Virginia Mathematics TeacherTEACHER (VMT) is published twice yearly by the Virginia Council of...

VirginiaMathematicsTeacher

A Resource Journal for Mathematics Teachers at all Levels.

Volume 37, No. 1 Fall, 2010

Spirals ofIsosceles Right Triangles

Volume 37, No. 1 Fall, 2010

VirginiaMatheMatics teacher

The VIRGINIA MATHEMATICS TEACHER (VMT) is published twice yearly by the Virginia Council of Teach-

ers of Mathematics. Non-profit organizations are granted permission to reprint articles appearing in the VMT provided that one copy of the publication in which the material is reprinted is sent to the Editor and the VMT is cited as the original source.

EDITORIAL STAFFDavid Albig, Editor, e-mail: [email protected] Radford University

Editorial Panel Bobbye Hoffman Bartels, Christopher Newport University; David Fama, Germana Community College; Jackie Getgood, Spotsylvania County Mathematics Supervisor; Sherry Pugh, Southwest VA Governor’s School; Wendy Hageman-Smith, Longwood University; Ray Spaulding, Radford University Jonathan Schulz, Montgomery County Mathematics Supervisor

MANUSCRIPTS & CORRESPONDENCEFor manuscript, submit two copies, typed double spaced. We favor manuscripts on disk or presented electronically in Word. Drawings should be large, black line, camera ready, on separate sheets, referenced in the text. Omit author names from the text. Include a cover letter identifying author(s) with address, and professional affiliation(s).

Send correspondence to Dave Albig at: Box 6942 Radford University Radford, VA 24142

Virginia Council of Teachers of MathematicsPresident: Beth Williams, Bedford County SchoolsPast-President: Carolyn Williamson, Retired from Hanover County Public SchoolsSecretary: Debbie Delozier, Stafford County Public SchoolsNCTM Rep.: Margaret Coffey, Fairfax County Public SchoolsMath Specialist Rep.: Corinne MageeElected Board Members: Elem. Rep: Sandy Overcash, Virginia Beach City Schools; Meghann Cope, Bedford County Schools Middle School Reps: Anita Lockett, Fairfax County Public Schools; Alfreda Jornegan, Norfolk Public Schools Secondary Reps: Ian Shank, Hanover Public Schools; Cathy Shelton, Fairfax County Public Schools. 2 Yr. College Rep: Joseph Joyner, Tidewater Community College 4 Yr. College Rep: Joy Whitenack, Virginia Commonwealth; Maria Timmerman, Longwood University Membership: Ruth Harbin-Miles

Publicity: Laura Rightnour, Hanover County Public Schools

Treasurer: Diane Leighty, Powhatan County Public Schools

Webmaster: Jennifer Springer, Charlottesville City Schools

Webpage: www.vctm.org

Membership: Annual dues for individual membership in the Council are $20.00 ($10.00 for students) and include a subscription to this journal. To become a member of the Council, send a check pay-able to VCTM to: VCTM c/o Pat Gabriel; 3764A Madison Lane, Falls Church, VA 22041-3678

Printed by Wordsprint Christiansburg225 Industrial Drive, Christiansburg, VA 24073

TABLE OF CONTENTS

Grade Levels Titles and Authors ................................................................. Turn to Page

General President’s Message .............................................................................1 (Beth Williams)

General Symmetry from Kindergarten to Calculus .............................................2 (Christine Latulippe) General Math Web Resources for Students, Parents, and Teachers .................5

General VCTM 2010 Math Beauty Contest Winners ..........................................5

General The Pascal-Fermat Correspondence: How Mathematics is Really Done .......................................................................................6 (Keith Devlin)

General Instructional Games with Calculators ....................................................9 (Wallace Judd)

General A Tale of the Test Journal ....................................................................11 (John E. Hammett III)

General Problem Corner ...................................................................................13 (Ray Spaulding)

Grades K-6 Is Elementary Education a Concern of Mathematics Association of America Members ........................................................22 (Patricia Clark Kenschaft)

General VCTM Fall Academy of Sweet Briar ..............................................23, 43

Grades K-8 Addressing Parents’ Concerns about Mathematics Reform ...............24 (Hendrickson, Siebert, Smith, Kinzler, Christensen)

Grades 8-12 Writing-to-Learn in Mathematics .........................................................28 (David L. Fama)

General Affiliates Corner ...................................................................................29

Grades 9-12 For Your Information - PUBLICATIONS ..............................................30

Grades 9-12 Delving Deeper: In-Depth Mathematical Analysis of Ordinary High School ..........................................................................31 (Dick Stanley and Jolanta Walukiewicz)

Grades 9-14 Empirical Approaches to the Birthday Problem ...................................37 (Alfinio Fores and Kevin M. Cauto)

Call for Articles (Fall 2011) ..................................................................40

ABOUT THE COVER: See Problem 203 in the Problem Corner to find out how this spiral of triangles was constructed and to find a related problem.

Virginia Mathematics Teacher 1

GENERAL INTEREST

President’s MessageBeth Williams

Mathematics Is . . .“Mathematics is much like the Mississipp. There are side-shoots and dead ends and minor tributaries; but the mainstream is there, and you can find it where the current-the mathematical power-is strongest. Its delta is reserach mathematics: it is growing, it is going somewhere (but it may not always be apparent where), and what today looks like a major channel may tomorrow clog up with silt and be abandoned. Meanwhile a minor trickle may suddenly open out into a roaring torrent. The best mathematics always enriches the mainstream, sometimes by diverting it in an entirely new direction.”

Ian Stewart, From Here to Infinity, Oxford University Press, Oxford, 1996, p. 11Reprinted from The American Mathematical Monthly, February 2010

Copyright by The Mathematical Association of America

Welcome back to a new school year! Do you know the many benefits of VCTM membership? As I step into this new position as your VCTM President, I would like to take a moment to introduce myself. Since 1980, I have worked in school buildings teaching mathe-matics. My 24 years in elementary classrooms have given me a love of and fascination for finding the connections between mathematics and all other subjects. My work as a Mathematics Specialist has given me respect for the depth of understanding needed at every level to teach mathemat-ics well. This year I step into a new professional role, lead-ing elementary mathematics and science instruction for my school division. This will require much collaboration with colleagues to promote the use of high level tasks and in-quiry based instruction as we all work to incorporate the new 2009 Mathematics Standards of Learning. For the past nine years, I have become increasingly im-pressed with what the VCTM membership offers. First, the fabulous professional development opportunities that we provide for our members, like our Teacher Academies. Back in the fall of 2007, VCTM implemented a series of Teaching Academies based on a model from NCTM’s pro-fessional development statement. The goal of these acad-emies was “to aid individual teachers and school districts in their effort to implement standards-based programs.” We are excited to continue this tradition with our fall academy at Sweet Briar College on October 1st and 2nd. The theme of this Academy is “Making Sense of Change.” All of the

sessions will be focused on incorporating the new 2009 Mathematics Standards into our teaching practice. If you have not yet registered, I urge you to do so. A registration form and “Conference At a Glance” details are available on our website www.vctm.org and in later pages of this journal. Another benefit of belonging to VCTM is that member-ship gives you access to many grants and scholarship op-portunities. Many of these offering are listed in our new newsletter. The first edition has already been sent out and we hope that everyone has received one. That mailing in-cluded an events calendar that all of our members need. If you did not receive this mailing, we may not have your cor-rect address. A quick email to [email protected] will put you on this mailing list. Don’t miss out on having this valuable information! One of the best perks of membership is the opportunity to create a network of colleagues. The beginning of a new school year is the perfect opportunity to invite others into our organization. Share VCTM members’ talents and ex-pertise with teachers just beginning their careers and with those new to our buildings. Bring them to our Annual Con-ference in Richmond in March of 2011, or to an Academy nearby. Become a mathematics mentor and share our journal ideas. Use this fresh start to try something differ-ent and to support your colleagues in their learning of new content.I wish you a wonderful school year!Beth Williams

2 Virginia Mathematics Teacher

GENERAL INTEREST

Symmetry from Kindergarten to CalculusChristine Latulippe

Geometry should not be thought of only as definitions and two-column proofs, but as a new and exciting way of looking at the world around us. Examples of symmetry are abundant and varied and useful as a method of engaging students to examine their worlds and their view of geom-etry differently. The four goals within NCTM’s Geometry Standard include:

instructional programs from pre-kindergarten through grade 12 should enable all students to apply transformations and use symmetry to analyze mathematical situations (NCTM 2000).

As with the other goals, this remains the same across the grade bands, but the tools, processes, and vocabulary of symmetry grow and develop as students progress through school. Similar ideas can be found in the California Math-ematics Standards at various grade levels. In general, if students have a strong understanding of symmetry and the language of symmetry, the more easily they will be able to use these tools to analyze mathematical situations. Early on we familiarize students with the idea of some-thing looking balanced, and humans are in fact naturally drawn to symmetrical things. There are three primary types of symmetry transformations.

► A first type of symmetry is reflections, which are often in-troduced as flips, folds, or mirror images. Consider fold-ing a piece of construction paper in half to cut a child’s valentine heart.

► A second type is rotations or point symmetry; these can be through of as turning an image to match it with itself, as with a child’s pinwheel, or the letter S.

► Third are translations, which can be through of in terms of wallpaper border patterns, a strong of paper dolls, or working at a computer to copy, slide, and paste an im-age.

With strong development of these basic ideas, symmetry can be a very useful mathematics tool through calculus. Recognition of symmetry is key in the early grades, and fortunately there is symmetry all around us to drawn upon for examples. Something as simple as solving picture puz-zles by turning pieces (rotations), flipping them over (reflec-tions), and experimenting with new arrangements (trans-lations) can be a rich exploration of symmetry. In nature, art, and architecture, it is especially easy to find examples of symmetry: reflection symmetry is most common (a cat’s face, the capitol dome in Sacramento), but rotation sym-metry is also present (a starfish or quilt). Especially in an elementary classroom where many subjects are woven together in a successful lesson, symmetry has a natural link to social studies, art, or science lessons. As teachers, we can help students develop the vocabulary that supports

their recognition and creation of symmetry, subtly laying strong foundations. By grades 3 through 5, students are expected to develop greater precision when they describe symmetry and con-gruence. Shapes with multiple lines of reflection are often introduced as this age, and students can begin exploring angles of rotation and terms such as vertical and horizon-tal when discussing reflection symmetry of specific figures. Humans are comfortable looking at symmetric designs, making the images available in magazines a rich starting point for a symmetry lesson. A trip to the local shopping center can also yield another collection of images in a very short amount of time. An easily accessible resource for devel-oping symmetry vocabulary is the alphabet. Consider the uppercase alphabet, in a sans serif font like Arial. Which letters will ap-pear the same if you fold them or turn them? Changing the fonts or using lowercase letters creates an entirely new examination and re-sults. Depending on the font you use, a capi-tal K may or may not have a horizontal line of symmetry, and a capital S may or may not have rotation symmetry. In addition to the more traditional forms of the alphabet, I have found that students at all levels are delighted and inspired by the “inversions” created by Scott Kim. An in-version is a word or name written so that it reads in more than one way, and Kim’s inversions range from basic to advanced. His book, Inversions, and web site www.scot-kim.com, include examples and resources for teachers. This word art can be used to introduce or to review the ideas of symmetry; printing them on overhead transparen-cies can help students to more fully understand the ideas of flipping and turning. Building on the ideas of logos and the alphabet, we want students to create their own symmetrical designs and patterns, not just identify them. As students to create a real word that has either line or rotation symmetry, or both. Have a logo-creating contest with four categories of winners: designs with no symmetry, designs with only rotation symmetry, designs with only reflection symmetry, and designs with rotation and reflection symmetry. By the middle school years, it is expected that students will be familiar with rotation and reflection symmetry and will begin a more in depth investigation of symmetry as it relates to congruence, similarity, and translations. Con-sider a set of Tangram pieces. The two large triangles are congruent to one another and the two small triangle pieces are congruent to one another. Each pair share the same shape and size. The small, medium, and large triangles are similar to one another―they have the same shape, equal corresponding angles, and the lengths of their correspond-ing sides are proportional to one another. Another way to illustrate similarity is to consider manipu-lating images with the computer. When scaled incorrectly,


images will be distorted, and not similar; when scaled cor-rectly, two images will be similar. Working with a coordinate grid or Cartesian plane in the middle grades allows us to expand our symmetry discover-ies to working with variables and general patterns. Begin-ning with sliding Tangram pieces about on a coordinate grid, we can discuss translations like moving a figure three units to the right and six units up. We can also record the coordinates of the vertices of the original object and its im-age, and look for patterns in a chart, resulting in the transla-tion (x, y) → (x + 3, y + 6). In high school, with the introduc-tion of vectors, these two moves can be recorded as one. Electronic manipulatives like those packaged with NCTM’s Navigating through Geometry in Grades 9-12 (Lott 2001) are another interactive way to have students examine re-flections, rotations, and translations of figures and explore properties. When students become more comfortable with symmetry transformations, they can explore the effects of combinations of symmetries. For example, if we look at an equilateral triangle, there are three possible rotations and three possible reflections, each of which creates a figure congruent to the original triangle. Combining these rota-tions and reflections introduces the question: “Is there a single transformation that would have the same result?” Organized charts of these results can be used to introduce algebraic ideas such as closure, commutativity, and an identity. A very useful presentation of these topics can be found in Kaleidoscopes, Hubcaps, and Mirrors: Symmetry and Transformations (2004) from the Connected Math-ematics Project. By high school, the same NCTM goal is taken to new depths, applying symmetry to the analysis of broader math-ematical situations, and expanding the middle grades ex-perience through the use of matrices, vectors, and dynamic geometry tools. A convenient way to review the basic lan-guage of symmetry is with images of car company logos and hubcaps. Another springboard for the discussion of symmetry, which many students may be familiar with is the inversions or ambigrams created by John Langdon for the book Angels and Demons. John Langdon also has a book, Wordplay: The Philosophy, Art, and Science of Ambigrams, and web site, www.johnlangdon.net, that provide step-by-step hints for creating your own symmetric word art. The California Algebra I standard 21 includes “graph quadratic functions,” a task that students typically consider disconnected from identifying a parabola’s axis of sym-metry and finding its roots. In the middle school years, we discuss the property of a reflection line-the corresponding points on the original and image will be an equal distance from the reflection line. If students can identify the axis of symmetry for a quadratic equation, and recognize it as a line of reflection, any brief table of values they create has double the power when graphing. Visually, a function and its inverse create a design with reflection symmetry where the line y = x is the line of reflection or mirror line. To graph the inverse of a function, students can use the same prin-ciples as when graphing a parabola―image points will be equidistant from the line y = x as are the corresponding im-age points. When students graph a parabola, or the inverse

of a function, they are creating designs with line symmetry, in much the same way elementary-aged children create valentines. (See Figure 1.)

