorem - is that they tend torely on brute-force and give little insight into whythe theorem is true. Unfortunately some results e.g.finding large primes or factoring large integers in-trinsically require such methods, and whilst it maybe true that a computer proof may bring little in-sight, its very existence may inspire people to findmore elegant, shorter, or illuminating proofs.

Taking a longer-term view, the availability ofcomputer assistance may encourage mathematiciansto a more precise syntax and to express more for-mally what is in their minds (de Bruijn, 1970). Sucha development may, in turn, aid the teaching of theart of constructing proofs and so lead to the develop-ment of "expert systems" to undertake a least someaspects of mathematical work (including all the rou-tine algebraic manipulation, computation, etc.), inpartial fulfillment of Leibniz's dream of a rationalcalculating device.

One final point: Since every proposition that isprovable has among its many proofs one of minimallength and since the proofs of any given length are(at most) finite in number there must be true theo-rems of mathematics that cannot be demonstratedby traditional discourse within the longest humanlifetime. It would appear then that there are math-ematical theorems that can only be proved with theaid of computers if we are unwilling to wait too long.

1.3 Experimentation in Mathematics

Certain branches of mathematics have alwaysbeen open to experimentation but the arrival ofcomputers means the scope for experimentation inmathematics has been greatly increased. In some of

the sections above we have indicated cases where ex-periments have been used to provide data on whichconjectures and, in some cases, theorems have beenbased. Euler, remarking on the necessity of observa-tion in mathematics, said "The problems of num-bers that we know have usually been discovered byobservation, and discovered well before their valid-ity has been confirmed by demonstration "

The sheer speed of computers means that calcu-lations which would once have taken a lifetime cannow be completed in hours, or even minutes. Add tothis the fact that the results can often, if required,be presented in graphical form rather than as a listof numbers and we see that the interpretation of theexperiments may be made much easier. The case ofthe iteration of complex-valued functions illustratesthis point.

Of course, when a constraint is relaxed, there isa danger of excess. The ability to perform calcula-tions does not mean that everything can or shouldbe calculated. There is a balance to be struck andthis must be guided by experience - not to mentionthe cost of the computations. The effort and costinvolved need to be combined with the probabilityof success, in the sense of solving a problem or un-covering some useful fact. Computation for the sakeof computation is not to be encouraged.

Although experimentation in pure mathematicshas its uses it is, perhaps, in the area of statisticsthat it is particularly valuable. We take two exam-ples.

Simulation

Even before the availability of the moderncomputing technology, experimental sampling andMonte Carlo methods have played a role in statis-tics for studying the performance of statistical tech-niques under the assumption of probability models.The computer has enhanced this aspect on a largescale. One famous example is the Princeton RA-bustness Study (Andrews et al, 1972) where sets ofestimators under a system of different modelling as-sumptions are studied by means of computer simu-lation. The results have stimulated new mathemati-cal research into robust estimators (e.g. asymptotictheory) but on the other hand they cannot merelybe interpreted as conjectures that can and shouldbe validated by mathematical proof, but they havean importance in itself and have already influencedthe practice of analyzing data.

More generally, computers have given a ma-jor impetus to the idea of mathematical modellingwherein a physical or logical situation is embodiedin a mathematical model whose operation may then


be simulated on a computer. Thus we no longerneed to place physical models in a wind tunnel butinstead simulate the model on a computer. Similarlywe do not have to build a new telephone system tosee if it works since we can first simulate the systemon a computer.

Exploratory Data Analysis

It is sometimes stated that the computer has ledto an unwelcome shift from hard thinking to a sense-less computation of examples and experimentation.A balanced picture would say that the computerhas led to broader variety of "types of rationality"to approach problems and it is necessary to judge inevery situation which approach is more reasonable.

The classical paradigm for applying statistics isto think first very hard and then construct a proba-bilistic model and an adequate design for gatheringdata. But this strategy is not feasible in quite a lotof situations where little is known about the dataand the underlying system of interest. In connec-tion with the numerical and graphical capabilitiesof computers a new methodology of data analysis,called Exploratory Data Analysis (Tukey, 1977), hasbeen developed. The computer has made it possibleto experiment with several models for a data set, toconstruct a variety of interesting plots of the data togain insights into patterns, structures and anoma-lies of the data and to develop conjectures concern-ing the features of the system underlying the data.Such a type of exploratory mathematics would notbe practicable on a large scale without using com-puters.

1.4 Iterative methods

Methods of solving systems of linear equationsare traditionally divided into (i) direct and (ii) in-direct, or iterative, methods. The direct methodsinclude Gaussian elimination, the indirect methodsinclude the Gauss-Seidel method. Direct methodshave the advantages (a) that they will always pro-duce the solution provided that it exists, is uniqueand that sufficient accuracy is retained at everystage, and (b) that the solution is found after aknown number of operations. They have the disad-vantage that very sparse systems of equations, suchas arise in finite difference approximations to differ-ential equations, may become rapidly less sparse asthe elimination process proceeds so raising the stor-age requirement from a multiple of n (for n equa-tions) to something like n2. Iterative methods, onthe other hand, may fail to converge to a solutionand, if they do converge, it is not obvious how manyoperations they will require to produce the desired

4,

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accuracy. They have, however, the very consider-able advantages that they are very well suited tocomputers and preserve the sparsity of the coeffi-cient matrix throughout.

Direct methods of solution of nonlinear systemsare rarely available; there is, after all, no directmethod for solving the general polynomial of eventhe fifth degree and so iterative methods are gen-erally used. As in the case of linear systems, con-vergence may not always occur, though conditionssufficient to ensure convergence are usually known;and although in some cases the number of iterationsnecessary to produce convergence to a specified ac-curacy may not be easily predicted, it is frequentlynot a matter of great importance alid, if time is lim-ited, accelerating techniques can often be used.

The revival of interest in iterative methodsbrought about by the use of computers has led tosignificant advances in the study of functions whichare iteratively defined, e.g. by a relation of the type

= F(Z,,)

where Z0 is a given complex number and the func-tion F(Z) may contain one or more parameters.Some functions of this type, such as

Zn+i = z + C

were studied over 60 years ago by Julia (1918) andFatou (1919), but attracted relatively little interestat that time. In the case where the function F(Z)involves one complex parameter C and we define theset. of points Kc to be those points Z such that theiterated sequence of points given by

Z, F(Z), F(F(Z)), ...etc.

does not go to co, then the boundary of Kc is calledthe Julia set associated with F(Z) and C. Only re-cently, thanks to the availability of computers and,particularly, of colour graphics terminals has the ex-traordinary nature of these Julia sets and their nu-merous spin-offs been appreciated. For example, theMandelbrot set is defined as the set of values of Cfor which Kc is connected. The relation above is afractal curve, the discovery of which, due to Man-delbrot, has inspired a great deal of exciting andattractive research by Dohady, Hubbard and manyothers (Devaney, 1989).

The enthusiastic study of fractals has grown veryrapidly in recent years and the ready vailability ofhigh definition computer graphics ha made it possi-ble for schoolchildren, as well as teachers, to producea wonderful variety of exotic pictures based uponiteration of simple functions of complex variables(Peitgen et al, 1992). Even more recently the work

of Barnsley on iterated function schemes, which re-sult in remarkably lifelike pictures of ferns and trees,has aroused a lot of interest. The mathematical the-ory of fractals is much more demanding than theirproduction on a computer but good progress hasbeen made here, undoubtedly inspired by the com-puter graphics successes.

1.5 Algorithms

An algorithm is simply a procedure for solvinga specific problem or class of problems. The notionof an algorithm has been around for over 2000 years(e.g. the Euclidean Algorithm for finding the high-est common factor of two integers), but it has at-tracted much greater interest in recent years follow-ing the introduction of computers and their applica-tion rot only in mathematics but also to problemsarising in technology, automation, business, com-merce, economics, the social sciences, etc. (see alsothe chapter by Maurer). Computer algorithms haveSeen developed for many commonly occurring typesof problems. In some cases several algorithms havebeen produced to solve the same problems, e.g. tosort a file of names into alphabetical order or to in-vert a matrix, and 'n such cases people who wish touse an algorithm will not only want to be sure thatthe algorithm will do what it is supposed to do, butalso which of the several algorithms available is, insome sense, the "best" for their purposes. An algo-rithm which economizes on processor time may beextravagant in its use of storage space or vice-versaand the need to find algorithms which are optimal,or at least efficient, with respect to one or moreparameters has led to the development of complex-ity theory. Thus the Fast Fourier Transform hasreduced the time complexity of computing Fouriertransforms from order n2 to order n log n, which isof considerable practical importance for large val-ues of n. More recently the problem of designingalgorithms which can be efficiently run on severalprocessors working in parallel has attracted consid-erable interest.

Algorithms which are ideal on a single proces-sor may be highly inefficient, or even fail entirely,on parallel processors. The search for algorithmsfor the efficient solution of mathematical problemson systems of parallel computers is a major area ofresearch and conferences on this topic are held regu-la:ly. The problems are mathematically challengingand are also of considerable practical importance.With parallel computer systems now being readilyavailable, courses on parallel computing are beingtaught at undergraduate level which, five years ago,would have been possible in very few places.

A final point is this: the growing importance ofalgorithms suggests an enlarged role for proof by al-gorithm in which a constrictive proof of an existencetheorem is obtained by exhibiting an algorithm toconstruct the object posited.

1.6 Symbolic Mathematical SystemsThe possibility of using a computer to manipu-

late symbols, rather than numbers, and so provideusers with packages for algebraic manipulation andindefinite integration was appreciated from the ear-liest days of computers. Packages such as ALPAKand Slagle's SAINT (Slagle, 1963) both date fromthe early 1960's. Not only were such packages avail-able, they were used, Around 1960, Lajos Tokacsused ALPAK to carry out some very tedious alge-braic manipulation involving 1200 terms to find thesecond moment in a problem in queueing theory, ofimportance to Bell Laboratories. No one had hadthe courage or energy to do this by hand. When thesecond moment was finally found it reduced to justthree terms, after which a shortened mathematicalderivation was obtained and a general theory de-veloped. Two points are worth noting: After thebrute- force use of ALPAK the nature of the solu-tion inspired mathematicians to find a more elegantderivation - in support of our remark in Section 1.2;secondly, without the use of a symbolic manipula-tion package it is unlikely that this work would havebeen done at. all.

Another early system, FORM AC, was utilized tohelp with the solution of the restricted case of the 3-Body Problem and, more recently, it has been usedto check that two 752-term polynomials, occurringin the theory of plane partitions, are identical.

Some symbolic manipulation packages are gen-eral, but many more are applications specific. Wehave mentioned CAYLEY which is widely used forthe study of finite groups both at research leveland as a teaching aid. Other specific systems in-clude MATRIX, REDUCE (Fitch, 1985), MAC-SYMA (Pavelle and Wang, 1985); Maple (Char,1988); Mathematica (Wolfram, 1988); many moretraditional algebra systems are surveyed in Pavelleet al (1981). These are further discussed in the chap-ter by Hodgson and Muller.

1.7 Computers and Mathematical Communi-cation

Whilst it affords great personal satisfaction toprove (or disprove, or conjecture) a result., the math-ematical community only gains if that result is com-municated to others. This communication may takevarious forms (though the distinctions are not. rigid).


epistolary - where A writes a letter to Bcommunicating the result;

proscriptive - where A writes the result ona wall (literal or metaphorical) for others to read;

privately published - the usual form is a de-partmental technical report, whose existence is an-nounced;

publicly published - journals or books.This communication may be received either di-

rectly by the person who is going to use the result,or indirectly.

The advent of computed-aided typesetting andcamera- ready copy has obviously changed the vi-sual form of mathematical comunication (partic-ularly the publicly-published) and its economics.This has consequences for mathematicians (espe-cially editors) who may need to read the input tosuch type-setting systems. But computer technol-ogy is capable of changing and is changing, far morethan this.

Epistolary. Computer networks have revolu-tionized this method of communication by allow-ing "letters" to be sent via electronic mail insteadof physical mail As more and more mathematiciansare linked by such networks, they will replace mostwritten communication.

Proscriptive. In addition to the physical no-tice hoards in one's own department or elsewhereon which one can place proofs (or, more likely, an-nouncements of technical reports containing proofs),computer networks distribute electronic "bulletinboards" to various sites which "subscribe" to them.In some areas of computer science in North America,most results are announced on such bulletin boards.

Private Publishing. This is closely related tothe above. Such networks also distribute electronic"newsletters" to individual subscribers, which oftencontain lengthy articles in draft form, or state con-jectures or problems.

Public Publishing. This is the area whoseform has been directly least affected. Though thereis talk of it, no serious refereed journals distributedby electronic means exist.

REFERENCESAndrews, D.F. et al. [1972]: Robust Estimates and

Location, Princeton, NJ: Princeton UniversityPress.

Appel, K. and Haken, W. [1976]: The solution ofthe 4-color map problem, Sci. American (Oc-tober), 108-121.

Atkin, A.O.L. and Birch, B.J. (Eds.) [1971]: Com-puters in Number Theory, San Diego: Aca-

C)

1


demic Press.

Char, B.W. et al [1991]: Maple V Library ReferenceManual, New York: Springer-Verlag.

Churchhouse, R.F. [1973]: Discoveries in numbertheory aided by computers, Bull. IMA, 9, 15-18.

Churchhouse, R.F. [1980]: Computer arithmeticand the failure of the associative law, Bull.IMA, 16, 210-214,

Churchhouse, R.F. [1985]: Computer arithmetic II:some computational anomalies and their con-sequences, Bull. IMA, 21, 70-73.

Churchhouse, R.F. and Herz, J.C. (Eds.) [1968]:Computers in Mathematical Research, Ams-terdam: North Holland.

Devaney, R,.L. [1989]: An Introduction to ChaoticDynamic Systems (2nd Ed.) (A-W Studies inNon linearity), Reading, MA: Addison-Wesley.

Elkies, N.D. [1988]: On A4+1141-C4, Math. Comp.,51, 825-835.

Eaton, P. [1919]: Sur les equations fonctionelles,Bull. Soc. Moth. France, 47, 161-271.

Fitch, J. P. [1985] REDUCE, J. Symb. Comp., 1,211-227.

Ingham, A.E. [1932]: The Distribution of PrimeNumbers, Cambridge: Cambridge UniversityPress.

Julia, G. [1918]: Memoire sur l'iteration des fonc-tions rationelles, J. Math., 8, 47-245.

Lander, L.J. and Parkin, T.R. [1967]: A counter-example to Eulers's sum of powers conjecture,Math. Comp. 21, 101-103.

Leech. J. (Ed.) [1970]: Computational Problems inAbstract Algebra, Oxford: Pergamon Press.

Pavelle, R. and Wang, P.S. [1985]: Macsyma fromF to G, J. Symb. Comp., 1, 69-100.

Peitgen, Jiirgens, H. and Saupe, D. [1992]:Fractals for the Classroom. Part One: In-troduction to Fractals and Chaos, New York:Springer-Verlag.

Ramanujan, S. [1927]: Collected Papers, Cam-bridge: Cambridge University Press.

Slagle, J.R. [1963]: A heuristic program that solvessymbolic integration problems in freshman cal-culus, J. ACM, 10, 507-520.

Tukey, J.W. [1977]: Exploratory Data Analysis,Reading, MA: Addison-Wesley.

Wolfram, S. [1988]: Mathematica, Reading, MA:Addison-Wesley.

Zabusky, N.J. and Kruskal, M.D. [1965]: Interactionof "solitons" in a collisionless plasma and therecurrence of initial states, Phys. Rev. Lett.,15, 240-243.

Part IITHE IMPACT OF COMPUTERS AND COMPUTER SCIENCE ON THE

MATHEMATICS CURRICULUM

Anthony RalstonSUNY at Buffalo, Buffalo, NY 14260, USA

2.0 The Changing Science of Mathematics

In this section we will consider how computersand computer science should be causing changes inthe mathematics curriculum because of the chang-ing importance of various branches of mathematicswrought by cr::-nputers and computer science (seethe chapter by Steen). One aspect of this changeis that increasingly the knowledge of mathematicsimportant, to the user of mathematics is no longerthat of detailed knowledge but rather what mightbe called "meta-knowledge" about the characteris-tics and power of methods, often numerical, for thesolution of classes of problems (see the chapter byMascarello and Winkelmann). A related perspec-tive is that computers have hrought mathematicsmuch closer in philosophy to the classical naturalsciences where there has always been an interplaybetween theory and experiment. Now mathemat-ics, too, has a laboratory - the computer - on whichexperiments can be performed which lead to the-ories and on which theories can be tested. Thesepoints should he kept in mind in what follows. Al-though we shall not return explicitly to them, theyinfluence much of this section.

2.1 The Common Mathematical Needs ofStudents in Mathematics, Science and Engi-neering

(a) Preparation for UniversityMathematics

To provide a context in which to discuss the im-pact of computers and computer science on curricu-lum and pedagogy, it is necessary to agree first,, ingeneral, on the appropriate mathematics for the sec-ondary school student (see the chapter by Graf etal) and then to consider the university curriculum.Since thrr-.: are significant differences between dif-ferent parts of the world on when secondary schoolends and university instruction begins, the com-ments which follow will have to be interpreted inthe local context.

Algebra has traditionally been an importantsubject in high school. Since elements of abstractalgebra are likely to become increasingly importantin mathematics education, it is clear that algebra

19

will remain of central importance in the secondaryschool curriculum. The important thing, however, isnot to have students achieve great manipulative okillin algebra (e.g. in polynomial algebra) but ratherto teach them to consider algebra as a natural toolfor solving problems in many situations. Neverthe-less, the ability to use formulas and other algebraicexpressions will remain necessary.

In recent years there has been a trend towardreplacing much of Euclidean plane geometry withthose aspects of geometry more closely akin to al-gebra. This is useful as a preparation for universitymathematics but there is much feeling among math-ematics educators that the loss of Euclidean geome-try is a sad development. A consensus on how geom-etr: might best he taught at. school and universityis not yet available. It should be noted, however,that some computer scientists feel that the aspect oftraditional instruction in geometry concerned withteaching the meaning and construction of rigorousproofs can be achieved through material concernedwith the analysis and verification of algorithms (seethe chapter by Maurer).

For many parts of mathematics trigonometry isuseful preparation. But we note that much of thetedious work which was necessary in the past, bothnumerical and symbolic, can now be easily done onhand-held computers.

Next we mention calculus. In many countriesthis has been a secondary school subject for manyyears for most university-bound students while inother countries only the very best students begincalculus in secondary school. The main thrust ofsecondary school calculus has been to provide stu-dents with techniques, and to prepare those intend-ing to study mathematics at university with a firstintroduction to the concepts they will encounter atthe university level. Since all the techniques of sec-ondary school calculus as well as much of univer-sity calculus can now be done on hand-held devicesor on symbolic mathematical systems (often called"computer algebra" systems) on computers (see thechapter by Hodgson and Muller), calculus at the sec-ondary as well as university level must focus almostentirely on concepts and not on computation.


Various new subjects have become part of thesecondary school curriculum in recent years. Amongthese is probability which has come into the cur-riculum in many countries. Topics such as discreteprobability distributions, the binomial distributionand related topics are appropriate. So too, is anintroduction to data analysis and elementary statis-tics because of their importance in science as well asmathematics. Another subject, about which therewill be further discussion below, which we wouldlike to see more of in the secondary school curricu-lum, is discrete mathematics including elementarycombinatorics and graph theory as well as an intro-duction to induction and recursion. In this connec-tion it would be appropriate to introduce bOth thedesign and verification of a number of important al-gorithms such as those for sorting. Finally we notethat elementary linear algebra, particularly matrixalgebra and work with systems of linear equations,should certainly be considered for the secondaryschool curriculum.

(b) The University MathematicsCurriculum

The core of the university mathematics curricu-lum for many years has been the calculus and, to alesser extent, linear algebra. This is the case no mat-ter how much mathematics the student may havestudied in secondary school. Computers themselveshave an impact on both the content of this curricu-lum and its pedagogy. Not only do computers allowmore interesting and effective presentation of clas-sical subject matter but, in addition, as with thesecondary curriculum, they affect what subject mat-ter is important to students. For example, symbolicmathematical systems suggest a deemphasis on themore skill-oriented portions of the current curricu-lum.

Informatics (i.e. computer science) itself alsoimplies changes in the content of the core curricu-lum. This is essentially because informatics is ahighly mathematical discipline whose problems re-quire almost universally the tools of discrete ratherthan continuous mathematics. Thus, there is nowa strong argument. to provide a balance in thecore curriculum between the traditional continuousmathematics topics and topics in discrete math-ematics (Ralston 1981, Ralston and Young 1983,Ralston 1989). For university courses aimed at abroad spectrum of mathematics, science and en-gineering students, this balance may well containnearly equal portions of the continuous and the dis-crete. For those courses aimed at specific studentpopulations, the balance might. be weighted morein the direction of the discrete for informatics and

social and management science students, might beabout equal for mathematics students themselvesand surely should be weighted more toward tradi-tional continuous mathematics for physical scienceand engineering students. It needs to be empha-sized, however, that all groups of students needsome exposure to both the continuous and discreteapproaches to mathematics. Whether students areexposed to calculus first and then discrete math-ematics or vice versa will depend on the studentpopulation and on institutional convenience.

The actual content of the discrete mathemat-ics component is still quite variable. However, thediscrete component normally contains at least some"traditional" discrete mathematics (e.g. combina-torics, graph theory, discrete probability, differenceequations) as well as perhaps some abstract algebraalthough the latter may follow in a later course aftercompletion of the core courses.

We note also the importance of mathematicallogic in the core university curriculum. Althoughtraditionally an advanced undergraduate or a post-graduate subject (at which levels there will be acontinuing need for specialized courses), logic is soimportant in informatics that it needs to be intro-duced early in the university mathematics curricu-lum. Moreover, with the increasing need for peoplein the scientific and technical professions to han-dle information in a precise manner, logic has greatvalue for a wide variety of students. Logic is an im-portant constituent of many discrete mathematicscourses (see below). But it can also be consideredas a subject for a course by itself which would fol-low the introduction in discrete mathematics. Sucha course can usefully be given a distinctive computerflavor as described by Schagrin et al (1985).

As a final matter, we stress the importance ofusing the paradigms of informatics (e.g. an algo-rithmic approach, iteration, recursion) in the teach-ing of mathematics at all levels. Although theseparadigms may seem most easily applicable to dis-crete mathematics, there is considerable scope fortheir introduction into the classical continuous cur-riculum.

The reader may be surprised to find no mentionof numerical analysis here (or hereafter in this doc-ument) because this subject is the one that mostobviously combines the continuous and discrete ap-proaches to mathematics. But we take the posi-tion that numerical analysis is now such a well-established subject in the mathematics curriculumthat it does not need to be discussed in the contextof this report. This is however, not to say that thesubject matter of numerical analysis is no longer af-

Impact of Computers on the Mathematics Curriculum 21

fected by advances in computing; developments in,for example, parallel computing are having great im-pact on numerical analysis.

2.2 A Discussion of Particular CurriculumAreas on Which Computers and InformaticsHave an Impact

Although discrete mathematics and calculus arediscussed separately in what follows, it should beemphasized that there is no intellectual reason toconsider them as separate subjects. Indeed, they aremutually supportive and ideally would be taught inintegrated courses (see the chapter by Seidman andRice). However, for at least some years to come,such integrated courses will be relatively rare, notleast because of the lack of textbooks for integratedcourses.

(a) Discrete Mathematics CoursesWe begin with a discussion of what topics in dis-

crete mathematics should be contained in coursesintended for mathematics students as well as forstudents in the social and management sciences. Al-though the topics to be listed below cover a broadspectrum, it is possible to design a coherent coursecovering these topics if the course is built aroundthemes such as algorithms and their analysis andinductive and recursive thinking.

A Discrete Mathematics Syllabus1. Mathematical Preliminaries - Sets, functions, re-

lations, summation and product notation, ma-trix algebra, an introduction to proof and logicconcepts.

2. Mathematical induction including its applica-tion to algorithms and recursive definitions.

3. Graphs, digraphs and trees including path,searching and coloring algorithms, tree traversal,game trees and spanning trees and applicationsin a variety of areas.

4. Basic Combinatorics including the sum andproduct rules, permutations, combinations andbinomial coefficients, inclusion-exclusion, thepigeonhole principle and combinatorial algo-rithms.

5. Difference equations (i.e. recurrence relations)including first order equations, constant coeffi-cient equations and the relationship of recur-rence relations to the analysis of algorithms, par-ticularly divide-and-conquer algorithms.

6. Discrete probability including random variables,discrete distributions and expected value.

7. Mathematical logic including the propositionalcalculus, Boolean algebra, the verification of al-

gorithms and an introduction to the predicatecalculus.

8. Infinite processes in discrete mathematics: Se-quences, series, generating functions, approxi-mation algorithms.In addition, other possible topics depending

upon local needs and desires are:9. Algorithmic linear algebra including the use of

Gaussian elimination as an entree to abstractlinear algebra and an introduction to linear pro-gramming and applications of linear algebra.

10. Decision mathematics including such things asqueueing theory and packing problems.

11. Algebraic structures such as rings, groups etc.12. Finite state machines and their relation to lan-

guages and algorithms.

And, of course, there can be extensions of all theabove topic areas to more advanced subject matterif desired and appropriate.

Since the Strasbourg conference in 1985, at least40 books have been published from which a courseon the above lines can be taught (see, for example,Epp, 1990 and Maurer and Ralston, 1991).

The experience of those who have taught suchcourses is that, despite the potpourri of topics listedabove, these courses can be made interesting andsatisfying if a consistent, coherent approach is takenwhich emphasizes algorithmic, recursive and induc-tive thinking.

Following a course from a syllabus like thatabove, a variety of advanced courses in discretemathematics can be contemplated although onlythe largest institutions would be able to offer allof these. Indeed, each of the subject areas listedabove suggests one or more advanced courses whichwould build on the introductory material in a firstdiscrete mathematics course. Most of these coursesare currently in a process of evolution as the sub-ject matter in the first discrete mathematics coursechanges and develops and as the applications of dis-crete mathematics grow and diversify. A programwhich combines a carefully constructed introductorydiscrete mathematics course with several advancedcourses will give the student a firm basis for study-ing informatics as well as providing a basis for pro-fessional work in modern applied mathematics andother fields in science and engineering.

-e?..)


(b) Calculus in the Computer Age(i) The Role and Relevance of Calculus

Among the key factors which compel change inthe teaching of university mathematics courses are:

- the substantial experience with minicomput-ers and microcomputers and programming packageswhich many students have had before coming to theuniversity;

- the growth of new areas of applied mathemat-ics such as the analysis of algorithms and computa-tional complexity.

One result of this is that many students have at-titudes and expectations which lead them to believethat the most challenging and meaningful mathe-matical problems today are related to computersand informatics. This cannot help but influencehow we must motivate mathematics students andall other students in mathematics courses.

In considering the place of calculus in the com-puter age, we cannot forget that it is one of hu-mankind's great intellectual achievements. Everyeducated person should be aware of it. Its his-tory exemplifies the "unreasonable effectiveness" ofmathematics better than any other branch of math-ematics. And its effectiveness is as great today asit has ever been. But this does not excuse teachingcalculus as is so often the case now with an empha-sis only on the execution of mechanical procedures- and paper-and-pencil procedures at that. Insteadcalculus needs to be taught to illustrate the uniqueways of thinking it epitomizes.

The realm of applications of calculus remains im-mense. They are, indeed, increasing due to the in-creasing mathematization of heretofore qualitativesciences like biology. In constructing calculus mod-els of phenomena and then solving the resultingequations, there is often an interplay between thesemodels and their discrete counterparts with the cal-culus models representing the limiting behaviour ofthe discrete models. It is now more important thanever to include this interplay in calculus (and dis-crete mathematics) courses because inevitably thesolution of most problems in calculus involves the(di. crete) computer. The discretization necessaryto solve problems of calculus with a computer oftenhas not borne a close relationship to the underlyingdiscrete model. But the increasing power of com-puters means that more and more frequently it ispossible to have computer models which mirror veryclosely the discrete models from which the continu-ous model was initially abstracted.

There already are powerful software tools whichcan be used in the study of calculus. These in-

clude symbolic mathematical systems and a varietyof graphical packages. Advances have taken placeso rapidly in these areas that it is now the case thatvery powerful symbolic and graphical systems areavailable on hand-held computers (e.g. the HP-48S)as well as on microcomputers. One result of this isthat an understanding of functions, variables, pa-rameters, derivatives etc. and the ability to inter-pret formulas and graphics is becoming more im-portant to the student than skills in executing the(numerical or symbolic) procedures of calculus. Inthe teaching of calculus to all students the need isclear for a shift from an emphasis on calculationaltechnique to one which emphasizes the developmentof mathematical insight.

(ii) The Content of Calculus Courses

If functional behaviour and representation areto be the focus of the calculus course, then continu-ous functions and discrete functions (i.e. sequences)must be emphasized and motivated by a wide vari-ety of mathematical models. (Indeed, mathemat-ical models and their applications in a variety ofdisciplines should be an important part of calcu-lus courses.) (Note: it can be argued that se-quences belong more properly in the discrete math-ematics course discussed previously. This only illus-trates the need to bring the discrete and continuouspoints of view together into an integrated sequenceof courses as soon as possible.

An important theme in calculus courses shouldbe the contrast between the local and global be-haviour of functions. Local behaviour is, of course,derived by studying the derivative for continuousfunctions (and the difference operator for discretefunctions). And similarly the integral (and summa-tion) operators are used to derive global informationabout functions. Undoubtedly it will remain neces-sary to develop some ability to do formal computa-tions with derivatives and integrals. But the majoremphasis should be on numerical algorithms (par-ticularly for integrals) and on how derivatives andintegrals can be used to understand the behaviourof functions.

A topic such as the Taylor series representationof a function should be used to show how good localinformation can be obtained using low-degree Taylorpolynomials and interpolating polynomials, anotherarea where the analogy between the continuous andthe discrete may be usefully shown.

Finally, there should be a balance in the cal-culus course between traditional topics and oneswhose importance has greatly increased because ofthe advent of computers and informatics. Thus,

Impact of Computers on the Mathematics Curriculum

for example, the 0() and o() notations, which en-able the asymptotic growth rates of functions tobe compared to standard functions like polynomialsand logarithms, are not always taught in calculuscourses, but they should become so.

This discussion is intended only to provid, theflavor of how an orientation toward computationshould change the approach toward teaching mostof the standard calculus topics.

(iii) Computers for Learning andTeaching Calculus

Computers enable teachers to modify theirmethods of teaching calculus (and, of course, muchother mathematics also) in order to meet better theneed of their students. Computer graphics is a pow-erful medium in which to provide examples - andnon-examples - of continuous functions, discontin-uous functions, the area under a curve, directionfields and nowhere differentiable functions as well asin many other areas. Well-designed software (therestill isn't nearly enough of this) can be used bystudents to discover and explore the concepts men-tioned above as well as such fundamental conceptsas slope and tangency (see also Section 2.3). But theeffective use of such software requires that teacherssometimes depart from a lecturing style and go in-stead to guiding and interacting style with smallgroups of students or individual students.

Well-designed software will also permit enhance-ments by students through the writing of (usuallyshort) programs. This is just another way in whichstudents can be actively involved in their own learn-ing although it is important that the use of thecomputer does not become the message instead ofthe mathematics which it is supposed to illustrate.Thus, programming per se should not play any sig-nificant role in a calculus course.

Another impact of the computer in calculus maybe to change the order in which topics are taught.For example, it is becoming increasingly commonto introduce limits at the very start of a calculuscourse. Tangent functions and area under a curvecan be motivated and defined graphically. Whena formal definition of a limit is needed, studentswill be ready for it. As another example, differ-ential equations can now be treated much earlierin the curriculum than was previously possible be-cause of the ease of understanding made possible bynew graphics systems (see the chapter by Tall andWest). They can be introduced right after differ-entiation and before integration. Studies are nowunder way to discover whether such reorderings willlead to a greater or more rapid understanding of

23

fundamental concepts and theorems.To take full advantage of the use of computers in

teaching calculus, it will be necessary to change thestandard classroom environment. Classrooms needto be provided with large monitors or screens onwhich the monitors may be projected. Both insideand outside the classroom, students need adminis-tratively easy and user-friendly access to comput-ers and software. Teachers will need private com-puter facilities in order to prepare course material.A prerequisite for this is in-service training so thatteachers may become comfortable with computersand then fluent in their use and aware of possibili-ties beyond what may be available in the particularsoftware on which they have learned.

Finally, we note the value of using computersin the classroom to teach mathema:s. The desir-ability of this for calculus and related subjects isparticularly clear since the dynamics of computergraphics is ideally suited to help explain a subjectwhich is essentially about change. Indeed, it is ironicthat only static technology - the chalkboard and theoverhead projector - are still used so widely to teachcalculus. There is a considerable amount of softwareavailable now which can be used in the classroom toteach calculus (e.g. Flanders (1991)) and differen-tial equations. There is much less software availableto teach discrete mathematics in the classroom butthere are numerous aspects of discrete mathemat-ics (e.g. induction and recursion) for which suitablesoftware would be valuable in the classroom. Wecan expect to see the development of such softwarein the near future.

2.3 Exploration and Discovery inMathematics

The idea of using computers to enable studentsto explore mathematics and discover mathematicalpatterns for themselves is not a new idea (Steen,1988). However, the advent of powerful and avail-able computer systems makes this point so impor-tant in teaching mathematics today that we devotean entire section to it.

First, why should exploration and discovery beimportant components of the educational process inmathematics? The answers parallel the reasons whywe teach mathematics in the first place:

active learning leads to better retention and un-derstanding and more liking of the mathemat-ics we teach because the mathematics is seenas a basic component of human culture; it alsoleads to more self-confidence in the ability to usemathematics to solve problems;

3 k)


exploration and discovery helps to teach peopleto think;discovery provides the greatest aesthetic expe-rience in mathematics, the "oho" of seeing orproving something is what makes mathematicsattractive;exploration and discovery are perhaps the bestways for students to see that mathematics is souseful;

discovery enables the student to see a familiaridea applicable in a new context, thereby en-abling a grasp of the power and universality ofmathematics.Computer technology may be used to assist in

mathematical exploration and discovery in a varietyof ways; for example:

through visualization of a great variety of twoand three dimensional objects via computergraphics, students may explore questions anddiscover results by themselves.through computer graphical presentations of in-teresting geometries like "flatland" and turtlegeometry;via exploratory data analysis to, for example,draw conclusions from data (e.g. is it bimodal?are there outliers?), to transform data (e.g. bylogarithmic plots), to smooth data and to com-pare different sets of data.by graphical and numerical explorations of howto approximate complicated functions by simpleones;by applying the first step of the inductiveparadigm compute, conjecture, prove inmany, many different situations;by using symbolic mathematical systems to dis-cover mathematical formulas such as the bino-mial theorem;by designing and executing different algorithmsfor the same or related tasks.

This list could be made much longer. Readers willprobably be led to make their own suggestions.

There are various implications to using comput-ers to facilitate exploration and discovery:

we must start with easy tasks so that studentsfeel they are really succeeding on their own andare not being led step by step by the teacher;teachers need to be educated for this kind of in-structional mode; few teachers can become com-fortable with these ideas without explicit educa-tion; we note, in particular, that testing whathas been learned by the student is not easy. Butexperience has shown that success is not onlypossible but yields rich rewards. The difficulties

can be overcome; teachers can be trained to feelcomfortable with this mode of learning.

2.4 Some Speculation about the FutureAs mathematics becomes increasingly an exper-

imental science, it is inevitable that computers andcomputer science will have increasing influence onthe mathematics curriculum. Computer science willbecome a gradually greater focus of applications ofmathematics and this will affect what is importantin mathematics. At the same time the means bywhich all mathematics is taught. will be inextrica-bly entwined with computer technology. Althoughthe cost of this technology will continue to be aproblem for developing countries, the curricular in-ertia in developing countries is far less than thatin the developed countries. Developing countrieshave an unparalleled opportunity to use computersand the influence of computer science to modernizetheir mathematics curricula and their mathematicsteaching faster than will be possible in developedcountries.

REFERENCESDouglas, R.G. (Ed.) [1986]: Toward a Lean

and Lively Calculus, MAA Notes Number 6,Washington, DC: Mathematical Association ofAmerica.

Epp, S.S. [1990]: Discrete Mathematics with Appli-cations, Belmont, CA: Wadsworth.

Flanders, H. [1991]: Microcalc 6.0, Calculus Soft-ware for VGA, Ann Arbor, MI.

Maurer, S.B. and Ralston, A. [1991]: Discrete Algo-rithmic Mathematics, Reading, MA: Addison-Wesley.

Ralston, A. [1981]: Computer science, mathemat-ics and the undergraduate curricula in both,Amer. Math. Monthly, 88, 472-485.

Ralston, A. (Ed.) [1989): Discrete mathematics inthe First Two Years, MAA Notes Number 15,Washington, DC: Mathematical Association ofAmerica.

Ralston, A. and Young, G.S. [1983]: The Futureof College Mathematics, New York: Springer-Verlag.

Schagrin, M.L., Rapaport, W.J. and Dipert, R.R.[1985]: Logic: A Computer Approach, NewYork: McGraw-Hill.

Steen, L.A. [1988]: The Science of Patterns, Science,240 (29 April), 611-616.

Part IIICOMPUTERS AS AN AID TO TEACHING AND LEARNING MATHEMATICS

B. CornuIUFM, Grenoble, France

3.0 IntroductionMathematicians and mathematics teachers have

been provided with a new tool, the computer. Thereis no shortage of applications or interesting exam-ples which one can quote. But, like all tools, thecomputer by itself does not supply a solution tothe problems of mathematics education. There isno automatic beneficial effect linked to a computer:The mere provision of micros in a class - or lectureroom will not solve teaching problems. It is essen-tial, therefore, that we should develop a serious pro-gramme of research, experimentation and reflectivecriticism into the use of informatics and the com-puter as an aid to teaching mathematics. It will notsuffice to think only in terms of mathematics andthe computer, and of the production of softwarewhich amuses and interests mathematicians. Wemust also take into account types of knowledge andthe ways in which these can be transmitted, and at-tempt to study, in a serious epistemologically-basedmanner, various concepts and the obstacles whichthey present to learners. We must think of students,their development and the matching of new and oldknowledge. We must consiaer in depth the teachingpossibilities created by the computer. It is essential,above all, that we should move beyond the stage ofopinions, enthusiasms, and wishful thinking and en-gage in a true analysis of the issues. Only in this waywill we come to a true resolution of certain problemsof teaching. Such research, of necessity experimen-tal, will have to be critically evaluated. It must beshown how, in given circumstances, the use of thecomputer can facilitate the acquisition of a partic-ular concept. Finally, such research work will haveto be built upon and developed to provide a vitalcomponent in the training (whether formal or self-directed) of teachers and lecturers. Only then cancomputers have any large-scale effect on mathemat-ics teaching.

Certainly such research has been done in thepast. few years, and we can now see examples of usesof computers in education, based on a serious studyof the didactical problems to be solved. In suchuses, the computer is not a tool supplementary totraditional teaching; it is integrated in a pedagog-ical strategy, adapted to the actual obstacles thestudents have in learning. But much remains to bedone. Both the development of educaticnal researchand the evolution of technology have the potential

25

to effect major changes in teaching and learning inthe future.

Computers for mathematics teaching are not sowidely used as one could think. Appropriate soft-ware and strategies do exist; but they are usedby few teachers; one of the main problems nowis to help all teachers to use computers, not as anew experience, but as a common tool for teach-ing. This requires not only good training for teach-ers and good pedagogical products and tools, butalso good integration of new technologies in curric-ula and good long term pedagogical strategies.

3.1 A changing view of mathematics

There are many references in this book to theway in which the computer can lead to a changedview of what mathematics and mathematical activ-ities comprise. For example, as the experimentalaspects of mathematics assume greater prominence(see Section 2.3), and there is a corresponding wishto ensure that. provision should be made for studentsto acquire skills in, and experience of, observing,exploring, forming insights and intuitions, makingpredictions, testing hypotheses, conducting trials,controlling variables, simulating, etc. Examples ofhow such work can be carried out are found in laterchapters in this book. However, mechanisms need tobe found for disseminating information about fruit-ful experimental environments and how these canbe formed.

Yet, as we put new emphasis on the particularactivities listed above, it is also necessary to ensurethat such traditional activities as proving, general-ising and abstracting are not neglected or omitted.We will need to find an appropriate balance betweenèxperimental' and more formal mathematics.

The possibilities presented by the computer willactually help focus our attention on the kind andtypes of knowledge which we wish students to ac-quire. Not only are new possibilities offered to us,but also a greater incentive to identify more pre-cisely our educational goals.

If our aims of teaching change significantly soas to encompass and stress the 'process' of mathe-matics more than the 'products' of the mathemat-ical activities of others, then there will, of course,be a need to identify those parts of mathematics

C)


most suitable for our purposes. Topics and areas ofmathematics must be selected which encourage andfacilitate an experimental approach.

Finally, in this section we must stress two impor-tant, interrelated points. Many, indeed the major-ity, of our students do not intend to become math-ematicians. We must not lose sight of the implica-tions of this in terms of educational goals and em-phases. But, also, many of these may be studentsof the experimental sciences. This raises further im-portant issues, for experiments in mathematics dif-fer somewhat from those in the physical and natu-ral sciences. The techniques are often very similar,but in mathematics we have that extra, vital ingre-dient of 'proof'. Experiments are an essential andneglected part of mathematics, yet mathematics isnot an experimental science. The distinctions be-tween disciplines and ways of thought will have tobe displayed and observed.

3.2 Computers change the relation betweenteacher and student

Computers can affect the behaviour of students.This creates new interactions and relationships be-tween student, knowledge, computer and teacher.The role of the teacher in such situations demandsconsiderable thought.

(a) The mathematical activity of the stu-dent

Students will be better able to learn conceptualmaterial arid develop autonomous (as opposed toimitative) behaviour patterns with respect. to math-ematical ideas if they can be cognitively active inresponse to mathematical phenomena presented tothem. This activity should consist of the formationof mental images to represent mathematical objectsand processes. It should also include the develop-ment of skills in manipulating these objects and pro-cesses. In this way students can increase their abil-ity to think mathematically.

Inducing students to emerge from passivity andto think actively about mathematics is, however,not easy. One approach is to make use of the com-puter to supply sufficiently powerful and novel expe-riences to stimulate such behaviour. The action of acomputer program and the structure of data as it isrepresented in the computer can form useful mod-els for thinking about mathematical entities. Forexample, a "WHILE loop" whose body is a simplesum is a process that can represent the mathemati-cal entity

This expression, which troubles so many students,can then be thought of in terms of a simple, famil-iar and useful computer process. Again, in Pascal,representing a fraction as a record with two inte-ger fields (the second being non-zero) helps studentsthink about rational numbers as ordered pairs ofintegers, especially if they are given the experienceof writing programs to implement the arithmetic offractions without truncation.

More generally, many mathematical conceptscan be defined or described as procedures. Thisgives a more dynamic approach, and can help thestudent in understanding and in using these con-cepts. Algorithmics (see the chanpter by Maurer)gives many tools for introducing mathematical con-cepts in such a way.

Many examples of ways in which such ex-periences can be incorporated into mainstream,tertiary-level courses are available. Moreover, thesuccess of such initiatives would seem to be inde-pendent of several issues which in discussion tendto be overrated. An important factor in this ap-proach appears to be that students should write theprograms and so must be cognitively active aboutthe processes and data structures they are imple-menting. These experiences are then coordinatedwith classroom activity.

In their chapter, Mascarello and Winkelmanndescribe a course containing 'continuous' topics suchas multiple integration and ordinary differentialequations. Here the students wrote programs ina low-level language running on a microcomputer.These were interactive and the results were used forexperimentation and demonstration.

Of course, writing programs is not the only use-ful way in which students can use the computer.The use of complicated software packages for il-lustration of henomena that are very difficult todisplay otherwise can clearly broaden the students'awareness and add to their general understanding(see, for example, the chapter by Tall and West).They can, of course, also be used for explorationand discovery. Indeed, some would see the mostexciting opportunity offered by the computer to bethe way in which it can motivate students to exer-cise the process of discovery. Here we should onlystress the need to see exploration and discovery asessential mathematical activities to be practised.Traditionally, this has not been so - teaching andlearning have been almost wholly concerned withthe transmission and reception of accepted mathe-matical facts. However, now, for example, computersymbolic mathematical systems (see the chapter byHodgson and Muller) permit such rapid and flaw-

Computers in the Teaching and Learning of Mathematics 27

less processing of non- trivial examples that it iseasy first to look for patterns which suggest con-jectures and generalisations, and then to search forcounterexamples or machine-aided proofs.

Computers then can greatly assist us in extend-ing the range and the depth of students' mathemat-ical activities. In some approaches the students willwrite their own programs (and there will be an at-tendant risk that mathematical aims may becomeobscured by some of the programming problems);in others students will use prepared software. Bothapproaches have already been shown to be of great

. value; further investigations into both will now haveto be continued.

(b) The role of the teacher

The computer can be used in two distinct waysin the classroom. In one it is an aid for the teacher,an electronic blackboard more powerful than thetra-litional blackboard, the overhead projector, or acalculating machine but nevertheless a tool whoseoutput is almost entirely under the teacher's con-trol. In this role the computer does not upset thetraditional balance in the classroom. It will stilldemand effort on the teacher's part to select or pro-vide suitable software and it can give rise to ex-tra administrative problems; in return it should en-hance learning. However, it will not revolutionisethe classroom.

If, however, students are allowed and expected tointeract with computers, then the position changes,for this leads of necessity to a change of method-ology. The teacher no longer has total controlhis/her role can no longer be limited to exposi-tion, task-setting and marking. The format 'lecture-;:xamples, homework-exam' must be augmented by,for example, 'project (through interaction betweenstudent, machine and teacher) assessment on thebasis of a completed (and possibly debugged) as-signment'.

Probably the teacher must combine diverse usesof the computer. Some activities fit well with the'blackboard computer'; some others will be moreefficient if each student has the opportunity to in-teract with the computer.

Such a change would produce a revolution inmost class- and lecture-rooms. It demands thatteachers should not only acquire new knowledge,skills and confidence in the use of hardware andsoftware, but that they should also radically changetheir present aims and emphases, and accept a less-ening in the degree of control which they presentlyexert over what happens in their classrooms. Thislast demand means a sacrifice of traditional security,

at a time when teachers will still be fighting hardto gain new skills and acquire confidence in them.It would be foolish to underestimate the challengethis presents.

The acquisition of new skills will be time-consuming and constantly changing hardware andsoftware will make the process a continuing one. Formany mathematicians these new skills will be read-ily usable in their research work. Others may betempted particularly when universities and othereducational institutions are under pressure to feelthat such time would be more profitably spent inincreasing personal research output, rather thanin improving their teaching, particularly if this re-quires such a large step in the dark.

Computer usage is still actively avoided by manymathematicians and by many mathematics teach-ers. The problem at the tertiary level is particu-larly great, for the gulf between the traditional lec-ture often given to a hundred or more students andthe classroom/laboratory in which students interactwith computers is enormous. To bridge this gulf willneed considerable investment in both material andhuman resources. Time, assistance and in-servicetraining will have to be provided on a scale un-precedented at this level. Particular attention willhave to be directed at those teachers who still havemany years - even decades - to go before they re-tire from teaching. First, however, the necessity forchange will have to be accepted, and this will onlycome through clear, unequivocal demonstration ofthe benefits which can accrue from innovation.

The current problem now is to make all teach-ers able to use computers in teaching, or to knowwhy they will not use them! This leads to differentproblems:

The availability of computers in the teachers' en-vironment: Can they easily find and use a com-puter at home for preparing their teaching andelaborating activities? Can they easily find anduse a computer in the school? Are computerseasily available in classrooms?The user-friendliness of hardware and software:Will it take hours and hours to prepare a lessonwith computers, and will very specialised abili-ties to use such software be needed?The integration of the computer in the teach-ing strategy and in the learning environment.The computer is a tool among others, and itsuse must be integrated in a pedagogical strat-egy. Textbooks, homework and all the activitiesof the learner must take this into account.The computer does not only change the teacher's

role, but also the attitude and activities of the


students. The relationship is not only betweenthe teacher and the learner: The computer takesits place in the relation, and it also develops thegroup work and the project activities. Learningfrom pupil-pupil talk is one of the components ofthe learning environments provided by new tech-nologies. Different types of environments, differ-ent teaching methods and different strategies canbe used by teachers. Computers change the organ-isation of education, and give teachers the role of apedagogical-engineer in elaborating their strategy,in preparing their teaching, in choosing among theresources available and the tools and products theywill use.

3.3 Some particular uses of the computer inthe classroom

We have already remarked on the way in whichcomputers can assist in the introduction, develop-ment and reinforcement of mathematical concepts,in building up intuition and insight, etc. In this sec-tion we look at particular ways in which they canbe used within the classroom.

(a) Graphic possibilities

Many of the applications of computers in teach-ing make use of the possibilities provided for graphicdisplay. There can be no doubt about their value inproviding quickly good quality graphic illustrationswhich can help build intuition. The example of or-dinary differential equations such as x' = x2 t,whose .olutions cannot be written down in elemen-tary terms, is now widely known and used: Visu-alising the field of tangents and visualising manysolutions of the equation make the student betterable to understand the concepts which intervene inthis domain. Moreover, this allows them to discussexciting questions concerning the behaviour of solu-tions.

Where the computer scores over many other me-dia is that graphics capabilities now enable move-ment as well as static diagrams to be portrayed.This, of course, was true of the film. Yet now thepossibility of being able to change parameters adds acompletely new dimension to the teaching/learningexperience.

Much interesting and high quality graphic soft-ware is now available and allows visual representa-tions from areas such as calculus, differential equa-tions, linear algebra, numerical analysis, and geom-etry.

A famous example in geometry is that of Cabri-Geometre which allows pupils to draw geometrical

figures very easily, and to modify them by movingsome elements (points, lines, etc.), and see at thesame time how the figure evolves. Invariants andloci can be visualised in a very user-friendly way.Such software can be used by the teacher for demon-strating or by the students in an interactive way.

(b) Many types of utilities are available for usein teaching. Spreadsheets are the best known ex-ample. They provide a good environment for intro-ducing many concepts in arithmetic, algebra, andeven calculus. At a very elementary level, they per-mit interesting activities about the concepts of vari-ables, parameters, unknowns, etc. They also pro-vide nice illustrations of iteration. They are increas-ingly used in teaching.

