FOCUS_14_3.pdf - Mathematical Association of America

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.'", THE NEWSLETTER OF THE MATHEMATICAL ASSOCIATION OF AMERICA

Mathematics in the Technical Work Force

2 EdItorl8l: IJo;WeNeecl EdueadOBReform?

3 CD-ROM ,TextbOoks aDd'Cakulus

Susan Forman and LynnArthurSteen

Conventional wisdom about mathematicseducation has often focused on "three Ts":texts, tests, and teachers. Current politicalpriorities have added two more Ts: trackingand technical education. Although these latterare somewhat less visible, especially to mathematicians, they are no less important for thequality of mathematics education.

The campaign for advanced technical education is a centerpiece of the Clintonadministration's educational program. Theeconomic rationale for this priority is clearlyspelled out in Robert Reich's The Work of

Nations. The political rationale is centered inconcern for an internationally competitiveworkforce. The personal rationale can be seenin the long lists of college graduates (and Ph.D.recipients) who cannot find work suited to theireducation.

To improve the transition from school to work,Labor Secretary Reich and Education Secretary Richard Riley advocate greater emphasison programs in grades 10-14 that prepare students for technical careers in fields such as

Please see Technical Work Force on page 9

6 Student Attitudesin Calculus

Reflections on the Joy of TeachingAlan Tucker

10 CiviIizecIMathematics

12 PerscmaI Opinion

14 Contributed Papersin San Frandsco

16 The MAA Gopheris on its Way!

19 New MathematicalCurriculumInitiatives at theNSF

Remarks on receiving the MAAs 1993 National Award for Distinguished Teaching,Cincinnati, Ohio, January 1994.

I am very honored to be a recipient of theMAA's 1993 National Award for Distinguished University/College Teaching ofMathematics. Teaching collegiate mathematics has always been so satisfying for me that1have felt obligated to those around me in myacademic life-my students and fellow faculty-to pay them back in some way formaking my life so enjoyable. While teachingday in, day out can lose some of its joy andexcitement, one critical aspect of good teaching, I believe, is keeping the excitementconstantly alive. If teachers show that theyfeel good about teaching, their students willfeel good about learning, and this in tum enhances the pleasure of teaching.

At the same time that I am excited about beinga mathematics teacher, I am fairly humble aboutthe fact that 95% of the work in the educationalprocess is done by the students; I am just theircoach. I want very much to help them learn, butit is very hard to get into their heads to know theright things to say and the right times to saythem. Thus, I characterize my job as teachingstudents to show me how to help them learn. Toemphasize my role as coach and where the realresponsibility lies, I often compare developingthe mind to developing muscles. If I take a person into a room full of exercise equipment andI talk about how these machines can developthe body, my lecture, no matter how polished,will not build anybody's muscles. Mathematical learning involves growing many newsynapses (and related biological changes) in the

Please see Reflections on page 8

The MathematlealAssociation of America

1529 Eighteenth Street, NWWashington, DC 20036

• The MAA Gopher is here! See page 18.• Individual Subscriptions now available

for Math Horizons. See page 23.

FOCUS

FOCUSFOCUS is published by The MathematicalAssociation of America, 1529 EighteenthStreet Northwest, Washington, DC 200361385, six times a year: February, April, June,August, October, and December.

Editor and Chair of the MAA NewsletterEditorial Committee: Keith J. Devlin, SaintMary's College of California

Associate Editor: Donald J. Albers, MAAAssociate Executive Director, and Directorof Publications and Programs

Managing Editor: Harry Waldman, MAA

Production Specialist: Amy E.Stephenson, MAA

Proofreader: Meredith Zimmerman,MAA

Copy Editor: Nancy Wilson, Saint Mary'sCollege of California

Letters to the editor should be addressed to:Keith Devlin, Saint Mary's College ofCalifornia, P.O. Box 3517, Moraga, CA94575, e-mail: [email protected]

The FOCUS subscription price to individualmembers of the Association is $6.00,included in the annual dues. (Annual duesfor regular members, exclusive of annualsubscription prices for MAA journals, are$68.00. Student and unemployed membersreceive a 66 percent discount; emeritusmembers receive a 50 percent discount; newmembers receive a 40 percent discount forthe first two membership years.)