Figure 1.

A discussion of even and odd functions often brings up ideas of symmetry. Even functions, where f(-x) = f(x), are symmetric about the y-axis have reflection symmetry, where the line of reflection is the y-axis. Odd functions, where f(-x) = -f(x), have rotation symmetry of 180o about the origin; if we place our finger on the origin and rotate the graph upside down, it will look the same, much like the capital S discussed in the early grades. Examples of even functions are y = x2 and y = cos(x); examples of odd func-tions are y = x3 and y = sin(x). Recognizing properties of symmetry can save students time and energy in the long run when they apply them to mathematical situations, such as integrating to find the area under a curve. Because of the line of symmetry, in a well-defined even function on the interval from -a to a, the area to the left of zero matches the area to the right, so it is safe to consider only one half of the graph when integrating and double the resulting area. Be-cause of the symmetry of a well-defined odd function, the area from -a to a will be 0; the area in Quadrant III matches the opposite of the area in Quadrant 1, so they will add to zero. Refer to Figure 2 and Figure 3.


ConclusionIntroducing students to the idea of symmetry can help them to look at the world around them with a fresh set of ques-tions to be answered and patterns to be seen. The ability to recognize, create, and communicate about symmetry also provides students with another approach for their rep-ertoire of problem-solving strategies; examining a problem from a different perspective is often the precise tool needed to come up with a successful solution path. One final example for viewing the world of geometry and symmetry with open eyes is crossword puzzles. Collect three to five crossword puzzles from your local newspaper and consider them from a symmetry point of view. What do crossword puzzles have to do with symmetry?

ReferencesKim, Scott. Inversions. Berkeley, CA: Key Curriculum

Press, 1996.______. Inversions by Scott Kim. <www.scottkim.com/

inversions/index.html>(July 28, 2009).

Langdon, John. Ambigrams, Logos, and Word Art. <www.johnlangdon.net> (July 28, 2009)

______. Wordplay. The Philosophy, Art, and Science of Ambigrams. New York: Boardway Books, 2005.

Lappan, Glenda, James T. Fey, William M. Fitzgerald, and Susan N Friel, eds. Kaleidoscopes, Hubcaps, and Mir-rors: Symmetry and Transformations. Needham, MA: Pearson Prentice Hall, 2004.

Lott, Johnny W., ed. Navigating Through Geometry in Grades 9-12. Reston, VA: NCTM, 2001.

National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Res-ton, VA: NCTM, 2000.

CHRISTINE LATULIPPE, Cal Poly [email protected]

Reprinted with permission from The California Mathematics Council ComMuniCator, Vol. 34, No. 2, December 2009.

• COGITO, designed for gifted precollege students to develop their interest in mathematics and science, includes interviews with experts, profiles of young scientists, science news, Web resources, and searchable directors of summer programs, competitions, and other academic opportunities.

www.cogito.org

• Mathematical Quilts is a collection of more than 30 quilts based on mathematical principles and theorems. Students can explore mathematics and gain geometric insight.

www.mathematicalquilts.com

• Math in Daily Life is a chance for students to explore how mathematics plays a role in our daily lives with common situations, such as cooking or buying a car.

www.learner.org/exhibits/dailymath

• MathMastery provides animated interactive lessons, daily thematic word problems, and family activities for teachers and parents of students in grades 3-8.

www.mathmastery.com

• Multiplication.com worksheets include multiplication flashcards, charts, quizzes, and tests for teachers to print for classroom use.

www.multiplication.com/worksheets.htm.

reprinted from theNCTM News Bulletin, April 2007

WEB BYTES


GENERAL INTEREST

Math Web Resources for Students, Parents, and Teachers(National Education Association)

www.cut-the-knot.com/This site offers an engaging collection of games and puz-zles, math problems for all ages, fascinating facts and sto-ries, a math book store on the Web, and a host of other interesting things about math.

www.enc.orgThis site is provided by the Eisenhower National Clearing-house for Math and Science education. It offers informa-tion on such topics as Solve it Summer Math Programs for grades 4-8; national math and science standards and state frameworks; curriculum; Ideas that Work; and a review of Internet sites that provide activities, lessons, and materials for children of all ages.

www.figurethis.orgHere is a fun, family-friendly learning activity designed to help kids appreciate math and to engage parents in work-ing with their children. Math Challenges are stimulating, real-life-oriented queries−designed by teachers and built around high-quality mathematics. Challenges are targeted to middle-school students.

http://mathforum.orgThe Math forum offers a wealth of information and activities including a math library, problems of the week, discussion groups, and “Ask Dr. Math.” It provides individual centers dedicated to students, teachers, parents, and citizens.

www-history.mcs.st-and.ac.uk/history/index.htmlThis is a searchable archive of the wonderful history of mathematics that allows students, parents, and teachers to learn about the men and women who created mathematics from its inception over 4000 years ago to present day.

http://Illuminations.nctm.orgThis site has activities in which parents and children can explore math ideas and concepts through animation. Par-ents and teachers can learn more about the National Coun-cil of Teachers of Mathematics document Principles and Standards for School Mathematics and how the vision of high-quality mathematics it espouses comes alive in a K-12 classroom.

www.mathnotes.com/aw_span_gloss.htmlThis site offers a good mathematics glossary for Spanish-speaking students.

http://mathematicallysane.comThis site, sponsored by a national grass-roots group of teachers, administrators, teacher educators, parents, and mathematicians, addresses concerns about the future of mathematics education. MathematicallySane’s mission is to advocate for the rational reform of school mathematics. It seeks to: help educators, citizens and policy makers at all levels make a stronger case for better mathematics pro-grams; gather and disseminate diverse success stories; and provide a forum for reform minded mathematics edu-cators.

http://www4.nas.edu/onpi/webextra.nsf/web/proficiency?Open DocumentThis site provides information on a new report from the National Research Council, Adding It Up: Helping Children Learn Mathematics. This report urges for mathematics ed-ucation in this nation to be satisfactory, major reforms are needed in mathematics instruction, curricula, and assess-ment from pre-kindergarten through grade 8.

VCTM 2010 Math Beauty Contest Winners

K – 2 Shawn Harvey from Henry Clay Elementary School in Ashland, VA

3rd – 5th Charlotte Camp from St. Catherine’s School in Richmond, VA

6th – 8th Justin Tooley from Powhatan Junior High School in Powhatan, VA

9th – Algebra I Kristina Dickey from St. Catherine’s School in Richmond, VA

Above Algebra I Holiday Shuler from Langley High School in McLean, VA


GENERAL INTEREST

The Pascal-Fermat Correspondence: How Mathematics isReally Done

Keith Devlin

THE UNFINISHED GAME Known as the Unfinished Game problem, the puzzle asked how the pot should be divided when a game of dice has to be abandoned before it has been completed. The challenge is to find a division that is fair according to how many rounds each player has won by that stage. Today the (polished!) solution to the problem can be ex-plained to high school students in a few minutes (see the sidebar), but when you read Pascal’s letter you realize that it didn’t seem at all obvious to him how to solve it. Pascal begins his letter hesitantly:

I was not able to tell you my entire thoughts regarding the problem of the points by the last post, and at the same time, I have a certain reluctance at doing it for fear lest this admirable harmony which obtains between us and which is so dear to me should begin to flag, for I am afraid that we may have differ-ent opinions on this subject. I wish to lay my whole reasoning before you, and to have you do me the favor to set me straight if I am in error or to indorse [sic] me if I am correct. I ask you this in all faith and sincerity for I am not certain even that you will be on my side.

Just think about that. The two have already exchanged several previous letters on the subject, and still one of the greatest mathematicians of all time is not sure whether he has got it right. It turns out he hasn’t (at least not fully), al-though an alternative and much simpler approach suggest-ed by Fermat, which Pascal also summarizes in the letter, does work. (See the sidebar for Fermat’s simple solution.)

PASCAL’S SOLUTIONMuch of the nearly three-thousand-word letter is devoted to Pascal’s attempts to get his own approach to work. His ap-proach is so convoluted that it is difficult to follow. But that is precisely why I think it would be valuable to show stu-dents this historical document. It provides a close-up view of mathematical sausage in the making, in all its messy de-tail. In summary, Pascal approaches the problem by look-ing at the quantity e(a, b), which represents the share of the stake that player 1 should be given if player 1 requires a winning throws to win and player 2 requires b winning throws. Clearly, if the numbers a and b are equal, then e(a, b) = 1/2. The idea is to see how e(a, b) changes when each player wins one more throw. This idea leads to an algebraicexpression for e(a, b) in terms of e(a – 1, b) and e(a, b – 1), and Pascal solves the problem of the points by using recur-sion to calculate e(2, 3), the desired share in the particular game they considered. This solution requires some com-plicated algebra dependent on the theory of combinations Pascal worked out in connection with his famous triangle. Students can read Pascal’s entire account of his argu-ment in the August 24, 1654, letter on the Web (www.york.ac.uk/depts/maths/histstat/pascal.pdf). The argument has

A letter records how two of the greatest mathematicians of all time struggled for several weeks to solve a probability problem.

According to an old saying, there are two things to avoid seeing made—laws and sausages. See either process, and you will no longer like the product. Mathematics is the opposite. Few people ever see new mathematics being made, and yet, if they did, they might well like the product a whole lot more. The mathematics our students see presented in their textbooks is highly polished. The steps required to solve a problem are all clearly laid out, the methods having been honed to perfection by many generations of teachers and authors. The result is that students are denied what could be a valuable learning experience. Often when students meet a problem that differs only slightly from the ones in the book, they are unable to proceed, afraid to “play with” the problem for a few minutes to see whether they can find a way to do it, convinced that they simply do not have what it takes to do mathematics. No matter that the teacher makes suggestions—after all, mathematics teachers get their jobs precisely because they are among those rare people who are born miraculously able to see how to do it, right? But if students could see examples of the false starts and the erroneous attempts of the experts, they might be more in-clined to persevere themselves. The same type of reaction occurs at the college level, when students encounter advanced mathematics. They see Euclid’s proof that there are infinitely many prime num-bers or the classic ancient Greek demonstration that √2 is irrational, and they think they could never come up with the clever tricks those arguments use. A lot of the mystique about what it takes to do mathe-matics might be dispelled—surely to the benefit of math-ematics education—if our students occasionally saw how professional mathematicians measure up when they try a problem for the first time. Unfortunately, when mathema-ticians finally manage to solve a problem, they generally throw away the reams of false starts and failed attempts and show the world only the final, polished, and sanitized solution. Such sanitizing can give the impression that doing mathematics requires a highly unusual mind. One exception is a letter, never intended for publication, sent by one of the best mathematicians the world has ever seen, to a colleague of even greater stature. On Monday August 24, 1654, the French mathematician Blaise Pascal (of Pascal’s triangle) sent a letter to his countryman Pierre de Fermat (of Fermat’s last theorem), outlining the solution to a problem that had puzzled gamblers and mathemati-cians alike for decades.


also been reproduced in its entirety, together with a com-mentary (Devlin 2008). Pascal’s actual mathematics doesn’t really matter. He went off in a direction that doesn’t work well, and he be-came confused. He ends his letter with a plea for help:

Consequently, as you did not have my method when you sent me [your solution], [hence] I fear that we hold different views on the subject. I beg you to inform me how you would proceed in your research on this problem. I shall receive your reply with respect and joy, even if your opinions should be contrary to mine.

The moral of the tale is clear: Even professional mathe-maticians don’t necessarily get it right the first time, or even the second, or the third. The secret—but it really should not be a secret—is to just keep trying. Successful math-ematicians learn from their mistakes, sometimes work with someone else, and occasionally ask for help.

THE LETTER AS A CLASSROOM RESOURCE The educational benefit of students examining false starts and failed attempts to solve problems is well known and discussed in, for example, Brown and Walter (2004). The August 24, 1654, letter from Pascal to Fermat is particularly well suited for exposing high school and col-lege-level students to the process of actual mathematical discovery and problem solving for a number of reasons. First, the problem itself is a simple one that requires no mathematical knowledge to understand. Second, many accomplished mathematicians failed to solve this problem over several hundred years, some of whom went as far as to conclude that it was unsolvable. Yet when the problem was finally solved, the solution was an extremely short and simple one that required no mathematical techniques beyond counting. Moreover, the solution to the problem turned out to be pivotal in the devel-opment of modern society, leading directly to the develop-ment of modern probability theory, risk management, fu-tures prediction, and the insurance industry (Devlin 2008).How often can a scientific advance of such magnitude be made the focus of a middle school or high school math-ematics class? A class can begin by carrying out a practical exploration of the problem. Students can obtain an empirical solution by repeatedly tossing a coin or rolling a pair of dice. An obvious way is to have students work in pairs, with one student in the role of the player who has won two games, the other in the role of the player who has won one game. The student pairs then play out the remaining (“unplayed”) two rounds, recording which player wins three out of five. They should find that the player who starts out having won two rounds wins the imaginary five-game tournament three times as frequently as the other player. To gain some insight into one of the issues that chal-lenged Pascal, students can repeat the exercise with the amended rule that they stop playing as soon as one player has won three rounds. They will again find that the player who starts ahead wins roughly three-fourths of the time.

With the practical exploration behind them, students should have no trouble following Fermat’s argument. Even better, they can be asked to try to find a solution them-selves, either singly or in groups. Students can then be asked to try to follow Pascal’s own attempted solution. They can read Pascal’s own words as he tries to grasp the simple solution Fermat has sent him―the very solution the students have just discovered for themselves. The teacher should make it clear that the aim is not to fully understand Pascal’s intricate reasoning but to see just how much more complicated it is than Fermat’s. Students can also be asked to speculate exactly why a renowned mathematician like Pascal had such trouble fol-lowing Fermat’s reasoning. (No one knows for sure. Most likely part of the problem was that the very idea of counting hypothetical futures was entirely novel, although other fac-tors are possible [Devlin 2008].) Teachers may want to show the class an excellent video treatment of Fermat’s solution, a five-minute segment from program 6 (“Chances of a Lifetime”) in the PBS television series Life by the Numbers, first broadcast in 1998 (avail-able on DVD at www.montereymedia.com/science/). Pro-gram 6 also has other highly informative segments about probability theory. (The other five programs also provide valuable classroom resources, although so rapid has been the progress in real-world applications of mathematics that many are already quite dated.