(c) Databases are now more easily accessible.They suggest documentation activities, they allowstudents to look for sophisticated information andso develop project work. They also give teachers thepossibility to use or build large sets of exercises andactivities. The distant interrogation of databases isnow common and enlarges the resources for teach-ers.

(d) Artificial intelligence and problem-solv-ing tools are developing. The first step is to havesoftware and tools able to solve mathematical prob-lems. The second step is to produce software ableto help students in problem solving.

(e) Hypermedia and multimedia products:These allow the integration of different media, andtheir combination for educational uses. They al-low activities which are not 'linear', but in whichusers may build their own paths and organise theirown learning. They considerably enrich other ed-ucational tools, linking and making simultaneouslyavailable all existing types of software and other ed-ucational technology products. We surmise that inthe future this domain will provoke great changes inthe use of technology in education.

(f) Self-evaluation and individualised in-struction

One of the advantages of the computer is that ithelps the individualisation of teaching and learning.This is not only because the student can sometimeswork alone with the computer, but mostly becausethe computer can help to provide a teaching envi-ronment which matches the needs of each student

the way he learns, the right speed for her, theappropriate activities.

The computer can provide a tool for self-evaluation and can help students to take charge of


the organisation of their own work. It is a diffi-cult problem for students to judge how well theyare coping with a subject. One use of comput-ers is to enable students to test themselves. Ques-tion banks can be made available and instantaneousscoi e. given.

The advantages of Computer-assisted Learning(CAL) for individualised instruction have, of course,been argued for some twenty years, namely that thecomputer can offer non-threatening, individualisedresponses to students. There have, indeed, been sev-eral demonstrations of the value of CAL, for exam-ple, PLATO in the USA. However, as the cognitivecomplexity of what has to be learned increases, thedifficulties of producing adequate software becomevery great.

The problems become less pronounced when theaim of the program is to revise or to exercise and notto teach. Thus 'Recalling Algebra' and 'RecallingMathematics' (Kinch) are examples of software de-signed to help students prepare for the Entry LevelMathematics Exam at California State Universitywhich have been favourably received.

More and more, educational software includes a"counsellor", helping the student to make his or herway through the activities of the software, evaluat-ing him or her, and individualising the activities.

(g) Assessment and Recording

The computer can be used for testing students'progress. Some software employs the random gen-eration of test items. Such testing can, of course,go far beyond reliance on multiple choice items andcan measure responses other than correct and in-correct. Such newer testing procedures, which canbe designed to capitalise on the graphic potentiali-ties of the computer, can reduce testing time, allowtests to be broken off and resumed at any time, of-fer immediate summaries and analyses, and assignspecific help for identified deficiencies.

The obvious disadvantages include preparationcosts and the need to provide ready access to a com-puter. Open-ended testing of projects or personalproblem solving is at present difficult, but begin-nings are being made. Computer-assisted recordingalso has great potential.

A computer at home, or a computer easily usableat school, enables students to use individualised setsof data for homework or assessments

Very interesting examples of 'learning creditcards' are being experimented with: The card con-tains information about the learner, and gives himor her access to appropriate software and activities.

(h) Pocket calculators must be mentionnedhere. Even if their possibilities are small in com-parison with computers, they are improving veryrapidly: We now have calculators with graphicpossibilities and even with symbolic capabilities.And the permanent availability of pocket calcula-tors gives them great power. In many countries, theuse of pocket calculators in mathematics has beenintroduced into the curricula so that all teachers andall pupils use calculators.

(i) Student errorsRelated to the possibilities described above :s

that of investigating the errors which students makein learning mathematics. Such information can beused in two ways: To help the student remove mis-conceptions, which is its role in individualised CAL,or to help the mathematics educator to identify spe-cific points of difficulty and to design curricula withthese in mind. Errors are symptoms which allowus not only to identify stumbling blocks, but alsoto form an impression of the student's conceptions.The computer allows students to respond to theirerrors in a new way: They can identify and con-trol them themselves. Getting rid of them can evenbecome a motivation for learning.

One example of the use of the computer to detectand correct errors is found in Okon-Rinne's course-ware. This enables a student to choose a basic func-tion such as f(x) = lx1 and then to experiment withthe effects which translations and reflections have onit. Thus the graph can be translated vertically orhorizontally or reflected in the vertical axis. Simul-taneously the function changes to correspond to thenew graph. The intention is to detect, such commonerrors as confusing f(x) = Ix 21 with f(x) = lx+21,or f(x) = Ix+21 with f(x) = lx1+2. When an erroris detected, a tutorial subroutine is activated and af-terwards the student has the option of continuing orbranching back to an earlier unit.

3.4 Student responses to work with comput-ers

It is common to talk about the enthusiasm gen-erated in students by computer-based systems. Inmany experiments, it is claimed that this has re-sulted in students developing a new interest in thesubject and that the general level of student activityhad increased as a result of reacting with a computerpackage. Not only had activity increased, but so hadconfidence. Dubinsky typically reports (of a courseon discrete structures): 'This approach makes fora lively course in which students are responsive inclass and active outside class. In comparison with

t.t,


similar groups to whom I have tried to teach thismaterial, these students seem to be more prone tospeak in terms of sets and less confused by compli-cated logical statements'.

It must not be thought, however, that enthusi-asm can be automatically generated through the useof a computer. Much will depend on the studentsand the teaching situation; there are also negativeexperiences to report! One must also judge on howmuch students learn as well as the enthusiasm theyshow whilst engaged on the task.

Here one is faced with a new problem in teach-ing. Students can frequently appear fascinated bycomputer demonstrations or by working interac-tively with a computer, but what happens 'whenthe machine is switched off'? Will the students onlybe able to imitate what they have seen or will theyobtain a deeper understanding of concepts?

It is recognised that the value of much computerwork is largely dependent upon the follow-up activ-ities which 'must guard against the possibility thatthe machine is doing all the work and providing allthe answers'. Many traditional activities will stillhave to be carried out, thus suggesting yet againthat the computer's main contribution will be to en-hance student understanding and not to save timefor the lecturer. The introduction of the computeris unlikely to solve (or even ease) the problem ofoverloaded syllabuses.

3.5 The provision of software

The way software is conceived and designedevolves very quickly. The progress of technologyand the development of multimedia tools enrichesthe possibilities for pedagogical uses. The roles ofteachers, of pedagogs and of computer scientists insoftware design has evolved. Very user-friendly sys-tems allow any teacher to create teaching situationswith computers.

Current software resources may be considered inseveral categories:

(a) Sophisticated systems (in computer terms)such as the symbolic manipulation systems, largestatistical packages, etc., form the first cat gory.These systems have been developed in a 'goal-oriented' fashion, that is they seek to provi e solu-tions to specific mathematical problems. T ey havenot needed to consider to any great exte t 'peda-gogical design'. Interest in their use as pedagogicaltools is growing.

Commercial companies exist with an interestin marketing this type of software and research

mathematicians are involved in creating such sys-tems. As a result, sophisticated packages are self-perpetuating, Since they will exist, we need to un-derstand their pedagogical uses and the possiblydramatic effects they could have on current mathe-matics education.

(b) Less sophisticated in computer terms butstill very demanding in pedagogical design are thesoftware packages suitable for use on a microcom-puter. These packages attempt to aid the stu-dent's mathematical development and employ suchthemes as visualisation, simulation, exploration andproblem-solving. They may be used by studentsworking alone, in groups, or with a teacher. Manyindividuals and groups are writing such packages.Many are also provided by educational softwarecompanies.

A major problem arises here, The production ofpackages that can be recommended for widespreaduse as pedagogically sound and well-tested is an,xpensive, complicated task requiring considerableprofessional resources. It should involve fundamen-tal research based on the structured observation ofthe materials in use in parallel with the develop-ment of the materials. Thus the team may need toinclude mathematicians, educators, psychologists,computer scientists, graphic designers, publishersand editors. The financial needs of such a groupwould be considerable.

(c) General purpose programming languages canbe used as tools aiding students' mathematical de-velopment and are a readily available teaching re-source. Extension of such languages or even cre-ation of new ones expressly for this purpose wouldbe welcome.

This brief discussion of the present positionpoints out the need (i) to establish channels of com-munication so that researchers and educators areaware of resources currently available and (ii) to es-tablish structured research studies using currentlyavailable resources in order to gain and share un-derstanding of their use as pedagogical tools.

The emergence of software packages has raiseda new problem for mathematics teachers, that ofblack boxes, for they often/usually produce answerswithout giving any hint of the way in which theywere obtained. This may well conceal a wealth ofdeep mathematics. (It could, of course, be arguedthat the problem is not new, but merely heightened -for students have been employing algorithms whoseworkings they did not undefstand for centuries!).

How can students learn (be taught/encouraged)to look critically at the answers supplied? Howmuch should students be required to know about


the workings of black-boxes before being allowed touse them? For example, there are packages whichinvert matrices. If such a package uses floating-pointarithmetic, it can give answers which should not beaccepted at face value. At least students should bewarned about this or, better, should learn to recog-nise when this has occured.

3.6 Cultural, social and economic factors

We have written of the computer as an aid tomathematics teaching and learning. So is the over-head projector. The difference though between thetwo tools is not, however, solely the enormous dif-ference in the range of possibilities opened up bythe former. Equally, it springs from the enormouseffect which the computer is having upon societyoutside the confines of educational systems. As aresult society has expectations concerning comput-ers and their use, expectations which often have lit-tle basis in reality. Students too have expectationsabout their use. There are then enormous pressureson educators at all levels to use computers, not nec-essarily for their intrinsic value, but because societyexpects it, and not because to do so might be con-sidered old-fashioned and reactionary.

It will be difficultfor computers to be used effec-tively in education until society has become betterinformed about their power and limitations. Un-realistic expectations must be strongly discouraged.There is a danger that false advertising by computercompanies and software developers, and a pressurefrom various sections o society, could lead to ill-designed, over-optimistic innovation and, in turn,to a backlash comparable with that of the 1970sresulting from the hasty introduction of the 'NewMath'. Political decision makers in some countriesare 'pushing' computers and computer-related cur-ricula into education without adequate considera-tion of objectives and consequences.

It is important, therefore, to realise that:reasonable use of computers in education re-quires software programs and packages whoseeducational standards and qualities are compa-rable with the technical ones offered by the avail-able hardware,integrating computers into the curriculum mustbe coordinated with teacher/faculty in-service,professional development programs,educational budgets must be prepared to permitappropriate expenditure on hardware, software,and teacher development,no curriculum should remain stagnant for a longperiod.

Not all problems associated with computers ineducation can be anticipated. Many questionsneed to be answered through research initiatives di-rected at investigating the possibilities, limitationsand possible dangers of computer use in education.Some causes for concern are:

uniformity in students' thinking and reasoningcould arise from overuse of computers in thelearning process,standardisation of software development (in anattempt to form a commercial market) may leadto mediocrity and conformity,subtleties of communication between teachersand students could be impoverished by over-using computers,insensitive working with computers could ad-versely influence the total intellectual develop-ment of students (of their intuitive thinking, cre-ativity, perception, etc.).The case of developing countries demands spe-

cial attention. For them the provision and mainte-nance of hardware creates great problems. More-over, scarce resources must be husbanded carefully.The computer could offer special advantages tothem; on the other hand the absence or shortage ofcomputers could widen still further the gap betweenthem and the developed countries. Several confer-ences have considered the question of new technolo-gies in education for developing countries (see, forexample, Amara, Boudriga and Harzallah, 1986).

3.7 Conclusion

We are only experiencing the very beginning ofthe effect of computers on the teaching and learn-ing of mathematics. Gradually, we are beginningto take advantage of some of their more obviouspossibilities such as their quick and accurate pro-duction of graphical material, their quick and accu-rate (though not always precise) arithmetic, analyselarge quantities of data.

In numerous publications one can see examplesof mathematical situations for which the computerand informatics allow us to see and approach situa-tions from a new point of view. Obvious exampleswhich spring to mind are the many applications instatistics (dealing with vast quantities of data), inprobability (with all the possibilities opened up forsimulation by pseudo-random number generators);in geometry, too, there is a range of interesting ac-tivities production and processing of images, curveplotting, the transformation of images (translations,reflections, etc.), loci, exploration of images and fig-ures. The dynamic aspect dominates here: One can


visualise instantly the effect of varying a parameter.In linear algebra, an algorithmic approach furnishesa tool both for calculations and also for demon-stration. Here again the dynamic aspect plays animportant role: To see a matrix steadily assume adiagonal form is very different from obtaining theresult once and for all after a long and involved cal-culation. But it is above all in analysis that theopportunities to utilise informatics are richest andmost numerous. The study of numbers, of func-tions, of the solution of equations, observation andstudy of sequences and series (and in particular oftheir speed of convergence), integral calculus, differ-ential equations, asymptotic expansions, discretisa-tion, power series for functions, etc. In addition tothese `classical' fields where the use of the computerarises naturally, one has also seen developments innewer fields which have occurred largely because ofcomputers: Formal symbolic logic is a striking in-stance; discrete mathematics can provide us withother examples. The computer is not only an aidfor computation and demonstration, but a force fordevelopment.

In all of these cases, the contribution of the com-puter takes several forms. Firstly, it. is a calculatingtool allowing numerous and rapid calculations; italso serves to place renewed emphasis on numeri-cal methods, and thus on the study of algorithms;and, especially, it is a pedagogical tool for promot-ing teaching and learning.

However, let us reiterate, the act of using a com-puter does not automatically lead to an improve-ment. It is not a magic wand! Like all tools, it. canserve us badly; we must learn how to get. the bestfrom it.

The development of technology (computers be-coming smaller and cheaper) and the developmentof new tools (such as multimedia ones) will certainlyprovoke very large changes in education. Complexlearning environments and integrated software willbecome more and more available. The technologyage in education is still to come!

Computers are now widely to be found in schoolsand universities, but they are not always widelyused. Teachers are being trained in their use, butprincipally in techniques and programming, and thequestion of giving them a true pedagogical trainingis not totally solved. It is also necessary to bearin mind that if we wish to change the educationalsystem, then there will be a need simultaneously toreform both the training gi. 1 to those preparing toteach in schools and universities and also the con-tinuing education of existing teachers. Many inter-esting ai.;d rich experiments have been done, many

enthusiastic teachers have produced activities andtools, and have tried new pedagogical strategies us-ing computers. We now need to have ALL teachersable to use computers as a natural tool, and to in-tegrate them into their teaching.

At the same time there is the need to carry outmuch research and experimentation so that, we mayeffectively understand and control the impact of theuse of the computer on students' learning and ontheir conceptions and representations of mathemat-ical objects. Oily after such studies will we be ableto provide high quality software and, most impor-tantly, a new range of didactical activities, tasks andsituations to enhance !earning.

REFERENCES

Banchoff, T. et al (Eds.) [19831: Educational Com-puting in Mathematics, ECM87, Amsterdam:North Holland.

Amara, M., Boudriga, N. and Harzallah, K. (Eds.)[198(31: L'Informatique et l'enseignement desmathernatiques dans les pays en vole dedeveloppement, Proceedings of the first ICO-MIDC Symposium, Paris: ICOMIDC and UN-ESCO.

Cornu, B. [1992]: I, 'Ordinateur pour enseigner lesmathematigues, Paris: Presses Universitairesde France.

Hirst, A. and Hirst, K. (Eds.) [1988]: Proceedingsof the Sixth International Congress on Mathe-matics Education, Budapest, 1988, Budapest:Janos Bolyai Mathematical Society.

Johnson, D.C. and Lovis, F. (Eds.) [1987]: infor-matics and the Teaching of Mathematics, Pro-ceedings of the IFIP TC 3/WG 3.1 WorkingConference on Informatics and the Teachingof Mathematics, Sofia, Bulgaria, 16 - 18 May,1987. Amsterdam: North-Holland. ISBN 0-444- 70325 -X.

Tinsley, J.D. and van Weert, T.J. (Eds.) [1989]:Educational Software at the Secondary Level,Amsterdam: Elsevier.

7

LIVING WITH A NEW MATHEMATICAL SPECIES

Lynn Arthur SteenSt. Olaf College, Northfield, Minnesota 55077, U.S.A.

Computers are both the creature ani the creatorof mathematics. They are, in the apt phrase of Sey-

-'r Papert, "mathematics-speaking beings." J.Day. --alter, in his stimulating book Turing's Man[Bolter, .,984], calls computers "embodied mathe-matics." Computers shape and enhance the powerof mathematics, while mathematics shapes and en-hances the power of computers. Each forces theother to grow and change, creating, in ThomasKuhn's language, a new mathematical paradigm.

Until recently, mathematics was a strictly hu-man endeavor. But suddenly, in a brief instant onthe time scale of mathematics, a new species hasentered the mathematical ecosystem. Computersspeak mathematics, but in a dialect that is difficultfor some humans to understand. Their number sys-tems are finite rather than infinite; their addition isnot commutative; and they don't really understand"zero," not to speak of "infinity." Nonetheless, theydo embody mathematics.

The core of mathematics is changing under theecological onslaught of mathematics-speaking com-puters. New specialties in computational complex-ity, theory of algorithms, graph theory, and formallogic attest to the impact that computing is hav-ing on mathematical research. As Arthur Jaffe hasargued so well (in [Jaffe, 1984]), the computer rev-olution is a mathematical revolution.

New Mathematics for a New AgeComputers are discrete, finite machines. Unlike

a Turing machine with an infinite tape, real ma-chines have limits of both time and space. Theirs isnot an idealistic Platonic mathematics, but a math-ematics of limited resources. The goal is not just toget a result, but to get the best result for the least ef-fort. Optimization, efficiency, speed, productivity-these are essential objectives of modern computermathematics.

Computers are also logic machines. They em-body the fundamental engine of mathematicsrigorous propositional calculus. The first celebratedcomputer proof was that of the four-color theorem:the computer served there as a sophisticated ac-countant, checking out thousands of cases of reduc-tions. Despite philosophical alarms that computer-based proofs change mathematics from an a pri-ori to a contingent, fallible subject (see, e.g., [Ty-moczko, 1979]), careful analysis reveals that noth-ing much has really changed. The human practice

33

of mathematics has always been fallible; now it hasa partner in fallibility.

Research on the so-called Feigenbaum constantreveals just how far this evolution has progressed injust a few years: computer-assisted investigationsof families of periodic maps suggested the presenceof a mysterious universal limit, apparently indepen-dent of the particular family of maps. Subsequenttheoretical investigations led to proofs that are truehybrids of classical analysis and computer program-ming [Eckmann, 1984], showing that computer-assisted proofs are possible not just in graph theory,but also in functional analysis.

Computers are also computing machines. Byabsorbing, transforming, and summarizing massivequantities of data, computers can simulate reality.No longer need the scientist build an elaborate windtunnel or a scale model refinery in order to test en-gineering designs. Wherever basic science is wellunderstood, computer models can emulate physicalprocesses by carrying out instead the process im-plied by mathematical equations. Whereas mathe-matical models used to he primarily tools used bytheoretical scientists to formulate general theories,now they are practical tools of enormous value inthe everyday world of engineering and economics.

It has been just over fifty years since Alan Turingdeveloped his seminal scheme of computability [Tur-ing, 1936] in which he argued that machines coulddo whatever humans might hope to do. In abstractterms, what he proposed was a universal machine ofmathematics (see [Hodges, 1983] for details). It tooktwo decades of enginee;ing effort to turn Turing'sabstract ideas into productive real machines. Dur-ing that same period abstract mathematics flour-ished, led by Bou'haki, symbolized by the "gener-alized abstract nonsense" of category theory. Butwith abstraction came power, with rigor came cer-tainty. Once real computers emerged, the complex-ity of programs quickly overwhelmed the informaltechniques of backyard programmers. Formal meth-ods became de rutieur; even the once-maligned cat-egory theory is now enlisted to represent finite au-tomata and recursive functions (see, e.g., [Beckman,1984], [Lewis, 1981]). Once again, as happened be-fore with physics, mathematics became more effica-cious by becoming more abstract.


The Core of the CurriculumTwenty-five years ago in the United States

the Committee on the Undergraduate Program inMathematics (CUPM) issued a series of reports thatled to a gradual standardization of curricula amongundergraduate mathematics departments [CUPM,1965]. Shortly thereafter, in 1971, Garrett Birkhoffand J. Barkley Rosser presented papers at a meetingof the Mathematical Association of America con-cerning predictions for undergraduate mathemat-ics in 1984. Birkhoff urged increased emphasis onmodelling, numerical algebra, scientific computing,and discrete mathematics. He also advocated in-creased use of computer methods in pure 'math-ematics: "Far from muddying the limpid watersof clear mathematical thinking, [computers] makethem more transparent by filtering out most of themessy drudgery which would otherwise accompanythe working out of specific illustrations." [Birkhoff,1972, p. 651] Rosser emphasized many of the samepoints, and warned of impending disaster to un-dergraduate mathematics if their advice went un-heeded: "Unless we revise [mathematics courses]so as to embody much use of computers, most ofthe clientele for these courses will inst .01 be takingcomputer courses in 1984." [Rosser, 1972, p. 639]

In the first decade after these words were writ-ten, U.S. undergraduate and graduate degrees inmathematics declined by 50%. The clientele fortraditional mathematics migrated to computer sci-ence, and the former CUPM consensus all but dis-appeared. In 1981 CUPM issued a new report, thisone on the Undergraduate Program in Mathemat-ical Sciences ([CUPM, 1981], reprinted in [CUPM,1989]). Beyond calculus and linear algebra, theycould agree on no specific content for the core of amathematics major: "There is no longer a commonbody of pure mathematical information that every[mathematics major] should know."

The symbol of reformation became discretemathematics. Anthony Ralston argued forcefullythe need for change before both the mathematicscommunity [Ralston, 1981] and the computer sci-ence community [Ralston, 1980]. Discrete math-ematics, in Ralston's view, is the central link be-tween the fields. The advocacy of discrete math-ematics rapidly became quite vigorous (see, e.g.,[Kemeny, 1983], [Ralston, 1983,] and [Steen, 1984]),and the Sloan Foundation funded experimental cur-ricula at six institutions to encourage developmentof discrete-based alternatives to standard freshmancalculus. The impact of this work can be seen inthe growth of courses and publications: in the fiveyear period from 1985 to 1990, hundreds of courses

were created and over 40 new textbooks in discretemathematics were published.

Soon calculus itself came under scrutiny, as anatural force for counter-reformation. Critics ar-gued that the power of computation and the ubiq-uity of applications had changed fundamentally therole of calculus in the practice of mathematics (e.g.,[Douglas, 1986; Steen, 1988]). The National ScienceFoundation launched diverse projects to reshape thenature of calculus instruction. Virtually all of theseprojects feature supporting roles for the numeric,symbolic, and graphic power of computers.

The need for consensus on the contents of un-dergraduate mathematics is perhaps the most im-portant issue facing American college and univer-sity mathematics departments [CUPM, 1989]. Onthe one hand departments that have a strong tra-ditional major often fail to provide their studentswith the robust background required to survive theevolutionary turmoil in the mathematical sciences.Like the Giant Panda, these departments depend forsurvival on a dwindling supply of bamboo--strongstudents interested in pure mathematics. On theother hand, departments offering flabby compositemajors run a different risk: by avoiding advanced,abstract requirements, they often misrepresent thetrue source of mathematical knowledge and power.Like zoo-bred animals unable to forage in the wild,students who have never been required to mastera deep theorem are ill-equipped to master the sig-nificant theoretical complications of real-world com-puting and mathematics.

Computer Literacy

Mathematical scientists at American institutionsof higher education are responsible not only for thetechnical training of future scientists and engineers,but also for the technological literacy of the edu-cated publicof future lawyers, politicians, doctors,educators, and clergy. Public demand that collegegraduates be prepared to live and work in a com-puter age has caused many institutions to introducerequirements in quantitative or computer literacy.

In 1981 the Alfred P. Sloan Foundation initiatedcurricular exploration of "the new liberal arts," therole of applied mathematical and computer sciencesin the education of students outside technical fields."The ability to cast one's thoughts in a form thatmakes possible mathematical manipulation and toperform that manipulation ... [has] become essen-tial in higher education, and above all in liberal ed-ucation." [Koerner, 1981, p. 6] Others echoed thiscall for reform of liberal education. David Saxon,President of the University of California wrote in

a Science editorial that liberal education "will con-tinue to be a failed idea as along as our studentsare shut out from, or only superficially acquaintedwith, knowledge of the kinds of questions sciencecan answer and those it cannot." [Saxon, 1982]

Too often these days the general public viewscomputer literacy as a modern substitute for math-ematical knowledge. Unfortunately, this often leadsstudents to superficial courses that emphasize vo-cabulary and experiences over concepts and princi-ples [Steen, 1985]. The advocates of computer lit-eracy conjure images of an electronic society dom-

. inated by the information industries. Their sloganof "literacy" echoes traditional educational values,conferring the aura but not the logic of legitimacy.

Typical courses in computer literacy are fille('with ephemeral details whose intellectual life willbarely survive the students' school years. Thesecourses contain neither a Shakespeare nor a Newton,neither a Faulkner nor a Darwin; they convey nofundamental principles nor enduring truths. Com-puter literacy is more like driver education than likecalculus. It teaches students the prevailing rules ofthe road concerning computers, but does not leavethem well-prepared for a lifetime of work in the in-formation age.

Algorithms and data structures are to computerscience what functions and matrices are to math-ematics. As much of the traditional mathematicscurriculum is devoted to elementary functions andmatrices, so beginning courses in computingbywhatever nameshould stress standard algorithmsand typical data structures. As early as studentsstudy linear equations they could also learn aboutstacks and queues; when they move on to conic sec-tions and quadratic equations, they could in a par-allel course investigate linked lists and binary trees.

Computer languages can (and should) be stud-ied for the concepts they representprocedures inPascal and C, recursion and lists in Lisp--ratherthan for the syntactic details of semicolons and linenumbers. They should not be undersold as meretechnical devices for encoding problems for a dumbmachine, nor oversold as exemplars of a new formof literacy. Computer languages are not modernequivalents of Latin or French: they do not deal innuance and emotion, nor are they capable of per-suasion, conviction, or humor. Although computerlanguages do represent a new and powerful way tothink about problems, they are not a new fcrm ofliteracy.

Computer Science

In the United States, computer science programs

A New Mathematical Species 35

cover a broad and varied spectrum, from buginess-oriented data processing curricula, through manage-ment information science, to theoretical computerscience. All of these intersect with the mathematicscurriculum, each in different ways.

To help clarify these conflicting approaches,Mary Shaw of Carnegie Mellon University put to-gether a composite report on the undergraduatecomputer science curriculum. This report is quiteforceful about the contribution mathematics makesto the study of computer science: "The most im-portant contribution a mathematics curriculum canmake to computer science is the one least likely tobe encapsulated as an individual course: a deep ap-preciation of the modes of thought that characterizemathematics." [Shaw, 1984, p. 55]

The converse is equally true: one of the moreimportant contributions that computer science canmake to the study of mathematics is to de.. Jop instudents an appreciation for the power of abstractmethods when applied to concrete situations. Stu-dents of traditional mathematics used to study asubject called "Real and Abstract Analysis;" stu-dents of computer science now can take a coursetitled "Real and Abstract Machines." In the for-mer "new math," as well as in modern algebra, stu-dents learned about relations, abstract versions offunctions; today business students study "relationaldata structures" in their computer courses, and ad-vertisers tout "fully relational" as the latest innova-tion in business software.

An interesting and pedagogically attractive ex-ample of the power of abstraction made concrete canbe seen in the popular electronic spreadsheets thatare marketed under such trade names as Lotus andExcel. Originally designed for accounting, they canas well emulate cellular automata or the Ising modelfor ferromagnetic materials [Mayes, 1983]. Theycan also be "programmed" to carry out most stan-dard mathematical algorithmsthe Euclidean al-gorithm, the simplex method, Euler's method forsolving differential equations [Arganbright, 1985].An electronic spreadsheetthe archetype of ap-plied computingis a structured form for recur-sive proceduresthe fundamental tool of algorith-mic mathematics. It is a realization of abstractmathematics, and carries with it much of the powerand versatility of mathematics.

Computers in the Classroom

Just as the introduction of calculators upset thecomfortable pattern of primary school arithmetic,so the spread of computers will upset the traditionsof secondary and tertiary mathematics. This year


long division is passe; next year integration will beunder attack.

The impact of computing on secondary schoolmathematics has been the subject of many discus-sions in the United States (e.g., [Steen, 1987]). JimFey, coordinator of two assessments ([Corbitt, 1985;Fey, 1984]), described these efforts as "an unequivo-cal dissent from the spirit and substance of efforts toimprove school mathematics that seek broad agree-ment on conservative curricula." [Fey, 1984, p. viii]The new Curriculum and Evaluation Standards forSchool Mathematics [NCTM, 1989] of the NationalCouncil of Teachers of Mathematics as well as otherrecommendations from the U.S. National Academyof Sciences ([NRC, 1989; MSEB, 1990]) set expec-tations for school mathematics that employ calcula-tors and computers in every appropriate manner.

Teachers in tune with the computer age seekchange in both curriculum and pedagogy. But theinertia of the system remains high. For example,the 1982 International Assessment of Mathematicsdocumented that in the United States calculatorsare never permitted in one-third of the 8th gradeclasses, and rarely used in all but 5% of the rest[McKnight, 1987]. Recent data [NAEP, 1991] showsome improvement, but still fall well short of theNCTM recommendations.

Laptop computers are now commonthey costabout as much as ten textbooks, but take up onlythe space of one. Herb Wilf argues (in [Wilf, 1982])that it is only a matter of time before students willcarry with them a device to perform all the al-gorithms of undergraduate mathematics. RichardRand, in a survey [Rand, 1984] of applied researchbased on symbolic algebra agrees: "It will not belong before computer algebra is as common to engi-neering students as the now obsolete slide rule oncewas." Just five years after Wilf's article appeared,the same journal carried a review [Nievergelt., 1987]of the first. pocket calculator with symbolic algebracapabilities.

Widespread use of computers that do school andcollege mathematics will challenge standard educa-tional practice [Steen, 1990]. For the most part,computers reinforce the student's desire for cor-rect answers. In the past, their school uses haveprimarily extended the older "teaching machines:"programmed drill with pre-determined branches forall possible responses. But the recent linkingof symbolic algebra programs with so-called "ex-pert systems" into sophisticated "intelligent tutors"has produced a rich new territory for imaginativecomputer-assisted pedagogy that advocates claimcan rescue mathematics teaching from the morass

of rules and template-driven tests (see e.g., [Smith,1988; Zorn, 1987]).

It is commonplace now to debate the wisdomof teaching skills (such as differentiation) that com-puters can do as well or better than humans. Isit really worth spending one month of every yearteaching half of a country's 18-year-old studentshow to imitate a computer? What is not yet socommon is to examine critically the effect of ap-plying to mathematics pedagogy computer systemsthat are themselves only capable of following rulesor matching tempi, it wise to devise sophisti-cated computer sys.ems to teach efficiently preciselythose skills that computers can do better than hu-mans, particularly those skills that make the com-puter tutor possible? In other words, since com-puters can now do the calculations of algebra andcalculus, should we use this power to reduce thecurricular emphasis on calculations or to make theteaching of these calculations more efficient? Thisis a new question, with a very old answer.

Let Us Teach Guessing

Forty years ago George POlya wrote a brief pa-per with the memorable title "Let Us Teach Guess-ing" [POlya, 1950]. It is not differentiation that ourstudents need to learn, but the art of guessing. Amonth spent learning the rules of differentiation re-inforces a student's ability to learn (and live by)the rules. In contrast, time spent making conjec-tures about derivatives will teach a student some-thing about the art of mathematics and the scienceof order.

With the aid of the mathematics-speaking com-puter, students can for the first time learn collegemathematics by discovery. This is an opportunityfor pedagogy that mathematics educators cannot af-ford to pass up. Mathematics is, after all, he sci-ence of order and pattern, not just a mechanism forgrinding out formulas. Students discovering math-ematics gain insight into the discovery of pattern,and slowly build confidence in their own ability tounderstand mathematics. Formerly, only studentsof sufficient genius to forge ahead on their own couldhave the experience of discovery. Now with comput-ers as an aid, the majority of students can experi-ence for themselves the joy of discovery.

Metaphors for Mathematics

Two metaphors from science are useful for un-derstanding the relation between computer science,mathematics, and education. Cosmologists long de-bated two theories for the origin of the universethe Big Bang theory, and the theory of Continuous

Creation. Current evidence tilts the cosmology de-bate in favor of the Big Bang. Unfortunately, thisis all too often the public image of mathematics aswell, even though in mathematics the evidence fa-vors Continuous Creation.

The impact of computer science on mathemat-ics and of mathematics on computer science is themost powerful evidence available to beginning stu-dents that mathematics is not just the product ofan original Euclidean big bang, but is continuallycreated in response to challenges both internal andexternal. Students today, even beginning students,can learn things that were simply not known twentyyears ago. We must not only teach new mathemat-ics and new computer science, but we must teach aswell the fact that this mathematics and computerscience is new. That's a very important lesson forthe public to learn.

The other apt metaphor for mathematics comesfrom the history of the theory of evolution. Priorto Darwin, the educated public believed that formsof life were static, just as the educated public oftoday assumes' that the forms of mathematics arestatic, laid down by Euclid, Newton, and Einstein.Students learning mathematics from contemporarytextbooks are like the pupils of Linnaeus, the greateighteenth-century Swedish botanist: they see astatic, pre-Darwinian discipline that is neither grow-ing nor evolving. Learning mathematics for moststudents is an exercise in classification and memo-rization, in labeling notations, definitions, theorems,and techniques that are laid out in textbooks as somuch flora in a wondrous if somewhat abstract Pla-tonic universe.

Students rarely realize that mathematics con-tinually evolves in response to both internal andexternal pressures. Notations change; conjecturesemerge; theorems are proved; counterexamples arediscovered. Indeed, the passion for intellectual or-der combined with the pressure of new problemsespecially those posed by the computerforce re-searchers to continually create new mathematicsand archive old theories.

The practice of computing and the theory ofcomputer science combine to change mathematicsin ways that arc highly visible and attractive to stu-dents. This continual change reveals to students theliving character of mathematics, restoring to the ed-ucated public some of what the experts have alwaysknownthat mathematics is a living, evolving com-ponent of human culture.

REFERENCES

Arganbright, D.E. [1985]: Mathematical Applica-

A New Mathematical Species 37

Lions of Electronic Spreadsheets, New York:McGraw-Hill.

Beckman, F.S. [1984]: Mathematical Foundations ofProgramming. The Systems Programming Se-ries, Reading, MA: Addison-Wesley.

Birkhoff, G. [1972]: The Impact of Computerson Undergraduate Mathematics Education in1984, Amer. Math. Monthly, 79, 648-657.

Bolter, J.D. [1984]: Turing's Man: Western Cul-ture in the Computer Age, Chapel Hill, NC:University of North Carolina Press.

Committee on the Undergraduate Program inMathematics [1965]: A General Curriculumin Mathematics for Colleges, Washington, DC:Mathematical Association of America.

Committee on the Undergraduate Program inMathematics [1981]: Recommendations fora General Mathematical Sciences Program,Washington, DC: Mathematical Association ofAmerica.

Committee on the Undergraduate Program inMathematics [1989]: Reshaping College Math-ematics. MAA Notes No. 13, Washington,DC: Mathematical Association of America.

Corbitt, M.K., and Fey, J.T. (Eds.) [1985]: TheImpact of Computing Technology on SchoolMathematics: Report of an NCTM Confer-ence, Reston, VA: National Council of Teach-ers of Mathematics.

Douglas, R.G. (Ed.) [1986]: Toward A Leanand Lively Calculus. MAA Notes Number 6,Washington, DC: Mathematical Association ofAmerica.

Eckmann, J.-P., Koch, H. and Wittwer, P. [1984]:A Computer-assisted Proof of Universality forArea-preserving Maps, Memoirs of the Amer-ican Mathematical Society, 47:289 (January).

Fey, J.T., et al. (Eds.) [1984]: Computingand Mathematics: The Impact on SecondarySchool Curricula, Reston, VA: National Coun-cil of Teachers of Mathematics.

Hayes, B. [1983]: Computer Recreations, ScientificAmerican, 22-36, (October).

Ilodges, A. [1983]: Alan Turing: The Enigma, NewYork: Simon and Schuster.

Jaffe, A. [1984]: Ordering the Universe: The Role ofMathematics . Renewing U.S. Mathematics,Washington, DC: National Academy Press.


Kemeny, J.G. [1983]: Finite MathematicsThenand Now in Ralston, A., and Young, G.S.(Eds.), The Future of College Mathematics,pp. 201-208, New York: Springer-Verlag.

Koerner, J.D. (Ed.) [1981]: The New Liberal Arts:An Exchange of Views, New York: Alfred P.Sloan Foundation.

Lewis, H.R., and Papadimitriou, C.H. [1981]: Ele-ments of the Theory of Computation, Engle-wood Cliffs, NJ: Prentice-Hall.

Mathematical Sciences Education Board [1990]: Re-shaping School Mathematics: A Philosophyand Framework for Curriculum, Washington,DC: National Academy Press.

McKnight, C.C., et al [1987]: The Underachiev-ing Curriculum: Assessing U.S. School Math-ematics from an International Perspective,Champaign, IL: Stipes.

National Assessment of Education Progress [1991]:The State of Mathematics Achievement,Washington, DC: U.S. Department of Educa-tion.

National Council of Teachcrs of Mathematics [1989]:Curriculum and Evaluation Standards forSchool Mathematics, Reston, VA: NationalCouncil of Teachers of Mathematics.

National Research Council [1989]: EverybodyCounts: A Report to the Nation on the Fu-ture of Mathematics Education, Washington,DC: National Academy Press.

Nievergelt, Y. [1987]: The Chip with the Col-lege Education: The HP-28C, Amer. Math.Monthly, 94, 895-902.

Polya, G. [1950]: Let Us Teach Guessing, Etudes dePhilosophie des Sciences, Neuchatel: Griffon,pp. 147-154; reprinted in G. POlya, CollectedPapers, Vol. IV, Cambridge, MA: MIT Press,1984, pp. 504-511.

Ralston, A. [1981]: Computer Science, Mathemat-ics, and the Undergraduate Curricula in Both,Amer. Math. Monthly, 88, 472-485.

Ralston, A., and Shaw, M. (1980]: Curriculum '78:Is Computer Science Really that Unmathe-matical?, Communications of the ACM, 23,67-70.

Ralston, A., and Young, G.S. (Eds.) [1983]: TheFuture of College Mathematics, New York:Springer-Verlag.

Rand, R.H. [1984]: Computer Algebra in Ap-plied Mathematics: An Introduction to MAC-SYMA, London: Pitman.

Rosser, 1.B. [1972]: Mathematics Courses in 1984,Amer. Math. Monthly, 79, 635-648.

Saxon, D.S. [1982]: Liberal Education in a Techno-logical Age, Science, 218, 845 (26 November).

Shaw, M. (Ed.) [1984]: The Carnegie-Mellon Cur-riculum for Undergraduate Computer Science,New York: Springer-Verlag.

Smith, D.A.; Porter, G.J.; Leinbach, L.C.; andWenger, Ronald H. (Eds.) [1988): Computersand Mathematics: The Use of Computers inUndergraduate Instruction. MAA Notes No.9, Washington, DC: Mathematical Associationof America.

Steen, L.A. [1984]: 1 + 1 = 0: New Math for a NewAge, Science, 225, 981 (7 September).

Steen, L.A. [1991]: Twenty Questions for ComputerReformers in Demana, F., et. al. Proceedingsof the Conference on Technology in CollegiateMathematics, Reading, MA: Addison-Wesley,pp. 16-19.

Steen, L.A. (Ed.) [1988]: Calculus for a New Cen-tury: A Puinp, Not a Filter, Washington, DC:Mathematical Association of America.

Steen, L.A. [1987]: Who Still Does Math with Paperand Pencil?, Chronicle of Higher Education,A48 (14 October).

Steen, L.A. [1985]: Mathematics and ComputingEducation: A Common Cause, Communica-tions of the ACM, 28, 666-667.

Turing, A.M. [1936]: On Computable Numbers,rith an Application to the Entscheidungs-problem, Proc. London Math. Soc., 2nd Ser.,42, 230-265.

Tymoczko, T. [1979]: The Four Color Problem andits Philosophical Significance, Journal of Phi-losophy, 76, 57-85.

Wilf, H.S. [1982]: The Disk with the College Edu-cation, Amer. Math. Monthly, 89, 4-8.

Zorn, P. [1987]: Computing in UndergraduateMathematics, Notices of the American Math-ematical Society, 34, 917-923 (October).

WHAT ARE ALGORITHMS? WHAT IS ALGORITHMICS?

Stephen B. MaurerSwarthmore College, Swarthmore, Pennsylvania 19081-1397, U.S.A.

Overview. Roughly speaking, an algorithm is aprecise, systematic method for solving some class ofproblems. Algorithmics is the systematic study ofalgorithms how to devise them, describe them, val-idate them and compare their relative merits. Therehave been algorithms in mathematics since ancienttimes, but algorithmics is new. Only with the ad-vent of computers has it been possible to tackle suchlarge and complicated problems that a systematicapproach to algorithms is necessary. Because al-gorithms are now essential in almost all businessand scientific applications of mathematics (as wellas being increasingly important to mathematiciansthemselves and fundamentally important to com-puter scientists), it is important that mathematicseducation take algorithms and algorithmics into ac-count.

This paper has four sections. In Section 1, by farthe longest, we explain what algorithms are in muchmore detail, presenting many examples. In Section2 we do the same for algorithmics. In Section 3 wediscuss several reasons why the study of algorithmsand algorithmics is valuable in mathematics, andwe also discuss some counterarguments. Finally, inSection 4 we make some suggestions for incorporat-ing algorithms and algorithmics into the secondaryand tertiary mathematics curriculum.

1. What Are Algorithms?Algorithms turn input data into output data

through sequences of actions. For instance, an al-gorithm might take two integers and output theirproduct. The rules specifying the algorithm (includ-ing rules specifying what inputs are allowed) mustbe precise enough to satisfy

1. Determinateness. For each allowed input,the first action is uniquely determined, andmore generally, after each action in the se-quence the successor action is uniquely de-termined.

It doesn't do us any good to have an algorithm thatdoesn't stop, so we also require

2. Finiteness. For any allowed input, the al-gorithm must stop after a finite sequence ofactions.

Usually algorithms are devised to solve problems.Such algorithms must be appropriate for the pur-pose at hand:

3. Conclusiveness. When the algorithm termi-nates, it must either output a solution to the

39

problem for the given input, or it must indi-cate that it cannot solve the problem.

In some cases it is reasonable to relax these stringentrequirements; we'll take up this point later. Onecan also ask: how precise is precise? Just how arethe rules to be stated to make them precise? Goodquestion. It depends on who or what you are talkingto. We will also address this further. But let's turnimmediately to some examples.

Example 1: Arabic MultiplicationThe traditional paper and pencil algorithm for

multiplying two numbers expressed in arabic numer-als is brilliant. Too bad we all take it for granted.It's brilliant because it reduces a general problemto a small subcase how to multiply two single-digit integers and does so in a small amount ofspace. Here's the result of applying the algorithmto 432 x 378:

432378

34563024

1296

163296

Each row of intermediate calculation is obtainedby multiplying the top factor (432) by one digit ofthe bottom factor. If we expand out the first inter-mediate row in more detail, we get

4328

16

2432

3456

(1)

Of course, it's never written this way. To savespace, the "carries" are either all done mentally, orthey are marked with small digits as follows:

2 1

3 4 5 6

We include Display (1) to make the role of single-digit multiplications explicit. For instance, 16 is theproduct of the 2 in 432 by the 8 in 378.

Now, is this format precise enough for present-ing Arabic multiplication? Apparently so, becausesuch a format does seem to suffice for teaching thealgorithm to children (when presented with many

LA U


examples, lots of oral explanations, and hands-onpractice). And you do need to make use of dia-grams if the physical positioning of symbols on thepage is part of the algorithm.

Nonetheless, this is not the format we will usefor other algorithms, and it is not a good formatfor systematically .verifying the defining conditions1-3 above. So we now restate Arabic multiplica-tion using "algorithmic language", a language stylequite similar to a programming language. (We willassume basic familiarity with how such languagesare to be read, e.g., what a loop is, what an assign-ment is.) Actually, we restate only the part shownin Display (1) a multidigit number times a sin-gle digit number. The algorithm makes use of twoprocedures,

DigMult(a, b) which multiplies the singledigits a and b and returns Miand Mr, the left and right dig-its of the product.

DigAdd(a, b) which adds the single digits aand b and returns Al and Ar,the left and right digits of thesum. (At will be either 0 or1.)

Here's the algorithm:

Input ao, al, ..., am [the ones, tens,... , digitsof an (m+1)-digit number]

[the one-digit multiplier]Algorithm

carry 0

for j = 0 to mDigMult(ai, b)DigAdd(M,-, carry) [add any carry from

previous product]P3 Ar [jth digit of the product known]DigAdd(Mi, Ai) [needed in case the carry

affects the left digit]carry Ar [carry to the next single-

digit multiplication]endforif carry > 0 then Prn+i carry

Output P0, P1, , Pm and sometimes Pm +1[digits of the product]

This is no doubt hard to follow, but try carry-ing it out on the example above. Look at 432 x 7(the middle line of the first example), which showswhy the two lines before "endfor" are needed. Thatthis description is hard to follow should bring homethe point that the Arabic algorithm is really quite

subtle. (For instance, we don't include a step justbefore endfor to carry A1, because at this point Aiis always 0. Do you see why?)

The advantage of this formulation of the algo-rithm is that it is easier to verify that it is an al-gorithm. Is it determinate? Yes, because each lineleaves no doubt about what is to be done, and theorder of execution is also specified go down thepage, except when you get to the end of a loop, goback to the beginning. Is it finite? Yes, becausethe loop has only 5 lines, and the loop gets carriedout rn + 1 times. Does it solve the problem? Thisis not so obvious, but the specificity of the linesmakes it easier to present a proof when it is time toget around to that. (We will talk about algorithmverification later.)

Notice that this algorithm involves iteration:some subprocess is applied repetitively. In this casethe subprocess of multiplying two single-digit num-bers (and then carrying) was iterated. While analgorithm does not have to involve iteration (or arelated type of repetition called recursion), almostall algorithms of interest in mathematics do.

Example 2: Euclid's Algorithm

This one is much older than the first, and alsomuch simpler, but perhaps not so well known. It isthe classical Greek method for finding the greatestcommon divisor (gcd) of two positive integers. Itassumes you already know how to divide and findremainders. The algorithm keeps dividing and find-ing a remainder until the remainder is 0. Then theprevious remainder is the gcd of the original num-bers.

Here is a numerical example. Find the gcd of147 and 33. The quotient. of 147 divided by 33 is 4with remainder 15. That is,

147 = 33 x 4 -I- 15.

So any number that divides 147 and 33 also divides15, and conversely, any number that divides 33 and15 divides 147. Now, do the same operations to 33and 15 that we did to 147 and 33: 15 divides into 33with remainder 3. Thus a number divides 33 and 15if and only if it divides 15 and 3. But 3 divides into15 exactly. So the largest number dividing 3 and 15is 3 itself. Thus the gcd of 147 and 33 is 3.

In algorithmic language, Euclid's algorithm isthe following:

Input m, n [integers >= 0]Algorithm

num 4- m; denom nrepeat until denom = 0

quot 4 intim/dermal][integer part of num/denom]

rem 4 num quot*denomnum 4- denom; denom 4 rem

[update num and denom]endrepeat

Output num

For instance, for the numerical example above,initially num(erator) is 147 and denom(inator) is 33.Since 33 0 0, we enter the repeat loop, quot(ient)is computed as 4 and rem(ainder) as 15. Then(33,15) become the new (num,denom) pair. Sincedenom is still not 0, we traverse the loop again, and( num,denom) becomes (15,3). At this point, work-ing by hand, we immediately recognized that 3 di-vides 15, but a computer must "discover" this byfollowing the rules. Since 3 0 0, we enter the loopagain, and update (num,denom) to (3,0). Now de-nom = 0 and the algorithm quits, outputting num =3 as the gcd.

Notice there is no factoring in this algorithm.Another way to find gcd(m, n) is to factor in andn, and then take the product of all common factors.This second method is the standard one currentlytaught in elementary schools in North America. Forsmall values of in and n, the second method is oftenfaster than Euclid's method, but factoring very largenumbers is very hard. In general, Euclid's methodis the way to go.

Euclid's method is an algorithm. Clearly it isdeterminate. It is finite, because rem is always anonnegative integer and gets smaller with each it-eration, so eventually it must reach 0 and the algo-rithm stops. The algorithm is conclusive (correctlydetermines the gcd) for the reasons we argued in-formally above. A formal proof would be L math-ematical induction.

Example 3: Matrix Multiplication

Let A be an nix n rn-ttrix and B an n x p matrix.Call the entry of A in row i (down from the top) andcolumn j (from the left) aij. Similarly, B = [bid.Then their product AB is defined to he the in x pmatrix whose (i, k) entry is

Ea 7.7bk

.7(2)

For instance,

Algorithms and Algorithmics 41

7 61 2 3

4 5 6

1 4j 261 16,3 2

In particular, the (2,1) entry of the product is

4* 7 + 5 * (-5) + 6 *3 = 21.

How can we express the definition of matrix multi-plication as an algorithm?

Informally, you just go through each combina-tion of a row from A and a column from B and com-pute their product according to (2). Their productis a sum of real-number products, so we can com-pute it by keeping a running sum and successivelyadding real products until we are done. In algorith-mic language we have

Input A,B,m, n, pAlgorithm

for i = 1 to rnfork = 1 to p

Cik 4-- 0

[initialize the ik entry of C = AB]for j = 1 to n

Cik « Cik aii * bjkendfor

endforendfor

Output C

Example 4: Construct \FtAll the examples so far have been arithmetic or

algebraic. Here's one from geometry. By construct-ing a number r, we shall mean constructing a linesegment of length r, starting with a line segment oflength 1 and using a straightedge and compass. Toconstruct f2-, construct a unit perpendicular at oneend of the initial unit segment. By the PythagoreanTheorem, the hypotenuse has length 12-. Now it ispossible to construct f by repeating the process.Construct a unit perpendicular at the end of the seg-ment of length vr2-. The new hypotenuse will havelength V-3-. See Figure 1.


By induction, it should be clear that may beconstructed for al' positive integers n. Here is theconstruction in algorithmic language.

Input n, unit, line segment ABAlgorithm

for c = 2 to nConstruct BC I AB, with BC = 1AB +-- AC

endforOutput AB

[change names]

[segment of length Vii]

Is this determinate enough to be called an algo-rithm? It depends on the audience. If the readerknows well how to construct perpendiculars withstraightedge and compass, it is. If not, the line"Construct BC I AB" must be expanded.