Copyright © 1994 by The MathematicalAssociation of America (Incorporated).Educational institutions may reproducearticles for their own use, but not for sale,provided that the following citation is used:"Reprinted with permission of FOCUS, thenewsletterofThe Mathematical Associationof America (Incorporated)."

Second-class postage paid at Washington,DC and additional mailing offices.Postmaster: Send address changes to theMembership and Subscriptions Department, The Mathematical Association ofAmerica, 1529 Eighteenth Street Northwest,Washington, DC 20036-1385.

ISSN: 0731-2040Printed in the United States of America.Printed on recycled paper.

June 1994

EditorialDo We Need Education Reform?Do we need education reform? Is our education system currently in a "crisis"? Severalof the articles in this issue of FOCUS address these questions in one way or another.

To be honest, I do not know the answers. Certainly, things are not perfect-but then, theynever are (or were). Moreover, those of us in the education business should alwaysevaluate what we do, how we do it, and how successful we are. That much is part andparcel of being a professional, not just in education but in any walk of life. There arealways ways to improve, and we need to be constantly looking for those ways. But is oureducation system in need of radical reform, the kind of major overhaul that many arecalling for?

Most of us who have made mathematics the focus of our professional lives have beeneducated in the system that many now say is in need of complete change. If it was so bad,how come so many of us are doing what we do today?

One obvious difference-and maybe it is the principal difference-is the environmentin which children grow up. We went through the educational process at a time when thewritten word was still the principal medium for broadcasting information. We wereadapted to the concentration and time commitment necessary to obtain information frombooks. We learned at an early age that "technical" reading is not a passive activity, ratherit requires considerable involvement. The same thing is not true today. Audio-visualcommunication is, for many, the norm. Attention spans are shorter, and there is a greaterexpectation of "gloss". Whether you think this is a good thing or a bad thing is besidethe point; it is the way things are, and that is the environment in which we, as educators,must work.

The question is, do these changes in environment and expectation require just a modification to the education system we currently have, or is there need for radical reform?If the latter, how can we be sure we do not throw out the baby along with the bath water?As I said at the beginning, I do not know the answers, and I am wary of those that claimthey do. But not knowing the answers will not make the questions go away, nor will theissues become less acute.

-Keith Devlin

The above are the opinions ofthe FOCUS editor, and do not necessarily represent theofficial view ofthe MAA.

June 1994 FOCUS

CD-ROM Textbooksand CalculusRoland E. Larson

Textbook publishing appears to be on thebrink of a revolution-one that is likely tochange the way we think about classroominstruction. The static printed textbookmay be giving way to a new dynamic format: CD-ROM.

lntt r J( 11'0'1 I III ulu \'1 r 11111 I II

Wegave a briefin1roductionto the tangentline problem in Section 1.1. Althoughpartial solutions to this problem weregivenby Pierre de Fermat(1601-1655), Rell!Descartes(1596-1650). ChristianHuygens(1629-1695). and Isaac Barrow(1630-1677).credit for the firstgeneral solution is U3Ua11ygiven to Isaac Newton(1642-1727) andGottfriedLeibniz (1646-1716).

For more informationon the ereditingof mathematicaldiscoveriesto the first"discoverer,"see the articleMathematicalFirsts-WhoDOM It?

IsaacNewton

ity for interactivity, use of multimedia, andnonlinear organization.

Interactivity: Computer software is interactive if its responses depend on userinput. An automatic teller machine is interactive because the amount it dispensesdepends on the account holder's selection.

An interactive calculus textbook is onethat is capable of responding to user input.At the low end, an interactive calculustextbook could consist of a single instruc-

Trychanging thefunction. Can youfindvalues of xl forwhich thederivative is zero?

Trychanging thevalue of xl.Can you findthevalues forwhichthederivative is zero?

Graph of fInd tlngent It x =xl

Define derivative.

-1.2 -11.4 0.4 1.2

"Mathematic:al F1rsts- Who Done If'

hI')

Specify value of xl. (You caneditthis line.)

h(x):= g(xlj{x - xl) + I(xl) Define tangent line.