FERMAT’S SOLUTION TO THE UNFINISHEDGAME PROBLEM Two gamblers, Blaise and Pierre, place equal bets on who will win the best of five tosses of a fair coin. On each round, Blaise chooses heads, Pierre tails. But they have to abandon the game after three tosses, with Blaise ahead, 2 to 1. How do they divide the pot? The idea is to look at all possible ways the game might have turned out had Blaise and Pierre played all five rounds. Since Blaise is ahead 2 to 1 after round three, the first three rounds must have yielded two heads and one tail. The re-maining two throws can yield these combinations:

H H H T T H T T

Each of these outcomes is equally likely. In the first, the final outcome is four heads and one tail, so Blaise wins; in the second and the third, the outcome is three heads and two tails, so again Blaise wins; in the final outcome, the re-sult is two heads and three tails, so Pierre wins. This means that in three of the four possible ways the game could have ended, Blaise wins; in only one possible play does Pierre win. Blaise has a 3-to-1 advantage over Pierre when they abandon the game. Therefore, Blaise should receive 3/4 of the winnings, and Pierre should receive 1/4. This solution may seem “simple” today, but it definitely id not seem that way to the two mathematicians who worked it out. In fact, several world-class mathematicians had tried to solve the problem earlier and had failed completely. Some of them even went to far as to declare that the problem could not be solved. Now where have you heard that before?


FINAL REMARKS The Pascal-Fermat correspondence is an excellent teaching resource. It shows students that mathematics does not come easily, even to the world’s best mathemati-cians; that it can take time and effort even to understand a problem, let alone solve it; that the experts make mistakes; and that the principal requirement for doing mathematics is perseverance. Moreover, this resource does all this with a problem that is not only real but also one whose solu-tion was seminal in the development of modern society. There are, to be sure, other such examples, but they lack one feature that makes the story of the unfinished game such a valuable educational resource. The mathematics is short, simple (to today’s reader), and totally accessible to a middle school student. I stumbled on this superb educational example by ac-cident. Like many of my colleagues, I knew about the role Pascal and Fermat played in the establishment of modern probability theory, but until I researched the history, I never realized just how dramatic was the change in society their correspondence brought about, leading from a widely ac-cepted belief that mathematics could not be used to predict the outcome of future events, to the establishment of mod-ern predictive probability theory, risk management, actu-arial science, and the insurance industry, all within a single lifetime. Nor did I appreciate just how great was Pascal’s confusion nor how fully he displayed it in his letter.

I wrote this article to make this wonderful example more widely known among the mathematics education community.

BIBLIOGRAPHYBrown, Stephen I., and Marion I. Walter. The Art of Problem

Posing. 3rd ed. Hillsdale, NJ: Lawrence Erlbaum Associ-ates, 2004.

de Fermat, Pierre, and Blaise Pascal [the complete extant 1654 correspondence]. “Fermat and Pascal on Prob-ability.” Translated into English. www.york.ac.uk/depts/maths/histstat/pascal.pdf.

Devlin, Keith. The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern. New York: Basic Books, 2008.

Life by the Numbers. [PBS television series, WQED-TV]. www.montereymedia.com/science/.

KEITH DEVLIN, [email protected], is a mathematician at Stanford University in Palo Alto, California, whose educational interests focus on using different media to teach mathematics and to raise the general awareness of mathematics. He lectures frequently at schools, colleges, and public venues, has written a number of mathematics books for general readers, was an advis-er to CBS television for the first season of the NUMB3RS fictional crime series, and appears regularly on National Public Radio as “the Math Guy.” Richard Ressman

Reprinted with permission from Mathematics Teacher, copyright April 2010, by the National Council of Teachers of Mathematics. All rights reserved.


GENERAL INTEREST

Instructional Games with CalculatorsWallace Judd

The powerful idea of this article illustrates how to use readily available technology to enhance the learning of concepts in ways that are fun for students. It can easily be adapted to all levels from kindergarten through calculus.

This is a brief collection of calculator games that I have played with kids (and adults) over the last two years. Each game can be explained simply enough for first and second graders to understand, yet the games can be played with variations difficult enough to make most mathematics teach-ers stop and think. The mathematics requirements for all the games are minimal. Only the “Before” and the “After” games require a constant for addition and subtraction; all the others require at most a constant for multiplication and division, which is standard on most small calculators. None of the games require a machine memory, and all of them can be played in less than five minutes. The kids with whom I have played these games have enjoyed them very much. I hope you do, too.

Nim Teaches: Addition and place value concepts For: Two players and one calculator Object of the game: To get 67 on the display How to play: The first player pushes a single digit key (not zero), then pushes the + key. The next player takes his turn by pushing a single digit key (again not zero), then push-ing the + key. Players take turns until a player pushes the + key and the display reads 67. The player who pushes + and gets the display to show 67 wins. If a player pushes + and the display shows a number larger than 67, that player has gone “bust” and loses. Variations for primary grades: Use only the first row of digits─the 1, 2, or 3 keys─and 21 as the goal. Variations for junior high: Use the first column of digits─the 1, 4, or 7 keys─and go to 47.

Wipeout Teaches: Place value For: One number-giver and any number of players, each with a calculator. Object of the game: To remove one digit from the display without changing any of the other digits. How to play: The number-giver picks a number, which all players enter into their calculators, and says which digit is to be removed. Good numbers are those that have a pat-tern, so people can tell easily if any digit but the selected one has changed. Example: In the display, 876543, wipe out the 7 without changing any other digit. This is done by subtracting one number from the number on the display. So key in -, then the number to be subtracted, and press =. Does the display read 806543?

People can take turns giving numbers to each other and selecting the digit to be wiped out. Variations for grades 2-3: Limit the display to three digits. Variations for junior high: Use decimals in the display. Example: Wipe the 8 out of .567891.

Before Teaches: Counting and place value concepts For: Any number of players and a calculator that has a constant for subtraction. (To see if the calculator will work for this game, key in 1-=====. If the answer is 0 and does not change, the calculator will not work. If the answer changes from 0 to -1 and -2 to -3 to -4, . . . , then the calculator will work for this game.) Object of the game: To predict what number will show on the display when = is pushed How to play: Start the game by keying in the sequences 1 - =. Do not push C, the “clear” key, during the whole game. Check to see if the calculator is set by keying in 10 =. If the display does not read 9, something is wrong. Clear the machine and start over again. Key in a number and ask the students what number will come when the = is pushed. See who guesses correctly. Keep score if you want to make a competition out of the game. Example: Key in 45. Have students guess, then push =. The result should be 44. If the game is too simple, use starting numbers of 90, 110, or 1010. Two students can play, taking turns putting in numbers and see if the other can guess what will result after = is pushed. Variations for grades 4-6: Start with 10- = or with 100 - =. Variations for junior high: Use .1 - = or .01 - = as the start-ing sequence. A real stumper is .05 - =.

After “After” is a variation of “Before,” in which the constant is added to rather than subtracted from the number on the display. To set the calculator up, key in 1 + =, or 10 + =, or 100 + =. Then play exactly the same as “Before.” Example: Key in 1 + =. Then key in 45 and have students guess the result before the = is pushed. The result should be 46.

Solitaire Teaches: Basic mathematics facts and calculator functions For: One player and a calculator Object of the game: Using only the legal keys, to get the goal number on the display How to play: Pick a goal number and a set of “legal” keys. The game is to see who can get the answer in the fewest keystrokes. Example: Goal number, 17, and legal keys 5, +, -, x, ÷, or =. Using only these keys, get 17. One answer would be

5 + 5 ÷ 5 = + 5 + 5 + 5 =.


This solution takes 13 keystrokes. There are shorter solutions. Pick different goal numbers or vary the legal digit. Variations for primary grades: Use goal numbers that are multiples of the legal digit. Example: Using 2 and the keys +, -, x, ÷,or =, get 24. A first grader might do 2 + 2 + 2 + 2 . . ., until he got to 24. A shorter solution would be 22 + 2 =. Variations for junior high: Use negative goal numbers Example: Using 5 and +, - x, ÷ or =, get -3. Sophisticated solutions can be implied. Example: Using 5 and +, - x, ÷, or =, can you get to 26 in only ten keystrokes? (Try it before looking at the solution at the end of this article.)

Target K Teaches: Decimal place value and bracketing guesses For: A whole class and a calculator, or a small group or single student and a calculator Object of the game: To get the target number on the dis-play. The decimal part of the target number doesn’t count. For example, if the target is 500, a display of 400.4716 is correct. How to play: Put the multiplication constant into the ma-chine. (Warning: do not clear the machine after the constant has been put in, otherwise the constant will disappear.) The object of the game is to guess what number, when multiplied by the multiplication constant, will give the target number. Example: If the target number is 500 and the multiplica-tion constant is 17, then key in 17 x =. Make a guess, say 50 =, which gives 850―too large. Make another guess, say 20 =, which gives 340―too small. The correct answer must be between 20 and 50. When students find that 29 = is too small and 30 = is too large, they will begin trying decimals. Variations for primary grades: Limit the target numbers to exact multiples of the constant. For example, the target number could be 265 and the constant 5 x =. Still simpler is a target of 24 and a constant 8 x =. Variations for junior high: Pick a small target number and large constant.

The Big One Teaches: Decimal place value For: A small group or a single player and one calculator Object of the game: To get the display to read 1 How to play: One person sets up the calculator for the player(s) without the players seeing what was put in the calculator. The calculator is set up by keying a number, then ÷ =. After the calculator is set, players guess numbers and push = until the display is 1. Example: Select a “mystery” number between 1 and 100―say 2. Put it into the machine by keying 27 ÷ =. Push-ing the ÷ = keys makes the number 27 a constant divisor. (Caution―do not clear the machine at any time after the mystery number has been put in or the number will be wiped

out.) Give the calculator to the players, who try to guess the mystery number by keying in trail numbers and then pushing =. When the mystery number is guessed, the display will show 1 after the = key is pressed. If the number tried is not correct, the display gives clue by showing what the guess divided by the mystery number equals. Variation for primary grade: Limit the mystery numbers to between 1 and 10. Variation for junior high: Use three-digit mystery numbers.

Those are the games. Here are a few tips on how to in-troduce them to a class. Probably the best introduction is the play the game first with the entire class, or to play it with one person in front of the entire class. Even though the class cannot see the tiny digits on the calculator display, you can read the numbers out loud to them. This gives the students a feel for the game that they cannot get by just reading the directions at an interest center. After the rules have been explained and they have seen the game played once or twice, then let the students play in pairs or in small groups. After they have done that for five or ten minutes, students should be able to play the game independently. If you show them the game without letting them play it immediately, they have usually forgotten the rules or lost interest by the time their turn at the interest center comes. Although the kids get pleasure from playing any of these games just once or twice, the real benefits accrue to a student after playing the same game a number of times. Strategies become more sophisticated and generalizations develop. So introduce a single game and let the students play it briefly but regularly over a two- to three-week period. Generate enthusiasm through contests, posted problems about the game, or championship. Then introduce another game. This strategy allows these games to be real teaching aides, rather than simply amusing pastimes.

Note: The solution to the “Solitaire” puzzle is 5 x = x - + 5 ÷ 5 =.

For an extension of this idea, see the article by James E. Schultz, “The Constant Feature: Spanning K-12 Mathematics,” in the March 2004 issue of Mathematics Teacher, pp. 198-204.―Ed.

A word on the editorial approach to reprinted articles: Obvious typo-graphical errors have been silently corrected. Additions to the text for purposes of clarification appear in brackets. No effort has been made to reproduce the layouts or designs of the original articles, although the subheads are those that first appeared with the text. The use of words and phrases now considered outmoded, even slightly jarring to modern sensibilities, has likewise been maintained in an effort to give the reader a better feel for the era in which the articles were written.―Ed.

Reprinted with permission from Mathematics Teaching in the Middle School, Feb 2007 by the National Council of Teachers of Mathematics. All rights reserved.


GENERAL INTEREST

A Tale of the Test JournalJohn E. Hammett III

Matthew sat squirming in his seat in math class, fidget-ing absentmindedly with his mechanical pencil. “I studied for this test,” he thought to himself dejectedly, as he aim-lessly twirled his writing implement, staring at the last few test items as if they were suddenly written in some foreign language he didn’t speak or read. “I tried the practice prob-lems that were assigned for homework, did pretty well with them, but these types of word problems have always con-fused me,” he said silently. Matthew slowly realized that he wasn’t going to ace this exam like he had hoped, or like he’d promised his parents. He did what he could to attempt to solve those last few problems, and then got up to hand in his test paper to the teacher at the front of the classroom. “This one’s definitely going to be a test journal, Dr. Wright-Ng. I can feel it already,” Matthew said quietly to his instructor who watched him approach the front of the room with a somewhat frustrated look on his face. “I’ll keep thinking about those word problems, and be ready to tackle them again when you give the papers back. When do you think that will be,” Matthew half-heartedly inquired, not re-ally in any rush to reenter the murky waters he was wading through as he struggled to solve those last few problems. “I’m planning to get them back to you by the end of the week, so you can have the weekend to complete the jour-nal. Let’s see how you did, first, okay?” said Dr. Wright-Ng in an encouraging tone. “Okay,” Matthew said with some uncertainty as he returned to his seat. Dr. Jhernelle Wright-Ng was glad she had introduced the test journal into her tool chest of teaching implements, borrowing it from an enthusiastic and engaging professor who shared this idea and others about writing to learn tech-niques at a mathematics education conference in New Jer-sey. Dr. Wright-Ng had been pleased to find that a good number of her students benefitted from the second chance opportunity that the test journal writing experience had af-forded them. As she continued to proctor the test, her mind sifted through a number of questions that she initially had about the test journal and the answers she heard during the presentation by the professor in New Jersey.“What exactly is a test journal,” she had wondered? It’s a journal writing exercise under the larger umbrella of writing to learn techniques. What makes it so appealing is that it extends the shelf life of tests and exams beyond the time when these evaluations are retuned by the instructor. Al-though students might briefly engage with tests and exams when returned by their teachers, interest usually wanes fast and not much extra educational value is gained. How-ever, the test journal requires further interaction with the test items as part of the journal writing process beyond the test period itself and any brief review when they are re-turned. And without the requirement that students revisit the test items in detail, how many really do so voluntarily? Not many; probably not any. “How does the test journal work?” When students earn low grades or unsatisfactory scores on tests, their teachers