Example 5: Towers of Hanoi

Algorithmic approaches apply not just to tradi-tional mathematical topics, but also to any situa-tion where a systematic and repetitive approach isneeded for a solution. Towers of Hanoi (TOH) is agame played with a set of n rings (or disks) of differ-ent sizes and three poles. Initially the rings are allon one pole, from smallest on top to largest on thebottom. The object is to get them all to anotherpole, in the same order, making moves according tothe following rules.

1. Move only one ring at a time.2. A larger ring may never be placed on a

smaller ring.

TOH is often used by psychologists doing experi-ments with children. While it is easy to figure outsolutions for n = 3 or 4, for larger n most kids soonlose their way. University students often don't domuch better! The key to understanding why thegame can be solved is recursion reduce to the pre-vious case. Suppose we already know how to solvethe (n-1)-ring game. Regarding that subgame asan indivisible block, then Figure 2 shows how tosolve the n-ring game. This solution may be put intoalgorithmic language if we allow a procedure (recallDigMult in Example 1) to invoke itself. The pro-cedure H in the algorithm is first defined (in termsof itself) and then invoked by the (one-line) mainalgorithm. The poles are numbered 1,2,3. Note,therefore, that. if r and s are numbers of two differ-ent poles, then 6 r . s is the number of the thirdpole.

n 1

rings

Figure 2

Input num, Pinit, Pfin [number of disks,initial pole number, final pole number]

Algorithmprocedure H(n, r, s)

[move n disks from pole r to pole s]if n = 1 then Move disk on r to s

else H(n-1, r, 6r---s)[move all but bottom disk to nontarget pole]

Move disk on r to sH(n-1, 6r s, s)

[move other disks onto target pole]endif

endprocedure11(num, Pinit, Pfin)

[main algorithm invoke H]Output Solution to the game

That this is an algorithm is not so clear. It'snot clear how to start carrying out the call of H,since mostly it just calls itself again instead of mov-ing disks. It's also not clear that when it finishes(if it finishes), it has solved the game. But in factit is an algorithm, and once one develops a goodunderstanding of how recursion works, it is fairlyevident why. In any event, good programming lan-guages have recursion built in, and thus the algo-rithm above is easy to translate into such languages.

Exam' le 6: The Quadratic formula

The traditional formula for solving x2 +6x + c =0 seems simple enough; where's the algorithm andwhy bother with it? Well, there are several casestwo distinct real roots, one repeated real root, noreal roots and properly choosing between cases isan algorithmic matter. Even if the audience knowsabout complex numbers, if they want. to compute

solutions, there is the problem that most calculatorsand computers won't accept a request to take thesquare root of a negative number. So presenting thesolution process as an algorithm has merit.

Input a, 10,c [coefficients ofaxe + bx + c, with a 0 0]

AlgorithmD 4 b 2 4acif [three cases follow]

D > 0 then [two real roots]s 4

(b+s)/2ax2 4 (bs)/2a

D = 0 then x1 4-- x2 b/2a[one repeated real root]

D < 0 then [two complex roots]s

(b+is)/2ax2 (bis)/2a

endifOutput the roots, x1 and x2

If we want to be even more comprehensive, andallow input with a = 0, then we have to includeseveral more cases. Note that there are no loops inthis algorithm, but several if-statements (even moreif a = 0 is allowed). Many procedures in the ev-eryday world involve more multiple decisions thaniteration think of tax laws. Such procedures trans-late into algorithms with many if- statements.

Example 7: Numerical Solution of Equations

There is no formula for most equations f(x) = 0that need to he solved in real applications, so onemust use numerical approximations. A common ap-proach is the bisection method. If f(x) is continu-ous, and one can find input values a and 1) withf(a) < 0 and f(b) > 0, then there is at least oneroot in between. (f(a) > 0 and f(b) < 0 is just asgood, and below we cover both cases by the condi-tion f(a)f (b) < 0.) Try the midpoint c = (a + b)/2.It is unlikely that f (c) = 0, but the sign of f (c) tellsus which half of the interval [a, b] to look in further.Now iterate:


Input a, bAlgorithm

repeatc -- (a + b)/2if f(c) = 0 then exitif sign(f(c)) = sign( f(a))

then b c

else a 4 Cendrepeat

Output c

[f(a)f(b) < 0]

Now, this is not an algorithm, because it can goon forever. For instance, if f(x) = x2 2, a = 1 andb = 2, then it takes an infinite number of halvingsto converge to the root c = Of course, a cutoffcondition can be added:

endrepeat when labl < tolerance

for whatever tolerance you choose. Even wityh sucha condition, a real computer running this algorithmmay not terminate, because, if the tolerance chosenis very small, roundoff error may result in labl >tolerance no matter how many iterations are per-formed.

Nonetheless, it may be best to present this algo-rithm initially in the nonterminating form aboveit gets at the key idea of bisection without obscur-ing details, and it also ties in with the concepts ofinfinite processes and limits needed for a full math-ematical attack. So this is our first example thatsuggests why the three defining conditions at thestart of this section should often be relaxed.

Example 8: Sequences of Heads and Tails

An important role of mathematics is to guide usin making decisions under uncertainty. This can of-ten be done using probability theory, but often themost direct approach is simulation. To take a verysimple example, suppose we flip a fair coin until weget two heads in a row. How many flips should weexpect to take? If we actually carry out this ex-periment many times, we find out what to expect.Here is a algorithm to carry out the experiment onetime. Rand(0,1) is a command for flipping a coin;the output 1 means heads, 0 means tails. The algo-rithm could be run a thousand times inside a loopof a bigger algorithm, which could then analyze theoutput data in various ways (take the average, thevariance, draw graphs, etc).


Input (none)Algorithm

count 4-- headct 0

repeatflip 4- Rand(0,1) [0 or 1, at random]count count + 1if flip = 1

then headct headct + 1else headct 0

endrepeat when headct = 2Output count

[total flips to get 2 heads in a row]

Now, this algorithm violates our definition intwo ways. First, it is not determinate: actions arenot uniquely determined. Second, it is theoreticallypossible that it won't terminate we might get Osforever. Nonetheless, we certainly want to be ableto study such "algorithms". The hard part, actu-ally, is to get computers to perform such procedures,since computers really are determinate machines. Inother words, how can computers be made to producewhat appear to be random numbers? Fortunately,there are good answers, using "pseudo-random num-ber generators".

Example 9: A Calculator Exercise

Except in Example 1 we have not said anythingabout how our calculations are carried out; it couldbe by hand, by calculator, or by computer. In fact,what is easy to do depends on the device. In thisexample let us specifically consider hand calcula-tors, since one can hope that this product of mod-ern technology can be made available to studentsalmost worldwide.

Consider the problem:

evaluate a(bi + b2 + + bn).

How shall we do this? On my scientific calculator,which has parentheses buttons, I can do it exactlyin the order presented.

a x ( bi + 62 + + 6,1 ) = (3)

where each symbol now represents a button (excepta, b1, etc., may represent many number buttons, and... represents repetition). However, we can savetime if we multiply by a on the right:

bi + b2 + + b = x a = (4)

A "+" would do as well as the first "="; the pointis, the sum is computed as we go along, so once thesum is finished, we can proceed to multiplication.One button-push is saved. Also, if you have onlya simple 4-function calculator without parentheses

buttons, approach (4) is available while approach(3) is not.

Still other approaches are possible. Using thedistributive law, we could instead evaluate

abi + ab2 + + ab.The direct approach to this, using the fact that mul-tiplications are completed before addition (on mycalculator), is

a x bi + a x 62 + + a x by, =which involves considerably more button-pushes.But many calculators, mine included, have a fea-ture to shorten repeated multiplication by the samefactor: hit the x button twice. Thus the followingstring of steps displays first ab1, then ab2, and soon:

a x x bi = b2 = b3 =Now we want to add these up, but hitting + (or anyother operator on the main display) will cancel theeffect of x x . So instead we push M+, the memoryplus button, which does the addition in the hiddenmemory register. Finally, at the end, we push MRto remove memory:

a x x 61 M+ 62 M+ 67, M+ MR (5)

Perhaps this sequence looks sufficiently odd that apresentation in algorithmic language would help:

push a x x bi M+for k = 2 to n

push bk M+endforpush MR

A count shows that method (5) takes thesame number of button-pushes as the original ap-proach (3), and only one more than the best ap-proach (4). So this problem provides a good exam-ple of how the issue of relative efficiency of algo-rithms pertains to even very elementary mathemat-ics.

To close this section, let us emphasize that by al-gorithms we do not mean computer programs. Wemean procedures for solving problems presented in asufficiently precise form for careful analysis. Whilewe have written most of our algorithms in a stylewhich until recently has been associated only withcomputer programs, this is because that style is agood one for making key points precise. Our al-gorithm descriptions cannot be input directly toany computer. They omit all sorts of informationthat a computer would need to know about (how isthe data input and output, what type of variables

need to be declared, how much storage must be re-served?). Many computer scientists call this sort ofalgorithm description pseudocode, because it is notreal code for computers. But it is quite real forthe sort of communication that interests us herebetween humans and so we prefer to call it algo-rithmic language.

2. What Is Algorithmics?First, algorithmics does not mean performing

a lot of algorithms. Students worldwide have suf-fered too much rote repetition of mathematical al-gorithms over the years already. In the future, al-gorithms will be carried out more and more by ma-chines, or by person-machine combinations, so handcalculation except of the simplest sort should receiveless emphasis.

Algorithmics is the process of creating, under-standing, validating and comparing algorithms. Inshort, it is thinking about algorithms, not thinkinglike algorithms.

Here is another way to put this. The phrase "al-gorithmic mathematics" has two meanings, tradi-tional and contemporary [Maurer, 1984]. The tradi-tional meaning emphasizes carrying out algorithms,the contemporary emphasizes developing them andchoosing intelligently among different algorithms fo:the same task.

We now discuss the components of algorithmicsin more detail. It is standard to divide algorithmicsinto three parts, design, verification, and analysis.

Algorithm Design is the process of algorithm cre-ation. There are some general principles of algo-rithm design; it does not have to depend on un-teachable flashes of originality.

The most important idea, as in much of math-ematics, is to break a problem into pieces. If youcan find a small building block that you understand,try to iterate on that block. To sum a sequence ofnumbers, reduce to the case of summing two num-bers; create a running sum and add one more num-ber to it each time. To multiply two large num-bers (Example 1), figure out a way to reduce it tomany instances of multiplying two -,re -digit num-bers. To multiply two matrices (Example 3), firstuse the definition (2) to reduce this to many casesof a real-number calculation, and then use iterationto return this to single additions and single multi-plications.

Sometimes one does not immediately see how toreduce a large problem to small pieces. Then onetries to reduce it to slightly smaller pieces. Whatis the gcd of two large numbers in and n (Exam-ple 2)? Well, does some slightly smaller pair of


numbers have the same gcd? Yes, in n and nhave the same gcd as m and n, because anythingthat divides (evenly into) in and n divides in nand n, and anything that divides in n and n di-vides m = (mn) n and n. And if subtracting 11.from in once preserves the gcd, then subtracting asmany times as possible, leaving the remainder whenm is divided by n, also preserves the gcd. This isthe insight that leads to Euclid's algorithm.

The algorithm for Towers of Hanoi is also basedon reducing to a smaller case. You can solve thegame with n rings if you can solve the game withn 1 rings, as shown in Fig. 2.

There are, of course, many other principles of al-gorithm design, and whole university courses are de-voted to it. Here we'll mention two more, top downdesign and bottom up design. The former refers tooutlining the big picture first, and then filling in thedetails of the parts later. The latter refers to start-ing with small pieces and putting them togetherto do the whole job. While top down is generallythe better approach for involved problems, both ap-proaches have their roles.

Algorithm design is more or less the same thingas problem-solving methodology. Since mathemat-ics education is permeated with problem solving, al-gorithm design is rightly an important componentof a modern mathematics education. Practice in de-sign not only makes people more successful at solv-ing problems, but also it results in algorithms thatare easier to communicate to others and to verify.

Algorithm Verification is the process of confirm-ing that algorithms solve the problems they claimto solve; in other words, proving algorithms correct..Since loops are a primary aspect of algorithms, andsince a loop can be iterated any nonnegative integernumber of times, mathematical induction is the keymethod of verification.

Take Euclid's algorithm (Example 2). Let P(k)be the statement that, just before commencing thekth pass of the repeat loop, gcd(num, denom) isthe same as the gcd of the original m and n. ThatP(k) is true for all k > 1 is easily proved by in-duction, using the fact that gcd(m, n) = gcd(n, r)where r is the remainder when in is divided by n.(We argued this fact informally when Example 2was introduced, and again somewhat differently fiveparagraphs ago.) When the loop entrance condi-tion is tested for the last time, denom = 0, andso clearly gcd(num, denom) = num, and num is thevalue output. So by the induction, the output equalsgcd(m, n) and the algorithm is valid. A proof of cor-rectness like this is called a proof by loop invariant;the loop invariant is the statement you prove to be


correct each time you enter the loop.Or take the algorithm for Towers of Hanoi. We

may do induction on n, using the statement thatany call of procedure H(n, r, s) correctly moves nrings from pole r to pole s. Since the very definitionof H involves itself with n 1 rings, induction iseasy to carry out. In general, a recursive algorithmimmediately suggests an inductive proof.

The specifics of how to do induction for algo-rithms is not the point here. The point is that in-duction is the right tool. Mathematical induction,heretofore regarded in some quarters as a special-ized method for proving certain formulas for sums,must be viewed as a more central proof method inany curriculum that gives substantial emphasis toalgorithmics.

Algorithms need to be verified because more andmore our lives depend on them, and once they arein place (say in our bank, to maintain our accountrecords) they tend to get treated as black boxes. Tobe honest, algorithms used in the world at large arevery complicated, too complicated for humans tocarry out detailed mathematical proofs of correct-ness; and machine verification of correctness is stillin its childhood. Thus, empirical debugging tech-niques play a vital role.

But mathematical verification should not be dis-missed. First, big programs use many small build-ing blocks which can or have been verified. Second,the algorithms whose correctness you are primar-ily responsible for are the ones you create yourself,and knowledge of how to verify an algorithm can behelpful at the design stage. If you propose to in-clude a loop in your algorithm, and you know thatthe way to validate it is with a loop invariant, youwill devise the loop invariant before you write theloop, and then you can write it to be sure that theloop invariant is preserved.

Algorithm Analysis is the process of determininghow long an algorithm takes to run, and comparingthat run time to that of other algorithms for thesame problem and to absolute standards for thatproblem. "Run time" is a rough way to put it, sincethat suggests an actual machine (or person) to per-form it, and different machines (and persons) willperforni differently on the same algorithm. Usuallyone picks some salient feature, say the number ofreal-number additions if addition is the main oper-ation in the algorithm under consideration, and de-termines the number of repetitions of this feature asa function of the input size. This function is calledthe complexity of the algorithm, or its efficiency.

Take, for instance, our algorithm for matrix mul-tiplication (Example 3). If the two input matrices

are both n x n, then there are n2 entries to com-pute, and each entry requires n real-number multi-plications and n 1 additions. Therefore, the wholealgorithm takes n3 steps (if only multiplications arecounted), and 2n3 n2 (if additions and multiplica-tions are counted). Or take Towers of Hanoi. Theobvious thing to count is number of ring moves. Itturns out that, if there are n rings, the algorithmtakes 2" 1 moves. If t is the number of moveswith n rings, the recursive definition of procedure Hleads to the conditions

tn+i = 2t,, + 1, t1 = 1; (6)

the unique solution of these conditions is In = 2"-1.Calculations like these become valuable if the

number of steps appears large and one wonderswhether the problem will be tractable with the com-puting equipment. available. Suppose, for instance,that a problem requires n! steps when there aren input data. (Brute force approaches to the fa-mous Traveling Salesperson Problem take this manysteps, and the best exact methods known are in gen-eral not much better.) Then when n is merely 25,a computer that could do a billion steps a secondwould still take 50 million years to solve the prob-lem! In contrast, the same computer could play 25-ring Towers of Hanoi in only .003 seconds, and couldcompute the product of two 1000 x 1000 matrices ina second.

These efficiency calculations become even moreinteresting when you have more than one algorithmfor the same problem. Take Example 9 for comput-ing a(bi + + br,) on a hand calculator. The bestapproach we discussed takes n+2+C button-pushes,where C is the number of pushes needed to enterall of a, b1, bri; two others took n + 3 + C andthe fourth approach took much longer. On a handcalculator, each button-push takes time, so even asaving of one is significant. Furthermore, there arelots of other, elementary problems where differentcalculator methods make a considerable difference.Take the problem of evaluating a polynomial

p(x) xn xni

There are a great many multiplications involved, es-pecially if you don't have an exponential key. Butthere is another way to write this polynomial, bestunderstood in traditional notation if we use a nu-merical example. If

p(x) = 4x4 + 3x3 + 2x2 + r + 8,

then in nested form

p(x) = x(x(x(4x + 3) 2) + 1) + 8.

Calculating p(z) in this form takes many fewer steps(count them) and it's easy to carry out even with asimple four-function calculator and no memory. Bythe way, to make clear how both approaches work ingeneral, and to make the algorithms precise enoughto count the steps without confusion, the first thingto do is figure out how to state them in algorithmiclanguage.

Or take the algorithm for constructing VII (Ex-ample 4). It takes n- 1 right triangle constructionsto obtain 07. Noting this surely will inspire stu-dents to find a better way.

For Euclid's algorithm, the number of steps de-pends on the specific input, not just the size (num-ber of digits) of the input. And in the Two Headsalgorithm (Example 8), there is no input at all, butthe number of steps varies. In these cases one takesseveral measures of the algorithm's efficiency bestcase, worst case and average case. Average case isespecially important, but usually hard to analyze.

The hardest problem is to compare an algorithmto an absolute standard. The complexity of a prob-lem (as opposed to the complexity of an algorithm)is defined to be the number of steps needed by thebest possible algorithm for the problem. Problemcomplexity is the subject of much current researchit's hard to figure out the complexity if, as usual.you don't know what the best algorithm is. For in-stance, it is known that Arabic multiplication andstandard matrix multiplication are not the best al-gorithms for their problems, at least when n is quitelarge, but no one knows what. the best algorithmsare or how fast they are. Nonetheless, progress hasbeen made in finding bounds on problem complex-ity. And every once in a while the complexity ofa problem can be determined completely. For in-stance, it is not hard to show that the algorithm wegave for Towers of Hanoi is optimal.

In closing this section, we note that there areother ways to analyze the goodness of an algorithmthan speed. One can consider space complexityhow much storage is needed. One can also con-sider numerical stability. For instance, in solvinga quadratic (Ex ample 6), if b > 0 and 4ac is verysmall compared to 62, then (b - s)/2a is practically0, and roundoff error may swamp the computation.In this case it is better to set x2 to 2c/(b+ s). Alge-braically, the two formulas are equivalent, but lessroundoff error is introduced in the latter since b-F sis not near 0.

3. Why Study Algorithmics?We have already given our main reason: the use

of sophisticated algorithms to solve problems is al-


ready pervasive in the world, and so informed citi-zens need to know what can be done by algorithms,how it is done, and how algorithms can be assessed.Also, a fair number of people need to know how tocreate algorithms.

Mathematically, this is an extrinsic justifica-tion; algorithms are important, so students oughtto study them whether or not they are interestingmathematics or do good things for mathematics ed-ucation. Fortunately, there is equally strong intrin-sic justification.

First, introducing algorithmics in school raisesfresh questions about old material and allows forgreater student creativity. As Example 9 (calcu-lator efficiency) shows, even basic arithmetic is nolonger cut and dried. Too many traditional curric-ula consisted of many computational courses wherestudents were told the right methods, and a fewproof courses (say, classical geometry) where theywere ask, d to be creative, but in a narrow theo-retical way. In contrast, each question of the sort"devise an algorithm for ..." allows for many cor-rect answers (not all equally good) Even a student.who does not have a good theoretical grasp of theproblem at hand may come up with a correct algo-rithm.

Even incorrect algorithms can have worthy fea-tures. They may involve good heuristics imperfectbut insightful ideas that often lead to a reasonablygood solution in a reasonable amount of time. Also,an analysis of their flaws may be instructive and leadto interesting class discussions. For instance, sup-pose you want to pick a random set of two distinctnumbers from 1 to 10. What's wrong with picking anumber i at random from 1 to 9 and then picking anumber j at random from i+1 to 10? I once heard abusinessman say, speaking at a college graduation,that in the outside world one learns from one's fail-ures. While there may not be much to learn frommistakes in traditional rote calculations, there is agreat deal to learn from one's failures in devisingalgorithms.

A second intrinsic reason for studying algorith-mics is: it can help students understand traditionalmathematics better. You really have to understanda procedure well in order to "explain" it to a com-puter, or to write it in algorithmic language. Forinstance, to understand Arabic multiplication wellenough to describe it in algorithmic language (Ex-ample 1), you really have to understand place no-tation and the distributive law. And it's not justprocedures that come to be understood better, butabstract concepts as well. For instance, the functionconcept is concretized by seeing algorithms turn in-


puts into outputs. Real numbers are made moreconcrete as the student sees (Example 7) that suc-cessive rational approximations needed to computethem. Moreover, when students test their algorith-mic representations by running them on computers,they get instant feedback as to whether their con-structions are correct.

There are also arguments against studying algo-rithmics in school. The basic argument goes: thesort of questions emphasized in algorithmics are al-ready outdated, or soon will be. For instance, al-gorithmics puts great emphasis on the relative ef-ficiency of algorithms. But if one approach to aproblem takes n2 steps and another takes n steps,the difference in actual seconds will be unnoticeablefor the values of n used in any classroom. Or, whybother discussing methods for doing computationson a four-function calculator when soon all calcu-lators will be much more powerful? Indeed, what'sthe point of talking about the merits of differentmethods of polynomial evaluation, when on the nowpopular "algebraic" calculators, you can punch inthe definition of a function as a formula in x, thenpunch in a numerical value for x and finally just hitthe EVAL button? One doesn't need to know anymethod for breaking down the evaluation of the for-mula into small steps because the calculator does itall.

Generalizing, computing devices are gettingmore and more advanced in the sense that theycan respond to higher and higher level commands.When you think that, in the wings, there are ma-chines that will create proofs and create algorithmsfor solving problems, why do students need to beschooled in the ability to create algorithms them-selves?

I answer as follows. No doubt the level at whichit will be appropriate to do algorithmic analysis willchange over time. I really like to discuss differentmethods of polynomial evaluation with my classes,but one day (perhaps soon) this may seem as out-dated to them as if I were to explain the theory be-hind slide rules. But if we can draw any lesson fromthe history of computing technology, humankind,including students, will always use technology to itslimits, and its most powerful use will always involvethe interaction of human and machine. To pick asimple example, I am not worried that the differ-ence between an n2-step algorithm and an n-stepalgorithm will be lost on students. First, some stu-dent always tries to run a recently learned algorithmon data that is too large, and wonders out loud whythe machine sat spinning its wheels. Second, even ifmost students stick to small "textbook" data sets,

it is easy to show them that in the outside worldsome very large problems must be.solved where dif-ferences in algorithm efficiency are crucial. Sortingand searching (discussed later) provide good exam-ples; governments and large businesses must sortand search enormous data sets.

The issue, then, is to keep the algorithmic exam-ples up to date. This can be done if educators keepinformed about the latest research and the latesttechnology.

Sometimes the exact opposite reason is proposedfor not studying algorithmics. It is a theorem thatthere is no algorithm for determining which prob-lems are solvable by algorithms. (This is because the"universal Turing machine" cannot solve the "halt-ing problem".) So to emphasize algorithmics eithermisleads them about what algorithms can do or cutsthem off from problems that have no algorithmic so-lutions.

But we do not propose that only algorithmic ap-proaches to mathematics be studied. We only pro-pose that algorithmics receive much more attentionthan previously.

4. Suggestions For ImplementationTwo disclaimers: First, my knowledge of cur-

ricula worldwide is limited, and so I speak mostlyfrom an American viewpoint. Second, in a paperthis length, one can at most give illustrative exam-ples and broad ideas of how to implement algorith-mics. For more detailed ideas, appropriate at leastin North America, see [NCTM 1989, Kenney 1991).The suggestions below concern the primary and sec-ondary levels except for a few brief remarks aboutthe university level at the end.

Look at traditional computations more closely.Basic arithmetic, computations with polynomials,solutions of linear equations such things are oftentaken as routine and devoid of opportunity for freshthought. But from the viewpoint of algorithmicsthere is plenty to think about. Students can dis-cover traditional algorithms using design principles,and discover alternative algorithms. While they areunlikely to discover significantly faster algorithms,they can be told (or, at a higher level, shown) thatfaster algorithms exist, and that best algorithms areunknown.

Treat nontraditional computations related toclassical questions. In every country students learnclosed form solutions to certain sorts of equations,but they don't always look closely at how to eval-uate those solutions accurately, or discuss methodsfor approximately solving equations without solu-tion formulas. Students often learn to count per-

mutations and combinations, but they don't oftenconsider how to efficiently list all of them of a certainsize, or generate a random one. In short, classicalformulas that don't appear algorithmic raise algo-rithmic issues.

Introduce some new topics. There are wholefields of mathematics, with many applications, thathave an algorithmic flavor and are not representedat all in many curricula. Many of these are groupedthese days under the headings discrete mathematics,operations research and theoretical computer sci-ence. Here are a few examples, but at this point theyare little more than name-dropping, and one shouldrefer to texts in these fields, such as [Hillier andLieberman 1986, Manber 1989, Maurer and Ralston1991].

Difference equations is the study of inductivelydefined sequences such as the step-count sequencetn of Towers of Hanoi in Display (6). These includethe traditional arithmetic and geometric sequencesand series and much more. Computing terms ininductively defined sequences is immediately an al-gorithmic question, and conversely, analyzing algo-rithms reduces to analyzing difference equations.

Graph theory, in the sense of networks, is fullof algorithmic questions. If a graph represents anexisting road network, how do you find the shortestroutes between points (in distance, time, or what-ever)? If the network represents the possible linksbetween cities is a telephone network planned for adeveloping region, how do you decide which set oflinks will connect up the region at minimum cost?There are a variety of good (and not so good) algo-rithms for such problems, and many of these algo-rithms are not hard for students to discover.

Sorting and searching (e.g., alphabetizing andlooking through an alphabetized list for a word) arestandard computer science examples that wouldn'ttraditionally be thought of as having any mathemat-ical content clearly it is possible to sort and search,so what's the problem? But once again, there arelots of different methods, with various efficiencies,and various challenges to verify them and analyzethem.

Make computing power available to students.This is a tall order. No matter how rich the country,there are always newer and more powerful devicesone could want, and even in rich countries it may bea long time before there is one computer per studentin every class. But the point is, algorithmic ques-tions take on much more life when students havewhat they regard as powerful computing aids, andthen they discover they can still devise problemsthat aren't solved instantaneously. As discussed ear-


lier, even four-function calculators are very helpfulin bringing to life algorithm design and efficiencyquestions. With computers as well as calculators,one can start in the early years with such things asthe language Logo and Turtle graphics, and movein later years to computer algebra systems.

Introduce algorithmic language. Whatever com-puting power is available, precise methods for de-scribing algorithms are necessary if algorithms areto be an object of study and not just somethingstudents perform. There is no standard algorithmiclanguage, and perhaps different sorts of languagesare best for problems to be treated with differentsorts of machines (or by hand). Nonetheless, it isnot hard to devise useful language constructs.

Put more emphasis on mathematical induction.We have indicated how induction is the mainmethod for validating algorithms. Actually, induc-tion can be viewed more broadly, and as such is atthe foundation of algorithmics. There are inductivediscovery techniques (reduce to the previous case, orbuild up from small cases to find a pattern), induc-tive definitions, as in Display (6), inductive algo-rithm commands (loops and recursive procedures)as well as inductive proofs.

Eliminate the schism between solving and com-puting. Traditionally there is pure mathematics andapplied mathematics. Pure mathematicians provethat solutions exists, and applied mathematiciansfigure out how to find them. In algorithmic math-ematics, good computation methods are found si-multaneously with showing that solutions exist. Byputting these two issues together right from the ear-liest years, we help to overcome what has sometimesbeen an unfortunate two-class system in mathemat-ics and science.

A few words about the university level. Herethe schism between pure and applied has been par-ticularly acute. But it is breaking down. Manyresearch mathematicians in pure fields are findingalgorithmic questions interesting. Some algebraists,for instance, are now very interested in how classicalobjects in group theory can best be computed [Bee-son 1990; Mines, Richman and Ruitenherg 1988].This could filter into the classroom. Even in calcu-lus, some questions can be given a much more al-gorithmic flavor than they have been. The rules ofdifferentiation, instead of simply being a set of rules,can be viewed as the parts of an algorithm that de-termines the derivative for any elementary function(once elementary functions are given an inductivedefinition!). The rules of integration can be viewedas part of an algorithm that determines the integralof some elementary functions, and some discussion


can be added that integration is no longer an "art",because there is an algorithm for determining ex-actly when a function can be integrated in closedform. Those university mathematicians who havegotten interested in algorithmic questions should beencouraged to share with their colleagues their ideasabout how to introduce these new approaches in thestandard courses.

REFERENCES

Beeson, M. [1990]: Review of "A Course in Con-structive Algebra", Amer. Math. Monthly, 97(April), pp. 357-362. (See also Mines below.)

Hirsch, C. and Zweng, M. [1985]: The Sec-ondary School Mathematics Curriculum, (1985NCTM Yearbook), Reston, VA: NationalCouncil of Teachers of Mathematics.

Hillier, F. and Lieberman, G. [1986]: Introductionto Operations Research, 4th ed., Oakland, CA:Holden-Day.

Kenney, M., (ed.) [1991]: Discrete MathematicsAcross the Curriculum, K-I2 (1991 NCTMYearbook), Reston, VA: National Council ofTeachers of Mathematics.

Malkevitch, J. et al. [1988]: For All Practical Pur-pose, San Francisco: W. H. Freeman. Alsoavailable as 26 videotaped TV programs.

Manber, U. [1989]: Introduction to Algorithms, ACreative Approach, Reading, MA: Addison-Wesley.

Maurer, S. [1984]: Two meanings of algorith-mic mathematics, Mathematics Teacher, 77(September) pp. 430-435.

Maurer, S. [1985]: The algorithmic way of life isbest, College Math. J., 16 (January) pp. 2-18(Forum article and reply to responses).

Maurer, S. [1991]: Proofs and algorithms: a reply toGerstein, UME Trends, Vol. 3 (January) p. 8.

Maurer, S. and Ralston, A. [1991]: Discrete Algo-rithmic Mathematics, Reading MA: Addison-Wesley.

Mines, 13., Richman, F. and Ruitenberg, W. [1984A Course in Constructive Algebra, New York:Springer-Verlag.

NCTM [1989]: Curriculum and Evaluation Stan-dards for School Mathematics, Reston, VA:National Council of Teachers of Mathematics.

North Carolina School of Science and Mathematics(Barrett, G. et al) [1991]: Contemporary Pre-calculus through Applications, Providence, RI:Janson Publications.

Peressini, A. et al. [1992]: Precalculus and DiscreteMathematics, University of Chicago SchoolMathematics Project, Glenview, IL: Scott,Foresman.

Ralston, A. [1981]: Computer science, mathemat-ics and the undergraduate curricula in both,Amer. Math. Monthly, 88, pp. 472-85.

Ralston, A. [1984]: Will discrete math surpass cal-culus in importance? College Math. J., 15(November) pp. 371-382 (Forum article andresponses).

Ralston, A. and Young, G. S., (eds.) [1983]: The Fu-ture of College Mathematics: Proceedings of aConference/ Workshop on the First Two Yearsof College Mathematics, New York: Springer-Verlag.

ON THE MATHEMATICAL BASIS OF COMPUTER SCIENCEJacques Stern

Equipe de logique, Universite Pariset

Departement de Mathematiques et Informatique, Ecole Norma le SuperieureFrance

It is now clear to an anybody that a workingmathematician cannot ignore computers: as a con-sequence, it is commonly admitted that students inmathematics, and especially those who intend to beteachers in the field, have to be exposed to somehigh-level language (such as Pascal). Nevertheless,this is far from enough: the question of whether stu-dents in mathematics should be familiar with someparts of the theoretical foundations of computer sci-ence cannot be avoided because these topics are pre-cisely the parts of computer science close to mathe-matics and seem to be necessary in order to estab-lish connections between both fields that go beyondthe ability of using the computing power of modernmachines.

In France, following this line of ideas, the studyof algorithms and related topics has become, inmost universities, a significant part of the standardcurriculum leading to graduation in mathematics.Also, an optional test in computer science has beenoffered for a few years in the well-established "Con-cours d'Agregation de Mathematiques", which is akind of "teaching Ph-D", passed by most of theteachers for the age-group 17-22.

The author has recently published a book en-titled "Fondements Mathematiques de l'Informa-tique" [1990], which covers a large part of the re-quirements in computer science for undergraduateprograms in mathematics. The aim of the presentcontribution is precisely to present some generalideas that grew during the process of writing upthat book. These ideas are my personal views al-though I owe a great debt to many colleagues withwhom I have had inspiring discussions.

Before going into greater detail, let me make oneremark: Mastering some of the basic tools in com-puter science will not turn a mathematician into acomputer scientist. Instead, it should help to de-velop a different frame of mind, suitable to under-stand the specific features of computer science. Thisis most important for a mathematician because, as isshown in other contributions in this book, these spe-cific features will necessalily affect both the teach:ngand the practice of mathematics themselves.

Around the notion of computationComputation Theory is considered by many peo-

ple to be a very dull subject; nevertheless, it is the

51

first burden of the theory to provide a suitable crite-rion for drawing a limit between what is computable(or effective) and what is not. A simple way wouldbe to use the word computable for everything thatcan be processed on a real computer. Although thispoint of view is not completely meaningless, it re-mains rather vague and cannot be considered as agenuine mathematical notion because of its lack ofprecision. Furthermore, this point of view is noteven historically correct: a lot of outstanding workconnected with the subject of computation theorywas published before the first modern computer wasbuilt. For example, note the work of Turing [1936],Post [1936] on computation theory itself, and alsothe work of McCulloch and Pitts [1943] on the mod-elling of neuron nets, from which the theory of au-tomata grew.

It is precisely the theory of automata that we wepropose to choose as a starting point. Many reasonscan be put forward in order to justify such a choice.The theory is simple, established on firm mathe-matical grounds and provides various exercises inprogramming: for example, one can simulate an au-tomaton in a high-level language like Pascal or dis-cuss algorithms that compute the minimal automa-ton. Also, the concept of non-determinism, whichis of utmost importance in theoretical computer sci-ence, can be quickly and naturally introduced in asimple setting. Finally, the theory of automata hasseveral applications: to text editors and compilersin particular; this is not a minor argument.

Nevertheless, one can easily come to the con-clusion that automata do not provide a satisfactorymodel for real machines. This conclusion can bereached by writing down simple languages that arenot accepted by a finite automaton but also throughthe convincing observation that a central feature ofcomputers is completely wiped out, namely theirability to store data in a memory. We are thusback to our original problem of defining the notionof computable and it is reasonable, at this point,to require that this notion should be described us-ing various different techniques that come out tobe equivalent: this will ensure that a mathemat-ical invariant has really been found and this willmake Church's Thesis highly plausible. (Recall thatChurch's thesis states that the notion of machine-


computable function and the mathematical familyof recursive functions are identical).

Four distinct approaches can be taken.Adding a memory device to a finite au-tomaton. This yields the definition of a Turingmachine.Directly modelling actual computers. Thiscan be done through the notion of a random ac-cess machine (cf. Cook and Reckhow [1973])operated by a very simple language similar tomachine code.Defining a simple class of programs. Forexample one can define a restricted version ofPascal which uses only the integer type and thecontrol sequences if ... then ... else and while... do.Defining the class of (partial) recursivefunctions. This is a good opportunity to dis-cuss functional languages: recursive definitionscan be handled by using constructs that are ex-actly similar to those appearing in Lisp.The proof that all these definitions are actually

equivalent is a source of very interesting observa-tions. For example, the fact that the restricted ver-sion of Pascal can compute all recursive functionsproves the well-known fact that the goto statementcan be dispensed with. It may be worthwhile to notethat replacing while ... do by for only allows thecomputation of primitive recursive functions. Also,the simulation of a random access machine by a Tur-ing machine is a good exercise that shows how tohandle a sequential memory.

Once the notion of a computable function hasbeen given a precise definition, it becomes possibleto discuss decidability issues: By coding Turing ma-chines and constructing a universal machine, it doesnot require much more effort to state correctly the"hafting problem" and show that it is semidecidablebut not decidable (which means that a machine canfind positive answers in a finite computation timebut cannot do the same both for positive and nega-tive answers). It is not clear that the study of gen-eral recursion theory should be pursued. Still, onemay wish to present the semantics of recursive pro-cedures and the fixed-point approach to programsand develop the recursion-theoretic tools that areneeded, such as Kleene's theorem (which basicallystates that the name of a. recursive function can beused within its own definition).

Then, one can have a discussion on whether ornot the dichotomy decidable/undecidable is of prac-tical significance. This is a way to introduce Com-plexity Theory through 0:: constraints of time. Go-ing back to the various mathematical models of com-

putation, one can explain how a basic cost can beattached to the execution of each instruction, theoverall cost (or complexity) being the sum of allbasic costs. Thus, one can define the complexityfunction of an algorithm which measures its cost interms of the size of the data. Of course this com-plexity depends on the abstract machine chosen butone can check that, when one machine is simulatedby another, the complexity functions are polynomi-ally related. This allows the definition of the classP of polynomial time computations, which is a rea-sonable candidate for modelling a class of problemssometimes called feasible or tractable.

Around the notion of algorithm

Now that we are equipped with a theoretical no-tion of complexity, it is necessary to use it in con-crete situations. This can be done through a re-view of various algorithms. This review is, by nomeans, an exercise in programming style, even ifcorrect programs have to be written at some point.The emphasis should be on the design and analy-sis of algorithms, which are very closely connected.Of course, the rules of the game should be clearlystated and discussed, especially the choice betweenthe two main notions of complexity that are in use:worst-case analysis and average-case analysis. Thischoice depends on the underlying model: for exam-ple, average-case analysis is relevant when the prob-ability of "ill-behaved" cases is small. In both cases,the analysis is combinatorial in character and quiteoften yields non-trivial recurrence relations. In or-der to handle these, some specific tools are needed,like the statistics of permutations and distributionsand the use of generating series (cf. Knuth [1973]).Generally, such techniques (e.g. the use of singu-lar points of the generating series) only allow anasymptotic analysis and one may ask if this kindof information has any practical meaning: after all,the size of the data are bounded by the computingenvironment! It turns out that the asymptotic anal-ysis is actually relevant: When a given algorithmruns in time 0(n log n), for example, it is usuallytrue that the constant implicit in the 0 notation israther small and that the asymptotic behaviour isreached rather quickly.

The students should also get used to performingthe analysis of the complexity of an algorithm with-out going back to the original definitions, based onabstract models of computation. If the size of theintegers is bounded (which is often the case in prac-tical situations), the complexity is roughly the num-ber of machine instructions performed during execu-tion. This validates the use of the overall number of

JJ

comparisons as a measure of complexity for sortingalgorithms. When large integers are involved, thingsbecome a bit more complicated: a convenient wayis to multiply the number of instructions performedby n2, where n is the number of digits of the inte-gers used. This is to take into account the cost ofmultiplication as 0(n2).

Together with algorithms the specific data struc-tures used in computer science should be discussed:stacks, files, trees, graphs etc. It should be stressedthat this point of view is quite different from theone that was taken in the previous section: In com-putation theory, we considered simulations involv-ing basic manipulations on data structures and weclaimed that these manipulations were not costly,becauce we were interested in the general notion ofpolynomial time. In practical cases, a given polyno-mial time algorithm can be superior to another oneand, very often, the choice of a good data structuremay actually save a significant part of the runningtime.

The choice of algorithms that can be reviewed isquite large and depends on the mathematical back-ground of the students. At an advanced level, it isprobably more rewarding to give examples that usemathematics in a non-trivial way, such as:

The fast Fourier transform and its applica-tion to fast multiplication of integers (in time0(n log n log log n)).Basic algorithms for computer algebra.This can be an opportunity to demonstrate theuse of a computer algebra system, like Maple orMacsyma.The simplex algorithm for linear program-ming.Primality tests, at least probabilistic ones.Unfortunately, it is not possible to discuss these

algorithms in detail. and we will only briefly com-ment on the last example. As is well known, test-ing primality by sieving requires a large amount oftime and memory. In order to overcome this diffi-culty, one may try to use the mathematical prop-erties of prime numbers. For example, it is knownthat, whenever p is prime and a is not zero modulop, the so-called Jacobi symbol (2) is ec,ual to ate.This is not the case in general. More precisely, ifn is not prime, at most one half of the possible a'ssatisfy the equality

n= a 2 (mod n).

As was observed by Solovay and Strassen, the com-putations required to compute Jacobi symbols andexponentials modulo n can he performed efficiently.

Mathematical Basis of Computer Science 53

This makes it possible to recognize whether or nota given integer n is prime by picking random valuesof a and testing the above equality. If sufficientlymany tests are successful, n is declared to be prime.

At a more elementary level, examples can betaken from the following list (which is not exhaus-tive):

Sorting. This should include a comparison ofvarious algorithms and a discussion of quicksort,as an illustration of the power of the divide andconquer method.Searching, with an emphasis on the choice ofspecific trees as data structures.Pattern matching, because of the connectionwith automata.Graph algorithms, for the nice interplay be-tween discrete mathematics and computer sci-ence.

Graph algorithms can be a way to introduce!VP-complete problems. Indead, one can observethat computing shortest paths can be done in avery efficient way whereas no polynomial time al-gorithm is known for many graph problems, such asthe Hamiltonian path problem. This problem can bedescribed in very concrete terms as follows: Givena set of cities together with possible air connectionsbetween them, can one tour all the cities, visitingeach city once and returning to one's starting point?In order to handle this problem, one can

guess a plausible solution

check its correctness (in polynomial time)

All problems that can be solved in such a non-deterministic manner are called HP-problems, andan HP-complete problem is an HP- problem thatcan be used as a "subroutine" in order to solveall other HP-problems, with polynomially manyextra steps of computations. It is an open prob-lem (probably the most important problem in the-oretical computer science) whether or not an HP-complete problem can be solved through a polyno-mial time algorithm, and as a consequence, HP-complete problems are considered to be difficult:They can only be attacked by time-consuming tech-niques such as backtracking.

Of course, the class of HP-complete problemscan he given a formal definition through non-deterministic Turing machines. Once this is done,one can prove Cook's Theorem (Cook [1971]), stat-ing that the satisfiability problem for clauses of thepropositional calculus is HP-complete. More ex-amples of HP-complete problems can be given (cf.Garey and Johnson [1979]), such as:


The travelling salesman problemThe knapsack problemThe clique problemFinally, some indications can be given on how to

handle MP-complete problems and also on practi-cal applications of these notions, through the use of"one-way" functions.

Around logicMany authors now emphasize the role of logic in

the foundations of computer science. This is pre-sumably because of the deep connection that existsbetween computer programs and proofs. This con-nection was already implicit in the section on com-putation theory: The undecidability phenomenon isclosely related to Godel's Incompleteness Theoremsthat show the extreme limits of deductive mathe-matics.

It is therefore necessary to include a thoroughintroduction to logic in order to endow mathemati-cians with a synthetic view of computer science. Butit should be added that the interplay between math-ematical logic and computer science is such thatlogic cannot be taught now as it it was before theadvent of computers. This applies both to the for-mal presentation of syntactical objects and to thedevelopment of the theory itself.

From the formal point of view, it is extremelyhelpful to follow computer scientists and to considerformulas as trees and not only as strings of sym-bols, as was done classically. With this approach,a notion such as a free occurrence of a variable isgiven a clear, almost geometrical definition, whichwas not the case when it was introduced througha cumbersome recurrence. This can be quite im-portant considering the fact that syntax must hequickly understood by students who have not beenexposed to logic beforehand.

For the same kind of reasons, students have to bemotivated as early as possible. Indeed, this can bedone by discussing the aim of artificial intelligence:How to make correct inferences from a database ofknown facts. This is meaningful even in the simpleframework of propositional calculus and the diffi-culty of the problem can be understood by recallingthat the satisfiability problem is .APP-complete. Thesearch for solutions to the deduction problem thatare not brute search algorithms leads to the methodof resolution, which can be made very efficient in theparticular case of Horn clauses through linear res-olution. For the convenience of the reader, let usrecall that clauses are disjunctions of !iterals; liter-als are either positive, i.e. propositional variables ornegative (negation of such variables). Horn clauses

include at most one positive literal. The resolutionmethod is a way to derive a contradiction from a setof clauses by making systematic use of the tautology

(p V q) A (Tv r) q V r.

As far as the predicate calculus is concerned,it is almost compulsory to use a constructive ap-proach based on Skolem functions and Herbrand'stheorem. (Recall that Skolem functions ensure that,whenever a formula 3x4)(2, yl , , yn) holds, a pos-sible solution z of this can be computed by a termf (yi, , y).) Herbrand's theorem states that, pro-vided Skolem functions exist, any set of formulasfrom which no contradiction can be derived can berealized in a model whose domain is the set of closedterms). This provides both completeness and com-pactness by reduction to the propositional calculus.At this point, one should not avoid discussing un-decidability issues again: even if one starts with afinite set of formulas, one usually gets an infinitenumber of Herbrand clauses and therefore the Her-brand procedure does not necessarily come to a stop.

In the above setting, the search for a more ef-ficient procedure leads to Robinson's unification-resolution algorithm, and as in the case of thepropositional calculus, one has to restrict oneselfto Horn clauses if one is really concerned with ef-ficiency. As is well known, such a restriction en-ables the use of backtracking and this is basicallythe strategy of the Prolog language. The study ofProlog offers a very interesting application of logicin computer science. It shows that the views of ar-tificial intelligence can be turned into an actual pro-gramming methodology. Of course, it is clear thatProlog is a programming language and not a theo-rem prover and that completeness is lost, as a con-sequence of various features of the actual language,as the lack of the so-called "occur-check" and theuse of the "cut" primitive. In order to show howthe language works, simple programs can be writ-ten and discussed.

Now, Prolog is not the only example of application of logic to computer science and one canchoose to give an exposition of program verificationthrough Hoare's logic. Recall that this method isbased on cutting the execution path of a Pascal-likeprogram into loop-free pieces. To each cutpoint Ais attached a formula OA, whose free variables arethe actual variables of the program. Logic comesinto the picture in proving that, if execution leadsfrom A to B and if (bA is true at A, with the currentvalues of the varhbles, then (AB is true at B, withthe resulting values of the variables. This is usedto show partial correctness of the program, which

means that, if execution terminates, the final for-mula expresses that the result is as expected. Totalcorrectness can be proved along the same lines byusing a well-founded relation and proving that loopsdecrease values of the variables with respect to thiswell-ordering.

Finally, another topic where logic and com-puter science interact at the conceptual level isthe A-calculus, considered as another approach toprogram-correcness. Once again, it is based on theconnection between algorithms and proofs. Thistime one is talking about formalized proofs withinthe framework of intuistionistic logic (without use ofthe middle-third) and about systems of rules usingthe typed-A-calculus, where a proof yields a term t,which is, in a way, its algorithmic content (cf Kriv-ine and Parigot [1990]). In order to be more precise,let us recall that A-calculus builds terms from vari-ables, through the following rules:

if t and u are terms, then (tu) is also (applicationof t to u)if t is a term and x a variable, then Ax.t also(abstraction)The A-calculus can be considered as a kind of

machine language, a term being turned into a so-called normal form by reduction rules. In order toprogram a function with integer arguments (for ex-ample), one proves a formula (stating that the re-sult is an integer). This gives a term t and executionis just the reduction of the application of t to theterms denoting the arguments. Because of the wayprogramming is performed, correctness is ensured.Of course, the work on this type of programmingstrategy is only beginning and one should not con-ceal that the resulting programming style is highlyinefficient at this stage.

More on syntaxBecause of the orgaiization of our paper around

computation, algorithms and to ir,, we have notdiscussed some quite interesting Lonnections wheremathematics pr3vides the necessary background.For example, we mentioned that :ogical formulascan be considered as trees and the same is true ofcomputer programs. Now both usually appear asstrings of symbol;;. It is therefore very importantto be able to recover the full tree :structure from itsstring version. This is a part of compilation, calledsyntax analysis (cf Aho, Sethi, Ullrnan [1986]). Itturns out that the theory of context-free languagesis exactly the tiJol needed to perform syntax analysisefficiently.

Mathematical Basis of Computer Science 55

Conclusion

In this short paper, we have tried to describewhat we consider as the mathematical basis of com-puter science, to show how the chosen topics can beorganized and to motivate the choices that we havemade. Following the further developments of com-puter science, these contents will presumably haveto be expanded or modified. For example, it mayappear important to discuss boolean networks (tomodel VLSI) or to introduce tools for the study ofrelational databases. In any case, we feel that math-ematical tools for computer science will become apart of any advanced curriculum in mathematics.

REFERENCES

Aho, A.V., Hoperoft, J.E. and Ullman, J.D. [1983]:Data Structures and Algorithms, Reading,MA: Addison Wesley.

Aho, A.V., Sethi, R. and Ullman, J.D. [1986]: Corn-pliers, Addison Wesley, Reading, Mass.

Clocksin, W.F. and Mellish, C.S. [1981]: Program-ming in Prolog, Berlin: Springer-Verlag.

Cook, S.A. [19711: The complexity of theorem prov-ing procedures, Proc. 3rd Annual Symposiumon the Theory of Computing, 29-33.

Cook, S.A. and Reckhow, R.A. [1973]: Timebounded random access machines, J. Com-puter and Systems Science 7, 354-375.

Garey, M.R. and Johnson, D.S. [1979]: Computersand Intractability, A Guide to the Theory ofNP-Completeness, San Francisco: Freeman.

Hoare, C.A. [1969]: An axiomatic basis of computerprogramming, Comm. ACM 12, 576-580.

Kleene, S.C. [1939]: General recursive functions ofnatural numbers, Math. Anna/en, 112, 727-742.

Knuth, D.E. [1973]: The Art of Computer Program-ming: vol 3: Sorting and Searching, Reading,MA: Addison-Wesley.

Krivine, J.L. and Parigot, M. [1990]: Programmingwith proofs, J. Inf. Process. Cybern., EIK 26,149-167.

Manna, Z. [1964]: Mathematical Theory of Compu-tation, New York: McGraw-Hill.

McCulloch, W.S. and Pitts, W. [1943]: A logicalcalculus of the ideas immanent in nervous ac-tivity, Bull. Math. Biophysics, 5, 113-115.

Post, E. [1936]: Finite combinatory processes for-mulation I, J. Symb. Logic, 1, 103-105.


Robinson, J A. [1965]: A machine oriented logicbased on the resolution principle, J. ACM, 12,23-41.

Rogers, H. Jr. [1967]: Theory of Recursive Func-tions and Effective Computability, New York:McGraw-Hil).

Sedgewick, R.(1983) Algorithms, Rending, MA:Addison-Wesley.

Stern, J. [1990]: Fon(t.ernents Mathematigues del'Informatzgue, Paris: McGraw-Hill.

Wirth, N. [1986]: Algeritms ald Data Structures,Englewoori Cliffs, Prentice-Hall.

THE EFFECT OF COMPUTERS ON THE SCHOOL MATHEMATICS CURRICULUM

Klaus-Dieter GrafFreie Universita Berlin, Germany

Rosemary FraserUniversity of Nottingham, U.K.

Leo H. KlingenHelmholtz-Gymnasium, Bonn, Germany

Jan StewartUniversity of Nottingham, U.K.

Bernard WinkelmannUniversitat Bielefeld, Germany

IntroductionHistorical Sketch and TrendsThe three traditional cultural techniques (Kul-

turtechniken), which play the most important rolein our children's education are reading, writing andcalculating. From the time of their "definition"(perhaps 1200 years ago; Alkuin, an adviser ofCharlemagne, mentioned them) the sets of methodsestablishing these techniques have undergone greatchanges and so did the subsets which were accessibleat school levels. In our times the largest expansionoccurred in calculating, which developed into a tech-nique of solving problems formally with numbers,symbols, graphics and words. On one side, this isa result of extensive mathematical research, whichamong other results brought about powerful algo-rithms, easy to execute. On the other side this trendwas accelerated by the rise of powerful processors foralgorithms, namely computer systems together withtheir scientific background, informatics (i.e. com-puter science). These aids make a variety of formalproblem-solving methods accessible for school math-ematics and other subjects, which previously couldnot be executed by students and pupils. Algorithmsform one important class of these methods.

The development outlined above caused and stillhas a significant impact on school mathematics ed-ucation. At least three of the didactical dimensionsof the mathematics classroom are envolved: content,method and medium, to say nothing of the pupil -teacher relationship. Control on these impacts canonly be gained by integrating and organising theminto mathematics curriculum at all levels, since, asA. Ralston [1990] points out " .. only .. curricu-lum content can serve as a lever to change the en-tire mathematics education system". Computer usein mathematics education started as a very specialmethod with mostly special topics. Future com-puter use should be a standard method, applied inwhole strands of subject matter. This article will

57

give a review of some effective and successful stepsand some reasonable trends in the pursuit of thisgoal in school mathematics.

In addition, many of the examples of this pa-per indicate that the technology is already a signifi-cant factor in school classrooms, a factor that morethan deserves its place. The contribution that itcan make to the social and academic interactions is

vivid and, once experienced, always valued.Finally, just as children play out a wide range

of roles in being part of the community they are in,so too can computers. Thus we ask the reader toconsider the -omputer as a member of the classroomcommunity, one that is able to contribute to theday's activities in an appropriate fashion.

Considerations and concrete suggestions for theuse of computers in mathematics teaching dependon knowledge about and experience with such in-struments shared by teachers and mathematics edu-cators. Fifteen years ago these people had access tocomputers mostly as programmers in numerically-oriented languages. So computing power was mainlyused in secondary math education for numerical al-gorithms in the form of short Basic programs. Tenyears ago, another step but still in the algorithmicspirit was taken with Logo on various home com-puters with its underlying philosophy of exploringmathematics in specially designed microworlds andof learning mathematics by teaching it to the com-puter; Logo also included the use of geometry andsymbolic manipulations. Primary education was in-volved with these ideas, even kindergarten.

The proliferation of so-called standard softwareon personal computers in the last decade gave wayto new considerations and experiments, especiallywith spreadsheets, programs for data representa-tion, statistical and numerical packages, databases,CAD (Computer Aided Design)-software and corn-


puter algebra systems. But in the beginning suchsoftware was not very user-friendly, and afterwardsbecame too complex; the need soon became obvi-ous for special school adaptations which allowedeasy specializations, employed mathematical nota-tion similar to that used at school, and used power-ful and helpful metaphors, so that even users withlittle training and only occasional practice (as is typ-ical of school users) could succesfully handle them.This led to the creation of general and didacticalsoftware tools which sometimes also had a tutorialcomponent, thereby integrating some traditions ofcomputer-aided instruction (CAI). All these formsof using the computer came into being in sequencebut can now be found simultaneously in discussionsabout mathematics teaching.

Even if suitable hardware and software are nowavailable for ordinary schools, several necessary in-gredients are still missing: Teacher training is farfrom sufficient; hardware availability in most schoolsis still dictated by the needs of computer science andcomputer awareness courses and the concentrationof machines in special locations prevents or makesdifficult the natural, selective use of software - e.g. afunction plotter - during short episodes in the teach-ing process.

Influences on the Goals and Aims ofMathematics Teaching

In elementary schools children meet basic pro-cesses with patterns and numbers in the mathemat-ics classroom for the first time. There is a range ofuses of technology that have proved positive andstimulating in helping children to express them-selves and to progress in a confident and enjoyablefashion. In particular these can help to discoverypartly unconsciously of the importance of underly-ing structures as an aid to qualified communicationin language and problem solving. The computer iswell-suited to setting up structures - this will be il-lustrated in the examples that are discussed in detailin the section on Illustrative Software below. (Fora more comprehensive discussion of the influence ofcomputers on mathematics teaching, see the surveyby Fey, 1989.)

The emergence of multimedia technology meansthat our communication with computers and, in-deed, amongst ourselves will employ words, picturesand sound in equal partnership and will not be lim-ited to a fixed sequential presentation. Althoughthis article draws on the experience of using micro-computers in the classroom, it. will also be relevantto the more sophisticated interactive video deliverythat is now available.

At the secondary levels we consider two mainaspects which influence the goals and aims of math-ematics education: the (mathematical) preparationof students for their lives and occupationic, and therole of mathematics and its applications in society.

The students' preparation for their lives and oc-cupations starts in the first instance at school withits various disciplines. Since through the availabil-ity of computers, there are now strong tendenciesto introduce simulations into the school teaching ofscience, most notably in biology, or of introducingelements of statistics and data analysis into the mea-suring sciences and geography (cf. Winkelmann,1987), this is obviously a challenge to the teachingof mathematics: Mathematics should elucidate theprinciples, possibilities and possible pitfalls of thesemethods; ad-hoc-explanations of such methods bythe specific content-oriented disciplines are surelynot appropriate for giving the student a coherentappreciation.

It is important to realize that routine calcula-tions of all complexities will be done increasingly byubiquitously available machines which must be con-trolled at various levels by the users concerned. Thisrequires more insight, more breadth, more abilityto check consistency, but fewer routine algorithms.Such an emphasis belongs to the perennial goals ofmathematics teaching, of course, especially in thenew math movement. But now there is really thepossibility of leaving out some of the drill becausetechnology can take over. Even an insight into thefundamentals of computers and their programs maybelong to the preparation for life. This can often beshared with the other formal discipline, informat-ics/computer science, if it is implemented. It is hardto be more specific, since the determination of theelementary and more advanced cultural techniqueswhich are needed by the future citizens presupposesa futurist view of society which is notoriously hardto specify.

As to preparation for vocations, for universitystudies, fundamental ideas and experiences in al-gebra, geometry and fractals, analysis, data analy-sis and statistics, simulation and chaos would nowseem to be necessary in different kinds of studies.More specific preparations for special vocations areagain difficult to determine. For example, CAD(Computer-Aided Design which helps the construc-tion of planar, spatial and other objects on the com-puter screen) is necessary for an increasing numberof technical vocations, and this means the need fornew and different qualifications in geometry; butwhat is exactly needed and how to build a curricu-lum to fulfill the needs of the trades remains unclear.

1:u iJ

The same is also true for the other domains men-tioned in this chapter; therefore, it is not lazinessthat the descriptions above are so general and un-specific. The general direction of necessary changecan clearly be seen, but concrete decisions cannot bebuilt on scientific knowledge yet; we have to experi-ment and gather ideas, examples and proven resultsin concrete circumstances.

Mathematics education at school not only hasthe task of delivering to students the qualificationsasked for in vocations and daily life, but it shouldalso give insight into the role of mathematics in cul-

.ture and society, into the fundamental possibilitiesfor understanding and description offered by math-ematics, and into connected assumptions and lim-itations. In this respect, on the one hand todaythe greater part of the applications of mathematicsis transmitted by the computer and thereby influ-enced in its character, as will be discussed in someinstance below, and on the other hand the computeris fundamentally a mathematical machine and thusits proliferation is a tremendous amplification of themathematization of our lives.

Primary School

Computers and Calculators for YoungChildren

The greatest impact of computers on the learn-ing of school mathematics has occurred in secondaryschool. However, we wish to begin by discussing theprimary school curriculum for three reasons: .

a natural and basically positive attitude towardscomputers can only be achieved at this level.since primary school determines a student's life-long attitude toward mathematics, we must useall possible means - and the computer is one ofthe most powerful of these - to create a positiveattitude during primary education.it is necessary that teachers planning to use com-puters in secondary school and even in universi-ties understand what was done in primary schooland what the problems were there.The first major need to socialise with peer

groups and to share them arises when children moveout of the home into regular contact with othersat playschool or infant school. Here, also, seriouswork starts in developing spoken and written lan-guage skills, learning about the world and meetingbasic processes with patterns and numbers. Plentyof play and creative opportunities are provided toallow natural skills to flourish.

How can technology help in this busy activehappy environment of early childhood? Technology

School Mathematics Curriculum 59

is certainly part of the world that the children willgrow up in but one might feel it is not yet a part thatchildren need to meet directly. Indeed, there areconcerns expressed in some countries that it mightbe positively harmful to allow the use of technologybefore certain basic skills have been mastered.

In the next section we shall look at some exam-ples of use under 'content' headings although theyalso give rise to cross-curricula work. For ease ofillustration we shall take Language Development,Early Science and Basic Mathematics as our maincategories. The decision not to limit the primaryschool part of this article to mathematics is deliber-ate in view of the fact that most elementary schoolteachers carry a responsibility for the major part ofa total curriculum. It is thus important that theuse of computers be set in this context. However,the Language and Science examples also have a rel-evance to mathematical processes although this isnot made explicit.

Before looking at the specific examples, it is nec-essary to discuss the social situation that childrenfind themselves in. Basically, there is a teacher towhom they can turn and who organises their ac-tivities during the day; there is a group of childrenthat they work with, those they play with plus spe-cial friends that they confide in. Thus children con-tribute to a whole range of interactions sometimes aspart of a large class, at other times with a smallergroup, often just to one other person and, finally,they must frequently work things out as an individ-uzi. In short, the challenge that young children faceof being a member of the classroom community iscomplex and demanding.

Children need to develop good productive rela-tionships and for this they need effective verbal andnonverbal skills. Communication through body lan-guage and other nonverbal signals develop naturallyand requires no formal intervention. With the spo-ken and written word the structure of the language,although not formally expressed, begins to be un-consciously absorbed and then actively used to buildnew sentences and expressions. This somewhat sur-prising occurance indicates the importance of under-lying structures as an aid to communication. Thepossible role of the computer in this process wasmentioned above.

We shall analyse, albeit in a rather crude fash-ion, the roles played out by teachers, children andcomputers in the examples that follow.

Thus the focus of the following descriptions willbe to consider the quality of the communicationin the classroom community and to identify struc-tures and roles that enhance the interactions be-


tween members of the community.

Illustrative Software

1. Language DevelopmentWe describe here an extremely simple but power-

ful program called DEVELOPING TRAY. It allowsteachers to typ in pieces of text or poems they wishchildren to explore. The written material may be fa-miliar or unfamiliar; it may be related to a projectthey are studying or simply may have an interestinglanguage pattern or style. At first all the childrensee on the screen is the punctuation commas, fullstops etc. Even this stimulates discussion is it apoem or a piece of prose? They make their decisionand go to the `scratchpad'- a type of notebook builtinto the program to record their first predictionsabout the nature of the text. Now they must 'buy'letters. Every letter or letter pattern they choose toappear in the text costs them points. Every correctword they type in or correct guess they make ontheir note pad gains points. At first children tendto buy letters arguing about those which are 'thebest value'. As they see single letters and groups ofletters dotted about the screen, a pattern starts toemerge. The following letters at the beginning of atext:

0ce --o a ---emay suggest the familiar opening to a traditionalstory 'once upon a time'. The three letter wordt-e may be guessed as 'the'. The children can typein any missing letters. Correct guesses are not onlyaccepted by the computer but are also placed in therest of the text. This one 'h' typed in a correctposition places all the 'Ws' in the passage. Incor-rectly placed letters simply vanish from the screen.Thus the piece of writing is slowly revealed to theclass like developing film in a photographers's dark-room (hence its name). There is great excitementas a word or phrase is identified or as the rangeof words suggests the general content being writ-ten about. Prawns, shells, fish for example mightsuggest a passage about the sea; it could howeverhe about working on a trawler or in a fishmongersor part of a menu for a banquet. The children notonly have fun watching the text develop before thembut they also enjoy looking back at their notes on.their scratchpad to see whether their guesses wereright or wrong. It is not just an exercise in readingand comprehension; it is about collaborating andco-operating towards a common goal and it is fas-cinating for teachers to watch all the skills and in-teractions generated.

As a supporting structure into which the teacheror indeed the children may place any text for ex-

ploration, this software is independent of country,culture or age range. It offers a stimulus to explorelanguage from many different angles and from manydifferent content areas. It can be used by groups ofchildren just beginning to read or by groups study-ing an author's style or even to consider a math-ematical argument. In thin activity the computerplays the role of tasksetter and manager and pro-vides support to a rich and enjoyable learning expe-rience. It may be a task that an individual tackles,but equally small groups and large groups can com-bine their talents to find the hidden text. Teachersand children can work together if the text is notknown to them. Thus there is the opportunity forthe teacher to join in the activity as a fellow pupilrather than to share the role of a tasksetter withthe computer. The children find that they can usethe structure of language and their previous expe-rience in language to help solve the problem. DE-VELOPING TRAY stimulates communication andsupports the strengthening of the use of structurein language.

2. Early science

The following description by Anthony Paddle de-scribes work using EARLY SCIENCE. He considersa use of the computer that offers support to infor-mation structured in binary trees. A diagram suchas this is shown below.

BlueTit

Na

GreatTi'

Does it have a crestor its head?

tes

CrestedTit Has it get a

white patch onthe hack of itshead and neck'

Var-ni.t T:t

Several years ago, various versions of a computergame based on binary trees called ANIMAL ap-peared in magazines and books. A typical dialoguewith a computer running ANIMAL looks somethinglike this:

Are you thinking of an animal?Does it live in water?Does it fly?Does it walk on four legs?Does it go "BOING"?Is it a KANGAROO?

YESNONONOYESYES

Are you thinking of an animal?

Only the boldface answers are typed by theplayer; the questions themselves are stored in thecomputer's memory. Things become rather inter-esting when you think of an animal the computerdoes not know about:

Are you thinking of an animal?Does it live in water?Does it fly?Does it walk on four legs?Is it an elephant? (sic)

YESNONOYESNO

The animal you were thinkingof was a? MOUSE

Please type in a question thatwould distinguish an ELEPHANTfrom a MOUSE?

HAS IT GOT A TRUNK?

For a mouse the answerwould be? NO

In fact, the program starts each time knowingjust one question and two animals. All the oth-ers are added by the players in the same way asthe mouse. ANIMAL mimics a very simple learningprocess.

Clearly, the questions and animals are stored inthe form of a key (or, equivalently, a binary tree),ANIMAL combines the functions of a key-searchingprogram and key-building one. Although intendedas a 'try to fool the computer' game, it could be usedquite seriously as an identification aid. As such ithas definite advantages over a traditional printedkey, especially for children.

The first advantage is Oat only relevant ques-tions are displayed on the screen. A key containing1000 animals needs 999 questions but, in theory,

--astuaa


only ten of them need to be answered to identifyany one of the animals. In practice, reality tri-umphs over logic and keys cannot be designed thatwell; nevertheless, only a small fraction of the ques-tions are relevant at one time. The remainder aredistracting clutter, and it is easy to become hope-lessly lost in a large printed key. ANIMAL avoidsthe problem by avoiding the clutter. Secondly, AN-IMAL breaks down the highly abstract problem ofdesigning an identification key into simple, concretesteps. To add the mouse to the key it is not nec-essary to think of all the attributes of mice or tosearch for some essence of mouseness that will dis-tinguish mice from anything else. You are simplyasked to find one clear difference between a mouseand one other animal. The key-searching part of theprogram ensures that the other animal is the mostsimilar one already in the key, so that the mouse isinserted in the right place. This is not the only wayof breaking down the key-building problem nor, ifthe aim is to produce 'elegant' finished keys, is itthe best. Nonetheless it is easy and foolproof: Ifthe individual questions work, the whole key will.

ANIMAL was not designed as a robust educa-tional tool and suffers from a number of deficiencies.It is not possible to correct any of the questions oranswers once they are entered spelling mistakesare permanent. Nor is there a facility for saving akey on tape or disk, or for printing it out on paper.The language used by the program itself limits theuse (Àre you thinking of an animal?' is a built-inquestion). It would be awkward to use it to classifyplants or rocks.

There are now several elaborations of this idea,written for educational use, in which these problemshave been solved. THINK is one example which,while keeping the outward key format of ANIMAL,has become a sophisticated tool for the creation,correction and searching of binary trees.

A further development is offered by SEEK,which comes in a package with THINK, severalready-made keys and a program called INTREE fortyping in whole keys quickly. SEEK uses the com-puter's graphics to display the questions in binarytree form. The questions appear in boxes and, de-pending on whether you give a Y or N answer, youare led down a branch to the left or right into an-other box containing the next question or the an-swer. At any stage you can move back up the treeand down another branch, so that the whole treecan be explored. SEEK makes the structure of theinformation appear obvious. ANIMAL and its moredirect descendants appear, by contrast to producequestions from nowhere; they seem cleverer than


they are.

3. Using Key Handling Programs

Programs such as SEEK can be used in a surpris-ing number of ways with children. Obviously theycan be used to identify things if a suitable tree ofquestions is already available. At the other extreme,children can build their own tree from scratch, givena set of rocks, twigs or kitchen powders, for instance.

There are also strategies that fall between thesetwo. If a class is planning to go pond-dipping, a keyto the commonest animals might be created in ad-vance, using information from books. New animalscan be added one at a time as they are found, pos-sibly over a long period. This may well be the bestapproach with large, complex groups of things. Theinitial skeleton tree can be designed so that its mainbranches represent the major groups (nymphs, lar-vae, snails, worms, etc.) and the research involvedin creating it can give focus to the childrens' prepa-rations for the first outing.

In the classroom, identification exercises canprovide very effective frameworks for practice of ob-servational and experimental skills. A particularlygood example is the POWDER tree supplied withthe SEEK/THINK package. On the surface it issimply an identification key for common householdpowders, such as sugar, salt, washing powder, flourand baking powder. The questions, though, are notjust passive observational ones: Most of them askthe children to do something to the powder andwatch its reaction. In the next column is part ofthe key as produced by SEEK on a printer.

There is no one way of classifying things. Theremay be generally accepted ways for groups likeplants, animals or rocks, hit even these are subjectto constant argument among scientists. If childrenare to understand why things are classified the waythey are, they need to explore and compare differ-ent ways. It is here that programs like SEEK displaytheir real value. By taking care of the overall organ-isation of the tree, they let the children concentrateon close observation, comparison and the logical andlanguage aspects of choosing good questions.

Imagine that a group of children are trying toidentify some epsom salts using the POWDER tree.They will probably find that it is wrongly identifiedas a salt. If they decide to extend the tree theywill be asked to find a question to distinguish thetwo. This is no small challenge, finding the bestquestion may take a lot of time, experimenting anddiscussion. The first stage is to find out everythingthey can about the two substances by observing,

QUESTION YES NO

1 Feel your powder?Is it smooth orfloury? 2 3

2 Put some in ateaspoon andheat over acandle. Can yousee lots ofsteam?

BAKING 4POWDER

3 Look through amagnifyingglass to seeif it is lumpsor crystals. Isit crystals? 5 6

4 Put a drop ofiodine on yourpowder. Does it goblue/black?

5 Put some in ateaspoon andheat over acandle. Does itsmell liketoffee?

6 Put some inwater and shakeDo you get lotsof bubbles?

7 Put a teaspoonof powder on asaucer and addvinegar. Do youget bubbles?

FLOUR ICINGSUGAR

SUGAR 7

SOAP POLY-CELL

WASH- SALTING

SODA

practical testing and research into their uses. Theresult. may be quite a long list of differences, so thesecond stage is to decide on the best question to beadded to the tree.

`Does it dissolve in water?' is no good becausethe both do.

'Does it taste salty?' may be ruled out on safetygrounds (someone may try to identify somethingpoisonous).

'Do you buy it at the chemist?' requires prior

G 0-

knowledge and would be impossible to answer if youreally did not know what the powder was.

`Does it have big crystals?' does not have a clearanswer: It depends what you compare them with.Also the crystal sizes of both vary enormously.

`Does it have long, thin crystals?' is better, as is`Does it turn into white cake when you heat it overa candle?'

Some of these problems are quite subtle, andchildren are unlikely to spot them until they try the`bad' questions in a complete tree. Fortunately, allthe more recent programs let you prune and repaira tree without having to rewrite the whole thing;so children can learn from their mistakes and cor-rect them with a minimum of frustration. A goodway of identifying problems and sharing insights isto encourage groups of children to test each other'strees.

4. Mathematics

AUTOCALC is another example of a simpleprogram that promotes considerable discussion andsharing of processes. It enables children to articu-late their own methods and ideas and has provedan extremely valuable way to build their confidencein their mathematical abilities. The children arechallenged by the program to try out their men-tal arithmetic skills and to review and compare therange of possible processes. A large screen is neededat the front of the classroom. The screen presentsthe problems in the following format:

44

+ 29

After a delay the computer then supplies the an-swer to the calculation

44

+ 29

73

The mode of the program is to generate suchproblems by selecting random numbers according tothe parameters set at the begi- ling, using a chosenoperation and displaying the swers after a chosentime delay. The option screen used for defining thetype of problem to be set is shown below:


Autocalc Options

Type of problemDifficulty LevelTop numberMiddle numberBottom numberDelay time

SubtractionOwn1 to 201 to 10

0 to 202 seconds

This option setting provides simple subtractionproblems for young students.

Imagine a class of children working on the waysin which they àdd 9' to numbers. The computer isset to produce problems where the number is gener-ated between 0 and 99, the second number is fixed at9 and the time delay of 3 seconds before the answeris given has been set. Fifteen problems appear oneafter the other and the children attempt to calculatethe answer before it is displayed by the computer.To simulate the experience complete the followingproblems as quickly as you can:-

28 90 32 77 88 79 37 669+ 9+ 9+ 9+ 9+ 9+ 9+ 9+

Probably after the first try at this task the chil-dren will feel that they might be able to get theanswers in under 2 seconds so that they can haveanother go with reduced time delay. Some mighteven like to go at producing an answer in 1 second!After this activity the children are asked to say ex-actly how they got the answers. The following listof methods was the result of a class of ten year oldssharing their ideas:

1. Helen decided to add one to the 'tens' and thentake one awa 'rom the 'units'.

2. Jonathan wa.s happy to count on his fingers butdidn't always have time.

3. Susan added 3 three times.4. Jo subtracted '1' and added '10'.5. Anne worked out 'how many to the next 10's'.

This is then subtracted from 9 and the remainderaddede.g. 78 + 9 = 80 + = 87

6. Simon added 2 four times, then 1.7. Jane used different mathematics for different

problems.8. Michael just 'knew' the answers!

The children greatly enjoy sharing their methodsand trying out each other's ideas. They are also en-couraged to use calculators various tasks are given


that begin to expand their understanding of num-ber and to encourage them to feel confident enoughto move into estimating outcoming numbers as well.AUTOCALC can be run in a mode where the chal-lenge is not to do the calculation but to spot ev-ery time an incorrect answer is displayed. (This iscalled the 'oops' .mode.) The skills needed to suc-ceed here are now dependent on having a good graspof number bonds and relationships. Another excel-lent activity is to ask the children how many prob-lems they can make combining a number between0-99 and a number between 0-9 that has the an-swer of 5; e.g. 13 8 = 5; 1 x 5 = 5 etc. Afterexploring the problem in groups, the children offertheir solution. This brings out some fundamentalmathematical processes classifying sets within thesolution set setting the initial conditions in orderto limit the solution set to a finite set and manyothers. The children express these ideas in theirown language and, of course, they are not yet awareof the generalisation of such ideas. However it. is

at this point that we become aware that this simplecomputer program has given the children a stimulusthat has caused them to become true mathemati-cians. In sharing their mathematical processes andin valuing each other's ideas they will build up con-fidence in their own abilities to offer something tothe subject. In this way we can begin to remove thefear that so many people leave school with in regardto their mathematical abilities. A final stage to thediscussion of the '5's' problem is to watch the com-puter doing the same problem and to write reportto 'its parents' on its performance. As the computerapplies an extremely simple algorithm (it just keepsrandomly generating problems but only displayingthose that give the result. of 5), its performance iscertainly open to criticism. Here are some of thechildren's reports:

sJour- 0 k c ,

I 4i- .r,k Ljo:_ Should help (4: Ur' C4,./cl1-0 see fo con.; IP 900 313c:4143PIFO ercorage

c!-110)40 e 0344/n.9,

P,9/r),"

At.A?fte

elA4-1- y oust,

Jac vuud , 14e466.1. k'lL+,), /1.1diflon Aig k. itri_otA

; 5 0, cov1 C1l cA4k1k-111.P-1 FLA.A.-4

DEAR rtr 4d ns. C.

5u93rs+ Giveor o h, for chip. ic wc.i-from our sthooi (T1ernrry.

p.

Y"r Sov rnn: eq.nopyv.eig erAs k..7

TE;',cstiar X3 25/ Z.

witr hO.

p 4 it

8,c) )15 f, re Lieni.S en .

Sib 2'; j 10 ;Werent-

S b

Ault. 0

Critics might say that the activities promotedwith AUTOCALC are not valuable because theyare dealing with numbers out of context to any realproblem. However we hope that the examples here,which are only a minute part of the range of possi-bilities, show children becoming aware of their ownpower and thought processes and also taking over arange of 'teacher roles' at various stages. Feedbackto the teacher of the children's reasoning and theway in which they articulate this is a major contri-bution of AUTOCALC.

A few years ago Michael Girling (Her Majesty'sInspector) suggested that a definition for numeracymight be 'appropriate use of an electronic calcula-tor'. What number sense would one need in orderto qualify?

We suggest:

1. Instant command of single digit arithmetic2. Command of basic multiplication facts3. Skill in estimation4. Capacity to spot errors5. Capacity to select which operations are appro-

priate in any problemWith the exception of 5 all these points are

strengthened by the activities possible with AUTO-CALC.

Concluding RemarksThis section has taken just a few examples of

simple software to illustrate how computers canhave a stimulating and refreshing relationship withchildren. We are keen that the computer becomesan accepted assistant and friend of both teach-ers and children.The use of Logo, data banks and

word processing, have not been discussed here asmany books and articles are available to the readeron these topics. Such languages and systems canbe employed to stimulate discussion and exitemcntsuch as is described here. However, they can makeconsiderable demands on the users and we wouldrecommend that subsets of such systems are usedto start with. Slow progress is being made with im-plementing a curriculum that make effective use ofcomputers and calculators. This is due to the factthat there is not as yet a great deal of curriculumsupport materials to introduce the range of learningactivities that simple or complex computer softwarecan support. However, this material will graduallyemerge and there is certainly enough available toany enterprising school to offer children the advan-tages of a computer in their classroom.

Any school able to equip each classroom witha single microcomputer would gain experience andconfidence within a matter of months rather thanyears. Add to this provision a small laboratory forword processing etc. together with a collaborativestaff exploring possibilities together and the scenewill be set for an exiting time for children in such aschool.

Secondary School

Phenomena, Theories,Experimental Mathematics

In mathematical knowledge one can differentiatebetween facts on the one hand and the insight intotheir necessity and their connections on the otherhand, or between phenomena and theories. Thisdistinction becomes clear, for example, in the do-main of the geometry of triangles: Examples of phe-nomena are the observable facts, such as that thethree angel bisectors meet in one point and simi-larly for the perpendicular bisectors, that the sumof the inner angles equals 180 degrees, that two tri-angles which have the two sides and the enclosedangle equal have all other measurable parts equal,the formula of Pythagoras, etc. Most classical the-orems of school geometry belong here, but so alsodo more qualitative facts such as: If two sides arefixed in length, then the third side gets longer if theenclosed angle is made bigger (up to 180 degrees).There is now special software such as The Geomet-ric Supposer or Cabri Geometre which helps to findsuch facts by giving assistance in the making andsystematic variation of geometrical constructions.

In the domain of theory there is the logical or-dering of facts (local and global), the insight intothe necessity of observed facts, the determination


of the proper conditions under which the facts re-main true (the domain of validity), etc. As a con-crete example, let us look at the calculus (analy-sis).' Phenomena are: The graphs of functions, sayof f(x) = x sin 1/x, the fact that sin x/x tends to 1as x tends to zero, the divisibility of xn 1 by x 1

and the fcrm of the divisors, the formulas for thederivatives of elementary functions, the linearity ofthe integral, or the shape of solutions to a specificinitial value problem for a differential equation.

To the domain of theories, there belongs the def-inition and fundamental properties of the limit, thecompleteness of the real numbers, the definition ofthe integral, the limits of validity of theorems, andexplanations of facts by arguments.

It is interesting, that there may be different pos-sible theories, for example, Euclidean or Cartesiangeometry, with formalist or constructivist founda-tions. Or, in the case of analysis there are differ-ent possible non-equivalent theories, the classical

s-theory, non- standard analysis and differentconstructivist approaches. But all those differenttheories explain in different ways the same phe-nomena. And all the concrete applications of geom-etry or calculus only rely on the phenomena, not onthe underlying theories. In a similar way, comput-ers and mathematical software work exclusively inthe realm of the phenomena; they can only exhibitphenomena. And they are able to show the phenom-ena even to students who have not yet mastered thetheory.

This is the point in our argument: In a mathe-matics class using mathematical software, studentswill get to see and know a lot of mathematical phe-nomena. The mathematical theory then has to ex-plain these phenomena; thus mathematics shifts inthe direction of a science which orders, describes andmakes understandable facts that are already knownand obvious even without explanation. This is insharp contrast to classical teaching methodology, es-pecially it, such domains where it was hard to ap-proach the phenomena without theory or advancedtechnology.

Here is an example. In the study of functionsand their transformations, traditional teaching de-duced behaviour mostly from theory, since the ac-tual plotting of function graphs by hand was far tooexpensive, in terms of time and labour, in order tomake students see the facts, for example, the graph-ical translations connected with the transformationf(z) f(x + a). With the help of a function plot-ter they may observe those transformations, firstconnected with a concrete f and a, then system-atically explored with free chosen examples, and in


between also formulated as hypothesis and verifiedby arguments. In this way, the temporal order ofphenomena and theories reverses, and gets closer tothe usual habits of mathematics as a research ac-tivity. Of course, such an approach has often beenused with mathematical content where explorationof phenomena was cheaper.

The didactical paradigm just described has of-ten been referred to as "experimental mathemat-ics", but it has to. be stressed that theory is an in-dispensable part of it in order to be mathematics.Just playing around with a function plotter doesnot necessarily lead to insight. You normally needhints, ideas, hypotheses, questions in order to seesomething and get involved. (See Goldenberg, 1988for more specific considerations and examples.) Asa counterexample, using fractal generating softwaremay give spectacular pictures of great esthetic valuebat, if you stop at the phenomena, you won't get atmathematics with such software. You need at leastgeneral concepts such as self-similarity or symmetry,which are also needed for the better understandingand appreciation of the beauty of the pictures in-volved.

Software for Secondary SchoolMathematics

We shall discuss this for three content areasGeometry, Functions and Data Analysis.

GeometryTwo software packages for geometry education

were mentioned above: The Geometric Supposerand Cabri Geometre which allow constructions ofmost of the problems of Euclidean plane geome-try. A so-called draft mode allows the exploration ofconsequences of moving one point in a figure whilekeeping its connections to other points (see Fig. 1).Descriptions and examples are given in Schumann[1990]. Here we shall describe two other pieces of

Abb. 2.1 Abb. 2.2 Abb. 2.3

Figure 1

"teachware", which allow some unconventional ac-tivities which are closely related to the curriculumfor grades 7 and 8.

The elementary didactical philosophy is thatthere should be two levels of action in geometryclasses, when using a computer: On one level thepupils should learn the constructions manually withruler and compass, as usual. On another level theyimprove their competence with these constructionsby solving geometrical and applied problems withgraphics procedures on the screen which they per-ceive as efficient and comfortable tools. In par-ticular, this use of computer graphics in the earlyyears of secondary school has proved useful in threemodes:1. Using procedures for ruler-and-compass con-

structions which have already been understoodas building blocks for more complex construc-tions without the need to repeat the elementaryconstructions again and again.

2. Using procedures for constructions in wayswhich cannot be realized with ruler and com-pass.

3. Using procedures for large and technically dif-ficult constructions, which demand many itera-tions of elementary constructions.The Geometric Supposer fits in mode 1. We now

discuss two other software packages, SYMMETRICTURTLES and KALEIDOSCOPE, which illustratemodes 2 and 3.

SYMMETRIC TURTLES (Graf, 1988)

It is well known that Logo's turtle graphics canhelp at the beginning of geometry education. As atool which provides an extension of the ruler andcompass a "running turtle" has been developed.This follows the concept of Abelson's dynaturtle[Abelson and di Sessa, 1985], but without inertia.To some extent you can use it like a pencil, con-trolled with keys.

Keys 1, 2 .. 9 put it in slow or faster forward mo-tion on a straight line, key 0 stops it. Z or N lets theturtle draw or not draw when moving. A, S, D, Feffect small (5 degrees) or larger (15 degrees) left orright turns of the stopped or moving turtle. Q marksthe position of the turtle on the screen and deter-mines a number for this point. This point can bereached again via keys K or P. K turns the turtle inits actual position heading for another point. Thiscorresponds to putting a ruler through two points.P puts the turtle on an already marked and namedpoint. And so on.

This running turtle allows construction of manyfigures of interest in plane geometry. Besides this

turtle there are two more turtles, an "axial symmet-ric turtle" and a "central symmetry turtle" . Theyare controlled in the same way as the standard run-ning turtle. But they do not only draw the figurecontrolled but also the figure's symmetric image ina different colour relating to the x- or y-axis or thecentre point of the screen. This happens simultane-ously and pointwise. This mode of construction forsymmetries helps the user to recognise the proper-ties of the mappings immediately and more easily

-s7

Figure 2

Figure 3


than from the final picture. You see that a straightline remains a straight line, you see how the direc-tion changes under axial symmetry and how it re-mains the same under central symmetry. You alsosee that a straight line and its picture are parallelunder central symmetry, but have different direc-tions, etc. Figure 2 contains some examples. Unfor-tunately, the "dynamic" quality of the turtles can-not be seen from these figures:

Figure 3 shows how the following question canbe examined: "What happens when reflecting a tri-angle in different positions relative to the axis ofsymmetry or a point?"

Figure 4 gives a systematic answer to the ques-tion, "How can quadrilaterals be generated by re-flecting triangles?"

Figure 4

First, it is convenient to choose a side of a trian-gle as an axis of symmetry. Then with the turtle youget a kite. The special case of an isosceles triangleoccurs if the angle adjacent to the axis is 90 degrees.If this angle is greater than 90 degrees, then you geta quadrilateral which is not convex. You can alsoget a rhombus and square by starting with specialtriangles. But you never get a general rectangle ora parallelogram or a trapezoid. The central sym-metry turtle, however, applied on the centre of aside of a triangle produces a parallelogram imme-diately. This is an exciting discovery. The choiceof this special point of reflection is suggested by theexperiments shown in Figure 3. Again, no trapezoidoccurs. This fact can result in geometrical discus-sions. More details about these tools are given inGraf [1988]. Some reactions of teachers and studentteachers to this kind of teachware and some experi-ences in classes are also reported there.

KALEIDOSCOPE

In a paper by Graf and Hodgson [1990] it is


shown that the kaleidoscope can be a window tosome geometric concepts. These are elementary andrich at the same time. They are rich in the sensethat they offer not only a mathematical model ofthe kaleidoscope but also models for other worldslike planes to be tiled or even like a fictional kalei-doscope.

From a methodological point of view the mathe-matical problems connected with kaleidoscopes canbe worked on at the following five levels:1. Looking through the real kaleidoscope.2. Reducing the kaleidoscope to a model with two

or more real mirrors placed on a sheet of papercontaining some

3. Abstracting the mirrors and their reflections onstraight lines (axial reflections) constructed withruler and compass.

4. Transferring these constructions to a computergraphics display.

5. Using formal methods to describe the phenom-ena (and PROVE theorems!!!), for example,those of analytic geometry and linear algebra.Here we can only give a few glimpses on the soft-

ware for simulating kaleidoscopic phenomena on thecomputer and examples of patterns that can thus beproduced.a) Two-mirror kaleidoscopes: The main menu of-

fers a choice of four different types of kaleido-scope. Mode 1 leads into a dialogue about form-ing a kaleidoscope with an arbitrary angle. Theuser gives the positions of the axes and then theposition of the object to be reflected. The com-puter then displays the two axes and the object.It then constructs and displays one reflection af-ter the other until the pattern is complete. Thiscan be done with a pause after each image or inan automatic mode. Mode 2 allows one to se-lect a kaleidoscope with angle 45, 60, 72, 90 or120 degrees and then proceeds as above. Figure5 shows some steps in the development of a 60degree pattern and the same process for a 70 de-gree pattern. One of the many questions whichwill arise after such experiments is: How manyreflections are there before the pattern begins torepeat itself? Circular symmetry can be discov-ered and discussed after such experiments.

b) Polycentral kaleidoscopes: These are built froma greater number of mirrors and thus producegroups of images around several centres spread-ing in all directions. Forgetting about specialobjects between the mirrors and just regardingthe reflections of the triangles or quadrilateralsand so on shaped by the mirrors you discover

---.,r --. ----.<

,...-

.. -

> ::r

....: I.-

>-1

L.- .. -7,/,

.i., -,/ ,t/ Icl--........._1 v ' -1 - ' ' , ..:.."

Figure 5

another mathematical phenomenon: Coveringsof the plane. In the case of three equal mirrorsor "sides" you end up with a perfect tiling ofthe plane. Figure 6 shows the growing of sucha tiling. Obviously a discussion will arise fromthis about good and bad tiles.

Figure 6

'177 ,r77%

>.77

c) Fictional kaleidoscopes: The examples given sofar considered the transfer from real kaleido-scopes to mathematical models, combining ax-ial reflections and varying the type of kaleido-scope. Why not vary the mathematical modeland forget about reality? Central reflections(half-turns) will give us a model of some fic-tional kaleidoscopes having no physical counter-parts. One case is to look at a triangle again, de-termined by three centers of reflection, and seewhat happens after repeated reflections. We c:'nget a pattern extending throughout the plane(see Figure 7), leaving some blank spaces.

A new situation occurs if we start reflecting a tri-angle not about its corners but about the midpointsof its sides, and go on reflecting the images aboutthese midpoints. A new tiling of the plane devel-ops and what is really surprising for the beginner

this works well with any triangle. Next you canturn to quadrilaterals and again you discover imme-diately that a perfect tiling can be completed withany quadrilateral, even a non-convex one (see Fig-ures 8 and 9). So the fictional kaleidoscope bringsyou back to a real problem and the search for itscorrect mathematical solution.

Figure 7

Figure 8


Figure 9

We conclude this section about geometry withsome general remarks. There should always be avery careful examination of the advantages for learn-ing before the computer is used in some field ofmathematical education. There is no use in trans-ferring manual or mental activities (like construc-tions with ruler and compass) to the computer un-less this brings about more efficiency in learning.Another good reason for using the computer mayexist if the computer allows activities which the stu-dents cannot achieve with their hands or brains.Then the computer acts like an additional tool,increasing the traditional abilities of the students.SYMMETRIC TURTLES and KALEIDOSCOPEare good examples of such tools. They allqw

additional help in exploring mathematical prob-lems.a great variety of investigations with little efforteasy experimentationthe viewing of a very broad spectrum of complexgeometrical constructions which turn up whenstudying reflections of complicated figuresdoing manually impossible constructions like thepointwise simultaneous construction of two ormore figuresintroducing and using simple methods of CAD(computer-assisted design), a technique whichhas replaced manual technical drawing to a con-siderable extent.

Fly

tl

Functions

Function plotting software in acceptable qual-ity for use in schools is now available for nearlyall modern microcomputers, and there are now sev-eral sources of didactical material describing teach-ing units, giving hints and providing exercises whichcan be used by normal mathematics teachers. Thegeneral idea of a function plotter, to plot the graphcorresponding to a user given function, ran also beinverted, namely to plot the graph and let the user


look for the term. This is realized in the "Funktio-nen raten" (looking for the formula) part in Graphir.

For example, the program plots the graph of afunction say f(x) = 2x 3 but does not showthe term (Fig. 10). The user has to make use of theinformation given in the graph to guess the func-tion term and put it in. The computer reacts byplotting (in another color, if available) the graphcorresponding to the user's term in the same coor-dinate system. If the user has not got the correctsolution (Fig. 11), he or she can now see the dif-ference between the original and the guessed graphand use this information to debug, that is, to cor-rect any mistake. As many tries as desired can bemade. It is also possible to wipe out the screen andsee only the original function.

f Cx3.

Figure 10

C ?)

f Cx3,,x-3

Figure 11

The functions plotted by the program are of-fered in different sets, organized according to dif-ficulty and type of function (linear, quadratic, cu-bic, trizonometric, exponential, using absolute val-%les, etc.). The sets can be changed or augmentedwith a simple text editor by the teacher, according

to the needs of the students. A specific option letsthe program be used by two partners (individuals,groups): the first gives the term and the other hasto guess it from its graph.

The simple idea of the program gains its moti-vational and challenging character from the use ofa sophisticated function plotter, which comes closeto the accustomed appearance of terms and graphs,and from its deliberate gelerosity to an iriexoerienced user. It is simple to use. The user is notpenalized for wrong answers. And it has adequateerror control, not through comparing the user's termwith a predefined list of possible right terms, butby numerically comparing the graphs with a certaintolerance. So the software aims really to help usersto evolve and debug their knov,ledge about elemen-tary functions and their standard transformations.

The program is to be used mainly by individualsor small groups, in a wide variety of levels, grades 7to 12 and up. It may be used for drill and practice,and, of course, for remedial work.

Data Analysis

Statistical education as mathematics educa-tion in general often has to cope with the problemthat, in order to solve real problems, the necessarytechniques are taught and, in consequence, also un-derstood by students in isolation; their proper con-ditions of application, their region of validity, theirlimits are perhaps theoretically known, but seldompart of active knowledge. In order to overcome suchlimited understanding, one method is to confrontstudents with problems connected to themselves, sothat they don't take the methods as neutral, butof real importance. One of the goals of the soft-ware Times is just to give students some real data,connected to themselves, in order to analyze and todraw conclusions from the data and thereby aboutthemselves.

The software allows experiments with reactiontimes: The computer produces a specific signal andone of the students has to react in a specific way,for example, by pressing a specified button, and thecomputer measures the reaction time. The processrepeats, and the data are stored into a file bearingthe name of the student. Another student does thesame procedure, and the data is compared. Whichstudent is better? Is the arithmetical average a fairarbiter or is the median better? How should onejudge extreme values? The program offers severalmethods of comparing data, including some wellknown statistical techniques. It calculates diversequantities such as averages, variances, the plot of

one distribution of values against the other It doesQQ-plots, displays the data as time series, etc.1 Indefending their results, the students hopefully learnto judge cautiously, to see the techniques as helpfulbut normally not decisive tools, and the necessity ofproperly interpreting the data rather than automat-ically drawing conclusions after a routinely appliedtest.

General Tools and Methods

Besides studying software for specific mathemat-ical areas like the ones just discussed, it is importantto consider software which supports specific mathe-matical methods which have importance in differentareas. Here algorithms in their original sense (thinkof Euclid's algorithm) are most familiar and wereintegrated into mathematics education even beforethe advent of computer systems. First we shall givean example of an algorithmic strand, which fits thecurriculum for the German "Gymnasium". Fromthis you will be able to see how this old mathemat-ical idea of algorithm can be extended to a num-ber of complex mathematical problems. Then weshall discuss the general problem of how to com-bine in class teaching students to understand andexecute mathematical methods and to solve math-ematical problems (from multiplication to integra-tion) by hand or brain with the need to tell themthat there are computers which can do these thingseasily if you just give the problem to them in theproper way. This is the black box/white box prob-lem. Finally, we discuss two more general methodsof growing importance in mathematics education (aswell as in mathematics research) simulation andmodel building.

The algorithmic strand.

Algorithms are patterns with a certain schematicbackground; although high mathematical inventionwas necessary for their discovery, only stupid andexact processing is needed for their application.With this didactic philosophy the teaching of con-cepts and theories of mathematics had priority atschools. The use of algorithms formed the center ofexercises, homework and control of achievement andso pupils were educated as if they were little com-puters. Related to this secondary role of algorithmsis the fact that several thousand years of history ofmathematics have not produced a uniform languagefor the description of algorithms. Now there is a

1 For a more detailed analysis and critical descrip-tion see Biehler and Winkelmann (1988).


continuous algorithmic strand which forms 15% ofthe curriculum in mathematics education during thenine years of German grammar school. (We beginat year five since we do not consider the four years ofprimary education.) The following list shows typicalalgorithms and also their related subjects.5 Relations between the fundamental arithmetical

operationsTransformation between numbers with differentbases: (10,2,5,16 etc.)Division algorithmSieve of EratosthenesOptimizing termsSummation of arithmetical series according toGaussFundamental operations with sets

6 Calculation with fractions (handling formalrules)Greatest common divisor and least commonmultiple (algorithm of Euclid in several varia-tions)Prime numbers, twins of prime numbers, distri-bution of prime numbers etc., factorisation ofnumbersArithmetic means, relative frequenciesDiagrams of descriptive statistics

7 Tables of proportionsCalculation of percents and interestRandom experimentsConstructive geometry in two dimensionsGeometrical mapping

8 Algorithm of HeronIterations for linear equationsSymbolic processing with equations

9 Solution of quadratic equationsGraphs of quadratic functionsCombinatoricsContinuation of geometry (similarity)

10 Several methods of integration of the circleDivision of polynomialsTrigonometric constructionDescriptive statistics

11 Experiments with sequences and seriesDiscussions of functionsAlgorithm of Newton with variationsRegula falsiMethods of optimisation


12 Methods of integration according to Simpson,Gauss, Romberg etc.Symbolic integrationAlgorithm of Gauss for systems of linear equa-tions with variationsOperations with matrices etc.

13 Stochastic simulationsSymbolic handling of limits (l'Hospital)Standard methods of inductive statisticsMethods for numerical, graphical and symbolicsolution of simple ordinary differential equations

This listing refers to a basic level of higher ed-ucation. For the advanced level ("Leistungskurs"with 6 lessons in the week) a lot of possibilities canbe added in the last three years such as:

complex numbers, special numerical methods,algorithms of the theory of graphs, fitting ofcurves according to Taylor and Gauss, interpo-lation of functions according to Lagrange andNewton, cubic splines, study of nonlinear iter-ations, mapping and representation in three di-mensions, constructive non-euclidean geometryetc.

Also in the content of an algorithmic strand, themethodological aspects need not be lost. Severalbasic formulations of computer algorithms are help-ful for mathematical comprehension, too. For ex-ample, pupils always have difficulties understandingthe usual notations of sums, double sums, etc. Thealgorithmic notation using for-loops or nested loopsremoves many difficulties in understanding the roleof summation index etc. The practice of program-ming recursions is helpful for understanding the log-ical basis of induction proofs, etc.

For nearly all of the subjects listed software isavailable, some of it with more options than areneeded in schools. Most teachers are pleased thatthey do not have to enter into the specialities ofgraphic representation and the other "higher" workof computer insiders. Still some of them rememberanother kind of work only a few years ago: For im-portant algorithms of mathematics (Euclid, Gauss,Newton, Simpson etc.), teachers themselves had towrite their own programs. The advantage of thiswas that they could develop the central ideas simul-taneously in their classes and in the programs. Thedisadvantage was that the handling of many pro-grams was not easy. Still, a further advantage wasthat the teacher could modify an algorithm usingthe (sometimes unusual) suggestions of the pupils.As an example, in Newton's method for the solu-tion of transcendental equations, you could take the

tangent of a function instead of the function itself.Or you could take tangents with the same constantslope as the first tangent (the method converges inmany cases). Alterations of this kind are generallyimpossible with acquired programs, which seldomallow such open didactical processing. Naturally,for these purposes the teacher needs a simple andtransparent computer language with natural key-words and sufficient mathematical operators as wellas a compiler which can understand the languagein the same sense as humans. Teachers need aswell a good cooperation with teachers and pupilsof computer science, who can construct good pro-grams according to their desires. Some programs inthe school market need to have a didactical dimen-sion so that, for example, the plotting of functionscan be stopped and continued using the intuition ofthe pupils. During algorthmic processing intermedi-ate suppositions about the results should be possiblewhich can be verified or falsified.

Symbolic Processing/SymbolicManipulation

In recent years symbolic processing for personalcomputers has entered into schools (see the chap-ter by Hodgson and Muller). Solving linear andquadratic equations, equations of third and fourthdegree, large systems of linear equations, simplificz.-tion of rational expressions with "towers" of doublefractions, division and simplification of polynomialscan all be done with symbolic algebra, often inte-grated with the direct processing of very large in-tegers. Where exact methods fail, approximationsare possible. Symbolic differentiation and integra-tion, symbolic vector analysis and, finally, the sym-bolic solution of ordinary differential equations offirst and second order together allow the possibil-ity of ignoring all the rules of school mathematicsin a traditional sense. These packages are made byprofessionals. Therefore, they often do not presentintermediate steps and some other didactical re-main. Some of the symbolic packages are not pro-grammable by the user. Nevertheless the union ofnumerical, graphical and symbolical tools has enor-mous power for schools.