can offer them the opportunity to revisit the exams in the form of a supplemental writing assignment: they write a test journal about the experience. Although students can vent frustration about their poor performance, the intent of this journal writing is not to be a diary entry. Instead, when the learners receive their graded tests back from their teachers, they review them thoroughly, re-visiting all questions or problems that they answered either incompletely or incorrectly. When the students get their papers back, they should clearly and unequivocally know which items were somehow unacceptable. The teachers score but don’t typically fully correct the submitted test items. This should expedite the initial grading, which is a perk for teachers; more importantly, by teachers not com-pletely correcting wrong answers, the students still need to think about their mistakes. Teachers can minimally mark the students’ work, indicating which questions are incom-plete and/or incorrect; they can also return scoring rubrics with the tests which could be very informative to the stu-dents. Instructors can even offer, if they so desire, some nominal suggestions or hints. No answer key is posted, distributed, or even reviewed when the tests are returned. The students then need to determine which ones were incomplete, which ones were incorrect, and which might have been both! The learners have the responsibility to review those test items, correct and/or complete them, and resubmit them with annotations or commentary. As they are completing their journal entries, the students identify in writing why those original responses were wrong; they also revise those responses, making sure they completely and correctly answer each test question that did not earn full credit. This activity works not only with open-ended problem-solving questions but also with short-answer ob-jective, multiple choice items; the test journal can even be used as a follow-up review of any preparatory testing done in advance of actual high stakes standardized testing. “Won’t reading these journal entries be a whole lot more work for the classroom teacher?” As previously suggested, initial grading can be streamlined because only minimal corrections should be made. And improved papers should be more readily reviewed, since the solutions should be more complete and correct. “How is each journal entry actually constructed?” A sug-gested format is as follows: students turn a piece of pa-per sideways or landscape orientation. About one-third is folded to make a wide margin. The students present their revised work in the main section, and offer corresponding commentary to accompany the completed and corrected answers in the aforementioned wide margin adjacent to the work. The original scored but uncorrected test is attached to the journal for reference purposes. “Who should complete a test journal?” These journal en-tries should be completed and submitted on a regular ba-sis whenever students perform poorly on tests. However, each teacher would set some level of expectation about


when a test merits a journal assignment and when it does not. For example, a teacher can require a journal from any student who fails a test and offer the opportunity as an op-tion for any student who earns a marginally passing grade. Students who do well on tests may be exempted from the assignment, because they have demonstrated mastery of the material. Finally, teachers can limit how many test journals students can complete in a given term or marking period; that way, students might need to decide for them-selves when to write these journal entries. “What types of responses do students offer to the errors they make when they write these journal entries?” In terms of the mathematical work, student entries can run the gam-ut from correcting careless miscalculations, to complete skipped parts of problems, to change direction altogether in their problem solving strategies. Students can use this op-portunity to identify and correct their own misconceptions and misperceptions. They can reach an intended correct and complete answer during this second iteration. Some-times, however, students still don’t reach closure: a correct and complete solution. This, in and of itself, is vitally impor-tant information for the teacher to receive. In terms of the verbal commentary, as might be expected, some remarks are positive in nature while others are nega-tive. Some students who effectively utilize the journal writ-ing experience are able to correct their mistakes completely and accurately (e.g., “Here is where I went wrong.”); others offer an embarrassed apology for their mistake (e.g., “My bad!”); still others ponder why they made the mistake in the first place (e.g., “What was I thinking?!”). Admittedly, how-ever, some students at least initially squander the opportu-nity; these learners temporarily cannot solve the incorrect or incomplete problems entirely. Sometimes they make the same mistakes (e.g., “Are you sure this isn’t right?”), make a new mistake (e.g., “A new answer; I’ll take it!”), or even give up in exasperation and frustration (e.g., “No clue!”). In the latter cases where the students remained unsuccess-ful in solving the problems, the teacher can return the test items an additional time for revision, or can opt not to ac-cept it anymore, regrettably ending up with a lost opportu-nity for the student.

“What if students admit they still have no clue as to how to answer a particular test question, even after the graded tests have been returned?” The teacher can encourage them to be persistent, and to get help from a classmate or a tutor. The purpose of this writing assignment is for the student to reach closure by arriving at an appropriate and correct answer for the test items they got wrong, even if help is needed in achieving that goal. “Aren’t students really cheating on this post-test as-signment by asking classmates to share their correct an-swers?” Not really, because the journal writers must still compare and contrast these best responses against their original answers. They must put the work into their own words and take ownership of the ideas. “What if the students fail to complete or correct any of the test items?” The teacher sets the ground rules as to whether the student can make another attempt or not. No matter what, the process identified a lingering gap in stu-dent knowledge; this is, as previously mentioned, an im-portant piece of information for the teacher. The journal serves, therefore, as a reasonable assessment tool for the teacher. “So, the test journal should be considered an as-sessment tool?” Yes. As another student approached Dr. Wright-Ng’s teacher desk to submit a completed test, she partially snapped back into reality. Still, as she shuffled the tests on her desk, Dr. Wright-Ng couldn’t help but realize that, for all the reasons she just reviewed, the test journal assignment does indeed extend the educational impact of the traditional paper-and-pencil test beyond both the date she originally gave the test and the date when she returned the graded exams to her class. A smile grew on her face as she noted, perhaps most importantly of all, that her students who complete test journals construct for themselves some degree of success out of their own failures. The test journals can form a silver lining to any dark storm clouds created when her students test poorly; these writing exercises take an otherwise nega-tive classroom experience and potentially turn it into some-thing more positive, more satisfying, and more instruction-ally beneficial.

JOHN E. HAMMETT III, Ed.W. is an associate professor of math-ematics at St. Peter’s College in New Jersey.

Mathematics Is . . .“Mathematics is persistent intellectual honesty.”

Moses Richardson, Mathematics and intellectual honestykthis Monthly 59 (1952) 73.

Submitted by Carl C. Gaither, Killeen, TX

Reprinted from The American Mathematical Monthly, August-September 2009Copyright by The Mathematical Association of America


GENERAL INTEREST

Problem CornerRay Spaulding









I take space to be absolute.- Isaac Newton

I take space to be something purely relative, as time is. - Gottfried Leibniz

__________

A theory has only the alternatives of being right or wrong. A model has a third possibility; it may be right but irrelevant.

- Manfred Eigen

Reprinted from The College Mathematics Journal, September 2009Copyright by The Mathematical Association of America


GRADES K-6

Is Elementary Education a Concern of MathematicsAssociation of America Members

Patricia Clark Kenschaft

Last year in a “nice” white suburban town a fifth grade teacher was observed drilling her students in adding frac-tions by adding across the numerators and then adding across the denominators. Is this teacher an outlier? I fear not. Some years ago I went to a fifth grade class in one of New Jersey’s wealthiest districts. “Where is one-third on the number line?” I began. All those friendly white faces fell to the floor, so I repeated the question. “Near three?” the teacher guessed. She is one of the highest paid fifth grade teachers in the country. You can read about my adventures during seven years of helping elementary school teachers mathematically in “Ra-cial Equity Requires Teaching Elementary School Teachers More Mathematics” (Notices of the AMS, February 2005, also online at http://www.ams.org/notices/200502/feaken-schaft.pdf). I was inspired by the teachers’ eagerness to learn and their ability to do so, but distressed at their lack of mathematical knowledge. The teachers had emerged from a flawed system. Too often elementary school teachers teach incorrect “mathematics” and also communicate to their students that mathematics is too difficult for ordinary mortals. “If my teacher doesn’t understand this, I can’t either.” Such intel-lectual and emotional damage is so devastating that even a teacher who is mathematically competent will find it very difficult to undo. High school teachers and remedial college faculty must overcome much more than lack of knowledge. It seems to be that the critical path toward improving our entire math education system is helping pre-service elementary school teachers before they damage children. Mathematicians may have some reluctance to teach these courses, based partly (justifiably) on the difficulty of doing so, but also (less justifiably) on the perceived lack of intel-lectual challenge. One memorable semester, I taught three first grades each Wednesday morning, introductory calculus each Wednesday afternoon, and abstract algebra each Wednesday evening to graduate students. One evening I found myself saying, “When we were discussing this topic this morning in first grade...” The class roared in laughter, but I continued. The fundamental topics of abstract algebra are presented in first grade! Furthermore, the pedagogical approaches that reached the children were useful for graduate students. After the final exam that semester, the graduate students stood around and one said, “I think we learned a lot more this semester because you were also teaching first grade!” I eventually was able to teach pre-service elementary school teachers. Grappling with basic mathematical con-cepts with adults who don’t love math is very different from exploring them either with first graders or with graduate students. But they too can be enticed to reexamine con-cepts that they had been taught were “stupid questions.” I used Thomas Parker and Scott Baldridge, Elementary

Mathematics for Teachers (Sefton-Ash Publishing, 2003, see http://www.singaporemath.com), a text accompanied by five Singapore children’s texts, and was quite smitten. There may be other fine teacher-preparation programs, but I know there is at least once. Some teachers have told me that they are ordered by their superiors, “Teach only one method! More than one confuses the children.” It is hard to estimate the mathematical dam-age done by this widespread admonition. When elementary “mathematics education” consists of inculcating children with algorithmic skills, never to be questioned or varied, what does that do to citizens’ ability to think mathematically? Another pernicious aspect of elementary mathematics education is key words. One widely used test-prep program advocates, “When you see ‘each,’ multiply.” Administrators claim that such drilling improves test scores. One Montclair State University student intending to become an elementary school teacher insisted that because American small trucks had an average gas mileage of 20 mpg in 1999 and sedans had an average of 28 mpg, altogether they must have an average of 48 mpg. She was a pleasant person who knew she was outvoted, but no matter how many of her classmates tried to explain why the average must be between 20 and 28, she clearly felt betrayed. “‘Altogether’ means ‘add,’” she kept saying, incredulous that she had been taught wrong all these years. Persuading college students to abandon key words has been more challenging for me than leading them to enjoy mental math. One actually said, “How else are you sup-posed to learn?” Pre-service and in-service teachers can learn to think, even after decades of faulty teaching and administrative admonitions. Furthermore, they want to learn - in my experi-ence, without exception. Once, after exploring the multipli-cation algorithm using base ten blocks, a teacher became angry: “Why wasn’t I taught this before? I’ve been a third grade teacher for thirty years, and I could have been such a better teacher if someone had let me in on this secret thirty years ago!” Most memorable, perhaps, was a whole day that I spent with 28 Newark third grade teachers, two each from 14 schools. I changed the (math) subject every 40 minutes, and I’ve never had a more rapt class. After the first break they came to me with a question on a standardized third grade test “that none of us can answer. Can you?” I could and did. It was a combinatorics problem, reasonable for third graders but much harder after you have been taught never to think in a math context. The Conference Board of the Mathematical Sciences (CGMS), an organization composed of 16 mathematical organizations, released The Mathematical Education of Teachers in 2000. It recommends that future elementary school teachers take four courses in mathematics: (1) num-


ber and operations, (2) geometry and measurement, (3) data analysis, statistics, and probability, and (4) algebra and functions. States are not hastening to adopt the CBMS standards as requirements for certifying teachers, and insti-tutions of higher education are even less eager to mandate requirements above the states’. Wouldn’t it be great if American children emerged from elementary school either knowing algebra or ready to learn it? Children in some countries do. As I signed in at Janu-ary’s Joint Mathematics Meetings, I noticed “Romania” on the name tag of the woman checking me in. I confirmed she had put her native country after her first name. “Did you take calculus in eighth grade?” I asked, motivated by reports from two other Romanian immigrant desk clerks I’d met in the past two years. “No, we had integral in ninth grade.” “But you had differential calculus in eighth grade?” She nodded. I don’t want to suggest that we should model our educa-tion program on that of any other country, but wouldn’t it be nice if college math professors could teach only calculus

and up? One prerequisite for this pleasant possibility is that our elementary school teachers learn the mathematics we want them to teach. Members of the MAA are pivotal in remedying this situa-tion, both politically in getting appropriate state requirements and professionally in providing willing, competent teaching. What can you do to help?

PAT KENSCHAFT is Professor Emerita of Mathematics at Mont-clair State University and now teaches mathematics to pre-service elementary school teachers at Bloomfield College, also in New Jersey. She can be reached at: [email protected].

Reprinted with permission from MAA Focus, The Newsmagazine of the Mathematics Association of America, copyright Aug/Sept. 2009. All rights reserved.

Readers will be interested in “Racial Equity Requires Teaching Elementary School Teachers More Mathematics” in Notices of the AMS, Feb. 2005, 52;2, available at http://www.ams.org/no-tices/200502/fea-kenschaft.pdf

VCTM Fall Academy at Sweet Briar

Conference at a Glance

Saturday Pre K‐2 3‐5 6‐8 9‐12 College Math Specialists

8:00‐9:00 Vendors

9:00‐10:15 Number and

Number Sense Algebraic Thinking Data Analysis Geometry Geometry Float

10:30‐11:45 Computation and

Estimation Decimals

Computation /

Algebra Statistics Float Coaching Roles

Friday Pre K‐2 3‐5 6‐8 9‐12 College Math Specialists

7:30‐8:30 Registration / Vendors

8:30‐9:45 Place Value Computation and

Estimation

Number and

Number Sense Statistics Float

Building a

Program

10:00‐11:15 Computation and

Estimation Fractions

Computation and

Estimation

Number and

Number Sense

Mathematical

Literacy Float

11:30‐12:45 Algebraic Thinking Number and

Number Sense Geometry

Geometry /

Algebra Float Coaching

Lunch / Vendors

2:00‐3:15 Number and

Number Sense Singapore Math

Geometry /

Algebra Algebra Number

Analyzing Student

Work

3:30‐5:00 Keynote Speaker


GRADES K-8

Addressing Parents’ Concerns about Mathematics ReformHendrickson, Siebert, Smith, Kinzler, Christensen

Although NCTM’s Standards documents have been around for more than a decade, teachers still frequently encounter resistance when attempting to implement reform-oriented instruction and curriculum materials that are aligned with the Standards. Unfortunately, some of the strongest critics of reform are parents. Most have never experienced the type of mathematics instruction that the Standards recommend. The open-ended, conceptually oriented tasks that students bring home are different from their previous experiences and may be confusing. Parents’ own anxiety toward and traditional beliefs about mathematics can further heighten their concern about the mathematics their children are now doing. Our experience suggests that teachers can do several things to help ease parents’ concerns about Standards-based mathematics instruction and curriculum materials. During the past three years, we have worked with hundreds of parents as we have assisted in implementing the adop-tion of reform-oriented mathematics curriculum materials in twenty-six elementary school and five junior high schools. Two types of settings-evening meetings with groups of par-ents and informal, one-on-one conversations with parents-have been particularly productive in our efforts to address parents’ concerns. This article shares our experiences with both types of settings and describes what we did to make these settings successful for us and for the parents with whom we worked.

Addressing Parents in Groups: Evening Meetings We knew when we started reforming mathematics instruc-tion that we would have to provide support not only for teach-ers but for parents as well. We anticipated that parents would not naturally understand the purpose of the new curriculum materials or the assignments that their children were bringing home. We decided that evening meetings at each school would provide a forum for supporting parents that was within the bounds of the resources available to us. Our objectives for these meetings were to introduce the new curriculum, explain the purpose for the changes, and answer parents’ questions about the new curriculum. We began holding the meetings about one week after the start of school and completed all the presentations by mid-October. Our format for the evening meetings underwent many changes as we attempted to better meet parents’ needs. About halfway through the twenty-six presentations, we developed a format that seemed to yield the best results. This format consisted of a general presentation, a handout on homework, parent visits to classrooms, and a question-and-answer period with district and school representatives.