Enlightenment through Black BoxesIn a recent article, Buchberger (1990) asks,

"Should students learn integration rules?", giventhat now there are computer algebra software sys-tems available which solve any integration problemmuch more quickly and more reliably than any stu-dent could ever do with paper and pencil. Buch-berger immediately generalizes the question for all

those areas of mathematics which are "trivialized"by modern software, especially computer algebrasystems. He answers for mathematics and com-puter science majors with his "White Box / BlackBox Principle": Students should learn the theoriesand algorithms of such an area first, using the soft-ware only for subordinate tasks (e.g. partial fractiondecomposition) but, after having studied the area,all calculations from this area should be left to thecomputer.

For schools and general mathematical softwarethe situation is more complicated: Numerical andgraphical oriented software doesn't trivialize an areaof mathematics, but may provide profound help insolving problems; school mathematics does not onlyprovide mathematical theories and algorithms butalso their intended applications, their modes of useand the translation schemes needed in using themoutside of pure mathematics. High school studentsare just not future mathematicians, but could heregarded as future users of mathematics as well,who obviously should have a different attitude to-wards mathematical tools. So here we do not give arecipe but rather some considerations which mighthelp in coping at school level with the problem ofusing ready made software which cannot be made"translucent", since the details may be too com-plex, or totally hidden from the user, or just notworthwhile studying for secondary school students.

First of all, using ready-made mathematics, evenif not fully understood, is to be seen as taking part inmathematics as a social enterprise. It may he lookedon as part of teamwork: Users rely on professionalmathematicians and programmers. But the coop-eration is anonymous since the user can't talk to"coworkers ", and users have to know a lot in orderto use the black box correctly and with beneficialresults. But knowledge about black boxes (proce-dures, algorithms, etc.) can be of various kinds:

Logical or external. The user knows the math-ematical specification of the result the software de-livers, but doesn't know the method by which it hasbeen achieved. This is the classical black box andis usually the case with the use of computer algebrasystems or simple calculators. A symbolic integra-tion can be understood (and independently checked)even if the internal Risch algorithm isn't understoodor its existence even known. The cosine of a numbercan be interpreted correctly ^s the best approxima-tion within the domain of machine numbers to thecorrect real number, etc.

Analogous. If a complete specification of the re-sult of the software is not available, an analogousknowledge of a similar algorithm may often help.


The graph of a function, as displayed by a functionplotter, is different from the graph of the functionas normally defined within mathematics. But theexperience of doing function plotting and a reflec-tion on the possible pitfalls (e.g. vertical asymp-totes, discontinuities or the proper determinationof maxima) may help in understanding results andbecoming aware of possible limitations. For the nor-mal student it is not worthwhile to learn the specialtricks and algorithms programmers of function plot-ters use to give reasonable results even in difficultsituations. Analogous knowledge is needed in gen-eral in the use of numerical software possible pit-falls, trade-offs between step widths and obtainableaccuracies, between reliability and speed, etc.

Algorithmic. Here the user knows on a cer-tain level the specific algorithmic approach usedby the software, for example, that the numerical in-tegration software uses Simpson's rule, which theuse had applied in some hand calculations. But fora suitable use of the software, the user has to havesome more general knowledge, too the approxi-mation character and the order of the algorithm, itsdomain of validity, in what circumstances to switchto other algorithms, etc.

All three kinds of knowledge have their specialvalue, and in most circumstances they should com-plement each other. There is no a priori best way ofenlightening a software black box. Of course, math-ematics teaching has the duty to enlighten blackboxes, to make them grey at least, but in whichway and to what extent has to be decided in view ofthe intended use of the software, the kind of knowl-edge to which this new knowledge is to be addedand connected, and to tl . overall goals of mathe-matics teaching in the specific age group and schoolsystem in particular.

On the Concept and Importance ofSimulations

How does one simulate a dynamical process?Such a process is described by specifying the transi-tion from one state of the system to the "next" state;mathematically this is done by (systems of) differ-ence or differential equations. In order to simulatesuch a process, one first has to specify all param-eters, initial states and possible external influencesnumerically, and then follow the evolution of thestates numerically, replacing all mathematical oper-ations which have no direct arithmetical translationby numerical approximations. Some ending condi-tions have to be efficiently specified, too, for exam-ple, the maximum number of states to calculate, inorder to prevent never ending calculations.

u tJ


The resulting numbers normally quite a lotcan be given as tables or graphics. If the concretechoice of parameter values or initial states is notdictated by the situation but just ad hoc in orderto be able to start the simulation run, the wholeprocess will have to be repeated with other valuesfixed that means defining another scenario toget an overview over the behaviour of the system ina range of scenarios.

From this description we have a geberal infor-mal definition: By simulation (in mathematics) weunderstand the effective operational translation ofmathematical objects or processes into numericaloperations. (Outside mathematics the concept hasto be extended to include the building of a mathe-matical model first.)

Simulation in this sense is a general mathemat-ical method which has always been used but hasgained importance enormously through the avail-abilty of effective numerical machines, especiallycomputers. As a method it is very often not dif-ficult to apply, and it can be a mighty instrument,especially if combined with other, more traditionalmathematical methods such as proof, construction,algebraic calculation, analysis, etc.

Here are some examples of simulations:Function plotting. The mathematical object is

the graph of the function, say of f(x) = sin x, whichis a subset of R2. For the simulation one has firstto fix boundaries, say from to:27r, then approxi-mate the interval 2r] by a4nite set of floatingpoint numbers, calculate approximations tothe sineof these numbers, determine screen pixels to corre-spond to the calculated values, connect those pixelsby the built-in "line-drawing" routines, and displaythe result. The fixing of parameters will becomeeven more apparent, if you simulate functions withparameters, say f(x) = sin(ax), a E R.

Stochastic simulation. The mathematical ob-ject is, for example, a stochastic variable with itsdistribution, mean and variance, say a uniformlydistributed variable transformed by some compli-cated process or function f. To simulate it, you takea finite number of uniformly distributed (pseudo-)random numbers, transform them by (a numericalapproximation to) f, take the resulting finite disztribution as an approximation to the distributionsought, and calculate its mean and variance.

Solution to a differential equation.2 The math-ematical object is the general solution to the givendifferential equation. To simulate it, one chooses

2 An indefinite integration is a special case of this,namely the solution of y' = f(x).

several different initial conditions, solves the result-ing initial value problems by numerical methods andplots the results. The emerging picture should givesome insights into the flow-lines of the differentialequation, its overall behaviour and possible loca-tions of critical points.

Simulations normally share a double experimen-tal character: First by the numerical approxima-tions woe errors can be only estimated since theassumptions of strict error control in most cases can-not be verified by numerical methods alone, and sec-ond by the fixing of the parameters, boundaries, etc.Simulations need to be complemented by some the-ory, however rudimentary, in order to lead to insightand understanding. Thus the plotting of the sine-function can only give a non-misleading intuition,if the continuity and periodicity are known or canbe abstracted by the consideration of a well chosensequence of (simulated) pictures with some zoom-ing or similar means. The insight does not comefrom the pictures. The intellect of the students hasto see the connections between the pictures and thenecessities behind them; but to see the facts givenby the simulation may strongly help the student tounderstand the facts given by some theory.

Model Building

The building of mathematical models is seenby many people as the heart of application ori-ented mathematics teaching. If done properly, theusual restriction to linearity assumptions will soonbe noticed as inappropriate, and the use of simula-tion software in order to explore the (mathematical)models developed becomes necessary.

Here we describe briefly dynamic model build-ing of simple growth processes in the mathematicsclassroom with the program Modus, which at themoment is being tested in schools in a preliminaryversion. As with most dynamic modelling tools, thecrucial concepts are the distinction of the main vari-ables as levels and flows. Levels can only be changedthrough flows; this property is described in formalmathematical language by use of difference or differ-ential equations, the flows being the derivatives ofthe levels. The model building is done by construct-ing structure diagrams, thus avoiding the necessityfor an abstract formal language. The students eas-ily develop linear and exponential models of growth.The step from linear to exponential growth is madeby changing the constant flow to a (linear) functiondepending on the level, thereby introducing a firstfeedback loop (see Fig. 12 and 13).

O

0O

CzIraFlou Level

Figure 12

Figure 13

Students soon detect that exponential growth ispossible and widespread only in the beginningphases of growth processes, be it population growthin a new environment, growth of individuals, devel-opment of first love, spread of rumors and epidemics,but growth eventually has to slow down. In modelbuilding, this is interpreted as a dependency of thegrowth rate on the level reached, thereby introduc-ing a new feedback loop, now negative, "hich yieldslogistic growth (Fig. 14). Even without any formu-

Figure 14


las for the logistic function, the growth behaviourcan be completely understood from the model itself,and becomes evident 13::, ...serving parts of the phasediagram being gun. ed dyrAmically (Fig. 15).

--,- w mi

. =rffmcgt, 1=11Z7nin

LEVELII=1Ni 1 S.P.ss r ter P. isms us. Lev.. 1 a

3 . 3 -

0

fLow,Eue.

L .0.1.

Figure 15

Students (grade 10 or higher) learn to use the ar-gument that from the model, the concrete quadraticdependency of the flow on the level is clear. A posi-tive flow causes the level to grow, which means a mo-tion in the phase diagram to the right, now causinga new flow, etc. The motion in the phase diagramslows down and eventually (virtually) stops, whenthe flow approaches zero. So the equilibrium can bedetermined from the model, and similar argumentswith starting points above equilibrium show the sta-bility property. Such qualitative understanding hasto be developed carefully since it is not easy, butit is more adequate than reasoning with formulaswhich immediately break down when the model isfurther refined.

Conclusion

Consequences for Organisation andCurriculum

Organisation problems related to the use of com-puters in schools have been solved only for demon-stration lessons by teachers. Overhead projectorswith transparencies for all explications and a specialoverhead display for all output from the computerinstead of the blackboard are standard in many Ger-man schools. But 10 terminals are not enough, par-ticularly if students of informatics classes occupythe stations for many hours. Supervision cannot al-ways be done by teachers. The ordinary homeworkof pupils of math classes using the computer is tooeasily pushed aside; only pupils with computers athome can help theirselves, provided that their equip-ment is hardware and software compatible with theequipment of the school. It is difficult too, to orga-nize access to 10 terminals during a written exam for

8


25 pupils in the class. Therefore, often theoreticalquestions related to algorithms (additional specialcases, restrictions, possibilities for application etc.)have to substitute for the real use of the algorithmsfor problemsolving in examination periods.

The cons luences for the curriculum are veryimportant. School mathematics was determined forcenturies by the number of accessible methods forsolving problems equations of no higher degreethan 2, systems of equations with 3x3 or 4x4 ma-trices etc. Application problems were selected care-fully so that powerful computational tools were notneeded. With the speed and the capacity of mem-ory of modern computers in schools (newer stan-dard: 80386 processors, 1.2 MB RAM), numeri-cal and graphical approximations for solving equa-tions of higher degree and handling matrices of 10or 20 columns and rows are no problem. Graphi-cal representation of large sets of higher functionsor of complicated geometrical situations are also noproblem as are the symbolic transformation of com-plicated rational terms or the symbolic solution ofdifferential equations with interesting initial condi-tions. With these tools teachers can leave the smallgarden of traditional school problems and amplifyenormously the orientation to modern application.

Let us demonstrate this with two examples.First, from pure mathematics: After teaching curvefitting by Taylor approximations or Fourier approx-imations in the classical manner with the usualdemonstrations you can continue with Pade approx-imations using rational functions and use these forgood approximations to functions with singularities(see Fig. 16).

Our second example is from applied mathemat-ics: The teacher can show how to compute ap-proximations to curves of highways in the student'sneighbourhood by parametric splines with the helpof the computers.

Thus various new fields are opened for the cur-riculum. Simulations in natural science and socialscience using systems of difference equations can beused to solve interesting environmental or economicproblems never before accessible in schools. Thetheor: of graphs or the theory of functions withcomplex variables are other examples of new ele-.nentary work with modern tools.

Speculation on the Future

During the first twenty years of computer usein schools, the mathemeics classroom was the firstplace where most students met a computer at all.So math teachers had to pursue an additional goal:

upper picture: 1/cos(x) apProxlmated by Taylor-eerie order 4 and e in x - 0(only t:, central par: of the graph

ftt:cd well)lower Picture: 1 /cos(x) aPPrOXIMatOd by Pade-term

order 4.4 In x - 0(Invisible fitting of the curve inthe central part end half of the per:-Pherica: parte including the s:ng..:lerftleel

Figure 16

Make students familiar with the basic structure andfunction of a computer system and teach ae-m howto manipulate it. This situation has changed rapidlyand will have changed totally in the near future sincemost students now get acquainted with computersin their daily lives, in their family and recreationalenvironments, perhaps in computer scier.ce educa-tion, and so on. This means the computer has anew importance in math education, a more fruit-ful one, more oriented towards mathematics. Thisis described as follows in a study of mathematicseducation for the information age to be realizedin the Japanese New Mathematic Curriculum [Fu-jita/Terada 1991]. In upper secondary schools pri-ority should be

"on giving to students opportunities to domathematics rather than improving theirtechniques. Students should understand thatcomputers are powerful tools for intellectualactivities by human beings. In this connec-tion, studying mathematics may be the firstas well as the best experience for students touse computers for properly intellectual pur-poses, namely, to study academic subjectswith computers. These experiences couldeven be regarded as a prototype of scientificresearch activities with computers. Somegood students will have chances to observe,to model and to analyze in a mathemati-cal manner various phenomena presented bycomputers. Furthermore, computer simula-tion is close to mathematical reality. On theother hand, computers are extremely help-ful in fostering students' mathematical liter-acy. Rich mathematical experiences offeredby computers, particularly those through op-erational work by students, will pave the wayfor the majority of students to grasp con-cepts and to understand fundamental factsin mathematics."

The New Curriculum plans three courses, Math-ematics I - III, in grades 11 - 13 with a total of 10units (1 unit requires 35 class hours of 50 minuteseach), covering a core of mathematics to be learnedby all students with Math III certainly to be learnedby all science and technology students. Three morecourses, Mathematics A, B, C, in grades 11 - 13, to-talling 6 units, are composed of four option modulesfrom which two modules are freely chosen for in-struction by teachers or schools. Module 4 of MathA, computation and computers, offers students thechance to get to know and become familiar withcomputers as a tool for mathematics. Module 4 ofMath B, algorithms and computers, deals with thepowerful function of computers in doing algorith-mic computations in mathematics. Math C is char-acterized by the key phrases "application minded"and "do math with computers" in the areas of ma-trices and linear computation, various curves, con-ics and polar coordinates, numerical computationand statistical processing. The study mentions thatthe newly introduced topics related to computers inJapanese high school .mathematics require certainpreparation for success, namely, purposeful textbooks, effective teacher training, quality softwareand relevant development of teaching materials andmethods. Indeed, the educational use of comput-ers in class is non-routine and should be exploredwith respective emphasis of its three aspects; the


teacher-initiated use, the student-initiated use andthe system-initiated use.

From the viewpoint of a computer-supportedcurriculum, teaching with computers in a classroomwill consist of the following six components:1) "trial", where learners are invited to the new

topic with fun applications offered by the com-puter.

2) "approach", where learners have heuristic andoperational experiences with the aid of comput-ers.

3) "teaching", where the teachers give a lectureand learners get supplementary review and as-sistance from computers.

4) "experimental understanding ", where learnersgrasp concepts and facts through inductive andexperimental recognition with the aid of comput-ers without being burdened by too much drill.

5) "exercise", where learners can perform adequateexercises at their level and using standard (butinteractive) CAI.

6) "survey", where learners review the topic whichthey have learned and are given chances to viewfurther developments and applications.The principal underlying purpose of the New

Japanese Curriculum is to cultivate "mathematicalintelligence" by aiming at two targets: Mithemat-ical Literacy and Mathematical Thinking. The as-pects from the curriculum mentioned above showthat computer systems are considered to be veryhelpful for both fields.

These two fields are also mentioned among theprinciples for the development of a new mathematicscurriculum in the USA by 2000 [Ralston 1990]. Inthis reference it is stated that "Mathematical educa-tion should focus on the development of mathemat-ical power not mathematical skills". As to informa-tion technology there is this principle: "Calculatorsand Computers should be used throughout the K-12mathematics curriculum; moreover, new curriculaand new curriculum materials should be designed inthe expectation of continuous change resulting fromfurther scientific and technological developments".Goals from these principles follow for the elemen-tary grades (1-6) as well as for the secondary grades(7-12). So "the teaching of arithmetic in elemen-tary schools should be characterized by : ... a useof computer software in the teaching and learningprocess, ... proper and efficient use of calculatorsfor most multi-digit calculations as well as calcula-tions involving negative numbers, fractions and dec-imals". One important example of computer usein the secondary curriculum follows from th , goal


that this curriculum should develop students' sym-bol sense. This means developing "the ability torepresent problems in symbolic form and to use andinterpret these symbolic representations", and "theability to identify the symbol manipulations neces-sary to solve problems expressed in symbolic formand to carry out manipulations using mental com-putation, pencil and paper, a symbolic or graphiccalculator or a computer".

It was noted above that new mathematics cur-ricula should be designed in the expectation of fur-ther technical and scientific developments. Mostcertainly these will occur in artificial intelligenceand telecommunications. In a survey on Technol-ogy and Mathematics Education, James Fey [1990]writes about artificial intelligence, expert systemsand tutors: "One of the very active areas of infor-matics research is exploring ways that computerscan be programmed to exhibit 'behaviour' that sim-ulates human information processing. There are anumber of projects in mathematics education thatare attempting to capitalize on this computer capa-bility to design programs that act, in various ways,like teachers. The most interesting work along theselines is producing intelligent tutors for an array ofmathematical topic areas including arithmetic, al-gebra, geometry and proof, and calculus. Thereare some preliminary indications that those tutorsprovide very effective adjuncts to regular teacher-directed instruction".

As for telecommunications, one might think thatthis will be important for general or social educationonly. It is likely, however, that the ability to commu-nicate about mathematical problems in a worldwidegroup of peers will develop new attitudes towardsproblem solving, different from the widespread "sin-gle attack" of scientists and students. Also, it canbe imagined that a feeling for the benefits of inter-national and intercultural understanding can growmore intense through cooperation in a "serious"field like mathematics or science, in addition to theeffects of leisure fields like music, movies, etc.

We want to conclude this article by pointing toone of the greatest problems in the changing of themathematics curriculum under the challenge of com-puter systems: We must convince the curriculummakers and those who put changes into effect aboutthe necessity and the advantages of this change. Wehope that this article will provide good argumentsto everybody who wants to tackle this problem.

References

Software

AUTOCALC, Shell Centre for Mathematical Edu-cation, University of Nottingham NG7 2RD,UK.

CABRI GEOMETRE, Baulac, Y., Bellemain, F.,Laborde, J.M., LaboraLoire L.S.D. IMAGTour Irma BP 53 X, 38041 Grenoble Cedex,1988ff. MacIntosh, IBM PC. Versions in otherlanguages such as English or German are avail-able.

DEVELOPING TRAY, Inner London EducationAuthority, NCET, 3 Devonshire Street, Lon-don.

EARLY SCIENCE in Exploring Primary Scienceand Technology with Microcomputers, editedby Jan Stewart, Council for Educational Tech-nology, NCET, 3 Devonshire Street, London.

GEOMETRIC SUPPOSER, Schwartz, J., Yerushal-my, M., et al, Sunburst Communications 1985ff. Apple II, MacIntosh, IBM PC.

GRAPHIX, Tall, D., van Blokland, P., Kok, D.,Duisburg: Co Met Verlag fiir Unterrichtssoft-ware 1989. ISBN 3-89418-862-6. IBM-PC. AnEnglish version of this program is available' forexample, through Sunburst under the title "AGraphic Approach to the Calculus".

KALEIDOSCOPE, Graf, K.-D., Information can beobtained from the author at Freie UniversitatBerlin, D-W-1000 Berlin 33, Germany.

MODUS, Walser, W., Wedekind, J., Duisburg:Co Met Verlag fiir Unterriclitssoftware 1992.IBM -PC.

SEEK, Longman, UK; also Shell Centre for Mathe-matical Education, University of NottinghamNG7 2RD, UK.

SYMMETRIC TURTLES, Graf, K.-D. (see KALEI-DOSCOPE)

TIMES in Teaching with a micro: Math 3, Phillips,R. et al, 1986, Nottingham: Shell Centre forMathematical Education. BBC Micro.

Books and PapersAbelson, H. and di Sessa, A. [1985]: Turtle Geome-

try: Computation as a Medium for ExploringMathematics, Cambridge: MIT Press.

Bich ler, R. and Vinkelmann, B. [1988]: Mathe-matische Unterrichtssoftware: Beurteilungsdi-mensionen and Beispiele in Schmidt, Giinter(Ed.): Computer im Mathematikunterricht,Der Mathematikunterricht 34, Heft 4, 19-42.ISBN 3-617-24022-4.

Buchberger, B. [1990]: Should Students Learn In-tegration Rules?, ACM SIGSAM Bulletin, 24,1, 10-17.

Dubinsky, E. and Fraser, R. [1990]: Computers andthe Teaching of Mathematics, selected papersfrom ICME-6, Nottingham, UK: The ShellCentre for Mathematical Education.

Fey, J.T. [1989]: Technology and Education: ASurvey of Recent Developments and ImportantProblems. Educational Studies in Mathemat-ics, 20, 3, 237-272.

Fey, J.T. [1990]: Technology and Mathematics Edu-cation in Dubinsky and Fraser, 73-79.

Fujita, H. and Terada, F. [1991]: A Coherent Studyof Mathematics Education for the InformationAge; a Realization in the Japanese New Math-ematics Curriculum, plenary Lecture at ICM1-China Regional Conference on MathematicsEducation in 1991, Beijing, China. (Printsavailable from the first author, Department ofMathematics, Meiji University, Japan).

Goldenberg, E.P. [1988]: Mathematics, Metaphors,and Human Factors: Mathematical, Techni-cal, and Pedagogical Challenges in the Edu-cational Use of Graphical Representation ofFunctions, Journal of Mathematical Behavior,7, 2, 135-173.

Graf, K.-D. [1988]: Using Software Tools as Ad-ditional Tools in Geometry Education withRuler and Compasses, Education and Comput-ing, 4, 3, 171-178.

Graf, K.-D. and Hodgson, B. [1990]: PopularizingGeometrical Concepts: the Case of the Kalei-doscope, For the Learning of Mathematics, 10,3, 42-50.

Johnson, D.C. and Lovis, F. (Eds) [1987]: Infor-matics and the Teaching of Mathematics, Pro-ceedings of the IFIP TC 3/WG 3.1 WorkingConference on Informatics and the Teachingof Mathematics, Sofia, Bulgaria, 16 - 18 May,Amsterdam: North-Holland.

Okamori, H. [1989]: Mathematics Education andPersonal Computers, Tokyo: Daiichi-HokiShuppan.

Ralston, A. [1990]: A Framework for the SchoolMathematics Curriculum in 2000 in Dubinskyand Fraser [1990], 157-167.

Schumann, H. [1990]: Neue Moglichkeiten des Ge-ometrielernens in der Planimetrie durch inter-aktives Konstruieren, in Graf, K.-D. (llrsg.):


Computer in der 5'clitile 3, Stuttgart: B. G.Teubner, 45-72.

Winkelmann, B. [1987]: Information TechnologyAcross the Curriculum, in Johnson, D.C. andLovis, F. (Eds), 89-94.

Winkelmann, B. [1992]: Themenheft Wachstum.Dynamische Systeme im Mathematikunter-richt, Soester Verlagskontor, Soest.

A FUNDAMENTAL COURSE IN HIGHER MATHEMATICS INCORPORATINGDISCRETE AND CONTINUOUS THEMES

S. B. SeidmanAuburn University, Auburn, AL 36849, USA

M. D. RiceWesleyan University, Middletown, CT 06857, USA

THE CURRENT MATHEMATICSCURRICULUM: TRADITIONS ANDCONCERNS

For many years, a crucial place in the mathemat-ics curriculum of the last year of secondary school orthe first year of university studies has been occupiedby the differential and integral calculus. The calcu-lus can be seen both as the culmination of the sec-ondary school mathematics curriculum and as thebeginning of serious study of mathematics at theuniversity. In some sense, the study of calculus hasbecome synonymous with the serious study of math-ematics. The central and essential position occupiedby calculus can be traced to at least two interrelatedcauses.

For mathematicians, calculus represents themethodology and techniques needed for the studyof functions, first defined on the real line, then onhigher-dimensional Euclidean spaces, and finally onthe complex plane. Thus, the study of the calculusallows students for the first time to acquire the for-mal abstract tools that are essential for the furtherstudy of much of higher mathematics.

On the other hand, calculus provides the foun-dation for many applications of mat hematics to thephysical sciences and engineering. These applica-tions date back to Newton's original developmentof the calculus in the seventeenth century, and sincethat time Z1 successful across avast coire. . ...! iciplines, even including (in re-cent years), the ;ogical sciences and economics.All of the calculus-based applications are based onmathematical models that can be regarded as beingcontinuous; that is, the quantities being modeledare real numbers (or elements of some Euclideanspace Rn).

Given both the central mathematical position ofthe calculus and its vital role in applications (notto speak of the interaction between these two fea-tures), it, is easy to see why the calculus has occu-pied such a fundamental and unassailable positionin mathematics curricula During the past severaldecades, however, the central role of calculus hasbeen seriously questioned, and the questions havebeen repeated with particular emphasis during the

80

last decade (Ralston 1981,1989, Kenney and Hirsch,1991). Just as a major motivation for the predom-inance of calculus in the curriculum has been thewide range of applications of continuous mathemat-ics, the challenge to that predominance has arisenfrom the steadily increasing interest in the applica-tions of discrete mathematics in many disciplines.

This increasing interest in discrete mathemati-cal applications can be primarily attributed to thewidespread use of computers. Computcrs are essen-tially discrete machines, and the mathematics thatis needed to use them is also discrete. As a conse-quence, the discipline of computer science is heavilydependent on a wide variety of discrete mathemat-ical ideas and techniques. Furthermore, the easyavailability of computers has encouraged the useand development of discrete mathematical models inmany disciplines. For example, operations researchmodels (linear programming, integer programming,etc.) are widely used and are based on a discretemathematical perspective.

It is natural to expect that the rapid growthof interest in discrete mathematics and its appli-cations, fueled by the explosive developments asso-ciated with computers, should have an impact onthe mathematics curriculum. Although this im-pact would have been significant under any cir-cumstances, its effect in the United States hasbeen mag,-:ified by other questions that have beenraised in recent years about the teaching of calcu-lus. Widespread dissatisfaction has been reportedwith the nature of the calculus courses and theknowledge of the students that have completed them(Lochhead 1983, Steen 1983, Douglas 1986, Steen1988). The computer is also directly influencing thecontent of the calculus course itself, both by en-couraging the inclusion of numerical methods andby suggesting that symbolic manipulation softwaremay make emphasis on techniques of differentiationand integration obsolete (Bushaw 1983, Wilf 1983,Nievergelt 1987).

In summary, both the nature of the calculuscourse arid the fundamental position that calcu-lus has occupied in the mathematics curriculum for

more than a century have come under serious chal-lenge. These challenges have come both from withinand outside the community of mathematicians, andthey can primarily be attributed to the increasinglybroad role that computers are playing in the variousscholarly disciplines represented within the univer-sity and in the wider world. In the next section ofthis paper, we will look at the responses that havebeen proposed to these challenges.

RESPONSES TO THE CHALLENGE OFDISCRETE MATHEMATICS

When any curriculum is confronted by a newtopic that should be included, there are essentiallytwo potential responses. The new topic can eitherbe encapsulated in a course that is added to the cur-riculum, or it can be incorporated as a fundamentalconstituent of a revised course. Most topics thathave been added to the mathematics curriculum inrecent decades have been added as new courses (e.g.abstract algebra and topology).

It was therefore natural that when mathematicsfaculties were asked to include discrete mathemat-ics in the curriculum, this was most commonly doneby developing new courses in 0. ..rete mathematics.Such courses were designed primarily for students ofcomputer science. There have been two fundamen-tal problems with this approach. First, the discretemathematics courses were too often taken by third-year students, so that the material was learned toolate to be of use in the data structures courses takenby first and second year students of computer sci-erwe. Second, when students were expected to usetheir discrete and continuous mathematical skillsin fourth-year computer science courses (for exam-ple, in the analysis of algorithms), most have foundit very difficult to combine these skills effectively.Many students do not see any connections betweendiscrete and continuous mathematics, and are un-able, for example, to apply calculus techniques toestimate growth rates of discrete functions or to es-timate the size of discrete sums. This inability tocombine discrete and continuous skills i:. also foundin students of probability, operations research andsignal processing.

Both of the above reasons suggest that discretemathematics should be incorporated as a compo-nent of the fundamental mathematics course thatis offered to all students in their first two years ofuniversity study. This suggestion was first madeby Ralston (1981), who proposed that the study ofdiscrete mathematics precede the study of cal-Ails.He argued that such an organization would benefitvirtually all students of mathematics, and not just

Discrete and Continuous Themes 81

those students concentrating in computer science.Ralston's proposal has led to substantial discussionin the United States on the proper place of discretemathematics in the curriculum (Ralston and Young1983). The debate has focused on whether discretemathematics should precede or follow the calculusin the curriculum of the first two years. Many ofthe arguments advanced on either side are adminis-trative in nature, dealing either with the demandsof other curricula (such as physics or engineering)or with articulation with other institutions (suchas high schools, junior colleges or universities thathave retained the standard curriculum). One resultof this debate has been the publication since 1985of over 40 discrete mathematics texts for freshmanor sophomore courses (e.g. Ross and Wright, 1988,Maurer and Ralston, 1991).

Whether calculus is placed before or after dis-crete mathematics, it is by no means clear thatstudents who have completed both courses will beable to combine their discrete and continuous math-ematical skills in an effective manner. This problemhas been recognized by some designers of proposedcurricula, and consequently their calculus propos-als generally include some discrete aspects, such asextended discussion of numerical methods and sub-stantial use of sequences (see, for example, Bushaw1983).

Another possibility, which has been given littleserious attention, would be to develop a new, uni-fied curriculum that would interweave discrete andcontinuous themes throughout its courses. Whilethe first year of the curriculum would correspondto the calculus course, its real thrust would be thestudy of functional behavior and functional repre-sentation. The course would consider discrete func-tions (sequences) along with continuous functions,and would constantly emphasize analogies and par-allels between discrete and continuous situations.Thus the first year of the curriculum would be pri-marily continuous, but with a strong discrete flavor.The second year of the curriculum would focus onstructure, and would be primarily discrete, but witha strong continuous flavor.

This paper will argue that a curriculum unifyingdiscrete and continuous themes is not only feasible,but has the potential of providing students with abroad, powerful perspective embracing the mathe-matical ideas and techniques that are needed for thestudy of computer science. This perspective wouldalso yield a strong mathematical foundation for thestudy of engineering, the physical sciences, and in-deed for the study of higher mathematics itself.

Furthermore, the development of such a cur-


riculum would force a reexamination of the top-ics taught in the conventional calculus course. Asmentioned above, various recommendations havebeen made to remove or include particular top-ics. Although each such recommendation has beencogently argued, no consistent rationale has beengiven for the collection of topics that together makeup the proposed calculus course. The first-yearcourse outlined below has a consistent theme - func-tional behavior and representation - and each topicto be included in (or excluded from) the courseshould be judged on the degree that it matches thecourse's perspective.

In the following section, a detailed outline anddiscussion will be given only for the first year of theproposed two-year curriculum. At the conclusion ofthe paper, we shall return to the second year of thecurriculum, as well as to the larger issues raised bythe question of articulation with other curricula.

A FIRST-YEAR '7URRICULUM INCOR-PORATING DISCit..3TE AND CONTINU-OUS THEMES

The fundamental thrust of the proposed first-year curriculum is the behavior and representationof functions. Roughly, the first semester is devotedto tools for the description and analysis of functionalbehavior, with the focus shifting to representation offunctions in the second semester. Before presentinga more extended discussion of the benefits to beachieved by including both discrete and continuoustopics, it will be useful to give an annotated outlineof the first semester curriculum.

A. Functions1. Number and Relations

A knowledge of set concepts and notation is as-sumed. Inequalities will be emphasized.

2. Functions and OperationsThe function concept and functional notationwill be introduced, stressing the algorithmic in-ter -etation of the function symbol f. Discus-sion will include domain and range, operationson functions (arithmetic operations, composi-tion, translation), and graphs of functions. Use-ful functions will be introduced [polynomials,rational functions, exponential functions (de-fined on the integers), absolute value, floor, ceil-ing].

3. ModelsAlgorithms and elementary complexity analy-sis will be introduced (including binary search).This will allow discussion of the function LIg(n)

Models demonstrating the need to constructfunctions and to perform curve fitting will beincluded.

B. Behavior of discrete functions1. Sequences: Iteration and Recursion

This section will discuss a variety of sequencesincluding geometric squares, the Fibonacci se-quence and the sequence generated by the Eu-clidean algorithm.

2. Difference OperatorsThe difference operation A will be introduced asa function on sequences. The recursion scheme

Uk = Auk

will be treated in order to emphasize specialfunctions defined on the integers. Formulas forhigher differences will be discussed.

3. SummationThe primary topic here will be the binomial the-orem, bc'th in its standard form and in the ex-pression for (1+ Al". The second form will allowvarious formulas for finite sums to be presented.

4. Order Notation (0, o) and Limits of Sequences

C. Behavior of cont -mous functions1. Limit Heuristics

Limits of functions will be discussed only interms of limits of sequences. The continuity con-cept will be introduced. The operator

o f f(x + h) f(x)

will be introduced. Analogies to the discretedifference operator discussed above will be pur-sued.

2. First DerivativeThe derivative will be defined, and interpretedusing tangent lines. It will be shown that differ-entiable functions are continuous.

3. Differe:itiation RulesPowers and roots; product, quotient rules.

4. Monotone Functions and Local ExtremaA rigorous treatment will be postponed. Curvesketching will be introduced here and the use ofgraphing calculators will be stressed.

5. Second DerivativeConcavity will be discussed and applied tocurve sketching again using graphing calcioa-tors.

6. Extreme Values

Maximum-minimum problems will be solved.Examples will also demonstrate the use of piece-wise linear functions.

7. Related RatesThe chain rule will be presented, and related rateproblems will be solved.

D. Estimation and error1. Mean Value Theorem

Monotone functions will be discussed more rig-orously, and the MVT will be applied to globalestimation of functions.

2. Solution of EquationsNewton's method will be discussed from both ge-ometric and iterative perspectives. An elemen-tary treatment of error estimation will be given,and critical values will be estimated.

3. InterpolationInterpolation of functions by straight lines andparabolas will be discussed using the differenceoperators developed above.

4. ApprozimationSecond-order Taylor polynomials will be used toapproximate functions, and the estimated errorwill be computed. Analogies will be drawn be-tween interpolation and approximation and be-tween differences and derivatives.

E. Integration1. Introduction

The summation operator for sequences will beintroduced. Its relation to the difference oper-ator will be discussed. It will be treated as anaggregation operator, and used to motivate thediscussion of area.

2. The Definite IntegralThis will first be introduced using a piecewiselinear definition. This definition will then beapplied to step functions. The area definitionwill then be presented, and applied to parabolasusing the results on finite sums obtained above.Some elementary properties of the definite inte-gral will be presented, including the mean valuetheorem for definite integrals.

3. The Indefinite IntegralThis will be explicitly computed for step func-tions, piecewise linear functions and parabo-las.

4. The Fundamental Theorem. of CalculusThis will be derived from the mean value the-orem for definite integrals. The chain rule willbe applied to investigate some properties of theintegral of 1/z.

5. Evaluation of Integrals: Analytic Techniques


Substitution techniques will he discussed, as wellas the use of integral tables and symbolic calcu-lators.

6. Evaluation of Integrals: Numerical TechniquesThe trapezoidal rule and Simpson's rule will bediscussed. It will also be shown how integralscan be estimated using inequalities, and howsums can be estimated using integrals.

7. Applications of Integration: AggregationThe applications to be treated include work andvolume.

8. Applications of Integration: ModelingThe primary theme here will be the recognitionof Riemann sums in differing situations. Exam-ples will be taken from arc length and fluid flow.The basic point will be that when a model gen-erates a discrete (Riemann) sum, it can then beapproximated by a definite integral.

Although this annotated outline gives a goodoverview of the first semester of the proposed course,it is too brief to show how the interweaving of dis-crete and continuous themes can lead to major bene-fits. The following examples are meant to be typicalof the perspective that will be possible within thiscourse structure.

Example 1: At the beginning of the course, thediscrete exponential function, f(n) = 2n, will be in-troduced, along with its one-sided inverse, g(n) =max{k12k < n) = L1g(n) J. The function g(n) is vi-tally important in computer science; for example,g(n)-1- 1 is the worst-case number of comparisons ina binary search of a list of length n. The growth rateof g(n) is important, and is usually treated (via cal-culus) using L'Hospital's rule. We suggest a discreteapproach, based on the binomial theorem. Clearly29(n) < n, so that g(n)/n < g(n)/29("). To de-termine the behavior of g(n)/29(n) as n co itis sufficient to consider powers of 2 since g(n) isconstant between successive powers of 2. Since forn = 2k, g(n)/29(n) = k/2k, it is only necessary tolook at the behavior of k /2k as k cc. By the bino-mial theorem 2k = (1 + 1)k > k(k 1) /2, and hencek/2k < 2k/k(k 1) = 2/k(k 1), which gives theresult that g(n)/29(n) 0 and, therefore, so doesg(n)/n. The simplicity of the discrete argumentshould aid the student in learning, understandingan assimilating the growth rate of the continuouslogarithm.

Example 2: The syllabus outline has referred toanalogies between the discrete difference and sum-mation operators on the one hand, and differen-tiation and integration on the other. For exam-ple, the difference operator is defined on the se.-

JU


quence {71,} by slur, = un+1 u. If we de-fine the falling factorial function on the integers byz(m) = x(x 1). -(x in + 1) then it is easy to seethat Ax(rn) = AY') = m(m-1)x("1-2),and finally that Amx(m) = 7n! and Arn+lx(m) = O.Thus the behavior of the difference operator (and itsiterates) on the polynomials x(m) is strongly analo-gous to the behavior of the differentiation operator(and its iterates) on the polynomials {x"}. Further-more, since each collection of polynomials providesa basis for the vector space of polynomials of degreeat most n, an example has been introduced whichwill be useful in a later course in linear algebra.

One further benefit of the use of difference oper-stors is the natural observation that 42n = 2", ormore generally that Lik" = (k 1)k". This suggeststhat exponential functions, whether discrete or con-tinuous, may have a special role to play with respectto difference or derivative operators, and serves tomotivate the later observation that d /dx(ex) = ex.

Example 3: The first two examples used dis-crete ideas to motivate continuous concepts that areto he introduced later. In this example, continuoustechniques are used to obtain a discrete result. Theidentity giving the sum of a geometric progression,

n-1k x"

x -1k=0

can be differentiated using the quotient rule to ob-tain the identity

n-1

LJk=1

Using this identity, it is immediate thatn-1E k2k = (n 2)2" + 2k=1

kxk =(n 1)x"+1 nr" + x

(X 1)2

and thatn-1E k2-k = 2 1

2"-1k=1

The last result yields00

k2k = 2k=1

since k/2k k on (see Example 1). Thisexample serves to remind students that continuoustechniques can be important in discrete situations.

These examples demonstrate that the proposedcourse does not merely insert a collection of im-portant discrete topics into the calculus course, butrather expresses a consistent approach to all of thesubject matter. The fundamental perspective is the

ryfathematics and Its Teaching

study of functional behavior, and both discrete andcontinuous functions are treated throughout. Eachclass of functions is used to develop tools and sug-gest analogies that will be useful for the study offunctions of the other class.

The second semester of the course further elab-orates the functional perspective. Rather than givea detailed, annotated outline, we shall discuss thetopics to be covered and describe how they relateto the themes developed during the first semester.The second semester is primarily devoted to mate-rial taken from two broad categories, special func-tions and representation of functions.

Exponential and logarithmic functions will betreated in depth. The natural logarithm will beintroduced using the definite integral, and its prop-erties will be investigated. The inverse of the log-arithm will be motivated using growth models andthe differential equation dy/dx = ky and the rela-tionship of this inverse to the exponential functionwill be motivated using difference equations and thediscrete logarithm. Finally, the properties of thefenction er will be developed. Numerical estimatesfc.: exponential and logarithmic functions will beused throughout the discussion.

The next major topic will be trigonometric func-tions. Here the primary motivation will come fromthe geometry of the circle and from models of cir-cular and harmonic motion, although discrete pe-riodic functions, such as mod n, will also be used.The properties of the trigonometric functions will bedeveloped. Integration by parts will be introducedand applied to the special functions. The specialintegrals leading to the inverse trigonometric func-tions will be introduced here. Mathematical modelssuggesting the use of trigonometric polynomials willalso be used.

Once the standard functions have been treated,it will be natural to discuss various forms of in-finitary behavior The discussion will begin witha reconsideration of infinite sequences, including apresentation of indeterminate forms and their ap-plications to order notation. The remainder of thissection will be devoted to improper imegrals andinfinite series, emphasizing the analogies betweenthese two forms of infinite summation.

At this point, the focus will shift somewhat fromfunctional behavior to functional approximation andrepresentation. Thus the next major topic will bepower series, with particular emphasis on the useof 4.aylor series to represent functions. Generat-ing functions for simple recursions will be discussed,and a certain amount of attention will be devotedto computational issues and the estimation of er-

ror terms. The constant theme will be the use ofTaylor series as function approximations to obtaininformation about functional behavior that wouldotherwise be difficult to obtain.

The final topic will be trigonometric series, withparticular emphasis on the representation of func-tions using Fourier series. The treatment of Fourierseries at this early point will require the introduc-tion of complex numbers, which will reinforce thestudents' geometric understanding of trigonometricfunctions. Furthermore, the availability of Taylorseries will permit an analytic as well as a geomet-ric discussion of the identity e=x = cos x + isin x.Finally, the early introduction of Fourier series willmake it possible to discuss discrete Fourier seriesand their applications at a far earlier point in thecurriculum than is presently possible.

Clearly, the focus on functional behavior andrepresentation has produced a first-year course thatis quite different from what is currently taught. Theessential core of the current calculus course has beenretained, but it is always made clear that it is therebecause it throws a powerful spotlight on functionalbehavior and representation.

Conversely, many traditionally taught topicshave been removed. This pruning was only possiblebecause the developers approached each topic withthe same question: How does this topic impact onthe main theme of the course?

Now that the course has been outlined, it re-mains to show how it will fit into the curriculum. Wewill also have to pay some attention to the second-year cour.e that will follow this course, and alsoto the political and institutional problems that itsadoption would pose.

IMPLICATIONS FOR THE CURRIC-ULUM

The first question to be addressed is the audi-ence to be served by the proposed course. It isclearly ideally suited for students of computer sci-ence, since it merges themes from continuous anddiscrete mathematics in a synergistic manner. Stu-dents who have successfully completed the coursecan be expected to handle the mathematics arising(for example) in the analysis of algorithms. It canalso he argued that this course would be well suitedas a first course for students of mathematics, thephysical sciences and engineering. For these disci-plines the major omission has been vector geometryand multivariate calculus. In many universities, alarge proportion of this material is treated in thesecond year, and it is not unreasonable to supposethat even more could be shifted to a third-semester


course designed for those students.Although much vitally important mathematics

can be subsumed under the general heading of"functions", an equally important heading is that of"structure". While the proposed course is intendedto give students the most important tools that comeunder the former heading, it does not address thelatter. For students of computer science, both head-ings are equally important, and thus an importantplace in their education must be found for "struc-ture". Much of the debate summarized above on theplace of discrete mathematics in the curriculum canbe seen as a debate on the place of "structure" in thecurriculum. Following on the first-year course thathas been outlined above, it is reasonable to developa second-year course focusing on "structure".

Such a course will not be described in detail here,but it is possible to discuss briefly what general top-ics might be included. The primary strands mightbe discrete mathematics, linear algebra and proba-bility theory. Discrete mathematical topics could in-clude relations, graphs, Boolean algebras and formallanguages. The discussion of linear algebra could in-clude some multivariate calculus, which could thenbe applied in the probability portion of the course.Just as with the first-year course, the topics in-cluded in the second-year course should be chosenbecause they illustrate vital structural themes or be-cause they are motivated by or permit the develop-ment of important applications.

The introduction of courses designed along theselines will not be a simple matter. The obstacles thatwill be found range from the need for new textualmaterials to the difficulty of articulating the newcourses with other courses and institutions on alllevels. It would be an unfortunate mistake, how-ever, to conclude that bacause of the certainty ofencountering what seem to be insuperable obstaclesto the introduction of a truly new curriculum, theonly possible strategy is one of incremental change.'The development and introduction of a curriculumintegrating discrete and continuous ideas is an ex-citing challenge, and one that should be taken up inseveral places. What is really needed is a collectionof design and development experiments, performedin out-of-the-way "protected" environments. Oncea new curriculum has proven its viability and worthin one or more of these experimental environments,it will be time to address the structural and institu-tional issues involved in transplanting the successfulcurriculum to less protected situations.


REFERENCESBushaw, D. [1983]: A two-year lower-division math-

ematics sequence in The Future of CollegeMathematics ed. A. Ralston and G.S. Young,pp 111-118. New York: Springer- Verlag.

Douglas, R. (Ed.) [1986]: Toward a Lean andLively Calculus, Proceedings of a Confer-ence/Workshop at Tulane University, Wash-ington, DC: Mathematical Association ofAmerica.

Kenney, M.J. and Hirsch, C.R. (Eds.) [1991]: Dis-crete Mathematics Across the Curriculum, K-12, 1991 Yearbook of the National Council ofTeachers of Mathematics, Reston, VA: CTM.

Lochhead, J. [1983]: The mathematical needs of stu-dents in the physical sciences in The Future ofCollege Mathematics, eds. A. Ralston and G.S. Young, pp. 55-69. New York: Springer-Verlag.

Mr, urer, S. B. and Ralston, k. [1991]: Discrete Algo-rithmic Mathematics, Reading, MA: Addison-Wesley.

Nievergelt, Y. [1987]: The Chip with the Col-lege Education: the HP-28C, Amer. Math.Monthly, 94, 895-902.

Ralston, A. [1981]: Computer science, mathemat-ics and the undergradu,ate curriculum in both,Amer. Math. Monthly, 88, 472-485.

Ralston, A. and Young, G.S. (Eds.) [1983]: TheFuture of College Mathematics, New York:Springer - Verlag.

Ralston, A. (Ed.) [1989]: Discrete Mathematics inthe First Two Years, MAA Notes No. 15,Washington, DC: Mathematical Association ofAmerica.

Ross, K. A. and Wright, C.R.B. [1988]: DiscreteMathematics, 2d Ed, Englewood Cliffs, NJ:Prentice Hall.

Steen, L.A. [1983]: Developing mathematical ma-turity in The Future of College Mathematics,eds. A. Ralston and G. S. Young, pp. 99-107,New York: Springer-Verlag.

Steen, L.A. (Ed.) [1988]: Calculus for a New Cen-tury, MAA Notes No. 8, Washington, DC:Mathematical Association of America.

Wilf, H.S. [1983]: Symbolic manipulation and algo-rithms in the curriculum of the first two yearsin The Future of College Mathematics eds. A.Ralston and G. S. Young, pp. 27- 40, NewYork: Springer-Verlag.

.9

TEACHER EDUCATION AND TRAINING

Bernard CornuInstitut Universitaire de Formation de Maitres, Grenoble, France

IntroductionDuring the ten last years, considerable progress

has been made in the development of computerhardware and software, and many valuable educa-tional experiments have been carried on. However,computers are not so commonly used as one mighthave expected. In many schools the computers arelocked in a special room, and it is not easy for teach-ers to use them. They must plan in advance, be surethe room is available, get the key and check and pre-pare the computers. Then they go to the computerroom with the pupils, and the time spent there isgenerally not totally "integrated" into the course.

Thus even when the computer is used, the im-pact on the learning is nog, clear. For some pupils,it is clearly useful and fruitful, but do we know whyand how? We all know very good and enthusias-tic teachers using computers, and they generally doit with success. But it is time consuming, it needsa great personal investment, and the conditions ofsuccess are not easy to reproduce or to transfer toanother situation.

However, computers are now very common in so-ciety; they are used in many domains of daily life. Inmany countries national plans for computer equip-ment in schools have been achieved, and so a lotof computers are available in schools. Much educa-tional software has been produced, and it is oftenof high quality. The use of computers does indeedbecome easier.

Five or ten years ago, the focus was on the devel-opment of hardware and software, and on originalexperiments in using computers in education. Nowit appears that teacher training is the next majorand unavoidable step but one which has not beensufficiently studied. Most countries are now askinghow to train all current and future teachers in theuse of new technologies for education.

Of course, training plans have already beentried. The first ones were generally training in com-puter science. Teachers from various subjects weretrained in compute. science, and one thought thatthey would then be able to use computers in theirteaching in an efficient way. It did happen but onlyin some cases! And it did not solve the pedagogicalproblems of the use of computers, which increasinglyappear to be essential.

The use of computers in education has reliedmainly on some enthusiastic teachers who spendnights and weekends writing programs and prepar-

87

ing activities for their pupils. These teachers some-how got the training they wished (even if they learnta lot by themselves!). But we now need to go fur-ther, and the way the use of computers was devel-oped with some teachers is certainly not applicableto the teachers. We need to imagine new ways suchthat all teachers will be able to use computers.

In most countries computer science is not yet aschool subject. Therefore, except in some particu-lar cases, we do not need to train computer scienceteachers, but we need to train teachers in all sub-jects in the use of computers and new technologiesin the teaching of their subject. Thus we need toreflect on the contents of such training.

The main problem, as noted, is the of general-isation. We know how to train some teachers butwe now need to train all teachers. We have donesome very specific and sophisticated training; wenow need training which can be easily generalisedand delivered to all. We must take into accountthe willingness and the abilities of the "standard"teacher, and design adequate training. The usualtraining for good and enthusiastic teacheth is cer-tainly not directly reusable.

This is both a pre-service and an in-service mat-ter. In the next ten years in most countries, onethird of the teachers wilf, be changed (because of re-tirements and the increasing numbers of teachers).So pre-service training will be ( fficient for this third.The other two thirds will ne d in-service trainingduring the same ten years.

In the long term, one must think about thelink between pre- and in-service training. In anever changing world, it is impossible to give futureteachers the abilities and knowledge they will needthroughout their careers. They will have to learn, tothink and to reflect continuously. Pre-service train-ing is not intended to avoid in-service training, but,on the contrary, to prepare for it! Increasingly in-service training should be considered a normal partof the job of a eacher. It should not be only forvolunteers, but for all!

The evolution of teaching

For several different reasons, teaching is goingto evolve:

Technology is evolving quite quickly. Hardwareis becoming smaller and cheaper and more and


easier to use. Software is also evolving, becomingmore user-friendly, and it is becoming possible touse computers with good software which requireslittle preparation.Pedagogy is evolving. One reason is because re-search in education provides better knowledge ofteaching and learning. And new tools arc begin-ning to be available for teaching. But pedagogyis also evolving because of the democratisation ofeducation. More and more children have accessto education, and so pupils are increasingly di-verse, and need pedagogy adapted to their needs.In short, teaching needs to be more individu-alised.Current and future teachers must be prepared

for this evolution. It is not enough to masterthe knowledge and some pedagogical strategies andtools. Teachers must be able to deal with all theevolution which will happen. and to adapt to manydifferent kinds of pupils.

The school of tomorrow may be quite different.It will be organized according to a variety of peda-gogical styles. There will be large rooms for largeaudiences; standard classrooms; rooms for groupwork; rooms for individual work; rooms for practi-cal work or workshops; resource rooms etc. Not onlywill the pupils be provided with a variety of rooms,but so also will the teachers. Teachers now havea rather standard way of working. They come toschool to give their lessons, and they stay at home toprepare the lessons or to mark homewc-7!... One canimagine that teachers will increasingly work withtheir colleagues and that they will need to havespecial tools and mnterials available for their work.Certainly offices must be provided for teachers, androoms for group work. They will also need labora-tories to prepare lessons using technology.

The school of tomorrow will be equipped withadvanced technology computers, multi-media re-sources, easily available in each room (perhaps withpermanent equipment, or possibly by plugging inportable machines); Resource centers will also benecessary in schools. Libraries with books, software,audio, video, and multi-media products. As is thecase already with other subjects , one can imaginethat in the near future, mathematics laboratorieswill be available in most schools.

The role of the teacher is also changing. Sincepupils are more and more diverse, the teacher has tointervene in many different ways, not. o, as a lec-turer, giving lessons and delivering knowledge to thepupils. In the classroom teachers must use differentpedagogical styles and different. kinds of activities.They must also work with small groups of pupils and

sometimes individually with pupils. Activities withthe pupils may occur not only in the classroom, butalso in other rooms of the school such as a resourcecenter, laboratory, a room for small groups etc.

Altogether the teacher has to be a counselor, ad-visor, organizer, leader and a manager. The task isnot only mastering of teaching, but also masteringthe management of learning.

The way teachers work every day is evolving.They will probably be in the school all day and allweek long and will use various tools in preparinglessons and in teaching. They will work togetherwith colleagues, and even teach together with col-leagues. Their personal work will also evolve and bemore diverse. The evaluation of pupils is going to bemore and more complex, and the role of the teacherin evaluation will be more important. Evaluationitself is becoming more precise and more technical;the use of evaluation in training and in individu-alization of education will be a major role for theteacher.

Teachers will also have to be involved in the elab-oration of pedagogical tool". The evoluticn of teach-ing needs new tools, but also new ways in designingthe tools. Textbooks, software, video and audio doc-uments and resources for pupils will all have to bebetter adapted to specific pupils or groups of pupils.Their elaboration will need more techniques, moretechnology and more professionalism.

Team or group teaching will become more fre-quent. Teachers will work and reflect together andthis will soon be considered as a normal componentof the job of a teacher. As intellectuals, teachersmust continue training and reflection throughouttheir professional life.

Thus teaching can no longer be considered onlyas an art; it is a profession with all the componentsof the professionalism. And this has consequencesfor the education and training of teachers. We musttrain professionals!

A good professional must have access to the bestand most efficient tools and must be prepared touse these tools, to choose the tools to be used andto adapt to new tools. Once again we note thatcontinuous training is a natural part of the job of ateacher.

Will the computer be able to replace the teacher?The answer seems to be no. Teaching and learn-ing are very complex processes, and, although tech-nology brings new tools, the main didactic actor isthe teacher who manages the learning process andadapts it to each pupil and who insures theization of the knowledge and its compatibility withthe "external world". Among the productions of the

pupils in the class, those must be identified whichdeserve status of knowledge. Thus the teacher in-stitutionalizes knowledge.

Of course, personal and individual moments arepossible and necessary in learning, and the com-puter can make them more efficient As well, someparticular topics can be learned "automatically"with a computer (for example, typing; training inrepetitive skills such as computation, learning "byheart", etc.). But teaching and learning need in-teraction between the teacher and the pupils, andamong the pupils; the computer must be used in away that facilitates this interaction.

Teacher training

Which competencies?

In talking about teacher training we need first t.odetermine the competencies which are necessary fora teacher. Which teachers do we need for tomorrow?Which kinds of teachers do we want. to prepare?

First, we certainly need teachers who have mas-tered perfectly the knowledge they will have toteach; teachers must be competent. in their subject.But this is not enough. They must. not only be good"in" their subject; they must also be good "about'.their subject. They need to know about the originsand evolution of their subject, about its history andepistemology. They need to know about the role oftheir subject in society and about its applications.They need to know about the "philosophy" of theirsubject.

We need teachers who are able to communicateknowledge and to make pupils construct their ownknowledge. Teachers must he educated in the peda-gogy and didactics of their discipline. They mustknow about the obstacles to learning, they mustknow about the errors students may make and theirrole in dealing with these errors. They must knowabout she conditions which facilitate learning andthey must know about evaluation.

We need teachers able to manage and lead theirclasses. They must know about groups and individ-uals. They must have some knowledge of psychol-ogy.

We need teachers able to advise and orient theirpupils. Thus they must know the educational sys-tem, its place and its role in society so they needsome knowledge of sociology.

We also need teachers trained in the technicalaspects of their job, able to speak loudly and clearlyenough and able to use technical tools etc.

In general then, teachers have many differentroles and must be competent in each of them.

Teacher Education and Training 89

But what about new technologies and comput-ers? They are linked with each of the aforemen-tioned competencies. One must think about the roleof the computer with respect to the subject itself.What is the influence of the computer on mathe-matics, on the way mathematicians work and onthe mathematics which is taught in schools? Whatis the place of computers in the way mathematics isused in society? What is the influence of computerson the pedagogy and didactics of mathematics? Onevaluation? What is the role and the use of the com-puter in class management, in individualization, inthe organization of the teaching? How does it affectthe psychology of the pupil? What technical helpcan the computer bring to the teacher?

Of course, there are no definite answers to thesequestions; the education of teachers must makethem able to ask these questions and reflect aboutthem. Education cannot give definite competencies,but it must. give an aptitude to evolve; it must givethe basic tools necessary to be able to build one'sown strategies, one's own answers.

We now try to list some of the ^,ompetencies ateacher needs in computers and computer science,remembering that our purpose is not to train com-puter science teachers, but mathematics teachers.

Basic tools: such as word processing, spread-sheets, data processing, and also other techno-logical tools such as video and the overhead pro-jector. This is certainly a very important point:If we want ALL teachers use new technologies,they must be totally familiar with the most corn-mor q.nd easy to use; it is an absolute necessitythat ,,achers be able to use computers for ele-me.-itary applications. This is the way to makethe computer part of the "daily life" for teachers.Technical elements: to be able t.o use the hard-ware, to manipulate the main accessories, toidentify elementary troubles, and to deal withthe technology in the school; teachers need a ba-sic level of "familiarity" with technique.Elements of computer science. But to whatextent? Teachers certainly need to know justenough in order to "understand what happens";but the links between mathematics and com-puter science are so strong that it is certainlyuseful to know about some fundamental con-cepts (as well as some concepts of algorithmicssee the chapter by Maurer).Mathematics and informatics. Mathematics isevolving and changing under the influence ofcomputers and informatics. Therefore, teachersneed to maintain their mathematics knowledgeand to practice mathematics from an informatics


viewpoint. Mathematics is becoming more ex-perimental, more algorithmic, more numerical;teachers must be able to follow the evolution ofmathematics, and to acquire new competenciesand new attitudes and to be able to carry outnew activities in mathematics.Using existing resources. Teachers must be ableto know what exists different software, differ-ent tools, different strategies for teaching. Theymust be aware. of new products which appear.They must be able to choose among existing re-sources according to the needs of their pupilsand according to their pedagogical choices. Theymust be able to advise pupils which productsthey should use.Pedagogy, didactics and the computer. One ofthe main problems of the use of computers formathematics teaching is the integration of thecomputer activities into the pedagogical strat-egy. Too often, computer activities are justadded to the usual lessons. An optimal use ofthe computer needs not only good knowledgeof the hardware and software to be used, butalso mastering of the problems of learning. Ateacher should be aware of what we now knowabout how pupils learn; the computer should bejust a tool to implement new strategies and newsolutions to learning problems. It can be an ef-ficient tool, fo,. example for individualization ofthe learning and for evaluation, but only if indi-vidualization or evaluation problems are solvedin pedagogical and didactical terms. Technologydoes not replace pedagogy. So, training in newtechnologies cannot be independent of trainingabout pedagogy and didactics.Didactical engineering. Teachers have to elabo-rate the situations needed for pupils. Since theyhave a large number of tools at their disposaland a large number of choices in terms of strat-egy, a teacher needs to have the characteristics ofa "didactical engineer", i.e. they must have theability to use the results of research or theoret-ical statements and transform them into usableproducts.

Which methodology for training?

The methods used in teachers training are atleast as important as the contents of the training.It is well known that teachers usually teach, not asthey were taught to do, but by reproducing the waythey were taught. If you only use lectures to trainteachers (even if you lecture about active methodsfor teaching), they will then mainly give lectures totheir pupils.

So the most important thing in educating teach-ers how to use computers in teaching is not to givelectures on "how to use computers", but to actuallyuse the computer in the training. This is true forall new technologies. You should use the overheadprojector in the training, rather than give a lectureon "how to use the overhead-projector".

If you want to convince teachers that pupils canlearn better with the computer, just make theseteachers or future teachers actually learn somethingwith the use of computers.

This means that the training should include ac-tive parts, even if some theoretical aspects are alsonecessary. One often says that problem solving is agood way to learn mathematics; similarly, the solv-ing of teaching or learning problems is a good wayto learn about pedagogy, and the solving of teach-ing or learning problems using new technologies iscertainly a good way to learn about the use of newtechnologies in education.

Teacher training should not be only an accu-mulation of knowledge. As already noted, teach-ers should be prepared to evolve and adapt to newsituations.

Among the different methods which can be usedfor training teachers, "training by research" is prob-ably one of the best. It does not mean that all teach-ers should be researchers. But they should be ableto use the methodology of research, and this canbe learnt through group activities reflection, inno-vation, preperation of documents and of situations,etc. Teachers will need to learn to work in teamswith colleagues. To be prepared for suth activities,they need team activities in their train ng!

Teachers should also be trained to communicate,to read, to write (for their pupils; for their col-leagues; for publication) since this will also be acomponent of their job.

Teachers should be prepared for a diversity ofpedagogy. There exist many different pedagogicalstrategies, many different pedagogical styles. Toooften, one is convinced that one of these methodsis the best. But it is better to be able to deter-mine, in given conditions, at a given moment, withgiven pupils what is the appropriate method to en-able them to learn a specific topic. This impliesthat in the training itself many different methodsand strategies will be used lecturing with onecomputer in the room, used mainly by the teacher("blackboard computer"); collective activities in aroom with one computer for each student or groupof students; individual activities on computers; self-evaluation using computers etc.

Diverse software must also be used in teacher

Jr

training utilities, basic tools such as word-processors, languages, tutorials, open-ended soft-ware, multi-media tools etc. Giving teachers accessto the maximum of diversity increases their freedomin their own professional activities.

Which contents?

This is again a question without a definite an-swer. Of course, the answer should not be the samein pre- and in-service training. But, in fact, thetopic of computers is new for most in-service teach-ers, and they need training which is close to that inpre-service.

We can hope that in the future students will haveacquired the necessary elements about new tech-nologies in their previous studies, for example usingcomputers for word-processing.

The first question to be asked is: Do the teachersI train need computer science? Will they considercomputer science as a subject in itself? Should theyonly learn informatics as it impinges on their mainsubject, and through its use in their subject?

Many different answers to these questions havebeen attempted. Some countries have tried to trainteachers by giving them a full year of training incomputer science. This produces "specialists", butthe reinvestment for other teachers was not easy.Many countries organize sessions for teachers. Hereagain the diversity of what is offered to the teachersis certainly a good thing lectures on specific topics;one week or two weeks sessions; a course over oneterm or one year etc.

In pre-service training, there should certainly bespecific modules in order to prepare future teachersfor the use of computers. As a possible example,here is the contents of a course we have given formany years at Grenoble University, both to futureteachers and to in- service ones. This course lastsfor 150 hours (5 hours a week during 30 weeks).The title is: "Informatics for mathematics teach-ing". Each week, 2 hours are devoted to lecturesand 3 hours to practical work. The course is di-vided into three parts:

Informatics and algorithmics. Students learn thebasic use of a computer; they also learn elementsof algorithmics, including recursion, proof of pro-grams, evaluation of algorithms and data pro-cessing. They use three different languages forprogramming: Pascal, Logo, and Prolog.Mathematics from an informatics viewpoint. Inorder to use the computer in mathematics teach-ing, it is necessary to use it for mathemati-cal activities, and therefore to reconsider some

Teacher Education an" Training 91

mathematics con-epts with the help of comput-ers. Every year in the course we choose differentmathematics topics in the curriculum of univer-sity studies (not in the curriculum of secondcryschools because we want the students to be ableto accomplish by themselves the transfer of theseactivities to the field of secondary school math-ematics).Pedagogical and didactical viewpoint. In thispart, we use and analyse various existing tools(software, textbooks with computer studies in-tegrated into them etc.). We try to combinethe fundamental notions of pedagogy and didac-tics of mathematics together with technology.We also try to make the students solve teach-ing problems using computers. (For example: Imust prepare a lesson about linear equations fortomorrow. What will the content of the lessonbe? What software will I use? What will be theactivities of the pupils? Here is another example:I must prepare a course about linear equations,but I have six months to prepare it. How will Ido it?).During the year the students have to produce a

personal project which takes the form of a piece ofsoftware they design and experiment with.

Research, innovation and training

The development of the use of new technologiesin mathematics teaching makes it necessary that re-search be carried on in several domains researchabout mathematics learning; research about com-puters in mathematics teaching; research leading toappropriate software; research about teacher train-ing. This research may take several forms funda-mental research, applied and experimental research,innovation. Too often, there is a gap between thefundamental results of educational research, andproducts which are usable in teaching and in train-ing. We need to develop applications and implementations of the results of the research and we needpedagogical products based on research.

Innovation and research can contribute to teach-er training. Indeed, the participation of teachers orfuture teachers in innovative activities is a good wayfor training them.

The participation of teachers in elaborating andexperimenting with pedagogical products is neces-sary, but not sufficient. Designing good softwareneeds computer scientists and software specialists aswell as specialists of pedagogy and teachers (prac-titioners). It is a professional matter which needsprofessionals.


Two other tracks need to be explored:Teacher training is becoming more complex, andwe need courses and training activities adaptedfor this purpose. Courses for teachers and futureteachers must he developed. We also need toreflect about the specific competencies which ateachers trainer must have. In fact, in manycountries the first problem to be solved beforewe are able to train ALL the teachers is to trainteacher trainers.In order to diversify the tools usable in teachertraining, it would certainly be interesting to de-velop computer tools and software for teachertraining.

Conclusion

We have done a lot of experimenting with newproducts and new strategies, and the most enthusi-astic teachers have shown both their efficiency andtheir limits. The problem now is to generalise theuse of new technologies, so that ALL teachers areable to use them as they wish, or to know why theydo not want to use them.

Two conditions seem to be essential in order tohelp all teachers and future teachers use computers:

Make the computer actually available and us-able; make it a "daily life" tool; make it reallyuser-friendly. This means that it is necessarythat schools be well equipped. The aim shouldbe that each teacher or future teacher has a com-puter (either one the teacher owns (special plansmay need to be set up for purchasing comput-ers at reasonable prices; loans may need to beobtained for future teachers) or the institutionmust own computers and make them availablefor teachers and future teachers).Actually use the new technologies in teachertraining, and not just train about "how to use"them.In no case can the technology replace the peda-

gogy. A bad teacher using computers will certainlystill be bad! So training and education are neces-sary, but not only from the viewpoint of technology.We need coherent training, integrating both techno-logical and pedagogical approaches. Teachers musthe ready to evolve and adapt, and must retain theability to ask questions. At each instant they shouldask whether education or technology is the drivingforce.

Teacher training is a continuous process. Pre-service and in-service training are strongly linked,and both are necessary. No longer can a teacher beprovided with all the abilities and knowledge needed

at the beginning of a cvreer; training never ends,reflection never ends; in-service training should beconsidered as a natural component of the teacher'sjob. We must never forget that teachers are profes-sionals, and need professional training.

Many countries use a "cascade model" forteacher training. The education ministry organisesa course for a number of selected trainers; after-wards, each of them trains a number of other train-ers, who then train teachers (or trainers who train

). Suer: a model can be efficient, but may alsonot be! The main characteristics of good training

motivation, activities, understanding must bepresent at each stage of the "cascade". And thismodel can apply only to very specific, precise, andlimited training.

"Training plans" have been set up and imple-mented in many countries for training teachers inthe use of computers. They have only partly suc-ceeded. One reason for this is that they are gen-erally too restricted as to technology. A trainingplan should be more global, aiming not only at solv-ing new technology problems, but aiming at solvingteaching and learning problems. New technologyproblems should not be treated in too isolated acontext.

REFERENCESBall, D. et al (Eds.) [1987]: Will Mathematics

Count? Computers in Mathematics Educa-tion, an AUCBE report.

Bosler, U. and Squires, D. [1989]: Training teachersto design educational software in EducationalSoftware at the Secondary Level, (Tinsley, J.D. and van Weert, T. J., Eds.), Amsterdam:Elsevier.

[1988]: Pre-service teacher education, Proceedingsof ICME6 (Hirst, A. and Hirst, K., Eds.), Bu-dapest: .1. Bolyai Math. Soc.

[1988]: Information Technology in Computer Edu-cation, IBM Denmark.

[1990]: The Teacher Today: Tasks, Conditions,Policies, OEDC.

THE IMPACT OF SYMBOLIC MATHEMATICAL SYSTEMS ON MATHEMATICS EDUCATION

Bernard R. HodgsonUniversite Laval, Quebec, Canada

Eric R. MullerBrock University, Ontario, Canada

Symbolic manipulators, that is, computer pro-grammes with the capability of carrying out sym-bolic computations, for example, in calculus or lin-ear algebra, are now widely available. While theseare well-established tools in many areas of math-ematics, science and engineering, it must be recog-nized that they are still in their infancy with respectto their use in mathematics education. They repre-sent an ineluctable challenge to current approachesto the teaching of mathematics and there is a beliefamong some members of the mathematical commu-nity that electronic information technology, throughthese symbolic capabilities, will exert a deep influ-ence on how and what mathematics is taught andlearned (for example see Page (1990)). However noclear pattern has yet emerged on how such an influ-ence is to be articulated.

This paper will discuss certain aspects of the im-pact of symbolic manipulators on mathematical ed-ucation in the upper secondary years and the firstfew years of university. It is by no means intended togive the final word on such a vast field as much workis in progress and the technical environment (com-puter hardware/software and calculators) is con-stantly improving. The aim of this paper is ratherto examine some of the major issues and to indi-cate general trends which have developed since the1985 ICMI Study on "The Influence of Computersand Infoviatics on Mathematics and its Teaching".The influence of symbolic manipulators on more ad-vanced (senior) mathematics courses will not he ex-plored. This is not intended to belittle their impactat this level but rather to concentrate on those yearswhere these systems must be implemented in orderto benefit the largest possible number of studentsin mathematics courses. The influence of these sys-tems and their mathematical foundations (see forexample Davenport, Siret and -Fournier (1988)) willhe thrust into the upper level courses by more capa-ble and interested students as they progress throughthe system.

Section 1 defines Symbolic Mathematical Sys -terns in broad terms and presents an example oftheir potential use in mathematics education. Sec-tion 2 raises some general concerns related to theimpact of these systems on mat hematics educationwhile Section 3 discusses implementation of someof the required changes in secondary and university

93

mathematics education. The Appendices providethe following additional information: (1) referencesdealing with the technical aspects of some of thebetter known Symbolic Mathematical Systems, (2)further illustrations of the capabilities of these sys-tems, and (3) references to current projects aimedat the integration of such systems into mathematicseducation.

1. Symbolic Mathematical SystemsThe term Symbolic Mathematical Systems is

used to define calculator and microcomputer sys-tems which provide integrated (1) numeric, (2)graphic, and (3) symbolic manipulation capabili-tiesl. Numerical computations have always beenincluded in the domain of both the calculator andthe computer. This capability is usually thoughtof as the ability of doing decimal arithmetic. Forexample, if 1/3 + 1/9 is input, then the approxi-mate solution 0.444444 (to some prespecified num-ber of digits) is provided. Symbolic MathematicalSystems have the ability to perform rational arith-metic, that is, to give the exact answer 4/9 if theinput is 1/3 + 1/9. The user must request the dec-imal approximation if it is desired. Graphing is amore complicated numerical activity. Calculatorswith graphic capabilities (for example th. Casio

n00G, Hewlett-Packard dP -48SX or Texas In-

It should be noted that in a r lich more gen-eral context, the expression "symbolic computa-tion" could be construed as referring to varioustypes of symbolic objects, for example as describedby Aspetsberger and Kutzler (1988): geometric ob-jects (computational geometry), logic objects (auto-matic reasoning), programmes (automatic program-ming). The concerns of this paper are limitedto computations involving algebraic expressions, sothat typical topics of the field are symbolic differ-entiation and integration, calculation of sums andlimits in closed form, symbolic solution of systemsof equations and of differential equations, polyno-mial factorization, manipulation of matrices with orwithout numeric entries, arbitrary precision rationalarithmetic computations, etc. These are sometimesmisleadingly called "Computer Algebra Systems"but they can do much more than algebra as will heillustrated by the examples in this article.


struments TI-81) as well as microcomputer graph-ing programmes are available. To many mathe-maticians and mathematics educators symbol ma-nipulation by calculators (for example the Hewlett-Packard HP-28S or HP-48SX) and microcomputerprogrammes (for example Maple, Mathematica, De-rive to name a few) was a most unexpected de-velopment. It is the one capability which has thepotential of producing the most radical changes inthe teaching of mathematics at the secondary schooland university levels.

To convey a feeling for some of the capabilitiesof Symbolic Mathematical Systems and how theycould be used in a calculus class, consider the fol-lowing example of a session with a specific system(namely Maple but this particular choice is not cru-cial). Such an example could be done in class, orcould be structured as part of a laboratory exer-cise. The example illustrates the numeric, graphicand symbolic manipulation capabilities of the sys-tem and shows the system can be used in a modewhich requires no programming by the user, butonly the knowledge of a few command words. Forease of understanding lines starting with a # (andin italics) are external comments, lines starting witha ::: are the user's input and the lines in bold arethe (Maple) system's response.

#The task is to explore the derivative of ln(x) using#the definition of the derivative. First the limit of# (In(1)-1n(4))/(1-4), called y, as t approaches the# integer value 4 is explored from a numerical#point of view, by computing the value of y around#t = 4. Clearly the value at t = 4 does not exist.

y:= (1n(t)-1n(4))/(t-4);ln(t) ln(4)

t 4y :=

#A1 t = 3.99subs(t=3.99,y);

100 ln(3.99) + 100 ln(4)#Evaluation using floating-point arithmetic of this#last displayed expression then gives

evalf(");.250313

#At 1 = 3.999evalf(subs(t=3.999,y));

.250031

#At = 4.01evalf(subs(t=4.01,Y));

.249688

#At t = 4.001

evalf(subs(t=4.001,y));

.249969

#Looks as though the function is approaching 0.25#as I approaches 4. Does the grapi support this?#A plot of y for 3.5 < t < 4.5 is obtained.

plot(y,3.5..4.5);

0. 65

0.26

0.255

0.25

0.245

0.24

0.2353:6 3.8 4 4.2 4 . 4

# Yes it does and the graph indicates by a hole that#the function is not defined at t = 4, where y is#approximately equal to 0.25. One repeats this#experimentation with a few more integer and#rational cases, for example 5, 3/2, 7/3. Then the#symbol manipulation capabilities can be used to#evaluate the limit directly,

limit(On(t)-1n(3/2))/(t-3/2),t=3/2);

23

#suggesting that the limit of#(ln(t) ln(a)) /(t a) as t approaches a is 1/a#for all a>0. This is confirmed by the system.

limit(On(t)In(a))/(ta),t=a);1

a

#Which is also confirmed by the differentiation#capability of the system.

diff(In(a),a);1

a#Does the derivative have the properties expected?#Plot the function and its derivative on the same#graph (in some judiciously chosen interval .9.

plot({1n(t), 1/t}, 0.5..10.5);

Symbolic Mathematical Systems and Mathematics Education 95

#range. This is visualized with the following plot,#where we notice that the graph of y is#completely contained in the specified window.

plot( {y, 0.24, 0.26}, 3.6963..4.3379);

#Yes In(i) is a monotonically increasing#function ane the derivative is shown to be#positive in the chosen range. Measurements along#the axes appear to confirm the previously#computed points (4,1/4) etc. The formal#definition of the limit can also be explored, namely,#L is the limit of y as t tends to c if for every#eps > 0 there is a del > 0 such that if#0 < 1tal < del then Iy LI < eps.#Consider the case explored earlier where it was#conjectured that the limit of y as t tends to 4 is#I/4. Select eps = 0.01 > 0; is there a del such#that for 0 < It 41< del, then ly 1/41<#0.01? The condition ly /RI < 0.01 can be#rewritten as 1/4-0.01 < y < 1/4+0.01.#This is solved using the system. (Maple solves#different types of equations: algebraic, numeric,#differential, etc.) In this case we are#interested in the numerical solution of an#equation in one variable. Numerical procedures#for the solution of such equations often require#the user to specify an interval within which one#expects to locate the root. In this particular case#Maple does not require such a prompt and#provides the following:

fsolve(y=0.24,t);

4.33789986

fsolve(y=0.26,t);

3.696303966

#From these two values it is concluded (base' on#the continuity of the log function) that there is a#del, for example 0.2, such that when#0 < It 4 I < 0.2, then y is in the specified

#To demonstrate how incredibly sensitive and#accurate the limit procedure is, one can consider#the following.

limit(On(t)-1n(3.2))/(t-3.2),t=3.2);

undefined

# What happened? To resolve this apparent#anomaly the user must realize that elementary#functions involving numbers other than integers

or rationals are approximated (the calculator#mode), that is ln(3.2) is evaluated as shown by#the following output

y:=On(t)-1n(3.2))/(t-3.2);

111(0 1.163150810:= t 3.2

#and, because of the numerical approximation, the#limit of y as t approaches 3.2 does not exist.

For those who are not familiar with SymbolicMathematical Systems Appendix 2 provides furtherexamples of their capabilities. While special pur-pose packages have been created to cover specificaspects or topics within the mathematics curricu-lum, this paper is concerned with "full service" Sym-bolic Mathematical Systems which can become partof mathematics education across different coursesand at different levels. The more powerful systemswere originally created to help individuals performcomplicated yet algebraically routine mathematics.There is no evidence that the introduction of inex-perienced students to more dedicated (smaller spe-cially developed systems addressing one part of the


syllabus) has been more successful than introducingthem to the larger more sophisticated systems.

2. Mathematics Education ConcernsMathematics educators must continually make

decisions about what mathematics is to be taught,how it is to be presented and what student activi-ties are to be required or encouraged. To this de-cision making must now be added the role of Sym-bolic Mathematical Systems. These systems are afact of life and can no longer be ignored. Mathe-matics educators have the responsibility to decideconsciously whether this environment is to be in-cluded within the student's educational experienceand what should be the exact role of the Sym-bolic Mathematical System. This decision cannothe taken lightly for these systems can perform all themathematical techniques presently included in sec-ondary school mathematics programmes and mostof those included in the first two years of universitymathematics. The decision to include or exclude theexperience of a Symbolic Mathematical System hasfar reaching implications to the student, the teacherand to the curriculum. 'These are now considered inturn.

a) Implications to the studentThe magnitude of the experiences promised to

the student by Symbolic Mathematical Systems isillustrated by the following allegory:

A person explores her surroundings by walk-ing (pencil and paper) many interesting thingsare discovered, but situations in the neighbouringprovince are too far away to be experienced, so theuse of a car (standard scientific calculator) is al-lowed. As she drives along, local attractions areoverlooked in order to get to her destination. How-ever, even with this mode of transport, she can-not explore distant lands, so an airplane (SymbolicMathematical System) is provided. She lands in acountry where the language is not her own, customsare different as educators we would try to pre-pare her for this shock but there is nothing thatis quite like being there. What potential benefitawaits her! she can now explore concepts whichwere unknown before and she can contrast, com-pare and have a different view and appreciation ofher own culture and home environment. In this newland she continues to use the other modes of trans-portation, namely, walking and driving to enhanceher experience.

Mathematics education has many of the proper-ties of this allegory. Individuals develop their math-ematical understanding in various ways. Due to the

different roles played by the left and right hemi-spheres of the brain, it is most likely that the repre-sentation of mathematical concepts in complemen-tary modes such as numeric, graphic, and symbolicwill enhance the learning process. For the first timein the history of mathematics education SymbolicMathematical Systems offer the ability to move eas-ily and rapidly between these different representa-tions. It is expected that the use of paper and -;,en-cil will be retained by most students; however, oneshould not be surprised to find students who can op-erate completely within the computer environmentsince most systems now provide for easy interplaybetween word processing and Symbolic Mathemat-ical Systems.

b) Implications to the teacherFor the teacher Symbolic Mathematical Systems

are remarkable not only because they can be used todirectly perform rational, symbolic or graphic com-putations but, more importantly, because of whatthey suggest about. mathematics itself and aboutmathematics teaching. As Young (1986) puts it,"(...) we are participating in a revolution in math-ematics 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 talents; prob-lems arose that had not been previously visualized,and their solutions changed the entire level of thefield." Symbolic Mathematical Systems are partof this revolution. They can serve to help conceptdevelopment and, by permitting easy and efficientprocessing of non-trivial examples, they can stim-ulate exploration and search for patterns2, general-izations or counter-examples. The teacher must nowquestion the whole of mathematics education. Forexample, it is increasingly difficult to justify want-ing students to become good symbol manipulatorsunless it can be shown that such procedural skillsare essential to an understanding of the underly-ing mathematical concepts but no one has yet soshown. However this does not imply that students

2 "The rapid growth of computing and applica-tions has helped cross-fertilize the mathematical sci-ences, yielding an unprecedented abundance of newmethods, theories and models. (... ) No longer justthe study of number and space, mathematical sci-ence has become the science of patterns, with the-ory built on relations among patterns and on ap-plications derived from the fit between pattern andobservation." Steen (1988).

1u


no longer need develop "symbol sense" just as thearithmetic calculator has not reduced the need for"number sense". Suddenly the teacher is broughtto question both the content of the mathematicscourses and their presentation. As the former re-lates to curriculum more directly, the latter concernis addressed first.

The teacher must consider many factors whichaffect the learning of mathematics. An importantfactor is the social environment. Some students findit easier and more enjoyable to work on their ownwhile others prefer to work in groups. Some de-pend on the verbal or written or visual presenta-tion of mathematical concepts by those who under-stand them. Others find this distracting and preferto work directly from hooks. Computers provideopportunities to enhance these social environments.They also introduce a new factor the computer

which may, for some individuals, erect new bar-riers and difficulties. It is therefore important formathematics educators to provide alternative en-vironments for students to experience. Individualswill then be in a position to evaluate them and de-cide which provide the most opportunities for thedevelopment of their mathematical knowledge.

Symbolic Mathematical Systems can be inte-grated into mathematics education in a number ofdifferent ways. The three most obvious ones are:

(1) The teacher can use it as part of a lecture orclass presentation. This requires some projectionfacilities to allow the students to see what appearson the computer screen. For the mathematics in-structor the use of such a system in the classroomprovides very different class dynamics. Attentionhas to be paid to typing, errors, unexpected forms ofexpressions, graphs which appear different from thetraditional book presentation (cf. Muller (1992)),multiple answers, etc. Many mathematics instruc-tors find this situation difficult to handle. Perhapsthe central aspect in the successful integration of aSymbolic Mathematical System in the classroom isa necessary evolution of the role of the teacher whereintervention is no longer restricted to exposition. In-stead the teacher must become a "facilitator" cre-ating a context appropriate for a fruitful interactionbetween the student, the machine and the mathe-matical concept. The lecture-examples format mustbe replaced by a more open-ended approach. Al-though such a point of view is desirable even in acomputer-free classroom, it becomes essential whencomputers come into play. One of the reasons whyfilms and videos have played such a small role in themathematics classroom may be the mathematician'sbelief that you understand mathematics by doing it

and not by viewing it. Unlike film, Symbolic Math-ematical Systems provide an active environment re-quiring constant intervention and change of direc-tion. Nevertheless it would be naive not to realizethat many teachers will find the sacrifice of tradi-tional security quite threatening. This will be es-pecially true of mathematics teachers who sec theirrole as one of "professing" well-polished mathemati-cal knowledge. White (1989) has suggested that theuse of Symbolic Mathematical Systems "can be as-similated most easily in traditional teaching meth-ods and curricula." However, in practice, findingan appropriate role for the teacher may prove tobe a major barrier for the universal introduction ofSymbolic Mathematical Systems into the traditionallecture presentation and teachers should seriouslylook at alternative and/or complementary modes ofimplementation. Even though introducing an occa-sional Symbolic Mathematical System demonstra-tion into a traditional set of lectures is a start, whatis needed is a complete rethinking of the objectivesof those lectures.

(2) The technology can also be used in scheduledlaboratory sessions. This is probably the leastthreatening mode of introduction for the teacher.Laboratory activities can be developed and testedbefore the students try them. Students can be givenmaterials to prepare for the laboratory sessions andsupport can be provided for the students duringtheir scheduled laboratories. The physical labora-tory setup can vary. There are advantages to havingstudents working with their own system and advan-tages to having four to six students working togetherwith a single system. Activities appropriate for lab-oratory work with a Symbolic Mathematical Sys-tem should not he a simple duplication of activitieswhich can be achieved just as easily with pencil andpaper. What are appropriate activities? Clearlythe lack of sustained experience limits one's vision.Nevertheless it is suggested (cf. Muller (1991)) thatlaboratory activities should meet one or more of thefollowing general attributes:(a) they encourage exploration of mathematical con-

cepts;

(b) they probe inductive reasoning and/or patternrecognition;

they investigate interrelationships between dif-ferent representations algebraic, graphical, nu-merical, etc.;

(d) they involve problems which would be very diffi-cult and/or too time consuming to solve withoutthe technology.

One can visualize a situation where the lecture and

(c)


laboratory activities are merged and the lecture pre-sentation takes place in an area where students haveaccess to systems. Because students work at differ-ent rates with systems it is quite a challenge to lec-ture in the traditional way and have students work-ing independently or in groups. The lecture dynam-ics parallels the situation where one allows time forstudents to work independently on problems. De-vitt (1990) and others have used this method.

(3) It is important to prepare for the time when stu-dents will have easy individual access to SymbolicMathematical Systems. A consequence of techno-logical in-q,,,-,vements is that a calculator with inte-grated numeric, symbolic and graphic capabilities isno longer a dream and that such devices can onlybecome progressively more powerful and cheaper.Furthermore, one can expect that the difference be-tween portable computers and calculators will be-come less apparent. Denying the use of such calcula-tors/computers in structured mathematics instruc-tion does not solve the problem of their existence,and their access by a few more fortunate students.Every society believes that its students should beexposed to all environments which promise a richereducational experience. Of course many situationsarise where that society cannot afford to provide aparticular environment. Nevertheless this does notrelieve teachers from their responsibility to make ev-ery possible effort to provide them.

c) Implications to the curriculum

There is no doubt that Symbolic MathematicalSystems will have impact on the curriculum. Whatis in question is the magnitude of this impact. Thereis already evidence that traditional courses will haveto change if these systems are to be integrated in anymeaningful way. Even with relation to elementaryconcepts such as graphing, Dick and Musser (1990)observe: "This change in approach made possibleby these calculators marks a significant shift in howgraphing could be perceived by students. Instead ofas a final task to be completed, graphing can assumethe role as a problem-solving heuristic and a toolfor exploration." Thus the traditional calculus ap-proach of finding what the graph looks like is turnedaround to using calculus and numerical methods forlocating more accurately the properties which areknown to exist. Students rapidly come to appreci-ate both the exactness of non-numerical algebra andthe approximation techniques underlying numericalanalysis.

The decision as to what extent Symbolic Math-ematical Systems are to be included in the mathe-matics curriculum will vary according to the groups

of students being considered and their level. For in-stance, one could have requirements for a student ina university mathematics programme different fromthose for a student registered in a mathematics ser-vice course. In this respect there is much evidencethat shows that scientists from other disciplines (seefor example Lance et al. (1986)) serviced by math-ematics departments are interested that their stu-dents not be denied the use of Symbolic Mathe-matical Systems. Such scientists, often more open-minded than pure mathematicians with respect totechnological developments, simply perceive Sym-bolic Mathematical Systems as tools that can helpthem in their work and so are eager to use them. Itis therefore necessary to reassess the proper balancein the requirement of basic symbolic manipulationskills and in the choice of topics covered in the var-ious mathematics curricula.

Mathematics educators must make sure that inconnection with domains where Symbolic Mathe-matical Systems can play a role, their courses helpstudents acquire the appropriate intellectual skills.The required skills, while not really "new", are veryoften given little place in most traditional teaching:these are interpretive skills, needed to make math-ematical judgements, to appreciate the validity andlimitations of the tool being used, to assess the rea-sonableness of the computed "answer" (cf. Hodgson(1990)). Such skills, being much more demandingthan traditional algorithmic ones, will require thestudent to be confronted with a substantial numberof theoretical notions. Thorough understanding ofmathematical concepts is thus now surely as oreven more necessary in mathematics educationas it has ever been (cf. Hodgson (1987)).

Another issue which is important in a pedagogi-cal context is the extent to which the symbolic pack-age will act in a "black box" mode or on the contrarygive indications about how the "answer" to a partic-ular problem can be obtained. A White-Box/Black-Box Principle has been advocated by Buchberger(1990) in relation to the question: Should studentslearn integration rules? Buchberger's point of viewis essentially that in a stage where a certain math-ematical topic is being learned by the student, theuse of a Symbolic Mathematical System realizingthe pertinent algorithms as "black boxes" would bea disaster. So he calls for systems that would fea-ture the possibility to use an algorithm both as a"black box" (as is most often the case with exist-ing systems) and as a "white box", i.e. in a step-by-step mode in which the reduction of the prob-lem to subproblems is exhibited and in which theuser could eventually interfere. A similar view is

1U;


taken by Mascarello and Winkelmann (1992) in thisvolume. They claim that even if all the details ofthe internal functioning of a Symbolic MathematicalSystem are not usually essential to users, they mustnot remain totally hidden: understanding of mainideas and fundamental restrictions are necessary forproper use of (what they now call) the "grey" boxes.

Once these systems have been introduced intothe mathematics courses then the student evalua-tion must change to reflect this new environment.As less emphasis is placed on certain techniques andmore time is spent on concepts, the testing proce-dures must also change. Osborne (1990), Beckmann(1991) and others have started to address this issue.

It is clear that much experimentation and re-search are needed to establish how best to use Sym-bolic Mathematical Systems in the different courses,with the wide-ranging mathematical capabilities ofstudents, and with the various attitudes of teachers.Appendix 3 provides a list of ongoing projects whichare addressing some of these concerns.

3. Effecting curriculum changes

Generally curriculum changes in the secondaryschool system require much time to he implementedbut when they happen, they are universally imple-mented: this is a direct consequence of the highlycentralized administration of secondary school pro-grammes in almost all educational systems. Onthe other hand curriculum changes in universitycourses can be far more spontaneous, but they tendto be localized to a particular course or section ofa course, usually under the commitment of one ora few highly motivated individuals. Therefore theintroduction of Symbolic Mathematical Systems insecondary school and university mathematics edu-cation poses problems of a different nature. In theformer, to affect curriculum change one must con-vince a small group of influential curriculum mak-ers. For the latter, to ensure that the use of Sym-bolic Mathematics Systems becomes integrated incourses, it is necessary to expose the majority of fac-ulty members of the department to these systems.Kozma's (1985) study on instructional innovationin higher education supports this view. Ile con-trasted projects which were collaboratively devel-oped with those developed by individuals and foundthat the former were much more likely to be institu-tionalized. This section discusses some of the timeand effort consuming activities which are requiredwhen introducing Symbolic Mathematical Systemsin both upper secondary and university mathemat-ics education.

The number of different Symbolic Mathemati-

cal Systems is expanding rapidly. Some of themhave even been developed specifically for educationat secondary school or at the beginning of univer-sity education. In most systems, especially the morerecent ones, attention is being paid to make themmore user friendly, that is, easier to use. A list of ref-erences which review some of the better known sys-tems is provided in Appendix 1. While the choice ofa specific Symbolic Mathematical System appropri-ate for use in a given classroom context might reston various criteria (e.g. hardware facilities, level ofinstruction, topics to be covered, etc.), it is clearthat some basic requirements must be met by thosesystems. For instance the use of the software shouldbe transparent, that is students should spend theirtime thinking about the mathematics, and not howto operate the computer. Documentation shouldbe essentially unnecessary for users, so that whatneeds to be done at any point should be apparent(some on-line "help" facility might however be use-ful in this respect). The software should be robust sothat students' (sometimes unpredictable) behaviourshould not cause it to crash or hang up too easily. Itshould interact easily with some word-processor, ei-ther internally to allow preparation of "notebooks"integrating word-processed text inserted in the mid-dle of active symbolic software code, or externallyto facilitate preparation of reports by students. Butmost important of all the program, whether used ina tutorial or interactive mode, should be devised soas not to foster the myth of computer omniscienceand infallibility too often rooted in students' minds:while the computer brings in speed and reliability,it is the human being who has the intelligence andthe ability to reason and make decisions.

As the cost of basic microcomputer technologycontinues to drop, one would hope for an analo-gous reduction in the price of hardware necessary forsupporting Symbolic Mathematical Systems. Whilethis has happened in some cases, this is not thegeneral rule. Indeed, one should be aware thatthe general software development trend has beento demand more and more memory and disk space,thereby requiring more powerful and more expensivemicrocomputer units. Software developers tend tothink in terms of the latest available (or forthcom-ing) hardware facilities, and experienced users callfor more integration, namely word-processing, sym-bol and graphic manipulation, spreadsheets, etc.,all of which push up the requirements of the com-puter system. Thus the implementation of curricu-lum change involving Symbolic. Mathematical Sys-tems requires financial planning for the purchase ofequipment and software. Budgets must also be al-

f;


located to the maintenance of both hardware andsoftware. Mathematics departments generally havelittle experience in requesting monies. This hastended to be the prerogative of Science, PhysicalEducation, Fine Arts and other departments. Insecondary schools funds are sometimes allocated foriinplementation of curriculum changes, but these areunlikely to be sufficient. In a university setting itmay be worthwhile to run experimental sections toaccumulate evidence of improvements in traditionalindicators and, to obtain faculty and student atti-tudes and responses to these systems.

In a context where a lot of importance is givenin the literature to various symbolic software run-ning on microcomputers, it might be tempting tooverlook calculator technology. But. the calculatoris not restricted to school applications or to compu-tations on numbers in so-called "scientific" notation!There are a number of calculator projects reportedin the literature (see for example Nievergelt (1987)and Demana and Waits (1990)). It. is true thoughthat present calculators only have limited graphicand symbolic manipulation capabilities. But devel-opments in electronic technology strongly suggestthat such more powerful and user-friendly calcula-tors will most certainly be a reality in a not too dis-tant future. To equip a class or for individual use,calculator technology should thus be seriously con-sidered. This is especially true in situations where,for instance, electricity supplies tend to be unreli-able

Once the equipment. (hardware/software) hasbeen purchased, meaningful mathematics activitiesfor the students must be developed. Few such ac-tivities are available, although some recent publica-tions provide examples in calculus: see for examplethe Mathematical Association of America Notes Se-ries (P8) and the Maple Workbook (Geddes et al.(1988)) referenced in the Bibliography. But redefin-ing objectives for a course or building pertinent, ac-tivities is a daunting task. And for such a quest tohave a lasting effect, it should be undertaken notby one individual (with eventual loss of the effect,should that individual be away for a while), butrather by a group, for instance by a majority ofthe faculty members within a mathematics depart-inent. This raises the difficult question of how toreact to a possible lack of interest by some of thosefaculty members. After all, most are busy peopleand are not willing to invest large amounts of theirlimited time unless there is some evidence that theresult will be worthwhile. This is even more truewhen students' attitudes towards the use of Sym-bolic Mathematical Systems in the classroom are

not as positive as what could have been expected(see for instance Muller (1991) for an attitudinalsurvey of some teaching experience with a SymbolicMathematical System).

The principal word of warning is certainly thatimplementing the necessary curriculum changestakes a lot of human resources in the form of timeand dedication. It takes time to conceive the "newcourse", to develop meaningful students activities,to prepare new materials, to devise tools or assess-ment. And this must be done in contexts where of-ten no (or little) credit is given to those who embarkon such a task! Furthermore released time, super-vision, hardware and software all require financialresources in an area where administrators have notbeen used to allocating funds. Mathematics educa-tors must convince school or university administra-tions and funding bodies that such an investment isessential and is worth its value! And what is neededto support the argument is a critical analysis of con-trolled experiments, rat her than anecdotal reportingof experiences.

4. Conclusion

The introduction of Symbolic Mathematical Sys-tems into mathematics programmes should. be con-sidered within the broader context of the impact oftechnology on mathematics education. Mathemat-ics teachers who have successfully integrated othersoftware into their teaching of geometry, statisticsetc. as well as computer scientists can offer usefulinsights and pedagogical points of view. Most of theprojects aimed at. the integration of Symbo is Math-ematical Systems into mathematics teaching are ei-ther still under way or, if concluded, have resultswhich are difficult to interpret. For example, howdoes one separate the effects of a Symbolic Math-ematical System from other effects, such as thosegenerated by the enthusiam of those involved withthe experiment or the effects produced by the avail-ability of additional resources? It is most proba-bly too early to look for a significant impact, on thecurriculum (measured by the proportion of studentsin mathematics courses affected by the existence ofSymbolic Mathematical Systems). It appears to bethe consensus of those who are using these systemsin their teaching that the course is taught differentlybut that it retains a fairly traditional content.

Thus there are few proposals of changes inthe curriculum narrowly defined by course content.Some examples of proposals for change are: Tall(1985,1991) proposes a much greater visual com-ponent to calculus teaching; Willer (1990) sug-gests that the conceptual approach to calculus using


"Lipschitz-restricted" concepts of limit, continuity,differentiation and integration is a much more nat-ural one for students and one in which SymbolicMathematical Systems are easily integrated; Heid(1988) reports experiments in the resequencing ofskills in introductory algebra and calculus where at-tention to hand manipulation skills was drasticallyreduced; Artigue et al. (1988) traces the influenceof computers on the evolution of the teaching of dif-ferential equations; the texts of Hubbard and West(1991) and of Kook (1989) support this evolutionand emphasize the importance of visualization inthe study of differential equations.

It is anticipated that. many more such experi-ments will be reported in the near future as thereare many Projects on the way. Appendix 3 listssome of these projects for which information couldbe found. Ralston has constantly advocated cur-riculum reform at all levels of Mathematics Educa-tion in order to reflect the reality of today's tech-nology and prepare individuals for future technol-ogy; in Ralston (1990), he proposes a framework forthe school mathematics curriculum in 2000 whichis highly dependent on the use of technology. Yetteachers receive their mathematics education fromuniversity mathematics courses in which they makevery little use (if any!) of technology. How thencan they he expected to realize the importance ,ftechnology in Mathematics Education? The reformmust he spearheaded by the universities where thereexists a greater latitude for experimentation.

There is as yet, little evidence that SymbolicMathematical Systems have had a significant im-pact on the mathematics curriculum of secondaryschools and universities. It appears that the domi-nant reason for this lack of impetus on the curricu-lum is the education of teachers and faculty, thatis, the lack of experience in these systems by a largeproportion of mathematicians. In the university set-ting there is no evidence to suggest. that changes im-plemented by an individual in one section of a coursewill have any impact on the course as a whole un-less special effort is directed toward involving themajority of the faculty in a department. 'There aretoo many interests riding on the required introduc-tory mathematics courses to expect, that innovativechanges made by one individual will be able to per-meate the programme without the support. from themajority of individuals in that department..

In spite of the human and financial costs in-volved. there is no doubt, that Symbolic Mathemati-cal Systems must be introduced into the mathemat-ics curriculum. They probably constitute the singlemost powerful force compelling change in secondary

and university mathematics education in the nearfuture. They offer unprecedented opportunities todeepen and revitalize mathematics courses, focus-ing more on concepts and ideas than on mechan-ical calculations. While it is true that SymbolicMathematical Systems, whether on microcomput-ers or on hand-held calculators, can only becomemore powerful, more user-friendly and more widelyavailable, they offer right now an exceptional po-tential for progress in the teaching of mathematicsand there is no reason for mathematics educators todelay becoming seriously involved with them. Forsuch an evolution to happen, experiments must beperformed on a very large scale and results must beevaluated and widely disseminated.

Appendix 3 contains a (partial) list of projectspresently underway, in which Symbolic Mathemat-ical Systems are being used in the classroom bothat university and secondary school level. Hopefullythese projects can stimulate more mathematics ed-ucators to involve Symbolic Mathematical Systemsin their daily teaching.

Bibliography

Since the present volume updates the work andpublications of the 1985 ECM' Study on "The Influ-ence of Computers and Informatics on Mathematicsand its Teaching" held in Strasbourg, this bibliogra-phy is restricted to references which have appearedsince that meeting.

a) Proceedings

There are a number of conference proceedings,books of invited papers and series which providean overview of classroom and/or laboratory projectsand raise philosophical and cognitive issues of usinga Symbolic Mathematical System in mathematicseducation. References P1 and P2 are the outcomesof the 1985 ICMI Study.