General presentation We began every meeting with a forty-five minute general presentation on the new curriculum materials and their intended impact on children’s learning. The goal of this pre-

sentation was to show how the new curriculum could help children develop powerful ways of reasoning and thinking about mathematics. We often began with a brief testimonial by a teacher from that school, who talked about the exciting mathematics that her students had developed during the last few days or weeks. Next, a district leader presented several multi digit addition and subtraction problems and invited parents to do some of the problems and consider a variety of children’s solutions for the others. One example of multiple solutions that we used is the six different student solutions to 25 + 37 on page 85 of Principles and Standards for School Mathematics (NCTM 2000; see fig. 1). We found that by actually engaging parents in doing mathematics and sharing their solution strategies, parents were better able to appreciate the flexibility and understanding that come from invented solution methods. We followed parent participation with two video clips. The first clip was produced by the developers of the curriculum materials and showed examples of the children doing fraction activities. In our first few evening meetings, we discovered that many parents did not notice the children’s deep, insightful thinking; they noticed only that the children took a long time to get an answer. We found it necessary to point out what we saw and valued in the video and contrast that with traditional instruction so that parents could value what they were seeing. Our second video clip was much more successful in helping parents understand the benefits of the new curriculum. It showed individual interviews with two children, one from a traditional classroom and one from a classroom using the new curriculum and taught by the third author. Although each student achieved the same high score on the statewide mathematics test, the child using the new curriculum showed much greater understanding in his solutions than did the other child. The contrast in the outcomes of these two students’ learning experiences was clearly evident to many parents and evoked criticism of the statewide exams that had not disclosed the differences between these children’s understandings.

Handout on homework During the first few meetings, we encountered a lot of pa-rental concern about the homework from the new curriculum. Parents were unprepared for the open-ended, contextual-ized problems students were asked to solve, as well as the detailed reasoning and explanations that students were to employ in solving these problems. Parents’ difficulties were further exacerbated by the tendency for the problem contexts and methods in the early homework assignments to be an extension of what students had learned the year before. Naturally, because this was our first year using the curriculum, neither parents nor students were prepared adequately for these homework assignments. To address these issues, we prepared a handout in the form of a bookmark on parental involvement with mathemat-


ics homework. The content of the handout was adapted from the connected Mathematics Project Web site (2002) about communication with parents. The bookmark included four types of questions that parents could ask, depending on at what stage in the problem-solving process the student needed support. The categories include questions to help students think about how to start a problem (“What informa-tion seems to be important?”), how to get “unstuck” while in the middle of a problem (“Can you organize the information differently to show important patterns or relationships?”), how to critically analyze the solution (“Is there a way you can check to see if your answer is reasonable?”), and how to extend their thinking (“Is there another way to solve the problem?”). We distributed this handout after the general presentations.

Parent visits to classrooms After the general presentation, parents were able to go directly to their children’s classrooms and visit with the teachers. To help parents better understand the mathemat-ics curriculum, the teachers often prepared display tables with the textbooks, manipulatives, and samples of students’ work. This allowed parents the opportunity to actually handle the materials their children were using. Many teachers also

provided a brief presentation or activity in which they invited parents to participate in the mathematics games and class-room routines their children engaged in during mathematics instructions. After the activity, the teachers discussed how these activities and routines help children learn mathemat-ics, often sharing students’ work to demonstrate their points. The commitment of the teacher, the rich activities, and the discussion of mathematics learning often helped reassure parents that the new curriculum would be beneficial to their children.

Handling vocal parents through a question-and-answer period During the first few meetings, we encountered a small but vocal group of parents who opposed the reform curriculum. These parents often asked so many questions during the general presentation that we were unable to offer a coherent overview of the new curriculum. Moreover, the questions and comments from these parents were often inflammatory and emotionally charged. Their questions prevented us from achieving our goal of giving parents adequate information about the curriculum, because they interrupted the flow of the presentation and often stirred up unnecessary negative feelings and emotions.


To address this issue, we decided to accept parent ques-tions only after we had completed our initial forty-five-minute presentation. Furthermore, we attempted to anticipate the common questions that parents had and to address these questions systematically and coherently in the presentation and the handout on homework. We found that most parents were satisfied by the presentation and were eager to either visit the classrooms or go home. We therefore created a ten-minute intermission immediately following the general presentation. We invited parents to go directly to the class-rooms or stay for a question-and-answer period. Usually, 90 percent of the parents left immediately after the general pre-sentation. Some parents used the intermission to approach district and school leaders to ask questions. These informal conversations seemed particularly productive in addressing parents’ concerns. After all the parents who wanted to visit the classrooms had left, we held our question-and-answer session and stayed as long as there were questions. This left the vocal parents with a much smaller audience and prevented many of the antagonistic feelings that had been unexpectedly generated during the first few meetings.

Addressing Parents Individually: One-on-OneConversations During our many conversations with parents, we began to see common themes emerge in their questions. They typi-cally asked about the nature of mathematics, the learning of mathematics, and the implications of the curriculum for the traditional high school mathematics sequence, college entrance examinations, and employment.

Nature of mathematics Many parents were concerned about their children learn-ing the basic facts and traditional procedures, often because they viewed mathematics as consisting of a set of facts and procedures to be memorized and mastered. Common ques-tions from these parents included the following:

• How does the new curriculum address basic mathematics facts?

• Are the traditional algorithms taught in the new curriculum?• Why is it important to know more than one way to solve a problem?

Underlying each of these questions is the assumption that knowing and doing mathematics consists of using basic facts and traditional procedures to correctly compute answers to routine problems. To respond to these questions, we typically began by acknowledging the importance of computation in mathemat-ics. Then we attempted to help the parents understand that computational fluency is also important. By computational fluency, we mean that students could (a) demonstrate flex-ibility in the computational methods they chose, (b) under-stand and explain their methods, and (c) produce accurate answers efficiently. We tried to illustrate these principles with a problem that is more easily solved with an invented procedure than with a traditional one. For example, 376-99 is much easier to compute by first subtracting 100 from 376,

then adding 1 back than by suing the traditional borrowing procedure. Similarly, we also tried to point out that a flexible knowledge of basic facts might be more advantageous than rote memorization. For example, if a student forgets what 9 x 8 is, she might compute 10 x 8 and then subtract the extra 8. Lastly, we assured parents that children would learn powerful strategies for solving arithmetic problems and they may even invent the standard algorithms as they searched for more efficient ways to compute. Likewise, students would learn flexible methods for deriving basic facts and most likely would memorize many of the commonly used facts.

Learning mathematics Many parents were concerned about the types of activi-ties their children were using to learn mathematics. These parents typically encountered only traditional mathematics instruction when they were in school, and therefore they believed that mathematics must be taught and learned this way. Common questions included the following:

• Why change the instruction? It worked for me.• Why give only two or three problems? Doesn’t practice make perfect?

• Won’t all the different invented procedures be confusing?• Why do students have to show how they solved the prob-lem? Why do they spend so much time talking and writing about mathematics?

In response to these questions, we drew on the research base supporting reform-oriented curricula. Research has documented that traditional instruction fails to help students develop computation fluency and understanding (Boaler 1998; Erlwanger 1973; Sowder 1988). In particular, the teaching of traditional procedures often obscures the mean-ings of numbers and operations. Consequently, students do not learn why traditional procedures work or in what contexts they may be used. This makes it difficult for stu-dents to remember the procedures or apply them flexibly to problems that differ only slightly from the pages of exercises they completed for practice. Students’ lack of understanding also prevents them from knowing whether their answers are reasonable, leaving them insensitive to answers that are obviously wrong. Fore example, 14 - 9 = 15 can result from operating on the ones and tens separately and subtracting the lesser number, 4, from the greater number, 9. Because traditional instruction does little to help students develop understanding, numbers sense, or computational fluency, it leaves students vulnerable to mathematics anxiety and failure. Hence, developing understanding is an important goal of current reform efforts. We also point out to parents that much research has investigated the development of children’s understandings of number, measurement, and operations. Reform-oriented curricula are based on the findings of this research and are geared toward helping students develop understanding and computational fluency. The instruction is grounded in con-texts that are familiar to students and that allow them to build on the knowledge and intuitions they have developed from their experiences outside school. Furthermore, grounding


problems in contexts familiar to children makes it possible for them to judge the reasonableness of their answers and solu-tion methods. Students are given fewer problems so that they have time to reason, build and test conjectures, try multiple solution strategies, and make connections between what they are learning and experiencing and what they already know. Because learning with understanding is now more important than speed of computation, students do not need as much practice as in traditional instruction. Furthermore, to help ensure that students are learning with understanding, a significant amount of instructional time focuses on sharing solution methods, both orally and in writing, so that students can organize their thinking through expression, receive help-ful feedback, and be exposed to new ideas. This process of allowing students to work for longer periods of time on context-rich problems and to communicate their solutions enables them to develop many different solution methods they can use efficiently and flexibly.

Implications for future learning and employment The last category of questions involves parents’ concerns about the long-term consequences of the new curriculum. Parents are worried that reform-oriented instruction will not prepare their children adequately for future mathematics courses, college entrance examinations, or real-world uses of mathematics. Parents are particularly concerned that if children take higher-level courses in traditional mathemat-ics, they will not have the requisite knowledge of traditional procedures and facts that they must have to keep up with their classmates. To respond to parents’ concerns about the long-term implications of using a reform curriculum, we focused on the understanding that students develop and the contextu-alized nature of their knowledge. Because reform-oriented instruction focuses on computational fluency, students who compete a reform-oriented curriculum are likely to have a better understanding of and more flexibility with the proce-dures they have developed for computation. Furthermore, the procedures they have learned may also include the traditional procedures. When students have not learned traditional procedures, they have invariably learned other powerful and efficient procedures on which they can draw to achieve the same results. Finally, because much of the mathematics is learned from solving problems situated in real-world contexts, students from reform-oriented courses are much more likely to be able to see how they can apply their knowledge to situations they encounter in their personal lives and employment.

Making These Ideas Work for You The strategies and ideas in this article are not meant to serve as a template for you to allow closely in your interac-

tion with parents. Instead we hope that you will view these strategies and ideas as a starting point for developing your own approach to working with parents. Because your situa-tion is undoubtedly different from outs, you will need to tailor your approach to meet the specific needs of your students’ parents. An important part of your success will depend on your ability to obtain feedback from parents, other teachers, the principal, and the district leaders. This feedback is crucial in helping you continually adjust and change your strategies and approaches to better meet parents’ needs. As you ad-dress parents’ concerns and help them see the benefits of understanding mathematics, we are confident that you, like us, will be able to relieve parents’ concerns and help them support implementation of reform-oriented instruction and curriculum materials.

ReferencesBoaler, Jo. “Open and Closed Mathematics: Student Expe-

riences and Understandings.” Journal for Research in Mathematics Education 29 (January 1998): 41-62.

Connected Mathematics Project. “Communicating with Par-ents.” Cited September 2002. www.math.msu.edu/cmp/ImplementingCMP/ParentCommunication.htm

Erlwanger, Stanley H. “Benny’s Conception of Rules and Answers in IPI Mathematics.” Journal of Children’s Math-ematical Behavior 1 (2) (1973): 7-26.

National Council of Teachers of Mathematics (NCTM). Prin-ciples and Standards for School Mathematics. Reston, Va.: NCTM, 2000.

Sowder, Larry. “Children’s Solutions of Story Problems.” Journal of Mathematical Behavior 7 (1988): 227-38.

SCOTT HENDRICKSON, [email protected], teaches intermediate algebra, precalculus, and Advanced Placement calculus at Lone Peark High School in Highland, Utah. SHARON CHRISTENSEN, [email protected], teaches prealgebra and algebra at Mt. Ridge Junior High in Highland, Utah. Scott and Sharon are also secondary mathematics specialists for Al-pine School District in American Fork, Utah. HEIDI KUNZLER, [email protected], is the elementary mathematics specialist for Alpine School District. Scott, Sharon and Heidi are interested in supporting teachers in the effective implementation of a standards-based mathematics curriculum. DANIEL SIEBERT, [email protected], and STEPHANIE SMITH, [email protected], are assistant professors of mathematics education at Brigham Young University. Daniel’s current research interests are preservice teacher education and literacy in the mathematics classroom. Stephanie is an experienced pre-K-12 teacher, and her research interests include teachers’ and children’s conceptions of mathematics and the learning and teaching of mathematics for understanding.

Reprinted with permission from Teaching Children Mathematics, copyright August 2004, by the National Council Teachers of Math-ematics. All rights reserved.


Many of the recent changes in the mathematics curricu-lum and pedagogy focus on increasing student proficiency in writing about mathematics. According to a recent study concerning the retention of information the percentage breakdown is as follows:

1. 10% of what they read

2. 20% of what they hear

3. 30% of what they see

4. 50% of what they both hear and see

5. 70% of what they discover

6. 80% of what they experience

7. 90% of what they teach

The accompanying research project was developed to incorporate the last three retention percentages: to dis-cover, to experience, and to teach. A project of this kind is actually a research essay designed to enhanceconceptual understanding. Any well-designed research project should have the fol-lowing objectives:

1. to promote the clarification and organization of ideas and concepts

2. to stimulate problem articulation and analysis

3. to enhance pattern discernment and recognition

4. to develop critical thinking skills

5. to generate creative thinking skills

6. to facilitate interaction between the instructor and the student, since writing is inherently an active and not a passive activity.

The following research project was designed for a course in precalculus mathematics. The students were urged to employ both critical and creative thinking skills in their in-vestigation of this topic. It should be noted that if you allow your students to use their imagination and creativity to ex-plore what is basically a critical thinking activity, then you will begin to realize the advantages of the metacognitiveapproach to learning—a method which utilizes activities that promote student awareness of their own thinking pro-cess.

In conclusion, once your students are exposed to math-ematical research projects, then they are better able to ap-ply these mathematical concepts to many real-world situ-ations. This confluence of formats (theoretical constructs/real-world applications) is a valuable learning tool which can be transferred directly into the workplace. At this point I would like to showcase a typical research project concerning the exploration of prime numbers. Basic concepts, background information, and project objectives are outlined in the following format.

Writing-to-Learn Project This research essay must be completed two weeks be-fore the final examination for this course. Be sure to employ both critical and creative thinking skills in your investigation of this research project. If you elect to utilize any Internet database, then be sure to list the web address for each citation.

A. Basic Concept You are about to explore the topic of prime numbers. A prime number is a natural number that has exactly two fac-tors, itself and one. The first five prime numbers are 2,3,5,7, and 11, where 2 is the only even prime number. Euclid of Alexandria (circa 300 B. C.) was a Greek mathematician who is remembered chiefly as the author of the “Elements” -- one of the most celebrated textbooks ever written. Euclid proved that there is no largest prime which implies an infi-nite number of primes. Eratosthenes (circa 275-194 B.C.) was one of the most versatile scholars and scientific writers of the ancient world. He was invited by Ptolemy to suc-ceed Appollonius as the head of the library at Alexandria, the most famous library in the ancient world. His greatest mathematical discovery was a method for finding succes-sive prime numbers.