P1) The Influence of Computers and Informaticson Mathematics and its Teaching. Support-ing Papers of the ICMI Symposium, IREAl,University Louis-Pasteur, Strasbourg, 1985.

P2) Howson, A.G. and Kahane, J.-P. (eds.), TheInfluence of Computers and Informatics onMathematics and its Teaching. (Proceedingsof the ICMI Symposium, Strasbourg, 1985).Cambridge University Press, 1986.

P3) Johnson, D.C. and Lovis, F. (eds.), Informaticsand the Teaching of Mathematics. (Proceed-ings of the IFIP TC 3/WG 3.1 Working Con-ference, Sofia, 1987). North-Holland, 1987.

IllCJ


P4) Banchoff, T.F. et al. (eds.) EducationalComputing in Mathematics. (Proceedings ofECM/87, Rome, 19871. North-Holland, 1988.

P5) Demana, F., Waits, B.K. and Harvey, J. (eds.)Proceedings of the Annual Conference onTechnology in Collegiate Mathematics. (1st:1988; 2nd: 1989; 3rd: 1990). Addison-Wesley,1990, 1991.

P6) Cooney, T.J. and Hirsch, C.R. (eds.) Teachingand Learning Mathematics in the 1990s. (1990Yearbook). National Council of Teachers ofMathematics, 1990.

F7) Dubinsky, E. and Fraser, R. (eds.) Computersand the Teaching of Mathematics: A WorldView. (Selected papers from ICME-6, Bu-dapest, 1988). Shell Centre for MathematicalEducation, University of Nottingham, 1990.

P8) The Mathematical Association of America hasissued three volumes in the MAA Notes seriesrelated to this field and is preparing a fourthone: a) Smith, D.A. et al. (eds.), Comput-ers and Mathematics: The Use of Comput-ers in Undergraduate Instruction. MAA NotesNumber 9, 1988. b) Tucker, T.W. (ed.), Prim-ing the Calculus Pump: Innovations and Re-sources. MAA Notes Number 17, 1990. c)Leinbach. L.C. et al. (eds.), The LaboratoryApproach to Teaching Calculus. MAA NotesNumber 20, 1991. d) Computer Algebra Sys-tems in Undergraduate Mathematics Educa-tion. To appear.

P9) The Notices of the American Mathematical So-ciety feature a regular column under the titleComputers and Mathematics (past editor: .1.

Barwise; current editor: K. Devlin).

b) Author Bibliography

In addition to the references mentioned in thetext, the following list contains a selection of someuseful papers or books.

Adickes, M.D., Rucker, R.11., Anderson, M.R. andMoor, W.C. [1991]: Structuring tutorials us-ing Mathematica: Educational theory andpractice, Mathematica J., 1 (3), 86-91.

Akritas, A.G. [1989]: Elements of Computer Alge-bra, New York: John Wiley.

Artigue, M., Gautheron, V. and Sentenac, P. (1988]:Qualitative study of differential equations: Re-sults of some experiments with microcomput-ers in Reference P4 above, 135-143.

Aspetsberger, K. and Kutzler, B. [1988]: Symboliccomputation A new chance for educationin F. Lovis and E.D. Tagg (eds.) Computersin Education, 331-336, Amsterdam: North -Holland.

Aspetsberger, K. and Kutzler, B. [1989]: Using acomputer algebra system at an Austrian highschool in J.H. Collins et al. (eds.) Proceed-ings of the Sixth International Conference onTechnology and Education, CEP ConsultantsLtd., vol. 2, 476-479.

Auer, J.W. [1991]: Maple Solutions Manual forLinear Algebra with Applications, EnglewoodCliffs, NJ: Prentice-Hall.

Ayers, T., Davis, G., Dubinsky, E. and Lewin, P.[1988]: Computer experiences in learning com-position of functions, J. for Res. in Math. Ed.,19, 246-259.

Beckmann, C.E. [1991]: Appropriate exam ques-tions for a technology-enhanced Calculus Icourse in Reference P5 above (1989 Confer-ence), 118-121.

Beilby, M., Bowman, A. and Bishop, P. [1991]:Maths Si Slats Guide to Software for Teach-ing (2nd edition), CT1 Centre for Mathemat-ics and Statistics, University of Birmingham,UK.

Bjork, L.-E. [1987]: Mathematics and the new toolsin Reference P3 above, 109-115.

Bloom, L.M., Comber, G.A. and Cross, J.M. [1986]:Use of the microcomputer to teach the trans-formational approach to graphing functions,Int. J. of Math. Ed., 17, 115-123.

Brown, D., Porta, H. and Uhl, J.J. [1990]: CalculusMathematica: Courseware for the Nineties,

Mathematica J. 1 (1), 43-50.

Brown, D., Porta, H. and Uhl, J.J. [199-]: Calcu-lus & Mathematica, Reading, MA: Addison-Wesley.

Buchberger, B. [1990]: Should students learn inte-gration rules?, SIGSAM Bull., 24, 10-17.

Capuzzo Dolcetta, I., Emmer, M., Falcone, M.and Finzi Vita, S. [1988]: The laboratory ofmathematics: Computers as an instrument forteaching calculus in Reference P4 above, 175-186.

Cromer, T. [1988]: Linear algebra using muMATII,Collegiate Microcomputer, 6, 261-268.

1f ai

Symbolic Mathematical Systems and Mathematics Education

Davenport, J.H., Siret, Y. and Tournier, E. [1988]:Computer Algebra: Systems and Algorithmsfor Algebraic Computation, Academic Press.

Dechamps, M. [1988]: A European cooperation onthe use of computers in mathematics, in Ref-erence P4 above, 19.-209.

Demana, F. and Waits, B.K. [1990]: Enhancingmathematics teaching and learning throughtechnology in Reference P6 above, 212-222.

Devitt, J.S. [1990]: Adapting the Maple computeralgebra system to the mathematics curricu/amin Reference P5 above (1988 Conference), 12-27.

Dick, T. and Musser, G.L. [1990]: Symbolic/graphi-cal calculators and their impact on secondarylevel mathematics in Reference P7 above, 129-132.

Dubisch, R.J. [1990]: The tool kit: A notebook sub-class, Mathematica J., 1 (2), 55-64.

Ellis, W., Jr. and Lodi, E. [1989]: Maple forthe Calculus Student, Pacific Grove, CA:Brooks/Cole.

Fey, J.T. [1989]: Technology and mathematics edu-cation: A survey of recent developments andimportant problems, Educ. Studies in Math.,20, 237-272.

Flanders, H. [1988]: Teaching calculus as a labora-tory course in Reference P4 above, 43-48.

Foster, K.R. and Bau, H.H. [1989]: Symbolic manip-ulation programs for the personal computer,Science, 243, 679-684.

Geddes, K.O., Marshman, B.J., McGee, I.J., Ponzo,P.J. and Char, B.W. [1988]: Maple CalculusWorkbook, University of Waterloo, Canada.

Gray, T.W. and Glynn, J. [1991]: Exploring Math-ematics with Mathematica, Reading, MA:Addison-Wesley.

Heid, M.K. [1988]: Resequencing skills and conceptsin applied calculus using the computer as atool, J. for Res. in Math. Ed., 19, 3-25.

Heid, M.K., Sheets, C. and Matras, M.A. [1990]:Computer-enhanced Algebra: New roles andchallenges for teachers and students in Ref-erence P6 above, 194-204.

Hodgson, B.R. [1987]: Symbolic and numerical com-putation: The computer as a tool in mathe-matics in Reference P3 above, 55-60.

Hodgson, B.R. [1990]: Symbolic manipulation sys-tems and the teaching of mathematics in Ref-erence P7 above, 59-61.

103

Hosack, J. [1988]: Computer algebra systems in Ref-erence P8a above, 35-42.

Hubbard, J.H. and West, B.H. [1991]: DifferentialEquations: A Dynamical Systems Approach,Part I: Ordinary Differential Equations, NewYork: Springer-Verlag.

Hubbard, J.H. and West, B.H. [1991]: MacMath:A Dynamical Systems Software Package, NewYork: Springer-Verlag.

Kocak, H. [1989]:Differential and Difference Equa-tions through Computer Experiments (2nd edi-tion), New York: Springer-Verlag.

Kozma, R.B. [1985]: A grounded theory of instruc-tional innovation in higher education, J. ofHigher Education, 300-319.

Lance, R.H., Rand, R.H. and Moon, F.C. [1986]:Teaching engineering analysis using symbolicalgebra and calculus, Eng. Educ., 76, 97-101.

Mascarello, M. and Winkelmann, B. [1992]: Calcu-lus teaching and the computer. On the inter-play of discrete numerical methods and calcu-lus in the education of users of mathematics,(in this volume).

Mathews, J.H. [1989]: Computer symbolic algebraapplied to the convergence testing of infiniteseries, Collegiate Microcomputer, 7, 171 -176.

Mathews, J.H. [1990]: Teaching Riemann sums us-ing computer symbolic algebra systems, Col-lege Math. J., 21, 51-55.

Willer, H. [1990]: Elementary analysis with micro-computers in Reference P7 above, 179-184.

Muller, E.R. [1991]: Maple laboratory in a servicecalculus course in Reference P8c above, 111-117.

Muller, E.R. [1992]: Symbolic mathematics andstatistics software use in calculus and statis-tics education, Zentralblatt Didaktik Math. (toappear).

Neuwirth, E. [1987]: The impact of computer alge-bra on the teaching of mathematics in Refer-ence P3 above, 49-53.

Nievergelt, Y. [1987]: The chip with the college ed-ucation: the HP-28C, Amer. Math. Monthly,94, 895-902.

Orzech, M. [1988]: Using computers in teaching lin-ear algebra in Reference P8a above, 63-67.

Osborne, A. [1990]: Testing, teaching and technol-ogy in Reference P5 above (1988 Conference),60-67.


Page, W. [1990]: Computer algebra systems: Issuesand inquiries, Computers Math. Applic., 19,51-69.

Ralston, A. [1990]: A framework for the schoolmathematics curriculum in 2000 in ReferenceP7 above, 157-163.

Shumway, R. [1990]: Supercalculators and the cur-riculum, For the Learning of Math., 10 (2),2-9.

Small, D. and Hosack, J. [1991]: Explorations inCalculus with Computer Algebra Systems, NewYork: McGraw-Hill.

Small, D., Hosack, J. and Lane, K. [1986]: Com-puter algebra systems in undergraduate In-struction, Coll. Math. J., 17, 423-433.

Steen, L.A. [1988]: The science of patterns, Science,240, 611-616.

Tall, D. [1985]: Visualizing calculus concepts usinga computer in Reference P1 above, 291-295.

Tall, D. [1991]: Recent developments in the use ofcomputers to visualize and symbolize calculusconcepts in Reference P8c above, 15-25.

Wagon, S. [1991]: Mathematica in Action, San Fran-cisco: Freeman.

White, J.E. [1988]: Teaching with CAL: A. math-ematics teaching and learning environment,College Math. J., 19, 424-443.

White, J.E. [1989]: Mathematics teaching andlearning environments come of age: Some newsolutions to some old problems, Collegiate Mi-crocomputer, 7, 203-224.

Young, G. [1986]: Epilogue in R.E. Ewing, K.I.Gross and C.F. Martin (eds.), The Mergingof Disciplines: New Directions in Pure, Ap-plied and Computational Mathematics, 213-214, New York: Springer-Verlag.

Zorn, P. [1987]: Computing in undergraduate math-ematics, Notices Amer. Math. Soc., 34, 917-923.

Zorn, P. [1990]: Algebraic, graphical and numericalcomputing in elementary calculus: Report ofa project at St. Olaf College in Reference P5above (1988 Conference), 92-95.

APPENDIX 1This appendix provides a list of some Symbolic

Mathematical Systems software reviews. It is im-portant to realize that it is extremely difficult toevaluate and benchmark this software. Furthermore

Mathematics and Its Teaching

many of the evaluations do not take into accountpossible classroom use and the use by neophytes.

a) The Notices of the American Mathematical Soci-ety (see reference P9 above) have recently includedan individual review of most Symbolic Mathemati-cal Systems:

Vol. 35, 1988

The HP-28S brings computations and theory backtogether in the classroom, Y. Nievergelt, 799-804.

Supercalculators on the PC., B. Simon and R.M.Wilson, 978-1001.

Mathematica A review, E.A. Herman, 1334-1344. (Also: Other comments on Mathemat-ica, 1344-1349.)

Vol. 36, 1989

MicroCalc 4.0, G. Gripenberg, 680.

The menu with the college education (A review ofDerive), E.L. Grinberg, 838-842.

Milo: The math processor for the Macintosh, R.F.Smith, 987-991.

Milo, Sha Xin Wei, 991-995.

PowerMath II, Y. Nagel, 1204-1206.

More on PowerMath II, P. Miles, 1206-1207.

Vol. 37, 1990

Review of PC-Macsyma, Y. Nagel, 11-14.

Review of True Basic, Inc. Calculus 3.0, J.R.Moschovakis, Y. Matsubara, G.B. White, 129-131.

Derive as a precalculus assistant, P. Miles, 275-276.

The right stuff, K. Devlin, 417-425.

Almost no stuff in, wrong stuff out, J.D. Child, 425-426.

Four computer mathematical environments, B. Si-mon, 861-868.

Vol. 38, 1991

Crimes and misdemeanors in the computer algebratrade, D.R. Stoutemyer, 778-785.

Periodic knots and Maple, C. Livingston, 785-788.

b) Other reviews are:

Symbolic manipulation programs for the personalcomputer, K.R Foster and H." Bau, Science,243, 679-684 (1989).

Derive: .4 mathematical assistant, E.A. Herman,Amer. Math. Monthly, 96, 948-958 (1989).


Mathematica: A system for doing mathematics bycomputer, L.S. Kroll, Amer. Math. Monthly,96, 855-861 (1989).

Math without tears, C. Seiter, Mac World 8 (1), 159-165 (1990).

Theorist, J. Rizzo, Mac User, 6 (6), 57-59 (1990).

Mathematica: A system for doing mathematics bycomputer, A. Hoenig, Math. Intelligencer, 12(2), 69-74 (1990).

Theorist, F. Wattenberg, Amer. Math. Monthly,98, 455-460 (i391).

Review of Maple in the teaching of calculus, E.R.Muller, College Math. J. 'to appear).

c) Reviews of software and iments on experi-ments on their use in teaching can also be foundin specialized newsletters. Some examples are:

Computer Algebra Systems in Education Newsletterpublished by the Department of Mathematics,Colby College, Waterville, ME 04901, USA.

Maths & Stats published by the CTI Centrefor Mathematics and Statistics (Computer inTeaching Initiative), Faculty of Education,University of Birmingham, Birmingham, B152TT, UK.

Computer-Algebra Rundbrief published by Fach-gruppe 2.2.1 Computer-Algebra der GI, c/oDr. F. Schwarz, GMD, Institut F1, Postfach1240, 5205 St. Augustin, Germany.

APPENDIX 2This appendix provides a limited number of ex-

arnples to illustrate some of the capabilities of Sym-bolic Mathematical Systems (the system used hereis Maple but this particular choice is not crucial).These systems are so powerful that it is impossibleto provide a complete overview of their capabilitiesin a brief text.

#The system can be used to do some elementary#number theory. For instance the command ifactor

returns the prime factorization of an integer.ifactor(123456780);

(2)2 (3)2 (5) (47) (14593)

#With such a tool available, it might be tempting to#venture into some calculations that are not#trivial to do either by hand or in a standard#computer environment. For example the prime#factors of the Mersenne number 267 1 were#given in 1903 by F. Cole. It reportedly took him#"three years of Sundays" to complete the

#calculations. What can Maple do with that#number?

ifactor(2-67-1);

(761838257287)(193707721)

#Done in just a fraction of a minute!! (But#needless to say it is very easy to give as an#input a number that would take "three years of#Sundays" for the system to do.)#Roots of equations can be found directly.

y:= x-3-4*x^2-7*x+10;

y := x3 4 x2 7 x + 10solve(y=0,x);

1, 2,5#Even with symbolic coefficients.

z:= a*x-2-2*b*xl-c;

z := a x2 2 b x + c

solve(z=0,x);

1/22 b + 2 (b2 c)1/2

a

1/22 b 2 (b2 a c)1i2

a#The example in Section 1 demonstrated that#these systems are able to compute limits and to#dzfferentiate. They can also find Taylor's Series,#sum finite and infinite series and integrate#and solve differential equations.

taylor(ln(x-2),x=1,4);

2(x 1) (x 1)2 + 2/3 (x 1)3+ 0((x 1)4)

sum(j^2, j=1..m);

1/3 (m + 1)3 1/2 (in + 1)2 + 1/6 In + 1/6

int(x^3*sqrt(x"2-9),x);

3/5 -(xz 9)1/2x2_54/5 (x2 9)1/2

(x2 9)1/2x4

deq:= diff(x(t),t) *t-3+x=0;

ddeq := (dt x(t)) t3 + x = 0

dsolve(deq,x(t));

x(t) = exp(1/2 )C

#The next few examples are taken from linear#algebra, namely, the solution of linear#equations and some properties of matrices and#vectors.

eqns:= a*x+b*y=e,c*x-I-d*y=f; vars:= x,y;


eqns:= x+b y=e,c x+d y= f}vars:= y}

solve(eqns,vars);

bfed afce{x=ad cb' adc 13}

A:= arraya[1,a,a-2],[1,b,V2],[1,c,c-2]D;

A := array(1..3, 1..3,

[1, a, az]

[1, b, b2]

[1, c, c2])

det(A);

b e2 b2c a c2 + a2c + a b2 alb

factor(");

( c + b)(a c)(a b)

#(The symbol " refers to the previously#displayed expression.)

al:= array([xl,yl,z]l); a2:= array([x2,y2,z2]);a3:= array([x3,y3,z3]);

al := array(1..3,

[xl.,y1,z1.])

a2 := array(1..3,

Ex2, y2, z2))

a3 := array(1..3,

[x3, y3, z3])

vol:= abs(dotprod(al,crossprod(a2,a3)));

vol :=abs(xl (y2 z3 z2 y3)

+ yl (z2 x3 x2 z3)

+ zl (x2 y3 y2 x3))

a:= array([[13,5],[5,2]]);

a := array(1..2, 1..2,

[13,5]

[5, 21)

c:= eigenvals(a);

c := 15/2 + 1/2 2211/2, 15/2 1/2 2211/2

#The decimal approximation to these two#eigenvalues gives

evalf(c[1]); evalf(c[2]);

14.93303438

.066965625

APPENDIX 3

There is as yet no single source which canprovide a comprehensive international listing ofprojects in the area of Symbolic Mathematical Sys-tems in Mathematics Education. Therefore, the fol-lowing list cannot be regarded as comprehensive:1: The Swedish ADM project (Analysis of the role ofthe Computer in Mathematics Teaching); see Bjork(1987).2: The Research Institute for Symbolic Computa-tion at the Johannes Kepler University, Linz, Aus-tria.3: The Computers in Teaching Initiative Centre forMathematics and Statistics (Development of classwork sheets to be used with Derive), see the Maths

Stats newsletter published by the CTI Centre,University of Birmingham, UK.4: A European Cooperation on the use of Computersin Mathematics; see Dechamps (1988).5: The National Science Foundation (U.S.A.) isfunding a number of different university projectsspecifically directed at integrating Symbolic Math-ematical Systems into the calculus curriculum. Thefollowing is a selection providing a one line state-ment together with the university and the principalinvestigator.

Developing a user friendly interface to Mapl,and incorporating use of system into teaching cal-culus, Rollins College, Winter Park; FL; DouglasChild.

Developing new calculus curriculum using Mapleon a VAX, Rensselear Polytechnic Institute, Troy,NY; William Boyce.

Developing a computerized tutor and computa-tional aid based on Maple, University of Rhode Is-land, Kingston, RI; Edmund Lamagna.

Developing an electronically delivered course us-ing the Notebooks feature of Mathematica, Univer-sity of Illinois, Urbana, IL; Jerry Uhl.

Developing a new calculus course emphasing ap-plications and using Mathematica, University ofIowa, Iowa City, IA; Keith Stroyan.

Developing a laboratot based calculus course us-ing Mathematica, Iowa State University, Ames, IA;Elgin Johnston.

Developing a new calculus course for liberalarts colleges using Mathematica, Nazareth College,Rochester, NY; Ronald Jorgensen.

Developing calculus as a laboratory course us-ing MathCad and Derive, Duke University, Durham,NC; David Smith.

Emphasizing computer graphics using Maple andemphasizing concepts via programming' in ISETL,


Purdue University, West Lafayette, IN; Ed Dubin-sky.

Porting the laboratory calculus developed at Dukeover to Mathematica, Bowdoin College, Brunswick,ME; William Barker.

Collecting, testing, and desktop publishing thebest materials being developed using Mathematica,University of Michigan at Dearborn, Dearborn, MI;David James.

More detailed informations about projects in theU.S.A. integrating Symbolic Mathematical Systemsin the calculus curriculum can be found in the re-ports contained in reference P8b above: Tucker,T.W. (ed.), Priming the Calculus Pump: Innova-tions and Resources. Mathematical Association ofAmerica (MAA Notes Number 17), 1990.

CALCULUS TEACHING AND THE COMPUTER. ON THE INTERPLAY OF DISCRETENUMERICAL METHODS AND CALCULUS IN THE EDUCATION OF USERS

OF MATHEMATICS

Maria MascarelloPolitecnico di Torino, 1-10129 Torino, Italia

Bernard WinkelmannInstitut fiir Didaktik der Mathematik, D-4800 Bielefeld, Deutschland

1. NEW POSSIBILITIESThe computer is a mighty mathematical tool,

not only for mathematical research, but even morein the process of applying mathematics and in theprocess of teaching and learning mathematics. Inthe following, we shall concentrate mainly on thenew possibilities which the computer presents in therealm of calculus for users and future users of math-ematics. By a user we mean somebody who is in-terested in mathematics rlerely (or mainly) throughthe use of mathematical models (in particular calcu-lus models) to solve (extra-mathematical) problems.Future users of mathematics are, for example, en-gineering students, but even those learning calculusin schools as part of a general education may beincluded under this rubric.

1.1 New possibilities for the userWe describe first the changes in the mathemati-

cal knowledge and habits of the user of mathematicsinduced by the availability of sophisticated math-ematical software to all who have to rely heavilyon mathematical problem-solving such as engineers,natural scientists, etc. During the past decade wehave seen the proliferation of mathematical softwaresystems for personal computers which have becomemore powerful and/or more user friendlyl. By rais-ing the standards in these two domains, such sys-tems are now in the hands of a rapidly growing num-ber of users, even if until now (1991) they have notyet reached the majority of the teachers of mathe-matics, at least at the secondary level. But, if thetrend continues, not only professional users of math-ematics, but also most students and teachers willsoon have regular access to such systems. However,

1 A typical example might be the realm of com-puter algebra systems: In the progress from MU-MATH to Derive there has been a big gain in userfriendliness, allowing the use of the system even byusers reluctant to program, but - at the same time -with a certain loss in functionality, e.g. in the solv-ing of differential equations. On the other hand, theprogress from MUMATH to Mathetnatica is mostlyin power, much less in user friendliness. See also thechapter by Hodgson and Muller in this book.

108

the integration into regular classroom teaching willstill be a problem.

The classic situation of the user of mathemat-ics could be described - in a somewhat oversimpli-fied manner - as a huge amount of passive mathe-matical knowledge contained in monographs, hand-books, recipes. Traditionally, this knowledge couldonly be used by being activated through the activemathematical knowledge of the user himsel or bydirect cooperation between the user and a mathe-matically more knowledgeable person. In contrastto this, the mathematical knowledge contained inmathematical software can have a far more activecharacter, e.g. in giving advice and help interac-tively, offering possibilities for exploratory experi-ments or answering questions, acting like a mathe-matical expert system. Even more common numer-ical software, which exists in the form of sophisti-cated procedures, is far more active than the recipesof the old-fashioned handbooks, since in many casesthese procedures are in fact polyalgorithms: Theydecide with considerable expertise which particu-lar algorithm should be invoked, depending oh thecircumstances2. So the demand for mathematicalknowledge on the part of the user has changed. Theemphasis has shifted from detailed knowledge of theadvantages and disadvantages of specific numericalmethods and of the algorithms themselves to somemeta-knowledge of the possibilities of numerical al-gorithms in general and their interaction with theconcrete application situation.

As an example let us look at the process of thesolution of ordinary differential equations3. This isindeed an example of great importance since suchequations appear in many applications and are atthe heart of applicable elementary calculus. So ifit is possible to master them at a more elementarylevel than hitherto was possible, this could even beregarded as the most appropriate goal for the teach-ing of elementary calculus at schools and colleges. Inthe education of engineers at technical universities

2 cf. Rice [1983], e.g. p.291f.3 cf. Winkelmann [1984] and the chapter by Tall

and West in this book.

or similar institutions, where differential equationshave always been part of the calculus sequence, evenbeginning calculus could concentrate more on appli-cations and so give the student a more realistic and,one hopes, a more motivating start.

In the pre-computer age an engineer or scientistwho had to handle differential equations was sup-posed to have detailed knowledge of diverse meth-ods for the analytic solution of various elementarytypes, to be able to master complicated analytic-algebraic formulas and to carry out lengthy error-free symbolic and numerical calculations. Now he orshe can use software which has this knowledge andability built in, since it can solve more elementarydifferential equations than a non-specialist mathe-matician call do4. i3ut in building up the modelthe user still has to understand fully the meaningand significance of the diverse quantities (variables)and of their derivatives and to be able to relatethese to each other in order to set up the differentialequation. And to give the details to the computerprogram, a thorough intuitive understanding of themathematical meaning of the identifiers which ap-pear in the modeling equations is needed, be it asvariables, parameters, initial values, names for (yetunknown) functions (dependent variables) and soon. If an analytic solution exists, the program willnormally present it as a somewhat confusing lengthyexpression which must be qualitatively interpretedto be understood, namely through looking for sim-pler special cases, for settings of specific parame-ters or initial values, for asymptotic patterns of be-haviour, etc. This process is guided by the intendedinterpretation of the solution in the context of theapplication model. If no analytic solution exists, theuser may give his equation to some ready-made nu-merical software. In this case he needs some knowl-edge to make reasonable explorative choices of thevalues of parameters and initial values; there shouldbe some experience with numerical phenomena (pit-falls of computations) and the ability to interpretthe numerical and graphical output of the computerand to use this interpretation interactively for newchoices of starting points for the next calculation.

In total, there can be observed a specific shiftin the spectrum of abilities, from precise algorith-mic abilities to more complex interpretations, so tospeak from calculation to meaning, which in a cer-tain sense is a reversal of the historical evolution. Inthis process the mathematics to be mastered tendsto become intellectually more challenging, but tech-nically simpler.

4 cf. Watanabe [1984].

Calculus Teaching and the Computer 109

What does this mean for the mathematical ed-ucation of the future user? Of course, there is nodirect way from the mathematical activities of theuser to the teaching process; the goal must not beconfused with the means. Understanding and abili-ties for complex interpretations can only be buk upby personal involvement of the student; she has todo full (but simpler) examples in all the main stepsherself, be it by hand-calculating, by using interac-tive symbolic systems or calculators or by program-ming in some suitable programming language. Thisseems necessary in order to get an awareness of themathematical situations, even if such activities areno longer part of the final application process. Andeven if today's sophisticated mathematical softwareneed not and cannot generally be fully understoodby the normal user, there must not be totally blackboxes; a principal understanding of simple cases, ofmain ideas or of fundamental restrictions can begained and seems necessary for proper use of thenow 'grey' boxes'.

On the other hand it is quite clear that extensivedrill in formal calculations, in fluent structured pro-gramming or even in the handling of some softwarepackage cannot be justified in view of the changedqualifications needed by the user.

1.2 New possibilities in the teaching-learning process

In the field of teaching methods the computer,if it has been loaded with the appropriate software,will function as a simplifying aid, almost as a su-per hand-held calculator which permits the pupil toovercome computational obstacles in the treatmentof more complex problems and to handle more re-alistic applications, e.g. in dealing with larger ma-trices, in the numerical solution of differential equa-tions, or in the symbolic treatment of more com-plicated formulas; this will serve to widen the po-tential scope of mathematics education in terms ofcontent. On the other hand, a computer equippedwith appropriate languages and environments canbecome an instrument for solving problems in the

5 Buchberger [1990] gives an argument for a muchmore strict procedure: first, the algorithms of thesoftware have to be completely understood by thestudent; afterwards he may use the software for allcalculations. But Buchberger has the algorithmsof Computer Algebra and mathematical majors orcomputer science majors in mind; his arguments donot extend to numerical software and typical futureusers of mathematics. See also the somewhat moredetailed discussion in the chapter by Fraser, Klingenand Winkelmami in this book.


hands of the student (interactive calculating or pro-gramming); in this case, the student tends to under-stand techniques more at the cognitiVe level, and nolonger mainly at the level of skill. Beyond that, thecomputer, with its possibilities for illustra' 'on andsymbolization, will provide opportunities for morecomprehensive and rapid mathematical experiences.

This presents problems and tasks as well as op-portunities for educators mainly on two levels. On amore technical level, there is the necessity to providemore suitable software with stir -; mathematicalfunctionality, educationally soul.0 help functions,user interfaces for the inexperienced user, and ac-companying explanations, hints and worked-out ex-amples for teachers. On a more fundamental level,the problem is to achieve a balance between thequantitative and qualitative relation of new and oldgoals and methods as well as to set up the righttrends for future developments.

The computer creates new opportunities for in-struction in analysis, e.g.

numerical and graphical illustrations6,more complex and more realistic applications.a language in which to describe traditional cal-

culus,CAL (computer-aided learning) in its various

forms.Some traditional motivations for treating con-

ceptually exacting analysis in school can, however,no longer be maintained. For instance:

calculations such as finding extreme values orareas can be easily done without analysis,

practical applications in physics or technologywhich used to rely on analytic methods now are rou-tinely done numerically on a computer by discretecalculations.

This results in a crisis: The legitimacy of tra-ditional analysis in school is challenged; educatorswill have to make clear to the general public, andthe teacher will have to explain to his pupils howand why the treatment of continuous analysis stillmakes sense nowadays.

In Section 3 we shall report on some experimentsconcerning the use of informatie tools in teachingbasic mathematical courses at the Politecnico (Poly-technics) of Torino, Faculty of Engineering Sciences.We emphasize that. the choice here has been tokeep the teaching of calculus reasonably traditional,while at the same time giving some basic notions of

6 See Tall [1986], the article by D. Tall in John-son /Louis [1987] and the chapter by Tall and Westin this book.

informatics in the main course of lectures and de-voting special laboratory sections to "calculus at thecomputer".

2. THE DISCRETE - CONTINUOUSINTERPLAY

2.1 General considerationsAlthough the role of applications of analysis has

been changed both by the growing number of disci-plines using mathematical models and by new meth-ods, particularly the extensive use of computers,an understanding of fundamental concepts in whichmathematizations take place remains indispensable.Examples are:

variable quantity, changefunctional dependencylocal rate of changeaverage valueaccumulation.

We shall refrain from discussing here how far tra-ditional mathematics education was able to attainthe goal of teaching these.

Now it is evident that these ceni,ral conceptsof mathematical applications can be implementedboth by discrete and by continuous conceptual-izations. Corresponding to such continuous con-cepts as function, differential equation, derivative,weighted integral, and integral, are the correspond-ing conceptualizations in discrete analysis, namely:sequence and time series, difference equation, dif-ference, arithmetical mean value, and sum. Thesediscrete concepts are often technically and almostalways intellectually much simpler than their con-tinuous counterparts.

In the following we will give some justifications,which are, in our opinion, crucial in answering thequestion now raised inevitably: "Why use the con-cepts of continuous analysis in teaching at all?" In(a) and (b), we state the problem, conceived as anepistemological question regarding the role of anal-ysis in applications and model building, and in (c)and (d) we introduce the argument which solves thedilemma.

(a) Insufficiency of continuous analysis for ob-taining concrete numerical results. Let us recallsome of the facts: Most integrations cannot be ex-ecuted analytically, but only numerically; this isall the more true for solving differential equations.Even tasks as simple as determining the extremesof a familiar function like x sin x require numericalmethods. School mathematics has hitherto confineditself in a rather unnatural way to problems involv-ing classes of functions which were solvable by ana-lytic methods. It has paid dearly for this with heavy

losses in orientation to problems of reality, contentand relevance. This is particularly true for classi-cal university courses in, for example, elementarydifferential equations7.

(b) Most concrete models using analysis have adiscrete basis. This is first evident in the social sci-ences or in population biology, where the quantitiesto be modelled are numbers of items or individuals,or monetary units, which cannot be subdivided atwill. But in physics, too, for instance, most modelsstart discretely: even disregarding the fact that theuniverse is finite in principle and structured in parti-cles, and that there are quanta (i.e. smallest units),it is a fact for quantities which are usually conceivedof as being continuous, and mathematized accord-ingly, that concrete models based, say, on results ofmeasurements, will start as discrete models simplybecaur continuous functions cannot be obtained asthe results of a series of measurements which yieldonly discrete sequences or time series. (This doesnot hold, of course, for modeling based on theoreti-cal approaches.)

(c) The continuous character of models usinganalysis is the result of the intended domain of valid-ity.8 Most mathematical models have a specific in-tended domain of validity, especially a certain scalelevel, even if this is not explicitly stated. A Eu-clidean line serves as a model for edges of solid bod-ies, e.g. of a shelf, only at a macroscopic scale. Ifwe look at such an edge through an electron mi-croscope, the edge doesn't look straight any more,and on the atomic scale, it loses its one-dimensionalcharacter too. Therefore, although the edge is wellmodelled by a line, we should not draw conclusionsfrom this model outside its intended domain of va-lidity. In an analogue sense, calculus models of dis-crete real phenomena typically are only intended forphenomena at scales where the discreteness doesn't

7 This is properly described in Artigue [1989].8 We have taken this argument from Rice [1988]

who writes under the sub-heading "Verifiable Hy-potheses: Does Mathematics Model Reality?": "...we can argue that the real world is inherently dis-continous everywhere, its 'microscopic' structure iseither discrete or random or both. In any case,the mathematical definition of continuity, deriva-tion, etc., do not apply because, at some fine scaleof examination, the functions are undefined or dis-crete or something intractable. The implication ofthis view is that the concepts of smoothness and be-haviors of functions are related to a scale and thatan adequate mathematical model must take this intoaccount." (p. 37).


enter. Calculus concepts such as limit, derivative,integral are not to be interpreted in the strict math-ematical sense, but they express certain invariances:The corresponding discrete concepts do not dependon the step size, provided it is sufficiently small (butyet in the intended scaling domain). This consider-ation gives sense to the use of calculus models insuch typically discrete domains as population dy-namics or economics. But of course, there are alsomodels, which do not show such invariances in theirintended scaling domain. These should not be mod-elled by calculus. Such situations arise in consid-erations about fractal phenomena: the length of acoastline (as a quantity of integral type) is typicallynot invariant with the measuring unit, but of coursethe assumed statistical self-similarity also holds onlyin a sensible scale, which certainly does not extendto the microscopic level.

(d) The transition from models to concrete nu-merical results cannot be accomplished in generalwithout continuous analysis. This is true, for onething, because of the rounding errors which in-evitably occur in numerical computing, and haveto be controlled by a more abstract model whichdoes not include the discretization error. A second,deeper reason follows from a closer look at the dis-crete aspects mentioned in points (a) and (b): It isthe case that the step widths used in (a) and (b) arebasically independent of each other, as is to be ex-pected from the argument in (c). The density of thevalues measured in the measuring process is gener-ally determined by practical considerations such asinformation content and "cost". One of the mostfundamental hypotheses for determining the stepwidth is that a diminution of the step width mayyield more exact results, but basically not resultswhich differ in principle. The phenomena which areto be observed and/or described are considered to beinvariant with respect to the step width used in theobservations provided it is sufficiently small. Thisfits in with the assumption that the correspondinglimits exist. It is only on the basis of this assump-tion that the measuring process can be carried outin a discrete way chosen by practical considerations.In this case, however, the phenomena concerned arebasically invariant with respect to the step width,and are thus best described in mathematical modelswhich do not explicitly contain a step width. Thefact that the step width with which the measureddata were obtained is only of marginal importancefor the model explains why step widths used, say,to solve numerically the corresponding differentialequations, will generally be completely independentof the step width used in measurement. Both are

i 6


independently determined by practical criteria suchas cost and the precision required.

This fundamental consideration has been refor-mulated here for the special case where the results ofdiscrete measurement are used as a starting point.It is true, in an analogous way, pointed out in (c),for all the other cases in which mathematizing andmodeling is done by analysis.

This behaviour is of course not valid for all math-ematical models in the sciences or other domains.But it is in a sense typical for calculus models: Ifthis behaviour is not observed in a specific situa-tion, then normally we should really use discretemodels, and if we - for technical reasons - neverthe-less use some calculus models, we should be awareof the improper use and of possible difficulties in in-terpreting results. This may happen for example ifwe try to consider "fractal" phenomena in nature,such as natural borders (of islands, leaves of trees,etc.). Here, for example, the application of formulasfor the length of a curve does not make much sense.

2.2 The context of dynamical systemsDynamical systems (systems of time-indepen-

dent explicit first order ordinary differential equa-tions) appear as rather natural mathematical mod-els for many situations in a variety of disciplinessuch as the physical, biological or economic sciences.Here typically we have to distinguish between situa-tions where a natural step width exists whose valueinfluences the phenomena, and situations in whichphis is not the case. In both cases, modeling with(discrete) difference equations is possible and ad-equate; but whereas in the former case, the stepwidth of the difference equation has to be equal tothat of the underlying situation, in the latter it maybe chosen as a free parameter which suggests thatthe use of differential equations might be more nat-ural.

As an example, consider the logistic growth ofa (biological) population. if the generations of thepopulation are distinct, as with certain bugs, theremay be observed oscillations and fluctuations of thepopulation, which are easily modelled and explainedin the context of a difference equation, but woulddisappear in the transition to the corresponding dif-ferential equation (if it were not explicitly mod-elled by including a time lag which would inducesimilar fluctuations but would exclude the result-ing equation from what is normally considered adifferential equation in mathematics). But if gen-erations are not distinct and population oscillationsare slow compared to normal reproduction times,modeling with (logistic) differential equations seemsadequate, even if there were only discrete points in

time where new offspring could be noticed.2.3 Symbolical, numerical and qualitative

solutionsLinear differential equations, and some others

which may be transformed to those, can be solvedexplicitly by closed formulas. From such formu-las one can - at least in principle - answer almostany question about the underlying dynamical sys-tem: asymptotic behaviour, stability and depen-dence on initial values and parameters. But thisis the exception, not the rule, since most dynamicalsystems arising from model building are essentiallj,nonlinear9 and do not admit any closed-form solu-tions. Numerical solution algorithms on the otherhand are generally not sensitive to nonlinearity, butthey share a double experimental character: in mostcases, the degree to which they approximate thetrue solution can only he estimated, not provenl°;and - more seriously - a numerical solution has astrict local empirical character. It does not by it-self allow any conclusions about other initial val-ues or parameters, which is catastrophic in applica-tions where such values are only estimated. So theynecessarily need to be complemented by theoretical,usually qualitative considerations about possible be-haviours of this or a slightly modified dynamical sys-tem, be it continous or discrete. So this describesanother complementarity between discrete numeri-cal and theoretical methods.

3. EXPERIMENTS IN USING INFOR-MATIC TOOLS

In this section we report on some experimentsconcerning the use of informatic tools in tet.ch-ing basic mathematical courses at the Politecnicoof Torino (Italy), Faculty of Engineering Sciences.These experiments refer in particular to the coursesMathematical Analysis 1 and Mathematical Anal-ysis 2 given to students of Mechanical Engineeringin the years 1980 to 1983, using pocket computers.This activity was continued in 1984 and 1985, inthe same courses, using such micro computers as theSharp MZ803 and IBM PC. At this second stage, theexperiment was concerned with a restricted numberof students, selected on the basis of a test.

The experiment was sufficiently successful sothat since 1986 all the students of the course ( about300) have been taught in the computer enhancedstyle. At the Politecnico of Torino an introductory

For an interesting account of nonlinear modelbuilding see West [1985].I° An exception is the so-called EEE-methods, seeKaucher/Miranker [1984], but use of these methodsis not yet widespread.

k ii

computer laborato_y is available for students; engi-neering students in the first two years have accessto the lab after the completion of a specific coursewhich prepares them for meaningful utilization ofthe available calculating devices and also suppliesthem with adequate knowledge of a programminglanguage.

Several instructors of the Engineering Facultyhave experimented with the use of the lab as an aidto the basic first two years mathematics courses, andfrom the resulting experience two didactic strate-gies have emerged. One was for the students them-selves to perform the actual writing of the software.The other was to use existing software. It was ob-served that the preparation of software is, even froma mathematical standpoint, an occasion for investi-gation of the topic at hand. However, a certain riskwas noted in the tendency toward interest in thecomputer itself to the detriment of time intendedfor dedication to mathematical reflection.

As far as concerns the use of already availablesoftware packages, the possibilities are many. Wehave readily available software written by colleaguesinstructing in analogous courses, that produced bystudents in previous courses, and, of course, soft-ware offered by the companies producing calculatingdevices.

While we refer to Boieri et al.[1984], to Mascar-elle -Scarafiotti [1987], [1988] and to Mascarello-Scarafiotti-Teppati [1989] for the general aims, thelist of the themes and the results obtained, weshould like to detail here some of the topics and con-tent, and to add some final comments, as a 'proof'of what we asserted in Section 2.

Let us begin by observing that, to carry out theexperiment in a useful way, it has been necessary torely on basic informatic arguments. To this end, inthe main course of lectures, the teacher, after giv-ing some notions of the theory of formal languages,then introduced machine-numbers and algorithmsfor floating-point arithmetic computations. At thesame time, in this first part of the course, someproofs of classical analysis results were presented incomputational form.

One of the most important experiments con-cerned the study of dynamical systems using micro-computers. More specifically, we began in Mathe-matical Analysis 1 with the study of discrete dynam-ical systems, which was introduced after the studyof sequences defined by recurrence formulas. As anatural continuation, in Mathematical Analysis 2we considered continuous dynamical systems, giv-ing a formal expression of the qualitative results.Finally, we returned to the use of microcomputers


to find numerical results; this was done in order tocheck the known results of the theory, and also toconjecture new results concerning open problems.

To be specific, we briefly list the contents ofthe exercise sessions concerning dynamical systems(Mathematical Analysis 2):

Cauchy problem for first order ordinary dif-ferential equations; solutions at the microcomputer,comparing the methods of Euler and Runge-Kutta.

First order systems of ordinary differentialequations, and in particular autonomous systems;visualization of the trajectories in the phase plane.

Second order ordinary differential equations;solutions on the microcomputer of some nonlinearequations of particular significance in applications,such as the pendulum and other equations of math-ematical physics.

A numerical approach and simulation on themicrocomputer of the trajectories for some problemswhich are still open in their qualitative aspects, asfor example the mathematical model of the Lorenzattractor.

Now we present some further details of some ofthe above, which appear to us particularly signifi-cant from the didactic point of view. i) The stu-dent, knowing the classical analytic theory of lin-ear equations with constant coefficients, and havingsome basic notions of the stability theory, is invitedto "solve" the equation i + ki + x = 0 on a mi-crocomputer and to visualize the trajectories in thephase plane (without any direct assistance from theteacher). Figures 1 and 2 show some drawings ofthe kind obtained by a student.

Figure 1: For the equation x + x = 0, x(0) =3, X(0) = 0 the Euler method converts what shouldbe a circle to an outward spiraling curve.

r-1- ti U


Figure 2: The true solution to i + i + 0.2x =0, x(0) = 3, i(0) = 0 is an inwards spiraling curve.The outward spiraling which results with the Eulermethod exactly compensates resulting in no spiral-ing effect.

A discussion with the students followed concern-ing the validity of the results obtained in this way;particularly surprising is the second picture, whereclosed trajectories appear for k 0 0. ii) The stu-dents "solve" on the microcomputer the pendulumequation x"+sin z = 0 by the Runge-Kutta method.Figure 3 shows the drawing obtained by one student.

Ax

zA

xy

Figure 4: Plot of a curve approaching theLorentz attractor as obtained by a student using themodified Euler method with x(0) = 0.00001, y(0) =0.000G1 and z(0) = 0.00001.

In Figure 4 there is a picture obtained by a, 1/t --,.. student (the completion of the program required

a certain informatic ability, due to the complica-tions arising from the 3-dimensional representation

Figure 3: Phase portrait of the undamped pen-dulum, z + sin x = 0 obtained by a student using aRunge-Kutta method.

We can observe that the picture seems satisfac-tory from a numerical point of view. Some qualita-tive aspects of the solutions are underlined by theteacher, as a check of the known results from thetheory. iii) The student is invited to simulate onthe screen the trajectories of the equation of theLorenz attractor:dx/dt = sx + sydy/dt rx y xz with s := 10,r = 28,6 = 8/3dz/dt = bz + xy.

son was attempted with known qualitative resultssince the existing literature on the subject seemsto be too far advanced for a second year engineer-ing student. However, a comparison was possiblewith what might be expected from the physicalphenomenon (such as fluid turbulence phenomena).What it is very important to emphasize is that atthis stage (end of Mathematical Analysis 2) studentswere able to evaluate correctly the results obtainedfrom the computer, namely, to take into accountthe discrepancies which may occur between numeri-cal solutions and analytic solutions, keeping in mindthat the final objective is the interpretation of thephysical phenomenon.

4. CONCLUSIONSOur considerations have shown that even today

when internally discrete digital computers are usedfor handling calculus models (so far as applicationsare concerned), continuous analysis cannot be dis-

pensed with when describing problems for whichanalysis has been classically used. This, however,need not lead to the conclusion that analysis educa-tion at school or universities should go on as before.Our discussion has shown the function of continuousanalysis in applications, and teaching must be donein such a way that this function is fulfilled. Thisrequires that the transition from the discrete to thecontinuous model and vice versa be experienced bythe students and that the respective particular pos-sibilities and limitations of the model be perceived.To us, it would seem dishonest to try to explainto the student the importance of analysis for appli-cations by means of unrealistic and oversimplifiedminimum-maximum tasks. Rather, it seems crucialto have the student at least begin to assess the use-fulness of the various components of the system ofanalysis, i.e. concepts, approaches, calculi, trans-lation schemes in practical applications. This goalshould be attained by appropriat4 problem solvingin the classroom; and formal explication should playa subordinate part. It remains to be seen how abalance between the individual components can heachieved. The following aspects, however, should beincluded in any case:

a) The teaching of analysis should include thetreatment and study of discrete models. This leadsto numerical computations. It does not necessar-ily imply explicit teaching of numerical mathemat-ics, but requires including important basic numeri-cal facts such as propagation of errors.

b) Building models is an important activitywhich must not be neglectA in favour of just in-terpreting models. In particular, this means thatthe techniques of finding suitable functions are asimportant as discussing functions.

c) The role and function of (continuous).calculusmust be developed in an appropriate way. It cannotbe used to obtain numerical results, save in excep-tional cases: it can, however, guide and direct theuse of numerical methods.

d) The recent development of computer sciencehas established techniques, in particular program-ming languages, which permit the precise descrip-tion even of complicated processes such as, for in-stance, the algorithms nece-s-...ry for symbolic dif-ferentiation. Mathematics teaching should increas-ingly make use of these results.

REFERENCES

Artigue, M. [1989]: Une recherche d'ingenieriedidactique sur l'enseignement des equationsdifferentielles en premier cycle universitairein Equipe de Didactique des Mathema- tiques

.1.. 2


et de l'Informatique: Seminaire de Didac-tique des Mathimatiques et de l'Informatique,Grenoble, Annee 1988-1989. Universite deGrenoble, LSD-IMAG, Institut Fourier.

Artigue, M. and Gautheron, V. [1983]: Systemesdifferentiels, Etude graphique, Paris: CEDIC.ISBN 2-7124-0722-9.

Buchberger, B. [1990): Should Students Learn Inte-gration Rules?' ACM SIGSAM Bulletin, 24,1, 10-17.

Boieri, P. et al. [1984]: Personal computers in teach-ing basic mathematical courses, SEFI AnnualConference: The Impact of Information Tech-nology on Engineering Education, Erlangen.

Johnson, D. and Lovis, F. (Eds.) [1987]: Infor-matics and the Teaching of Mathematics, Pro-ceedings of the IFI? TC 3/WG 3.1 WorkingConference on Informatics and the Teachingof Mathematics, Sofia, Bulgaria, 16 - 18 May,1987. Amsterdam: North-Holland. ISBN 0-444- 70325 -X.

Kauclier, E. and Miranker, W.L. [1984]: Self-Validating Numerics for Function Space Prob-lems, San Diego: Academic Press.

Mascarello, M. and Scarafiotti, A.R. [1987]: Speri-mentazione didattica nel Politecnico di Tor:ao:Supporto informatica ai corsi die AnalisiMatematica nel biennio della Facolta, L 'Edu-cazione Matematica, VIII, II, 2, 147-151.

Mascarello, M. and Scarafiotti, A.R. [1988]: Usingcomputers in calculus examples-classes for en-gineers, ECM/87 Educational Computing inMathematics, Eds T.F. Banehoff et al, Ams-terdam: North- Holland, 93-97.

Mascarello, M. and Scarafiotti, A.R. [1988]: Experi-ments in mathematical education at secondaryschool.and Politecnico of Torino (Italy), ICME6, Theme Group 2 Computers and the teach-ing of mathematics, Working Group B.5 Theeffects of technology and of computer scienceon a maths curriculum for the future, prepara-tory papers, Budapest, 88-93.

Mascarello, M.; Scarafiotti, A.R.; and Teppati, G.[1989]: Cultura e insegnamento: Esperienzesignificative appoggiate a metodi e strumentiinformatici, Proceedings of the congress /ICultura Mathematica e Insegnamento/, Uni-versita di Firenze, CDO, 253-260.

Rice, J.R. [1983]: Numerical 'Methods, Software,and Analysis, IMSL Reference Edition, NewYork: McGraw Hill.

p11111111111111111111111111111111111111111111111111111111111111111111111111111111111

OM. IMI111111111=,


Rice, J.R. [1988]: Mathematical Aspects of Scien-tific Software in J.R. Rice (Ed.), MathematicalAspects of Scientific Software, Springer, NewYork and in The IMA Volumes in Matheutaticsand Its Applications, 14, 1 - 39.