B. Sieve of Eratosthenes1. Begin by constructing a 10 X 10 array of the natural num-

bers from 1 to 100. The number 1. should be crossed out since one, by definition, is not a prime number.

2. Circle 2, the first prime number. Then cross out the re-maining multiples of 2 using a horizontal slash (–).

3. Circle 3, the second prime number. Then cross out the remaining multiples of 3 using a vertical slash (|).

4. Circle 5, the third prime number. Then cross out the re-maining multiples of 5 using a diagonal slash (/).

5. Circle 7, the fourth prime number. Then cross out the remaining multiples of 7 using a reverse diagonal slash (\).

GRADES 8-12

Writing-to-Learn in MathematicsDavid L. Fama


6. At this point, except for 2,3,5, and 7, all the multiples of 2,3,5, and 7 have been “sifted” out.

7. Circle the remaining numbers that are not crossed out. Therefore, the remaining circled numbers are the prime numbers less than 100.

C. Twin/Triple Primes Several pairs of primes in the list of primes less than 100 have a difference of two. For example, the pairs 3 and 5 or 5 and 7 each have adifference of two. These pairs are known as twin primes. Is there an infinite number of twin primes? The answer to this question is not known, but it is expected to be in the affirmative. Three consecutive odd primes that have a difference of two are known as triple primes. For example, the only known prime triplets are 3, 5, and 7.

Project Objectives1. List all the prime numbers less than 100.

2. In the table below, complete the list of all the twin primes less than 100. In addition, find the sums and products of these twin primes.

Twin Primes Sums Products

___and___ ____ ____

___and___ ____ ____

___and___ ____ ____

___and___ ____ ____

___and___ ____ ____

___and___ ____ ____

___and___ ____ ____

___and___ ____ ____

3. Determine the “pattern” concerning the sums of twin primes.

4. Determine the “pattern” concerning the products of twin primes.

5. The number 13 is a prime, and 31, the digit reverse of 13, is also a prime. Therefore, 13 is known as an emirp (prime spelled in reverse) since its digit reverse is a dif-ferent prime. The prime 31 is also an emirp. However, the prime 11 is not an emirp since its digit reverse is not a different prime. List all the emirps less than 100.

6. A special kind of prime number is the Mersenne prime, named after the French mathematician and monk Marin Mersenne (1588-1648).

a. Investigate the special property of the Mersenne prime, and then compose a research essay based solely on your research.

b. Investigate the role of Mersenne primes in today's tech-nological society, and then compose a research essay based solely on your research.

DR. DAVID L. FAMA is the Dean of Instruction at Royal Crest Academy in Front Royal, Virginia.

Affiliates CornerNVCTM: Week of October 18: Seminar/SocialContact Gail Chumura for more information at [email protected]

Battlefields CTM: Saturday, November 6: Math Professional Development Day,THEME: Changes in Math Ed at Benton, MS, ManassasContact Karen Mirkovich for more information at [email protected]


GRADES 9-12

For Your Information - PUBLICATIONSContent Area Mathematics fro Secondary Teachers: The Problem Solver, Allen Cook and Matalia Romalis, 2006. xi + 324 pp., $69.96 paper, ISBN 1-929024-95-9. Christopher-Gordon Publishers; (800) 934-8322; www.christopher-gordon.com

This book is a general overview of mathematics for the high school teacher. It is intended as a refresher on various top-ics that a teacher may have forgotten from college days or maybe never covered in a college course. Topics reviewed include algebra, geometry, trigonometry, functions, statistics, vectors, calculus, discrete mathematics, and linear algebra. In each section, key ideas are highlighted and followed by several problems with solutions―and then several more problems are left for the reader to solve. Answers to odd-numbered problems are given in an appendix. The problems range from elementary to challenging. Each chapter is about fifteen pages long, so the explana-tions tend to be sparse. If a teacher were to use this book as a way of learning new material, I am sure that reacher would soon be looking for a book solely devoted to the subject. However, for a quick refresher of a topic and for the opportunity to solve some related problems, the book would make a nice addition to any teacher’s library.

Mathematics Minus Fear, Lawrence Potter, 2006. 287 pp., $17.95 paper. ISBN 0-7145-3115-4. Marion Boyars Publishers, distributed by Consortium Book Sales; www.marionboyars.co.uk.

Combining mathematics history, algorithms, and examples in a light and enjoyable package, Lawrence Potter has composed a witty and informative overview of most of the mathematical content we should have learned in school. The text is supplemented by a collection of eighty-three puzzles. Mathematics Minus Fear uses a conversational manner to challenge the reader to revisit and clarify a wide range of mathematical topics, from the discussion of dif-ferent numerical systems to how to solve Sudoku puzzles. Most of the examples and content discussed were easily accessible to my high school students and provided some “Aha!” moments. The book is divided into four parts with an introduction and contains a wealth of engaging and easy-to-follow explana-tions for most algorithms and foundational concepts taught in the mathematics classroom. Using a set of characters from a fictional classroom described in the introduction, the author sets the tone for the discussion of a wide range of seem-ingly basic concepts and procedures that the mathematically gifted reader will appreciate and the mathematically chal-lenged one will understand and enjoy. The author uses lots of humor and ancient and contemporary mathematics his-tory to put concepts in context and explain their usefulness. I would have preferred to have the puzzles be more closely related to the topics being discussed, but I enjoyed the book and the puzzles nonetheless. Overall, this is a good addition to any school library. It can be a great resource to infuse some reading into the mathematics classroom.

LUIS LÍMA, Digital Harbor High School, Baltimore, MD

Mathematics Is . . .“Mathematics is pure language-the language of science.”

Alfred Adler, Mathematics and creativity, in Mathematics:People Problems, Results, vol. 2, Douglas M. Campbell and

John C. Higgins, eds., Wadsworth, Belmont, CA, 1984, p. 3

Reprinted from The American Mathematical Monthly, December 2009Copyright by The Mathematical Association of America


GRADES 9-12

Delving Deeper: In-Depth Mathematical Analysis of Ordinary High School

Dick Stanley and Jolanta Walukiewicz

In this article, we give an example of what “delving deeper” might mean with respect to standard, rather ordinary high school problems. The purpose is to illustrate the math-ematical depth that is potentially present, even in simple problems. We use what we call an extended analysis of a problem, which is an analysis from a mature mathematical perspective, with careful attention paid to mathematical reasoning and to using good mathematical habits of mind. (See Cuoco, Goldenbery, and Mark 1996.) Our intent is to foster powerful ways of thinking that are characteristic of mathematics and science. As a setting for a high school problem, we choose so-called motion problems. These problems have been the focus of many cartoons and stories, and they represent what some people see as obscurity and foolishness in school mathemat-ics. Moreover, these problems have often been trivialized in high school textbooks that have students memorize solution pattens to stereotyped versions of the problems. However, motion problems do have many virtues. They are excellent vehicles for illustrating good mathematics (functions, equations, and rates of change), as well as good applications or mathematics. Moreover, when they are taken seriously, these problems can lead to results that are mathematically interesting and surprising and that may be unsuspected even by those who have taught the problems many times. The following is one version of a classic high school mo-tion problem.

Car A sets out traveling 50 MPH. Car B starts three hours later and tries to catch up. If car B travels at 75 MPH, when does car B catch up with car A?

An initial analysis of this problem appropriate in a beginning algebra course might proceed as follows:

Simple solutionThe problem asks for the time required for car B to catch up with car A. We let tc represent this time, measured from the time that car B starts out. The distance traveled by car B until it catches up is then 75 • tc, whereas the total distance traveled by car A is 50 • (tc + 3). These two distances are the same, and setting the two expressions equal leads to a linear equation in the unknown, tc:

(1) 75 • tc = 50 (tc + 3)

Solving the equation gives the answer to the problem: tc = 6 hours.

A simpler solutionThe preceding solution is simple, but it still involves the

formal mathematical steps of defining an unknown, setting up an equation, and solving the equation. A simpler solution uses basic quantitative reasoning about the situation:

When car B starts, car A has been traveling for three hours and is therefore 3 • 50, or 150, miles ahead. Since car B is going 75 – 50, or 25, MPH faster than car A, it will catch up in 150/25, or six, hours.

Solving a problem by using basic quantitative reasoning in this way helps us back from the problem and reflect on it in a way that makes sense before we become locked into a more formal mathematical approach. It is a good strategy that can be applied to many problems. Moreover, since students may come up with this approach on their own, teachers must be able to recognize and support this kind of intuitive and tentative mathematical thinking.

A DEEPER LOOK AT THE MATHEMATICS UNDERLYINGTHE PROBLEM: A GENERAL SOLUTION In the remainder of this article, we approach the same problem but with a different attitude. Instead of being content with an answer to the problem, we see our work as actually starting with this solved problem. Our goal is to find what is mathematically interesting and general about motion prob-lems such as this one. We first note that, so far, the problem is a “numbers in-number out” problem: we are given numerical information–a time (3 hours) and two speeds (50 and 75 MPH)–and we have found a numerical answer ( 6 hours). We have used mathematics to solve the problem, but the answer itself is not interesting mathematically. Moreover, if we are given differ-ent numerical information as input, we will have to start over. A more powerful use of mathematics is to find a general answer by asking what the required “catch-up time” tc is for any speed vB of car B (in a situation where car A is traveling at 50 MPH and has a three hour head start). This problem is no more difficult to solve. We can repeat exactly the same derivation that we previously used to find the numerical an-swer but use vB in place of 75. The distance traveled by car B until it catches up is vB • tc, whereas the distance traveled by car A is still 50 • (tc + 3). These two distances are the same, and setting the expressions equal to each other leads to the same linear equation in tc that we previously obtained but with 75 replaced by the parameter vB:

(2) vB • tc = 50 • (tc + 3)

When we solve this equation for tc, we do not get a number. Instead, we get a function:

(3)


This result represents a general solution to the problem for any catch-up speed vB. The function given by equation (d) is more interesting mathematically than the number (6 hours), which was the previous answer. Surprisingly, this simple step is seldom taken in school mathematics. The tendency is to use algebra to find numeri-cal answers to problems without going on to use algebra to generalize these solutions. But working to find a general solution is a good mathematical habit of mind that provides many benefits. The form of the general solution reveals the meaning of the different parts of the expression in equation (3). The numera-tor is the distance (150 miles) traveled by car A before car B starts, whereas the denominator is the difference between the speeds (how much faster car B is traveling). Attaching specific meaning with respect to the problem situation to the different parts of a formal expression is another good principle that we will employ frequently. As an aside, we note the use of subscripts in the above expressions. The symbol vB starts for the speed of car B and tc stands for the catch-up time. Some educators believe that subscript notation is confusing to students, but we have found that students easily become accustomed to subscripts and that subscripts are worthwhile. Subscripts facilitate the process of building concise and meaningful notation and offer a helpful way of keeping track of what is going on in a problem analysis. Proceeding with the analysis, to get an answer for any given speed vB, we just plug the speed into equation (3) and do the arithmetic. For example, the answer to the original problem (6 hours) is found by substituting vB = 75 MPH into equation (3). Figure 1 is a graph of tc as a function of vB. On the graph, we can see the solution to the original problem tc = 6 hours, as the value of the function (3) at vB = 75 MPH.

Exploring the general solutionThe function (3) and its graph in figure 1 tell us exactly how the catch-up time tc depends on the catch-up speed vB. But rather than see them as an end in themselves, we can use them to further extend the analysis of the problem. By analyzing the function (3), we find that for vB close to 50 MPH,the catchup time tc is very large. This result makes sense with respect to the problem situation. However, as vB becomes very large, analysis of the function shows that the catch-up time tc approaches 0. This outcome also makes

sense: if car B were a ray of light, it would catch up with car A almost instantly. What if vB is less than 50 MPH? In the problem situation, it means that the pursuing car B is go-ing slower than car A and therefore falls farther and farther behind. Car B will never catch up. As another example, we can ask about the slope. The slope of the graph at any point vB can be found by using calculus. It is

(4)

We can find the approximate slope at any point by mea-suring on the graph. Here we are more interested in the meaning of the slope than in the calculus method for finding it. Experience shows that many people can derive equation (4) by using calculus, but surprisingly few can interpret the meaning with respect to the problem situation. The value of the slope at vB = 75 is about -0.25. What is the meaning of this number? First, we note that the sign is nega-tive. This result makes sense, since as the catch-up speed increases, less time is needed to catch up. Second, what are the units of the slope? We can see that they are “hours per mile per hour.” So the rate of change of the function at vB = 75 is about -0.25 hours per mile per hour. Therefore, for everyone 1 MPH that car B’s speed increases, the catch-up time decreases by about a quarter of an hour.

Delving deeper into the general solutionWe let the head-start time (that is, the delay between A’s start and B’s start) to be represented by th. From thinking about the situation, we know that if th increases, then tc also increases. Wondering whether this relationship is a propor-tional one is natural.

In a catch-up situation, is the catch-up time tc directly proportional to the head-start time th?

We also know that as the difference in speeds, vB - vA, increases, the catch-up time, tc, decreases. Asking whether this relationship is an inverse one is natural.

In a catch-up situation, is the catch-up time tc inversely proportional to the difference in speeds, vB-vA?

People’s intuitions seem to suggest that the answer to both these questions is yes. This response is accurate for the first question; but, perhaps surprisingly, the answer to the second question is no. To answer these questions, we need a better way to rep-resent the nature of the general relationships in the problem. We can replace all the numerical inputs to the problem with general parameters. This strategy leads to the general catch-up problem:

Car A sets out traveling at vA MPH. Car B starts th hours later and tries to catch up. If car B travels at vB MPH, after how much time tc does car B catch up with car A?

Although this problem is fully general, it is no harder to Figure 1. Catch-up time as a function of the catch-up car’s speed


solve than the original problem. In fact, by setting up and solving equation (1) exactly as we previously did, but re-placing numbers with parameters, we obtain the following:

(5)

A look at this formula gives a direct answer to our first question: for any given speeds vA and vB, the catch-up time tc is directly proportional to the head-start time th; the most constant of proportionality is the ratio of vA to the difference vB - vA. The formula also shows why the answer to our second question is no. Still, the catch-up time tc is inversely pro-portional to a related quantity. We consider the following equivalent way of writing equation (5):

(6)

The denominator of equation (6) is the proportional in-crease of the speed of car B over that of car A. Hence, the catch-up time is inversely proportional to the proportional increase of the speeds and not to the absolute increase of the speeds. In equation (6), the speed vA occurs twice. We can col-lapse them into one, thereby making it easier to see the contribution of this parameter:

(7)

From this representation, we see that the catchup time depends only on the ratio of the speeds. So, for example, if both cars increased their speeds by 25 percent, the catch-up time would not be affected. As a final step, we can work from equation (7) to represent the ratio of speeds directly:

(8)

This equation reveals a succinct and general statement about the problem situation: since A and B traveled the same distance, the ratio of their speeds equals the reciprocal of the ratio of the times that they spent traveling. Each of the four different representations (5) to (8) of the catch-up time tc with respect to the other three parameters helps us see different aspects of the relationships involving in this problem. Together, they complete a full process of generalization for this problem that began with the initial generalization in formula (3).