Tall, D. [1986]: Building and Testing a CognitiveApproach to the Calculus Using InteractiveComputer Graphics, Ph.D. Thesis in Mathe-matics Education, The University of Warwick,Faculty of Education.

Watanabe, S. [1984]: An experiment toward a gen-eral quadrature for second order linear ordi-nary differential equations by symbolic com-putation in J. Fich (Ed.), EUROSAM 84, In-ternational Symposium on Symbolic and Alge-braic Computation, Cambridge, England, July9 - 11, 1984, Berlin: Springer-Verlag, 13 - 22.

West, B.J. [1985]: An Essay on the Importance ofBeing Nonlinear. Lecture Notes in Biomathe-matics 62, Berlin: Springer-Verlag. ISBN 3-540- 16038-8.

Winkelmann, B. [1984]: The Impact of the Com-puter on the Teaching of Analysis, Int. J.Math. Educ. Sci. Tech., 15, 675 - 689.

Winkelmann, B. [1989]: Dynamische Systeme undDifferentialgleichungen. Einige mathemati-sche Anmerkungen zu dynamischen Systemenund ihrer Simulation, LOG IN 9, Heft 4, 19 -23. ISSN 0720-8642.

Oc'

GRAPHIC INSIGHT INTO MATHEMATICAL CONCEPTS

David TallMathematics Education Research Centre

Warwick University, Coventry, U.K.

Beverly WestMathematics Department

Cornell University, Ithaca, NY 14583, U.S.A.

The human brain is powerfully equipped to pro-cess visual information. By using computer graph-ics it is possible to tap this power to help studentsgain a greater understanding of many mathematicalconcepts. Furthermore, dynamic representations ofmathematical processes furnish a degree of psycho-logical reality that enables the mind to manipulatethem in a far more fruitful way than could ever beachieved starting from a static text and pictures ina book or roughly drawn pictures on a chalk boardor overhead projector. Add to this the possibility ofstudent -xploration using prepared software and thesum total is a potent new force in the mathematicscurriculum.

In this paper we report on the development of in-teractive high resolution graphics approaches to var-ious areas in mathematics. The first author has con-centrated initially on the calculus in the UK (Tall,1986, Tall et al, 1990) and the second is workingin the USA on differential equations with John H.Hubbard (Hubbard and West, 1990).

An interactive visual approach is proving suc-cessful in other areas, for example, in geometry ( TheGeometric Supposer, Cabri Geomare), in data ma-nipulation (e.g. Macspin, Mouse Plotter), in prob-ability and statistics (e.g. Robinson and Bowman,1987) and, more generally, in a wide variety of top-ics (such as the publications in the Computer Illus-trated Text series, which use computer programs toprovide dynamic illustrations of mathematical con-cepts).

New approaches to mathematics

The existence of interactive visual software leadsto the possibility of an exploratory approach tomathematics which enables the user to gain intuitiveinsight into concepts, providing a cognitive founda-tion on which meaningful mathematical theories canbe built. For example, the notion of a limit has tra-ditionally caused students problems (e.g. Cornu,1981, Tall and Vinner, 1981). The computer bringsnew possibilities to the fore; we may begin by con-sidering the gradient not of the tangent, or of achord as it approaches a tangential position, butsimply the gradient (or slope) of the graph itself.

117

Although a graph may be curved, under high mag-nification a small part may well look almost straight.In such a case we may speak of the gradient of thegraph as being the gradient of this magnified (ap-proximately straight) portion. For instance, a tinypart of the graph y = x2 near x = 1 magnifies to aline segment of gradient 2 (figure 1).

2f (x) =x

t-4

-

4

-4 -2 2

--2

magn.x1281.02

Figure 1: Magnifying a small part of a graph toshow its local straightness.

To represent the changing gradient of a graph,it is a simple matter to calculate the expression(f (x c) f(x))/c for a small fixed value of c asx varies. As the chord clicks along the graph forincreasing values of x, the numerical value of thegradient for each successive chord can be plottedas a point and the points outline the graph of thegradient function (figure 2). In this case the chordgradient function of sin x for small c approximatesto cos x, which may be checked by superimposingthe graph of the latter for comparison. Thus thegradient of the graph may be investigated experi-mentally before any of the traditional formalities oflimiting processes are introduced.

Such moving graphics also enable the student toget a dynamic idea of the changing gradient. Stu-dents following this approach can see the gradient asa global function, not simply something calculatedat each individual point.

The symbols dx, dy can also be given a meaningas the increments in x, y to the tangent. Betterstill, (dx, dy) may be viewed as the tangent vector,


a valuable idea when we come to the meaning ofdifferential equations.

-2

2

f(x).sinxfrom x=-0 to W

gradient function(f(x+c)-f(x))/c

for

c=1/10

Figure 2: Building up the gradient function of agraph.

Conceptualizing non-differentiable functions

In a traditional calculus course, non-differenti-able functions would not be considered until a verylate stage, if at all. However, if one views a differen-tiable function as one which is "locally straight",then a non-differentiable function is simply onewhich is not locally straight. For instance, the graphof lx 11 at x = 1, or 'sin x1 at x = 7r, has a "cor-ner" at the point concerned with different gradientsto the left and right. More generally, it is possibleto draw a function that is so wrinkled that it neverlocks straight anywhere under high magnification.

An example is the blancmange function bl(x),first. constructed by Takagi in 1903. First a saw-tooth s(x) is constructed for a real number x bytaking its decimal part d = z I NT(x) and defining

s(x) = d if d <ll

1 d otherwise.

The sequence of functions

bi(x) = s(x)b2(x) = s(x) s(2x)/2

6(x) = s(x) s(2"-1 x)/2"-1

tends to the blancmange function (figure 3).

Figure 3: Building up the blancmange functionadding successive half-size sawtooth graphs.

The process may be shown dynamically on a vi-sual display unit; we regret that it cannot be pic-tured satisfactorily in a book. But higher magni-fication of the blancmange function using preparedsoftware shows it can nowhere be magnified to lookstraight, so it is nowhere differentiable. This in-tuitive approach can easily be transformed into aformal proof of disarming simplicity (Tall 1982).

Visualizing solutions of first order differentialequations

In graphical terms, a first order differential equa-tion dy/dx = f(x, y) simply states the gradient ofa solution curve at any point (x, y) and a solutionis simply a curve which has the required gradienteverywhere. The Solution Sketcher (Tall 1991) orMacMath (Hubbard and West 1991) allows the userto point at any position in the plane and drawsa small line segment' of the appropriate direction.This line-segment may be marked on-screen andsuccessive line segments fitted together to build upan approximate solution curve. More broadly, it ispossible to draw a direction diagram with an arrayof such segments and to trace a solution by followingthe given directions (figure 4).

dy/dx=-x/y/ e' / / / ..- / -- -.- - --. , \ \ \

/ / / / / ..-. . --. \ ..... \ \ \ \ \/ / / / , ..- , .- , \-\\ \ \/ / I / / / - e _ -2...,....\ \ \ \ \

? / 1 / / / . - ----, \ \ , , \ \ \\/ I III/ 1 / -- - . \ \ 1, \ \ \ y

1 I I " 1 I / , \ I I

S_. I 1 1 / I I\ % \\1\ \ \---,'" / / III I\ \ .\" ' ' 2''''" ",\ ",'''.., ,- ..". , i , / / / /

improvedstep by step

step8.2

x=-2.6107y=3.8315dyrdx=8.8612

step no. 188

Figure 4: Drawing a numerical solution of a firstorder differential equation.

The differential equation

dyy = xdx

has implicit solutions of the form x2 +y2 = k, ratherthan an explicit global solution of the form y = f (x).At points where the flow-lines meet the x-axis, thetangents are vertical and the interpretation of dy/dxas a function fails, but the vector direction (dx, dy)is valid with dx = 0 and dy 0 0. Thus a first-orderdifferential equation is sometimes better viewed interms of the direction of the tangent to a solutioncurve rather than specifying the derivative.

Existence of solutions

There comes a time in every university courseon differential equations when honesty should com-pel the teacher to admit that cookbook methods forsolving differential equations are inadequate. Suchinnocent looking equations as

dy/dx = y2 x, dy/dx = sin(xy), dy/dx = e'

do not have solutions that can be written in termsof elementary functions. Students often mistakenlyconfuse this with the idea that the equations have nosolutions at all. However, if they are able to inter-act with a computer program that plots a directionfield and then draws solutions numerically followingthe direction lines, the notion of a solution takes ona genuine meaning: "Of course the equations havesolutions: we can see them!" From this cognitivebase it is possible to use the computer to analysesolutions in an entirely new way.

Qualitative analysis of differential equations

New forms of analysis emerge now that we cansee as many solutions as we wish all at the sametime. In figure 5, notice how the solutions tend to"funnel" together moving to the lower right-handside; in the upper right they spray apart (an "an-tifunnel"). Qualitatively descriptive terms such as"funnel" and "antifunnel" can be defined preciselyto give powerful theorems with accurate quantita-tive results (Hubbard and West 1991). For exam-ple, the equation dy/dt = y2 t in figure 5 has twooverall behaviours: solutions either approach verti-cal asymptotes for finite t or fall into the funnel andapproach y = Nit- as t +oo. In the antifun-nel there is a unique solution approaching y = +vwhich separates the two usual behaviours. Further-more, the qualitative techniques enable us to esti-mate the vertical asymptote for a solution throughany given point with any desired precision.

Graphic Insight into Mathematics 119

low t:-10.0

hi t:10.0

Figure 5: A family of solutions of a differentialequation, showing funnel and antifunnel behaviour.

S== 03 binary - 5 planets

(a)

03 binary - 5 planets

(b)

Figure 6: A numerical approximation to themany-body problem. (a) Masses in initial positionwith velocity vectors. (b) A little later under theaction of Newton's Laws.

m


Newton's Laws

The classical three body problem defies elemen-tary analysis, yet a computer program can cope withrelative ease. The program Planets (Hubbard andWest 1990) takes a configuration of up to ten bodieswith specified mass, initial position and velocity anddisplays the movement under Newton's laws (figure6). The data can be input either graphically withthe cursor, or numerically in a table. The programallows exploration of possible planetary configura-tions and it soon becomes plain that stability is theexception rather than the rule. One may wonderunder what circumstances stability occurs. Otherquestions arise, such as the reason for the braidedrings of Saturn that were a great surprise when firstobserved by the Voyager space flight. Nobody hadimagined such a behaviour beforehand, yet braidedbehaviour showed up in the very first experimentswith the Planets program.

Figure 7 shows a model of a possible orbit of atiny satellite around two larger bodies, alternatelyoscillating between revolving round one then mov-ing into a position of superior gravitational pull ofthe other and moving, for a time, to revolve roundthe other (Kocak 1986). Once again, computer ex-ploration shows vividly how three bodies move in acomplex pattern.

The theory of dynamical systems and chaosis a paradigmatic example of a new branch ofmathematics in which the complementary roles ofcomputer-generated experiments to suggest theo-rems and formal mathematical proofs to establishthem with logical precision go hand in hand.

Chaos has become not just a theory but alsoa method, not just a canon of beliefs butalso a way of doing science. ... To chaos re-searchers, mathematics has become an exper-imental science, with the computer replac-ing laboratories full of test tubes and micro-

GoPU,..1

.

dilli,...7.0

WillkeSitEallatfP1K-o''' le, wow-"'ANN rMP t.'

P.til

"..01,1.r.-4 F.....

-170

1../,-1:4,111.611, --..$10

ClearDimensionAlgorithmStep sizeTime)(tend.EquationParameterInttCondsWindoSizeJn.ps/FltUTILITIESUISUALAIDUnit

1. "mi... r it,'

Figure 7: A numerical plot representing a tinysatellite orbiting two larger bodies.

scopes. Graphic images are the key. "It'smasochism for a mathematician to do with-out pictures" one chaos specialist would say."How can they see the relationship betweenthat motion and this, how can they developintuition?". (Gleick 1987, pp. 38-39)

Systems of differential equations

The MacMath software of Hubbard and West(1991) draws solutions of systems of differentialequations dx/dt = f(x,y),dy/dt = g(x, y) in thex, y- plane and also locates singular points usingNewton's method, drawing separatrices for saddlepoints (figure 8).

Window Into

Figure 8: Locating singular points andtrices for saddle points.

separa-

In this way the computer may be used to drawsolutions of systems of differential equations that arefar too complicated to draw by hand. As a furtherexample, Artigue and Gautheron (1983) draw thesolutions of the polar differential equations

dr dO

di = sin r, = cos r

which exhibit limit cycles for r = kr (figure 9).

Figure 9: Limit cycles of simultaneous polar dif-ferential equations.

gci)

Generalizing the concept of visual solutions

A second order differential equation such as

d2x

dt2 t

no longer has a simple direction field in (t, x) space,because through each point (t, x) there is a differ-ent solution for each starting direction v dx/dt.However, this differential equation is equivalent tothe simultaneous linear equations:

dxdidv

dt= t-v

and in three dimensions, with coordinates (t, x, v),these equations determine a unique tangent vector(dt, dx, dv) in the direction (1, v, t). Hence the ideaof a direction field does generalize, but it must be vi-sualized in three-dimensional (t, x, v) space. Figure10 shows two solutions of the simultaneous differ-ential equation spiralling through (t, x, v) space andtheir projections onto the t x and t v planes, withthe t x projection giving solutions to the originalsecond order differential equation.

Visual exploration in geometry

Euclidean geometry traditionally served to in-troduce students to a deductive system. In manycountries (such as the United Kingdom) it has allbut disappeared from the mathematics curriculum.Computers now give the opportunity to manipulategeometrical figures to build up intuitions for pos-sible theorems (the Geometric Supposer, Schwartzand Yerushalmy, 1985, Cabri Geometre, 1987). Theinitial phase of study of geometry can now be anexperimental science, in which the student can usethe computer to construct a figure and experimentwith it.

Visual Data Processing

It is now possible to explore data visually, forexample, to see a line of best fit for data in two orthree dimensions. MacSpin allows up to ten cate-gories of data, from which any three can be selected

Graphic Insight into Mathematics 121

dx/dt=vdv/dt=-x

step=0.1.t=6.4x,0.1.9061v=1.4139

Figure 10: Two nearby solution curves for a pairof simultaneous differential equations.

rt Fichier Edition Li Minns Construct ins Transformation Dinerspoint sur Netinlet section de 2 objets

milieu d un bipointmedlottice dun bipointcliOlte parelleledi oite orthogonolecentre (run ref cleceicle cliconsctit a un tuengte

Figure 11: Cabri Geometre software for manip-ulating geometric figures.

and displayed. Though only represented as a projec-tion of three dimensions onto the two-dimensionalscreen, the data may be rotated and viewed dynam-ically from any angle to give a sense of depth that isnot visible in a static picture (figure 12). Individualpoints may be selected and inspected to see wherethe data originates to identify interesting informa-tion, such as outlying values. Rotating the data inthe figure suggests that it clusters together in a waywhich intimates that the three components are cor-related.

Modern spreadsheets, statistical packages anddata handling packages now include visual repre-sentation of data which encourages the user to ex-plore and communicate complex information in vi-sual ways.


I the EMI Oisp Mu Variables Subsets (cents Markets'OMIMPP.M Lone Milks E----11M1i Variables

Subsets3---L145717

,

..,

'flInrry1:1111 Events MEM

.

Joe MFredAngieSillNoreenCathieJimJanetFredaGeorgeAnnetteHarrNanceNic

ChrisSec

Figure 12: Manipulating data with three com-ponents to look for a visual correlation.

The ability to present and manipulate informa-tion visually is becoming widely available in manydifferent areas in mathematics. For example, Robin-son and Bowman (1986) introduce probability andstatistics using computer graphics with the inten-tion of giving a 'feel' for probability distributionsrather than elaborating mathematical detail. Moregenerally, the Computer Illustrated Texts (startingwith Harding 1985) are designed to use simple com-puter programs to provide interactive illustrationsof mathematical ideas which can be explored by thestudent in place of static pictures in a book.

Is programming essential?

We have not explicitly mentioned programmingfor the purpose of gaining insight into mathematicalprocesses. A body of expertise is growing in whichstudents are expected to write or adapt short pro-grams (usually in structured Basic, Pascal, or Logo)to carry out mathematical algorithms. From here itis often intended that they move on to prepared soft-ware that uses the underlying algorithms in a moreinteractive manner. The early computer-illustratedtexts assumed that the programming would be suf-ficiently simple that it would allow the student tomodify the programs, but this became an impossibleideal in later texts as more sophisticated programswere written that were too complex for the user tomodify. Programming requires a serious investmentin time and effort. However, it can pay vast divi-dends in gaining insight into the underlying math-ematical processes if the investment is sufficientlygenerous.

Dubinsky has evidence that having studentsmake certain programming constructions (in thecomputer language ISETL) can lead to their makingparallel mathematical constructions in their minds

and thereby come to understand various mathe-matical concepts (see, for example, Dubinsky andSchwingendorf 1991). Clearly a spectrum of ap-proaches may be possible with varying amounts ofprogramming, depending on the time and commit-ment available.

New Styles of Learning

Software is becoming widely available to givegraphical representations in calculus, differentialequations, geometry, data handling, numerical anal-ysis, and many other areas of mathematics. This isusually predicated on a new kind of learning ex-perience Q one in which the student may exploreand manipulate ideas, investigate patterns, conjec-ture theorems and test theories experimentally be-fore going on to prove them in a more formal con-text.

For instance, beginning calculus students mayinvestigate the gradients of functions such as sine,cosine, tangent, exponential and logarithm, andconjecture their formulas before they are derivedformally (Tall 1986, 1987). In differential equationsthey may explore problems at the boundaries of re-search (such as the rings of Saturn) and make themental link between the friendly world of (mostlylinear) equations that can he solved by formulas andthe strange world of those (usually nonlinear) thatcan not (Hubbard and West 1991).

This form of learning is not a replacement forformal deduction, but a precursor and a comple-ment to it. It enables the less able student to graspessential ideas that would previously be too diffi-cult when framed in a purely formal theory and forthe more able student to build a cognitive base forthe formal theory to follow. It enables a wide rangeof students to integrate their knowledge structurethrough their powers of visualization.

AcknowledgementThe authors are grateful to Professor John H.

Hubbard for his assistance in the preparation of anearlier version of this article.

REFERENCES

Alfors, D. and West, B. [1992]: Analyzer* calculussoftware (for the Macintosh computer), Read-ing, MA: Addison-Wesley.

Artigue, M. and Gautheron, V. [1983]: SystemesDifferentiels: Etude Graphique, CEDIC, Paris.

Bach, J.O. [1990]: Multi Mat (for I.B.M. compatiblecomputers; available in English), Copenhagen:Danish Mathematics Teachers Association.

0

Cabri Geometre [1987]: IMAG, BP 53X, Universitede Grenoble (for IBM and Macintosh comput-ers).

Cornu, B. [1981]: Apprentissage de la notion de lirn-ite : modeles spontanes et modeles propres,Actes du Cinquieme Colloque du Groupe In-ternational P.M.E., Grenoble, 322-326.

Devaney, R.L. [1990]: Chaos, Fractals, and Dynam-ics: Computer Experiments in Mathematics,Reading, MA: Addison-Wesley.

Dubinsky, E. and Schwingendorf, K. E. [1991]: Con-structing calculus concepts: cooperation in acomputer laboratory in The Laboratory Ap-proach to Teaching Calculus (Leinbach, L.C.,Ed.), MAA Notes Series No. 20, Washington,DC: Mathematical Association of America.

Gleick, J. [1987]: Chaos: Making a New Science,London: Penguin and Viking: New York.

Harding, 11. [1985]: Fourier Series and Trans -forms, A Computer Illustrated Text, Bristoland Boston: Adam Hilger (for the BBC, IBMand Apple Computers).

Hubbard, J.H. and Parmet, M. [1987]: 3D Analyzer;Complex Paint (for the Macintosh computer),Ithaca, NY: Cornell University.

Hubbard, J.H. and West, B. [1991]: DifferentialEquations: A Dynamical Systems Approach,New York: Springer-Verlag.

Hubbard, J.H. and West, B. [1991]: Mac Math: ADynamics Package (for the Macintosh com-puter), New York: Springer-Verlag.

Johnson, J.A. [1990]: Gyrographics (for I.B.M. com-patible computers), Stillwater, OK: CipherSystems.

Klotz, E. and Jackiw, N. [1991]: The Geome-ter's Sketchpad (for the Macintosh computer),Berkeley, CA: Key Curriculum Press.

Kogak, H. [1986]: Phaser: Differential and Dif-ference Equations through Computer Exper-iments (for IBM computers), New York:Springer-Verlag.

Mac Spin [1985]: D2 Software Inc., Austin, TX (forthe Macintosh Computer).

Phillips, R. [1988]: Mouse Plotter, Shell Centre,Nottingham (for the Archimedes computer).

Robinson, D.A. and Bowman, A.W. [1986]: In-troduction to Probability, Bristol and Boston:Adam Hilger (for the BBC and IBM comput-ers).

Graphic Insight into Mathematics

Schwartz, J. and Yerushalmy, M. [1985]: The Ge-ometric Supposer, Sunburst Communications,Pleasantville, N.Y. (for the Apple Computer).

Takagi, T. [1903]: A simple example of a continu-ous function without derivative, Proc. Phys.-Math., Japan, 1, 176-177.

Tall, D.O. [1982]: The blancmange function, con-tinuous everywhere but differentiable nowhere,Mathematical Gazette, 66, 11-22.

Tall, D.O. [1986]: Graphic Calculus (for BBCcompatible computers), Glentop Press, Lon-don.

Tall, D.O. [1987]: Readings in Mathematical Educa-tion: Understanding the Calculus, (collectedarticles from Mathematics Teaching, 10515-7),Association of Teachers of Mathematics, UK.

Tall, D.O. [199i]: Real Functions and Graphs(for BBC compatible computers), Cambridge:Cambridge University Press.

Tall, D.O., Blokland, P. and Kok, D. [1990]: A

Graphic Approach to the Calculus (for I.B.M.compatible computers), Sunburst Inc., USA.

Tall, D.O. and Vinner, S. [1981]: Concept imageand concept definition in mathematics, withspecial reference to limits and continuity, Ed-ucational Studies in Mathematics, 12 151-169.

Zimmerman, W. and Cunningham, D. [1990]: Visu-alization in Teaching and Learning Mathemat-ics, MAA Notes Number 19, Washington,DC:Mathematical Association of America.

13 u

Annotated References

This is a list. of some of the references earlier in this book of particular significance or usefulnesswith a brief annotation describing the contents of the article or book.

Aho, A.V., Hoperoft, J.E. and Ullman, J.D. [1983]:Data Structures and Algorithms, Reading,MA: Addison-Wesley.

This is a graduate text that presents thedata structures and algorithms that underpinmuch of today's computer programming. Itcovers these topics in the context of solvingproblems using computers and introduces stepcounting and complexity as an integral part ofproblem solving. It is very nicely written andcomprehensive.

Alfors, D. and West, B. [1992]: Analyzer* calculussoftware (for the Macintosh computer), Read-ing, MA: Addison-Wesley.

EDUCOM/NCRIPTAL distinguished soft-ware for 1990 for calculus and iteration offunctions of a single variable.

Artigue, M. [1989]: Une recherche d'ingenieriedidactique sur l'enseignement des equationsdifferentielles en premier cycle universitaire inEquipe de Didactique des Mathematiques et del'Informatique: Siminaire le Didactique desMathimatiques et de l'Informatique, Greno-ble, Annee 1988-1989. University de Grenoble,LSD-IMAG, Institut Fourier.

A didactical reflection on the prevalenceof the algebraic approach to differential equa-tions in undergraduate courses and construc-tion of a new course with emphasis on geomet-ric, qualitative and numerical elements.

Artigue, M. and Gautheron, V. [1983]: SystemesDifferentielles: Etude Graphique, CEDEC,Paris.

A pioneer work that deserves to be a clas-sic. Teaches qualitative methods in the studyof two-dimensional systems of autonomous dif-ferential equations with many beautiful de-signs.

Aspetsberger, K. and Kutzier, B. [1988]: Symboliccomputation A new chance for educationin F. Lovis and E.D. Tagg (eds) Computersin Education, 331-336, Amsterdam: North-Holland.

124

Considers a wide range of symbolic compu-tation activities with examples including com-puter algebra, computational geometry, auto-matic reasoning and automatic programming.

Banchoff, T. et al (Eds.) [1988]: Educational Com-puting in Mathematics, ECM87, Amsterdam:North Holland.

Proceedings of an international congresswith presentations about a number of math-ematical domains (differential geometry, cal-culus, dynamical systems, geometry, etc.) andreports about various teaching experiments.

Bolter, J.D. [1984]: Turing's Man: Western Cul-ture in the Computer Age, Chapel Hill, NC:University of North Carolina Press.

An innovative analysis of the philosophicalimpact of computers as "embodied mathemat-ics," of nature as information, and of humansas "information processors." Bolter drawsinteresting parallels with Athenian manlimited by finite mathematics, yet creatingprivate universes like the craftsman potter inPlato's Timaeus.

Bushaw, D. [1983]: A two-year lower-division math-ematics sequence in The Future of CollegeMathematics, A. Ralston and G.S. Young(Eds.), pp 111-118. New York: Springer-Verlag.

An article which presents an outline ofa two-year undergraduate sequence which in-tegrates calculus Mid discrete mathematics.Twenty-four different modules are listed aswell as the interrelations among them.

Clocksin, W.F. and C.S. Mellish [1981]: Program-ming in Prolog, Berlin: Springer-Verlag.

This is a textbook for teaching Prolog asa programming language. It briefly discussesthe logical foundations of the language butthe emphasis is on how useful programs canbe written using the Prolog systems that areavailable.

Computers and Mathematics, A column (past edi-tor: J. Barwise, current editor: K. Devlin) ap-pearing regularly in the Notices of the Ameri-can Mathematical Society.

Papers appearing in this column discuss allaspects of the influence of computers on math-ematics, both with respect to research and toteaching. Many Symbolic Mathematical Sys-tems have been discussed see Appendix 1 ofthe paper by B.R. Hodgson and E.R. Mullerin this volume.

Cornu, B. [1992]: L 'Ordinateur pour enseigner lesmathematigues, Paris: Presses Universitairesde France.

A multi-author book with chapters onmathematics and informatics, on various ex-amples of teaching mathematical concepts,and on the link between technology and ed-ucational research.

Devaney, R.L. [1990]: Chaos, Fractals, and Dynam-ics: Computer Experiments in Mathematics,Reading, MA: Addison-Wesley.

Paperback handbook full of explorationsfor either a classroom setting or individualstudy.

Douglas, R. (Ed.) [1986]: Toward a Lean andLively Calculus, Proceedings of a Confer-ence/Workshop at Tulane University, Wash-ington, DC: Mathematical Association ofAmerica.

The report of a conference sometimescalled the counterreformation (in relation todiscrete mathematics) - in which variousrecipes for improving the teaching of calculusare given.

Dubinsky, E. and Fraser, R. [1990]: Computersand the Teaching of Mathematics, Notting-ham, UK: The Shell Centre for MathematicalEducation.

This selection of papers from the Sixth In-ternational Congress on Mathematics Educa-tion contains those papers from ICME6 par-ticularly relevant to the use of technology inteaching mathematics.


Dubinsky, E. and Schwingendorf, K. E. [1991]: Con-structing calculus concepts: cooperation in acomputer laboratory in The Laboratory Ap-proach to Teaching Calculus, (Leinbach, L.C.,Ed.), MAA Notes Series, Washington, DC:Mathematical Association of America.

Describes an early version of the PurdueUniversity calculus project which emphasizescooperative learning. Contains examples ofcomputer assignments and examinations aswell as details of the computer laboratory set-up.

Fey, J.T. [1989]: Technology and Mathematics Edu-cation: A Survey of Recent Developments andImportant Problems, Educ. Studies in Math.,20, 237-272.

Provides an overview and analysis of re-cent progress in applying electronic informa-tion technology to the creation of new envi-ronments for intellectual work in mathematics.It discusses numerical computation, graphiccomputation and symbolic computation andcontains many concrete examples.

Foster, K.R. and Bau, H. H. [1989]: Symbolic Ma-nipulation Programs for the Person41 Com-puter, Science, 243, 679-684.

Provides a useful comparative summaryof the capabilities of a majority of SymbolicMathematical Systems. The article providesprices and types of hardware required. (Thesesystems are constantly changing and up todate information should be obtained from themanufacturers whose addresses are listed inthe article.)

Garey, M.R. and D.S. Johnson [1978]: Computersand Intractability, A Guide to the Theory ofNP-Completeness, San Francisco: Freeman.

This book includes a thorough introduc-tion to complexity theory and .VP-completeproblems. It shows how to recognize theseproblems and how to deal with them. It isa readable guide and offers an exhaustive listof NP-complete problems.

Goldenberg, E.P. [1988]: Mathematics, Metaphors,and Human Factors: Mathematical, Techni-cal, and Pedagogical Challenges in the Edu-cational Use of Graphical Representation ofFunctions, Journal of Mathematical Behavior,7, 2, 135-173.

1.1611..111.111.0.10.1.0.1.61111111/611111111111.1


Describes the work of a research group atthe Educational Technology Center (ETC) atthe Harvard Graduate School of Education.Many convincing examples show possible pit-falls and misunderstandings in the interpreta-tion of function graphs.

Gray, T.W. and Glynn, J. [1991): Exploring Math-ematics with Malhematica, Reading, MA:Addison-Wesley.

An introduction to Mathematica, writtenin the form of a dialog between the two au-thors, presenting exploration of various math-ematical concepts. A section is devotedto Mathematica's application to high school,college and university mathematics teaching.The book comes with a CD-ROM disk con-taining an electronic edition of the text in theform of Mathematica's Notebooks.

Held, M.K., Sheets C. and Matras, M.A. [1990]:Computer-enhanced Algebra: New roles andchallenges for teachers and students in T.J.Cooney and C.R. Hirsch (eds.) Teachingand Learning Mathematics in the 1990s, (1990NCTM Yearbook), 194-204, Reston, VA: Na-tional Council of Teachers of Mathematics.

Discusses the new roles for teachers andstudents when the computer enters activelyinto mathematics education. The teacher astechnical assistant. The teacher as collabora-tor. The teacher as facilitator and catalyst .

Responsibilities for evaluating student learn-ing, etc.

Hirst, A. and Hirst, K. (Eds.) [1988]: Proceedingsof the Sixth International Congress on Mathe-matics Education, Budapest, 1988, Budapest:Janos Bolyai Mathematical Society.

The 'state-of-the-art' on the use of corn-puters in mathematics education. Containsreports of plenary sessions on computerizationof schools and mathematics education, and onalgoritmic mathematics, and of a theme groupon computers and the teaching of mathemat-ics.

Hubbard, J.H. and West, B. [1991]: MacMath: ADynamics Package (for the Macintosh com-puter), New York: Springer-Verlag.

Twelve interactive and easy-to-use graph-ics programs for both iteration and differen-tial equations, with a handbook of sugges-tions for what to do with them; developedto accompany the authors' three volume textDifferential Equations: A Dynamical SystemsApproachalso being published by Springer-Verlag.

Jacobsen, E. [1989]: An International Perspective inSIGCUE OUTLOOK (Bulletin of the SpecialInterest Group for Computer Uses in Educa-tion), 20, 2, New York: ACM Press.

Provides a worldwide overview of micro-computers in education. The priorities of gov-ernments and current usages are illustrated byselecting representative countries in each con-tinent. UNESCO's role in the field of comput-ers and education is examined.

Jaffe, A. [1984]: Ordering the Universe: The Role ofMathematics in Renewing U.S. Mathematics,Washington, DC: National Academy Press.

A sweeping survey of contemporary math-ematics, "an ancient art, ... highly esoteric,and the most intensely practical of human en-deavors." Emphasizes the role of mathemat-ics in advancing computation, communication,physics, and engineering.

Johnson, D.C. and Lovis, F. (Eds.) [1987]: Infor-matics and the Teaching of Mathematics, Pro-ceedings of the IFIP TC 3/WG 3.1 WorkingConference on Informatic. and the Teachingof Mathematics, Sofia, Bulgaria, 16 - 18 May,1987. Amsterdam: North-Holland. ISBN 0-444- 70325 -X.

Many international contributions on a widescope of themes, from theoretical to concrete,from primary school to college level.

Kenney, M., (Ed.), [1991]: Discrete Mathematicsacross the Curriculum, K-12 (1991 NCTMYearbook), Reston, VA: National Council ofTeachers of Mathematics.

Discrete mathematics is the part of math-ematics where algorithmics are most natural.This yearbook contains articles are specific im-plementation ideas at various levels.

Klotz, E. and Jackiw, N. [1991]: The Geome-ter's Sketchpad (for the Macintosh computer),Berkeley, CA: Key Curriculum Press.

Developed at Swarthmore College as partof the Visual Geometry Project under the di-rection of E. Klotz and D. Schattschneider. Adynamic tool for exploring geometry (e.g., youcan construct a figure, distort it, and your con-struction follows the distortion.)

Knuth, D.E. [1973]: The Art of Computer Program-ming: vol 3: Sorting and Searching, Reading,MA: Addison-Wesley.

This is the third of Knuth's classical books.As with the other two, it should be consid-ered as a source for information and referencesand not as a basic textbook. Although thetitle may sound as if the book is mainly forprogrammers concerned with preparing sort-ing routines, it virtually covers all theoreticalaspects of programming.

Kook, H. [1986]: Phaser: Differential and Dif-ference Equations through Computer Exper-iments (for IBM computers), New York:Springer-Verlag.

The classic entry into the field of experi-menting with differential equations.

Koerner, J.D. (Ed.) [1981]: The New Liberal Arts:An Exchange of Views, New York: Alfred P.Sloan Foundation.

A position paper followed by ten responsesin which it is argued that analytic skills (e.g.,statistics, computation, applied mathem ics)are as crucial for liberal education as are tradi-tional literary, historical, and artistic studies.The computer "has altered the world in whichthe student will live as well as the manner inwhich he will think about the world."

Krivine, J.L. and M. Parigot [1990]: Programmingwith proofs, J. Inf. Process. Cybern. EIK 26,149-167.

This is one of the recent articles which de-velop the idea that proofs and programs arebasically the same object. Even though it ismainly written for readers with a strong back-ground in logic and computer science, it hasdeep insight on the subject.

LSW [1990]: Landesinstitut fur Schule and Weiter-bildung (LSW), Soest (Ilrsg.): Neue Medienim Unterricht - Funktionenplotter im Math-ematikunterricht. Soest: Soester Verlagskon-tor. ISBN 3-8165-1732-3.


This issue on the use of function plottersin mathematics teaching at schools gives gen-eral considerations on the didactical criteria,possibilities and limits of such tools, some con-crete examples for the use at various places inthe mathematical curriculum, and critical de-scriptions of some function plotters availablein Germany.

Malkevitch, J. et al., [1988]: For All Practical Pur-pose, San Francisco: W. H. Freeman. Alsoavailable as 26 videotaped TV programs.

An innovative "general studies" secondary-tertiary course which shows students manycurrent applications of elementary mathemat-ics and shows faculty how many new topics(often algorithmic) can be included in the cur-riculum.

Mathematical Sciences Education Board [1990]: Re-shaping School Mathematics: A Philosophyand Framework for Curriculum, Washington,DC: National Academy Press.

A detailed rationale for changing schoolmathematics, building on research related tothe role of technology and to the prpcess ofteaching and learning. Summarizes (with ex-tensive references) the relevant research litera-ture; poses open questions; and outlines goalsfor curriculum reform.

Maurer, S. [1984]: Two meanings of algorithmicmathematics, Math. Teacher, 77, 430-435.

Explains at length the difference betweenthe traditional and contemporary meanings ofalgorithmic mathematics. The two main ex-amples are polynomial evaluation (mentionedbriefly in Maurer's article here) and Gaussianelimination for solving systems of linear equa-tions.

Maurer, S. [1985] The algorithmic way of life is best,College Math. J., 16, 2-18 (Forum article andreply to responses).

The author presents a deliberately force-ful argument for abandoning the traditionaltheory/computation schism in favor of an al-gorithmics synthesis. The article is followedby numerous thoughtful responses, some quitecritical, and then a summary reply by the au-thor.

111111111111111111111111111111111111110111

128 Influence of Computers and Informatics on Mathematics and Its Thaching

Maurer, S. B. and Ralston, A. [1991]: Discrete Algo-rithmic Mathematics, Reading, MA: Addison-Wesley.

One of the most recent discrete mathemat-ics texts. It is intended for well-prepared first-year university students and stresses the algo-rithmic approach to discrete mathematics.

Mines, B., Richman, F. and Ruitenberg, W. [1988]:A Course in Constructive Algebra, New York:Springer-Verlag.

An example of what classical pure mathe-matics may look like if researchers take the al-gorithmic viewpoint to heart. Topics such asfactorization in polynomial rings are treatedby defining algorithms and proving them cor-rect. The review by Beeson (see referencesin Maurer, this volume) gives a thoroughoverview.

Muller, E.R. [1991]: Maple Laboratory in a Ser-vice Calculus Course in L.C. Leinbach et at(eds.) The Laboratory Approach to TeachingCalculus, MAA Notes Number 20, 111-117,Washington, DC: Mathematical Association ofAmerica.

Presents the development and implementa-tion of compulsory laboratories. Includes ex-amples of laboratory activities and providesdata on traditional indicators (failure rates,etc.) and student attitudes.

National Council of Teachers of Mathematics [1989]:Curriculum and Evaluation Standards forSchool Mathematics, Reston, VA: NationalCouncil of Teachers of Mathematics.

A key document in America's attempt tocome from way behind in mathematics educa-tion. Contains a detailed set of standards forschool mathematics, arranged in four groups(K-4, 5-8, 9-12), giving expectations and ex-amples in each curricular area. Builds onassumption of educating students for an in-formation society; advocates extensive use ofcalculators and computers throughout schoolmathematics.

National Research Council [1989]: Er ybodyCounts: A Report to the Nation on the Fu-ture of Mathematics Education, Washington,DC: National Academy Press.

A call for action issued by the NationalAcademy of Sciences to improve mathemat-ics education in the United States. Highlightshuman resource needs, learning through in-volvement, and curriculum priorities. Stresses,among other things, the way computers havechanged priorities for mathematics education.

Nievergelt, Y. [1987]: The Chip with the Col-lege Education: the HB28C, Amer. Math.Monthly, 94, 895-902.

Details the power of the PP-28C by pro-viding examples of its capabinties. Con-cludes that it "introduces one new elementinto the teaching of mathematics, namely aw-some computing power at both modest priceand size". (Which is even more true of themore recent HP-48SX.)

Okamori, H. [1989]: Mathematics Education andPersonal Computers, Tokyo: Daiichi-HokiShuppan.

This volume is an excellent survey of com-puter use in mathematics education in Japanfrom kindergarten to university. It covers re-search and practice and includes a number ofexamples of problem solving in the real world(e.g. mathematics of a lake, road mathemat-ics).

Page, W. [1990]: Computer Algebra Systems: Issuesand Inquiries, Computers Math. Applic., 19,51-69.

An educational-philosophical survey arti-cle on issues of special importance to all whoare involved with the instructional uses ofcomputers in the mathematical sciences.

Peressini, A. et al [1992]: Precalculus and DiscreteMathematics, University of Chicago SchoolMathematics Project, Glenview, IL: Scott,Foresman.

The Chicago project has developed a high-ly innovative mathematics curriculum for av-erage students in grades 7-12. This book isthe 12th year text. It includes algorithmics.There are several other innovative projects un-der way in America which will also result intexts.

Ralston, A. [1981]: Computer Science, Mathematics, and the Undergraduate Curricula in Both,Amer. Math. Monthly, 88, 472-485.

A.

An urgent appeal to mathematicians torecognize the fundamental mathematical re-quirements of computer science by giving dis-crete mathematics greater priority in earlyyears of mathematical preparation. The be-ginning of a decade-long effort to establish dis-crete mathe- ,atics on an equal footing withcalculus as an nortant foundation not onlyfor computing buu , for mathematics itself.

Ralston, A. and Young, G.S. (Eds.) [1983]: TheFuture of College Mathematics, New York:Springer-Verlag.

A report of the first conference at whichdiscrete mathematics as a possible alternativeto or coequal with calculus was considered.The papers discuss a wide variety of the rele-vant issues.

Rice, J.R. [1988]: Mathematical Aspects of Scien-tific Software in J.R. Rice (Ed.): Mathemati-cal Aspects of Scientific Software, New York:Springer-Verlag and in The IMA Volumes inMathematics and Its Applications, 14, 1 - 39.

Fundamental but concrete aspects of math-ematics, applications and the new role theseare taking when the users rely on ready-mademathematical methods.

Robinson, J.A. [1965]: A machine-oriented logicbased on the resolution principle, J. ACM 12,23-41.

This book gives an account of the impres-sive breakthrough achieved by its author to-wards performing deductive reasoning by amachine. It includes the presentation of theformalism of predicate logic, a thorough expo-sition of the resolution principle as well as adetailed account of a working computer pro-gram for showing "what follows from what".

Schmidt, Gunter (Ed.) [1988]: Computer im Math-ematikunterricht, Der Mathematikunterricht34, Heft 4. ISBN 3-617-24022-4, 19-42.

A special issue of a German journal aimedmainly at high school teachers. This issue con-tains three articles discussing the impact ofcomputers on mathematics learning, analysingsoftware for mathematics teaching, and de-scribing a possible use of recursion in teachingcalculus.


Steen, L.A. (Ed.) [1988]: Calculus for a New Cen-tury, MAA Notes No. 8, Washington, DC:Mathematical Association of America.

A report of a conference whose papers lookat the role of calculus and the teaching of cal-culus as we approach the 21st century.

Stern, J. [1990]: Fondements Mathdmatiques del'Informatique, Paris: McGraw-Hill.

This is an undergraduate textbook whichcovers computability, complexity, logic and thetheory of regular and algebraic sets. It is areadable introduction to the main tools andconcepts of theoretical computer science.

Tall, D.O. [1986]: Building and Testing a CognitiveApproach to the Calculus Using InteractiveComputer Graphics, Ph.D. Thesis in Mathe-matics Education, The University of Warwick,Faculty of Education.

Combining mathematical, psychologicaland epistemological studies with the develop-ment of suitable software, important insightsinto the nature of the learning of the calcu-lus are gained. The work is not applicationoriented, but the attempt to build up a trueunderstanding using discrete and continuousaspects serves the user of mathematics, too.

Tall, D.O. [1986, 1990]: Graphic Calculus I-III (forBBC compatible computers), London: Glen-top Press and, with P. Blokland and D. Kok,A Geometric Approach to the Calculio, (forI.B.M. compatible computers), Sunburst, Inc.,USA.

Interactive, extremely well designed, andeasy-to-use graphics programs for all sorts ofcalculus topics, including multivariable, withexcellent manuals for students and teachers.

Tall, D.O. [1987]: Readings in Mathematical Edu-cation: Understanding the Calculus, collectedarticles from Mathematics Teaching, 1985-1987, Association of Teachers of Mathematics,UK.

These are among the earliest writings onthe subject worldwide; they include many ex-cellent insights and suggestions.

Tinsley, J.D. and van Weert, T.J. (Eds.) [1989]:Educational Software at the Secondary Level,Amsterdam: Elsevier.


Proceedings of a 1989 IFIP working confer-ence with many examples of educational soft-ware and discussions of the trends of softwaredevelopment and evolution.

Wagon, S. [1991]: Mathematica in Action, San Fran-cisco: Freeman.

An example-based introduction to tech-niques, both elen rotary and advanced, of us-ing Mathernatica for mathematical computa-tion and exploration. "An underlying themeof this book is that a computational way oflooking at a mathematical problem or resultyields many benefits."

West, B.J. [1985]: An Essay on the Importance ofBeing Nonlinear, Lecture Notes in Biomathe-matics 62, Berlin: Springer-Verlag. ISBN 3-540- 16038 -8.

Fundamental aspects of nonlinearity, most-ly in the context of dynamical systems. A bittechnical in some parts.

Wilf, H.S. [1982]: The Disk with the College Edu-cation, Amer. Math. Monthly, 89, 4-8.

An early effort to alert mathematicians tothe power of symbolic computing systemswhich are now much more powerful than thoseof a decade agoand to the threat they posefor those who might continue the status quoin teaching undergraduate mathematics.

Zimmerman, W. and Cunningham, D. [1990]: Visu-alization in Teaching and Learning Mathemat-ics, MAA Notes Number 19, Washington,DC:Mathematical Association of America.

An authoritative collection from many ex-perts; topics include geometry, calculus, dif-ferential equations, differential geometry, com-plex analysis, linear algebra, iteration andstochastic processes.

Zorn, P. [1987]: Computing in UndergraduateMathematics, Notices of the American Math-ematical Society, 34, 917-923 (October)'.

A careful analysis of issuesphilosophical,pedagogical, practicalassociated with theintroduction of computers and computerlabs into undergraduate mathematics courses.Based on issues raised at a workshop of projectleaders who are seeking to make these changeson different campuses.

fl Ii r

Abstraction, 35Algorithm, 16, 20, 39f, 52f, 70-72, 73

proof by, 17Algorithm analysis, 46-47Algorithm design, 45Algorithm verification, 45Algorithmic language, 49Algorithmics, 45fAnalysis of Algorithms, 46-47Approximation, 84Arabic multiplication, 39Artificial intelligence, 28Assessment, 6AUTOCALC, 63-64

Black boxes, 72-73Blancmange function, 118Bottom-up design, 45Bourbaki, 3, 33

Calculator exercise, 44Calculators for young children, 59Calculus, 22f, 34, 80

computers in, 23C'AYLEY, 13Children, calculators for, 59

computers for, 59Classroom, computers in, 23, 35-36Clique problem, 54Complexity theory, 52Computability, 51Computation theory, 51Computer-aided design, 58Computer-aided instruction, 58Computer-aided learning, 110Computer algebra systems,

see Symbolic mathematical systemsComputer-assisted learning, 29Computer graphics, 23, 28, 117fComputer literacy, 34-35Computer science, 34, 37

theoretical, 49, 50Computers

experimental use of, 112-114for mathematics teaching, 25fin the classroom, 23instruction with, 110

Continuous functions, 82Courseware, 29

Index

131

3 c,

CUPM, 34Curriculum, mathematics, 75-76

in Japan, 76-77in the USA, 77

primary, 59fsecondary, 65funiversity, 20f, 82f

Curriculum change, 99-100

Data analysis, exploratory, 15software for, 70-71

Data structures, 53Databases, 28Difference equations, 49Differential equations, 74, 118

systems of, 120-121Discovery in mathematics, 23Discovery learning, 5Discrete-continuous interplay, 110fDiscrete functions, 82Discrete mathematics, 21, 34, 49, 80, 81fDynabook, 3Dynamical systems, 112

Early science software, 60-62Equations, numerical solution of, 43Errors, 82

student, 29Estimation, 82Euclid's algorithm, 40, 45, 47Euler's method, 113Experimentation in mathematics, 14, 65-66Exploration in mathematics, 23Exploratory data analysis, 15

Fast Fourier transform, 53Four colour theorem, 14Fractal, 16Function, 12, 82

blancmange, 118discrete, 82non-differentiable, 118plotting, 74recursive, 52software, 69-70Takagi, 13

Geometry, software for, 66-67visual exploration in, 121


(;raph algorithms. 53Graph theory, 49Guessing, 36

Hamiltonian path problem, 53Heuristics, 47Hypermedia, 28

Induction, 46, 49Instruction, individualized, 28Integration, 82Iterative methods, 15

Jacobi symbol, 53Journey in Mathematics project., 7Julia set., 16

KALEIDOSCOPE, 67-68Kcy handling programs, 62Kleene's theorem, 52

Language development software, 60Laptop computers, 36Learning, new styles of, 122Levels of research and development., 9Logic, 20, 54fLoop invariant., 45

Mac Math, 118Mathematician, student as, 2Mathematical communication, 17Mathematical induction, 46, 49Mathematical logic, 20, 54fMathematical metaphors, 36-37Mathematical modelling, 15Mathematics, discovery in, 23

experimentation in, 14, 65-66exploration in, 23new approaches to, 117-118science of, 19university, 19

Mathematics programs, 63Mathematics teaching, computers for,Matrix multiplication, 41Model building, 74-76Models, 111-112Moving target problem, 1Multimedia, 28

New mathemat les. 3curricula for, 6, 31

in Japan, 76-77in the USA, 77

Newton's Laws, 120Non-differentiable functions, 118NP-complete problems, 53Numerical analysis, 20Numerical solution of equations, 43

Operations research, 49

Partial correctness, 54Pattern matching, 53Plotting functions, 74Pocket calculator, 29Polynomial evaluation, 44, 46Primality tests, 53Primary schools:

curriculum, 59fsoftware for, 60f

Problem solving, 5Program correctness, 54Programming, 122Prolog, 54Proof, 13

by algorithm, 17

Quadratic formula, 42-43

Random numbers, 43Recursion theory, 52Recursive function, 52Representation, 84Resolution, 54Runge-Kutta method, 114

Satisfiability problem, 53Searching, 49, 53Secondary school curriculum, 65fSelf-evaluation, 28Simplex method, 53Simulation, 15, 73-74

25f stochastic, 74Smalltalk, 3Software, 29f

data analysis, 70-71for primary schools, 60ffor secondary schools, 66f

13

function, 69-70Soliton, 13Sorting, 49, 53Spreadsheet, 28Square-root construction, 41-42, 47Stochastic simulation, 74Student errors, 29Student-teacher relation, 26Symbolic mathematical systems, 17, 72, 93f

classroom sessions with, 97curriculum implications of, 99examples of use, 94-95, 105-106

individual access to, 98laboratory sessions with, 97software reviews of, 104-105

Symbolic solutions, 112Syntax analysis, 55

SYMMETRIC TURTLES, 66-67

Takagi function, 13Teacher, role of, 27Teacher-student relation, 26Teacher training, 89fTeaching, evolution of, 87Teaching assistant, macro as, 7Teaching-learning process, 109Theoretical computer science, 49, 50

Top-down design, 45

Total correctness, 55Towers of Hanoi, 42, 45, 46, 49Training methodology, 90-91Traveling salesman problem, 54

Turing machine, 52, 53

Unification-resolution, 54

University mathematics, curriculum for, 20preparation for, 19

Verification, algorithm, 45Visual data processing, 121Visual exploration, 121

kVIiite box, 7:/

Young children, calculators for, 59computers for, 59

1

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