STRUCTUREIf we had used general parameters th, vA, and vB in place of the numbers 3, 50, and 75, respectively, the representation in equation (7) could have been derived from the outset, without any more work. And although proceeding directly to a general solution would not be appropriate for students in beginning algebra, doing so is certainly appropriate for those who are already familiar with algebra. Proceeding directly

to a general solution is no more difficult than obtaining a numerical solution, and doing so has several advantages. For one, equation (7) is a solution to all problems of this type. Moreover, the expression on the right has a transpar-ent structure that allows us to give a dimensional check to our result. In this situation, the units do make sense: both the left-hand side and the right-hand side have the units of time. Since the ratio of speeds is dimensionless, the second fact on the right-hand side is a dimensionless factor. Paying attention to the role of structure in general and to the role of dimensionless factors in particular is another principle that serves us well. This principle goes against what we are often taught in mathematics classes, that is, to simplify arithmetic and algebra as we go along in a deriva-tion. The difficulty is that doing so destroys the structure of expressions. If we retain the structure of expressions, we can learn from them.

MODELING WITH FUNCTIONSThe initial generalization (3),

of the original catch-up problem seems natural, but our expe-rience has shown that it is not an easy idea to communicate. Many people who find that solving the original problem to find the catch-up time tc = 6 hours is straightforward are some-what baffled when asked to express the catch-up time as a function of the speed of car B, and not many people come up with equation (3) and its graph, figure 1. When asked to go beyond the initial numerical solution, more people come up with the graphs shown in figure 2. These graphs are distance-versus-time graphs of the motion functions of cars A and B:

(9) d = 75 • t,

which is the distance traveled by car B, and

(10) d = 50 • (t + 3)

which is the distance traveled by car A

Figure 2. A graphical approach to finding where the two cars meet

These functions represent a mathematical model of the situation. The problem is solved by finding out where these


groups intersect. Doing so involves solving the equation formed by setting the defining expressions of the functions equal to each other:

(11) 75 • t = 50 • (t + 3)

This equation is the same as equation (1), which was used in the first approach to the problem. The t-intercept of function (10) is -3 hours, where the negative sign indicates that car A started out three hours before car B. It results from defining the catch-up time tc as the time traveled by car B, so that car B starts out at t = 0. Modeling the problem situation in this way helps our understanding. It shows that solving an equation, such as equation (11), is equivalent to finding the intersection of the graphs of two functions, here the functions (9) and (10). The solution of the equation is the projection onto the t-axis of the intersection of these graphs. But we must also note that this model does not represent a generalization of the problem. Solving equation (11) for t gives the same single number that we obtained for the original problem (t = tc = 6 hours). In other words, the func-tions (9) and (10) and their graphs in figure 2 still represent a “numbers in-number out” approach, whereas the function (3) and its graph in figure 1 represent a true generalization. Nevertheless, the modeling approach in figure 2 and the generalization in figure 1 can be directly related in an inter-esting way, as we show in the following section.

RELATING THE FUNCTIONS MODEL TO THEGENERALIZATIONIf we think of figure 2 as a dynamic graph, we can use it to represent a generalization of the problem to any speed vB of car B. Specifically, we imagine the line graph of the function d = 75 • t in figure 2 being able to pivot at the origin (0, 0), thereby producing graphs of different slopes. At any particular slope, it is the graph of some function d = vB • t. The intersection of this graph with the fixed graph d = 50 • t can be found, as well as the projection of this intersection onto the t-axis. This dynamic pivoting allows us to generalize the problem to any speed vB. With an interactive geometry program, we can do more than imagine the graph as a dynamic one. We can actually carry out this process:

• In figure 2, construct the line that is the graph of car B so that it can pivot at (0, 0).

• Construct the rest of figure 2 with all parts fixed.• Have the program measure the distance from the origin

to the point on the t-axis where a vertical line though the intersection of the two graphs intersects the t-axis.

• Have the program graph this distance as a function of the slope of the pivoting line.

The result of the last step is the graph of figure 1. See-ing this relationship between the graphs of figures 1 and 2 can be a powerful way for students to tie together the many aspects of this problem and its solution.

FINDING THE DISTANCESo far, we have solved the problem of finding the catch-up time tc. Given the general expression for tc in equation (7), finding an expression for the catch-up distance, which we will call dc, is a simple matter. We need to multiply tc by vb. The result is

(12)

We can actually simplify the expression on the right a bit more by coalescing the two occurrences of the speed vB of car B into one. The result is

(13)

The structure of the expression for the catch-up distance dc is here revealed clearly. This distance is the product of a time, the head-start time th, and an expression with the units of velocity. The second factor on the right of equation (13) has the units of velocity, since it is the reciprocal of the differ-ence of reciprocals of velocities. We note that this structure is suggestive of the harmonic mean of two quantities, which involves the reciprocal of the sum of the reciprocals of the quantities. These sorts of structures appear in many appli-cations. Focal length of lenses and resistances in parallel are two examples.

A CONNECTION WITH GEOMETRY:THE MOUNTAIN PROBLEMThe catch-up problem itself has no interesting geometry, since all the motion happens along a single line. However, the intersecting graphs in figure 2 of the motion functions that model the problem do have a geometry. Moreover, it is the same geometry that occurs in the following classic geometry problem, the mountain problem:

We wish to find the height of a mountain. We take sight-ings on the mountaintop from two points, A and B, on a level plain. (See fig. 3.) At point A, we find that the slope of a line of sight to the mountaintop is mA, whereas if we move directly toward the mountain a distance D to point B, we find that the slope of a line of sight to the mountaintop is mB. What is the height H of the mountain in terms of what we have measured?

Figure 3. Finding the height of a mountain

In stating the mountain problem, we have gone directly to the general problem without using particular numbers. After the previous discussion of generalization, we belief that the reader will readily accept this general version. To show that the problem is, in fact, a classic one, we give here two illustrations of it from different times and cultures. See figures 4 and 5.


The mountain problem is a variation of the familiar problem of finding the height of a pole by measuring the distance to the pole and the slope of a line of sight to the top of the pole from some point on a level with the base. Although in the mountain problem we cannot measure the distance to a point under the top of the mountain, we can measure the baseline distance, D, and pin down the location of this measurement by finding the slopes of each of the two lines of sight. The height H of the mountain can be expressed in terms of the baseline D and the two slopes as

(14)

This relationship can be derived in various ways by using trigonometry, but it can also be found with a simple geometric argument that relies only on the definition of slope. Now comes the punch line. When we compare equation (14) with equation (13) for the distance dt in the catch-up problem, we see that they are exactly the same structurally. But when we think about it, this result is not surprising, since geometrically the graph of the motion functions is the same as a diagram of the mountain problem. And it makes sense that the slopes mA and mB in equation (14) correspond to the velocities vA and vB in equation (13), since velocity is represented by slope in a distance-time graph. What we have here is a case of isomorphism of problems. Two rather different problems–one a motion problem and the other a geometry problem–have the same mathemati-cal structure.

IMPLICATIONSThe extended analysis of the catch-up problem reveals the power of imaginative mathematical thinking. When a high school problem is analyzed by using only high school tools but in a thoughtful way, driven by curiosity and supported by good mathematical habits, there is often a reward. We can find interesting new mathematical relationships and discover connections with other problems in mathematics science. Carrying out an analysis of this sort illustrates what can be called a deep, or sophisticated, understanding of basic high school mathematics. Working in this way is an indispens-able aspect of innovative scientific thinking. Unfortunately, exposure to the kind of mathematical sophistication required to carry out such an analysis somehow slips through the crackers in a typical sequence of high school and university

mathematics and mathematics education courses. The kind of sophistication we mean involves the following:

• Giving an initial qualitative analysis of a problem, often by looking at extreme cases, to serve as a guide for a more detailed mathematical analysis.

• Going beyond a simple answer to a particular problem to a general answer to a type of problem

• Looking for ways to extend the result of an analysis of a problem so that it fits related problem situations

• Paying attention to units, to dimensional analysis,and to the role of dimensionless factors

• Trying to coax expressions into useful and revealing forms whose structure can shed light on the relationships in a problem

• Attempting to interpret symbolize expressions and their parts with respect to their meaning in a problem situation

• Getting to know a problem better by varying parameters in regular ways and in general, by encouraging mathemati-cal tinkering with the problem

• Looking for mathematical connections among geometric and algebraic approaches to problems

• Being alert for the possibility of isomorphism of problem types, that, different problem situations with identical mathematical analyses

• Above all, being mathematically curious by always seek-ing what is mathematically interesting about the situation in which a problem is set

Figure 4. Finding the height of a European castleSource: Bennett, J.A., The Divided Circle, a History of Instruments for Astronomy, Navigation and Surveying, p. 56, published by Phaidon Press. All rights reserved.

Figure 5. Finding the height of a Chinese cragSource: M. Ruth, Exploring Mathematics through History; fig. 8.4, “Measurement of a Chinese Crag,” p. 52. Reprinted with permission of Cambridge University Press. All rights reserved.


Practices such as these are sometimes subsumed un-der such general phrases as mathematical habits of mind or mathematical maturity. Although they are essential to any understanding of high school mathematics that can be called deep or sophisticated, they are seldom discussed in mathematical writing. These practices are also difficult to describe in a few sentences, as we have attempted to do in this article. They are much easier to illustrate through the device of an extended analysis of a problem. Analyses of similar problems in a similar spirit appear in Mathematics for High School Teachers: An Advanced Per-spective (Usiskin et al. 2003).

POSTSCRIPTSimilar deep analysis of other motion problems can yield results that are interesting and surprising in different ways. As an example, an analysis of the following problem has a deep structure that is related in a definite way to that of the catching-up problem. The reader is invited to explore this problem and find the connection.

In an earthquake, longitudinal S-waves and transverse P-waves travel outward from the epicenter. It is known that–

• S-waves travel at s = 3.4 km/sec, and• P-waves travel at p = 5.6 km/sec.

Suppose that at a seismographic station for a particular earthquake, the P-waves arrive at a time interval ΔT = 15 seconds before the S-waves. What is the distance D from the station to the epicenter?

REFERENCESBennett, J. A. The Divided Circle, a History of Instruments for

Astronomy, Navigation and Surveying. Oxford: Phaidon Press, 1987.

Cuoco, Al, Paul Goldenberg, and June Mark. “Habits of Mind: An Organizing Principle for Mathematics Curricula.” Journal of Mathematical Behavior 15 (December 1996): 375-402.

Engle, M. Ruth. Exploring Mathematics through History. New York: Cambridge University Press, 1995.

Usiskin, Zalman, Anthony Peressini, Elena Anne Marchi-sotto, and Dick Stanley. Mathematics for High School Teachers: An Advanced Perspective. Under Saddle River, N.J.: Prentice-Hall, 2003.

_______

The authors are grateful for the support of their colleagues: Patrick Callahan, Al Cuoco, Paul Goldenberg, Emiliano Gomez, Elena Anne Marchisotto, Anthony Peressini, and Zalman Usiskin.

EDITOR’S NOTEIn this article, Dick Stanley and Jolanta Walukiewicz take what, by itself, is a prosaic problem and transform it in various ways, producing new, mathematically deeper problems and a set of what might be called “mathematical good ideas”–ways to think that lead results that are mathematically more interesting than the numerical answer to any single problem is likely to be. The authors end by posing another problem for the reader to explore. The editors of “Delving Deeper” would like to add a metaproblem. Stanley and Walukiewicz have shown two problems that have the same deep structure:

Readers may find additional problems that have this same structure or that have the significant part. We invite them to

push the analogy to see in what ways the details of their problems are like the corresponding parts of other problems that have the same structure. Some of the results might be interesting as submissions to “Delving Deeper.” In another direction, readers might look at the general solution to the catch-up problem and ask what initial condi-tions would make such problems have “nice” (integer, say) solutions. See “Delving Deeper. Gauss, Pythagoras, and Heron,” by Bowen Kerins and the High School Teachers Group of the Park City Mathematics Institute in the May 2003 issue of the Mathematics Teacher for more on this theme. The process outlined by Stanley and Walukiewicz is well worth applying to other prosaic problems, and we welcome the opportunity to share results of those investigations with others through this department. The following is a favorite of our own for you to contemplate. Some years back, while thinking hard about students’ learning of linear algebra, we noticed structural similarities between the evaluation of

and that of

We wondered, given some suitable context (for example the determinant as a computation of area), what other analogies can be made between the two evaluations.

DICK STANLEY, [email protected], is a mathemat-ics specialist at the University of California at Berkeley, Berkeley, CA 94702. He is interested in finding more ways to let the beauty of mathematics emerge in high school and university courses. JOLANTA WALUKIEWICZ, [email protected], a mathemat-ics teacher at El Cerrito High School in El Cerriot, CA 94530, is interested in innovative methods of teaching mathematics and evaluating students’ learning.

Reprinted with permission from the Mathematics Teacher, copyright April 2004 by the National Council of Teachers of Mathematics. All rights reserved.


GRADES 9-14

Empirical Approaches to the Birthday ProblemAlfinio Fores and Kevin M. Cauto

Through stimulation, students perform experiments to de-velop probabilistic intuitions regarding the classic Birthday Problem. We describe two activities that help students see that repeated birthdays are not unusual. Using technology, they simulate the problem with different groups of virtual subjects. Because samples are generated easily with tech-nology, students can focus their attention on analysis and underlying concepts.

Introduction Many students do not have systematic opportunities to develop probabilistic intuitions. Developing a sense of the distribution of random outcomes requires at least two things. First, students need a large number of experiences with the same probabilistic situation, which is a rare event in most classrooms. Second, they need to encounter a va-riety of probabilist situations including equally likely events tossing a coin or one die) and unequally likely events (roll-ing various sums with a pair of dice). Often students form their conceptions about probability based on a very limited number of experiences, and they are frequently not aware that they have misconceptions and poor intuitions about probabilistic situations. It is important for students to bring these poorly formed notions to the conscious level so that they can modify them. One way to do this is to have stu-dents deal with a probabilistic situation where the result is unexpected. When students articulate their ideas before they experiment or analyze the situation, they maximize the potential that such situations will make them rethink their basic assumptions (National Council of Teachers of Mathematics [NCTM], 1989, p. 110). Students should actively explore situations by experimenting and simulating probability models. It is recommended that students perform experiments to develop probabilistic intuitions and concepts before dealing with theoretical probabilities, and stress the relation between experimental and theoretical probability (NCTM, 1989). This approach has the advantage of getting the students involved in their own learning (by doing experiments) which is one of the recommendations of the Professional Standards for Teaching Mathematics (NCTM, 1991). One approach to develop understanding of basic concepts of probability is to use simulations to construct empirical probability distributions (NCTM, 2000, p. 324). The use of technology can facilitate students’ learning of probability in at least two ways. First, students can generate a large number of simulations in a short time, so that they can observe the variability from one experiment to the next. Second, because the samples are generated fairly easily by the computer or the calculator, students can focus their attention on analyzing the data (NCTM, 2000, p. 254).

An unexpected situation: Repeated birthdays Often students find it surprising when in a given class-

room, there are two students with the same birthday. When you ask students what they think about the probability that in a group of 40 persons at last two have the same birthday (same month and day, not necessarily same year), many people think if should be fairly unlikely given that 40 is a small number compared with 365. They find it hard to be-lieve that the probability of repeated birthdays in this case is almost 90%. They also find it surprising that for groups of only 23 students the probability is already about 50% that two individuals will have the same birthday. Although people see fairly easily that in order to be certain that two people have the same birthday you need 366 persons in the group, they have not developed an appropriate intuition that it takes relatively so few people to have a high probability for a group to have repeated birthdays. For them, 23 is too small com-pared with 365 to think that the probability is about 50%. In this article we will describe two activities in which stu-dents conduct experiments with random numbers so they can see that repeated birthdays are not really that unusual. In a third activity, students use a calculator program to deal with the theoretical probability. We will make several as-sumptions to simplify the experiment. We will use the year with 365 days thus disregarding birthdays on February 29. We will assume that each day is equally likely for birthdays. In real life this is not quite true, in the U.S. the daily average of births is slightly higher during the summer months July - September, and lower in January (James, 2005). Rather than listing the birthdays by month and day, we will deal with numbers between 1 and 365. Thus 2 corresponds to January 2nd, 32 corresponds to February 1st, and so on, until 365 corresponds to December 31st.

First Empirical Approach We can use a computer or a graphing calculator to generate lists of random numbers. For example, with the calculator TI-84 Plus, pressing the MATH key, and moving the cursor to the PRB (probability) menu, the option randInt( will be shown.) For this function we can chose the range of random numbers and how many will be generated. So for example, randInt(1, 365, 23) will generate 23 random whole numbers between 1 and 365 inclusively). We can instruct the calculator to store the numbers in a given list by using the STO> key. So by typing randInt(1, 365, 23) STO>L2, the calculator will generate 23 random numbers and will store them in list 2. Once the data are stored in a given list, students can order them. To do so, students can press the STAT key and the EDIT menu choose SortA( and enter the list they want to sort, for example, SortA(L2). In Figure 1 two lists have been generated. After scoring, we see that list L1 has one entry repeated, 97. That means two people had the same birthday (97 corresponds to April 7). First simulation: 10, 20, 31, 63, 97, 97, 113, 122, 136, 152, 169, 179, 192, 212, 213, 222, 228, 294, 332, 342, 354, 360, 363 (repeated birthdays highlighted)


Second simulation; 2, 16, 18, 32, 45, 89, 93, 94, 99, 103, 111, 112, 117, 150, 220, 245, 283, 290, 299, 309, 310, 320, 350 (no repeated birthdays) Students in a classroom can generate store, and sort their own lists. As students go through their lists, they can see whether there are repeated numbers or not. When the teacher conducted this activity with a group of twenty students, each of whom generated a list of 32 numbers, it turned out that in 11 of the 20 cases there were repeated birthdays. When students generate longer lists, it be-comes obvious that the probability of repeated birthdays is fairly high. For ex-ample, when 20 students generate lists of birthdays for 41 members in a group, they will find that in the vast majority of the cases there is a repeated birth-day. In one case, twenty simulations of groups of 41 yielded twenty cases of repeated birthdays!

Another empirical approach In the next activity, students run a program that will simu-late adding people at random to a group until there is a repeated birthday (Flores Penafiel, 1990). For each group, the program adds a new birthday to the group and records it until there is repeated birthday. The calculator will display how many people were in the group when a birthday was first repeated, and does this for 50 groups. The calculator commands are on the left, the explanation for each line on the right.

PROGRAM:CUMPLE1For(N,1,50,1) start of loop for 50 groups365→dim(L4) list for 365 birthdaysFill(0,L4) zero for each day0→P zero people in the gorupRepeat max(L4)>1 loop group; instructions executed until one birthday is repeated(1+int(365*rand ))→R new birthday added at random(1+L4(R))→L4(R) one is added to birthday tally1+P-P number of people is countedEnd end of loop for groupDisp P number of people in group displayedPause pause between groupsEnd end of loop for 50 groups

The table below shows the results of running the program. We see that in this sequence of 50 experiments, in all cas-es it took 45 or few people to have a repeated birthday. Of course, running the program again will give slightly differ-ent results. Students can determine the median of the data in the table (22). This is another way to see that the prob-ability of repeated birthdays fro groups of more than 23 is greater than 50%. The graph in Figure 2 represents results of doing the simulation 200 times. Students can see that, indeed, in 107 of the cases (more than 50%), the groups were 24 or less when the repeated birthday occurred.

Theoretical probability After students have done several simulations to see the experimental probability for repeated birthdays, they can deal with the theoretical probability of the situation. In or-

Figure 1. Two Lists of Ordered Birthday

der to compute the probability of repeated birthdays, it is easier to think first about the probability that there are no repeated birthdays among a group of people. That is, we will compute the probability of the complementary event first. If there are two people, the probability of not having a repeated birthday is 364/365. If a third person is added, we need to multiply If a fourth person is added, we need to multiply the previous result by Thus, we can compute the probability of different birthdays in a recursive way. If we know the probability for a group of n people, we

Figure 2. Distribution of size of groups when birthday was repeated.


can compute the probability of n + 1 people multiplying the product corresponding to n people by If Q is the prob-ability that in a group of n + 1 people there are no repeated birthdays, than Therefore, the prob-ability of having at least two people with the same birthday is 1 - Q. Students can run a short program to calculate the prob-ability of repeated birthdays for successive numbers of people in a group. Here is a program for the TI-84 calcula-tor. For each one of 100 people added in succession, the program computes the probability that the new person’s birthday is not the same as any of the people already ac-counted and multiplies it by the probability that there are no repeated birthdays among people already in the group. It displays the number of people and the probability of the complementary event.

PROGRAM:BIRTHDAY1→QFOR (N, 1, 100, 1)Q*(365-N)/365→QDISP N+1, 1-@, ″ ″PauseEnd

Students can run the program and see what are the prob-abilities for successive numbers of people. In Table 2 we display the results for a few numbers, including 2, 4, 23, 30, 41, 50, 57, and 70. Students will see that for 70 people the probability of repeated birthdays is more than 99.9%, even though 70 may seem relatively small compared to 365.

Concluding remarks A common misconception about random samples is that the outcomes are “spread out” more of less evenly among the possible results. Students often think that the distribu-tion of the sample reflects pretty closely the whole distri-bution. Thinking about 23 people compared to 365 days, students may think that there are a lot of slots available, so that it is very unlikely that two people will have the same birthday. However, one of the understandings that students need to develop about random samples is that sometimes they come in clusters, and are not always uniformly or sym-metrically distributed among the possible results. Doing simulations like the ones described above can help stu-dents develop such understandings.

ReferencesFlores Peñafiel, A. (1990). El mismo compleaños: Explo-

rando el azar con una microcomputadora. Educación Matemática, 2 (1), p. 58-60.

James, M. S. (2005). Tables. Births and deaths by month, 1995 -2002. Retrieved July 16, 2009, from http://abc-news.go.com/Health/Science/story?id=990641

National Council of Teachers of Mathematics. (1989) Cur-riculum and Evaluation Standards for School Mathemat-ics. Reston, VA: National Council of Teachers of Math-emtics.

National Council of Teachers of Mathematics. (1991). Pro-fessional Standards for Teaching Mathematics. Reston, VA: National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics. (2000). Prin-ciples and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics.

AFFINIO FLORES, [email protected], is professor of mathemat-ics education at the University of Delaware. He teaches mathe-matics and mathematics education courses to undergraduate and graduate students. He encourages the use technology to help de-velop understanding of mathematical concepts. KEVIN CAUTO, [email protected], is a graduate of the University of Delaware with a bachelor’s degree in mathematics education and classical studies. He recently became a teacher in Palisades School Dis-trict in Pennsylvania. Kevin enjoys working with kids of all ages.

Reprinted with permission from the Ohio Journal of School Math-ematics a publication of the Ohio Council of Teachers of Math-ematics, Fall 2009.

Table 2. Theoretical probabilities for repeated birthdays.


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VCTM 2011 William C. Lowry Mathematics Educator of the Year Award

To: VCTM Members From: Brenda P. Barrow

Virginia Principals VCTM William C. Lowry Mathematics Educator of the Year Committee

Math Department Heads 1311 E. Bayview Blvd.

University Department Heads/Deans Norfolk, VA 23503

Email: [email protected]

Phone: 757 – 617 – 0984

Each year the Virginia Council of Teachers of Mathematics may recognize a classroom teacher on the elementary, middle,

secondary, university and math specialist/coach level for his/her outstanding work in the field of mathematics. One teacher

selected from each of the five categories may be awarded the VCTM William C. Lowry Mathematics Educator of the Year

Award. All awards will be announced in the spring of 2011.

Past winners and current elected VCTM Board members are not eligible for nomination.

The qualifications for this award are as follows:

* The nominee must be a current member of VCTM.

* The nominee must have a minimum of five years teaching experience and be a current classroom teacher, work with students as

a math resource teacher or be a math specialist.

* The nominee must have made notable accomplishments in teaching mathematics.

* The nominee may be nominated by a sponsor or may make a self-nomination. (Anyone who is a member of VCTM, a school

division superintendent, a school principal or headmaster, a supervisor, director of instruction, a college dean or department head

or the president of any NCTM affiliated group may sponsor a candidate.)

• Details about the nomination and information needed from the nominee will be mailed to the nominee.

You are encouraged to nominate an outstanding mathematics educator that you feel is deserving of this award.

Complete the form below and return it to the address on the form. Electronic nominations are acceptable.

The awards committee will contact the nominee upon receiving the nomination to request additional information.

Nominations must be postmarked or electronically submitted no later than October 1, 2010.

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

Nomination Form


Nominee information – Please PRINT or TYPE.

Date: ______________________

Name of Nominee: ______________________________________________________________

Home Address: ___________________________________________ Email: ____________________________

____________________________, VA ________ Home phone: ( ____ ) _________________

Nominee’s Position and School: ______________________________________________________________

Nomination Category: Elementary ____, Middle ____, High ____, University ____, Math Specialist ____

Nominee’s School Address: _________________________________________________________________

___________________, VA __________ School phone: ( ____ ) ________________

Sponsor Information - Please PRINT or TYPE.

Name: ____________________________________ Position or Title: __________________________________

School Division, College or University: ____________________________________________________________

Business Address: ______________________________________________________________________________

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A letter of recommendation DOES NOT have to accompany the nomination. The nominee will ask that you submit a letter to

him/her that can be included in the response packet with the other two letters of recommendation that he/she must submit.

Nominations must be postmarked or electronically submitted on or before October 1, 2010.

Please mail to: Brenda P. Barrow 1311 E. Bayview Blvd. Norfolk, VA 23503

Electronic nominations are welcome. Send to: Brenda Barrow at this email address. [email protected]

THANK YOU FOR MAKING THE NOMINATION!




Virginia Council of Teachers of Mathematics

Fall Academy Sweet Briar College October 1st and 2nd, 2010

Registration Form

Please Print

Name _________________________________________________________________________________ Address _______________________________________________________________________________ Street City State Zip email (Please include)_____________________________________________________________________ Home Phone (_____)__________________________School Phone (_____)________________________ School ______________________________________________________________

School Division ______________________________________________________________ Level (please circle all that apply): K-2 3-5 6-8 9-12 College

Mathematics Specialist

Pre-registration Deadline is September 1st, 2010. It must be received by the 1

st.

Registration fee includes lunch on Friday.

Registration fees are non-refundable. Registrations may be transferred.

________ FREE REGISTRATION FOR PRIMARY SPEAKER (One speaker per session) ________ $20 membership renewal for primary speaker ________ $50 Conference Registration fee for VCTM members ________ $70 Conference Registration fee for non-member or renewal of membership. This price

provides a 1 year membership.

Make check payable to VCTM and return to

Diane Leighty, Treasurer

VCTM

PO Box 73593

Richmond, VA 23235

[email protected]

Thank you! We look forward to seeing you in October!


VCTM Membership Form

Please print the information below.

Name: ____________________________________________________________________________

Address: __________________________________________________________________________ Street City State Zip

School: ____________________________________________________________________________

School Division: ____________________________________________________________________

Job Title: __________________________________________________________________________

E-Mail Address: _____________________________________________________________________

Home Phone: _______________________________ School Phone: ___________________________

Remember! We are a volunteer organization—teachers supporting teachers. Please help in any way you can!

Check Committee interests: Check leadership interests:

_____Publications _____Committee Chairperson

_____Publicity/Promotion _____Officer

_____Scholarship Check conference interests:

_____Nominations/Elections _____Speaker

_____Professional Development Grant _____Program Committee

_____First Timers Grant _____Conference Assistant

_____Mathematics Activities

Please check the appropriate membership renewal:

_____$20 Individual One-Year Membership _____$10 Student Membership

_____$39 Individual Two-Year Membership _____$20 Institutional One-Year Membership

_____$55 Individual Three-Year Membership _____$500 Lifetime Membership

I wish to donate $_____ to the VCTM Scholarship Fund to support the education of Virginia’s future math educators.

Total enclosed: $_____Make check payable to VCTM and return toPatricia Gabriel, VCTM Executive Secretary

P. O. Box 714 Annandale VA 22003-0714

Virginia Council of Teachers of MathematicsP.O. Box 714Annandale, VA 22003-0714 Pat Gabriel, Exec. Secretary

Non-ProfitOrganizationU.S. Postage

PAIDBlacksburg, VAPermit No. 159

DATE AND NOTE POSTVCTM FALL ACADEMYMaking Sense of Change

Putting the new Standards into PracticeSweet Briar College

October 1-2

VCTM ANNUAL CONFERENCE 2011Richmond

March 11-12, 2011

2010 Regional NCTM ConferencesDenver, CO Baltimore, MD New Orleans, LAOctober 7-8 October 14-15 October 28-29

NCTM 2011 Annual MeetingIndianapolis, IN

April 13-16, 2011Calling Virginia Authors: Virginia residents whose articles appear in the VMT will be granted free member-ship in the VCTM for one year. To qualify, the manuscript must be at least two typewritten pages in length.

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