Specify x-range for thegraph. (You caneditthis line.)

references, current journal articles, solutions to exercises, and alternativeapproaches.

By itself, the storage capacity of a CD isnot a compelling reason to change fromthe comfort of print. In fact, initially thecalculus reform movement talked aboutdeveloping a "lean" course. (My secondarticle on multimedia and nonlinearityexplains how a CD-ROM calculus textcan be both lean and voluptuous.) Muchmore compelling are a CD-ROM's capac-

Figure 1 This interactive exploration was written with Mathcad. In the initial screen the user is asked toestimate graphically the x-values at which the derivative off(x) = xl - 2x is zero. Each time the user edits the value ofxl, the graph updates to reflect the new input. After exploring this function. the user canedit the function f to obtain an entirely new exploration.

In terms of currently affordable technology, the disadvantage of a CD-ROMtextbook is that it is simply not as convenient as a printed textbook. Even mostportable computers weigh more thanprinted calculus textbooks! Users of aCDROM textbook are restricted as to wherethey can use the text-reading in bed, forinstance, is probably not a realistic option. Users are subject to computer orbattery failure, and screen resolution is notas good as typeset resolution. In fact, if aCD-ROM calculus textbook is simply anelectronic page turner, then the inconvenience of relying on a computer makes theprinted version the better choice.

The advantages of a CD-ROM calculustextbook are its storage capacity and itspotential for interactivity, use ofmultimedia, and nonlinear organization. Theremainder of this and the next article willconsider each of these four advantages.

Storage Capacity: An immediate advantage of a CD-ROM textbook is itsenormous capacity for storing data. Asingle CD can contain many thousands ofpages of text and art. In CD form, a calculus textbook can easily include historical

Up to now, college publishers have madeonly a few ventures into this new format.The reason is clear: the cost of producinga major textbook can easily transform this"cutting edge of technology" into the"bleeding edge of technology."

Now, however, the call for educationalreform has increased the likelihood thatCD-ROM textbooks could be commercially successful. With that possibility, wecan expect the appearance of several newCD-ROM products.

This article, the first of two, explores theadvantages of a CD-ROM calculus textbook.

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June 1994

would explain how these four constantsare related to horizontal shifts, verticalshifts, and reflections. A discovery modeallows students to deduce their own conclusions.

In practice, students seem to be more successful in courses that strike a balancebetween the storytelling mode and thediscovery mode. If a CD-ROM calculustextbook is used as an alternative to aprinted textbook, interactivity should formonly part of the picture. Much of the CORaM textbook should be comprised of"noninteractive" material, such as definitions, examples, and theorems.

Interactivity has consequences that reachbeyond the environment of a single student studying calculus. In a traditionalclassroom, an interactive CD-ROM textbook can be used as a lecturing device. Ina nontraditional classroom or a math lab,an interactive textbook provides a perfectvehicle for group activities. In a tutorialcenter, an interactive textbook is capableof providing unlimited extra practice.

The screen shown in Figure 3 is taken froman interactive calculus textbook. It is thefirst of a group of five screens that explorethe conceptof a tangent line.The first three"pop-up" screens walk the user through

The Slope ofa Tangent Line

Explore It Yourself

Analytic Approach

Numerical Approach

Graphical Approach

Explore It First .Thegraph of f (x) = x 2has manytangentlines, Oneof thetangentlines-the one at the point(t, t)-is shownat theright.

Howcould you findthe slope of this tangentline?A3withmany problemsin mathematics, youcan approachthe problemin three ways,

Figure 3 The screen design for an interactive textbook needs to be very different from the page design ofa printed textbook. For instance, this screen is the first ofa group offive screens that explore the concept of tangent lines. Rather than having a lot of information on a single screen, the design calls for"morsels" of information on several screens. This type ofdesign allows an interactive textbook to beused as a classroom lecturing device, and it also provides "rewards" for individual users. Each time theuser moves on to a new screen, he or she is rewarded with new visual stimulation.

ery" mode, as shown in Figure 2.

In this exploration, the user is asked tochange the values of the constants a, b, c,andd. Each time the user edits one or moreof these values, the graph updates to reflect the new input. A storytelling mode

What is thedomain of g?What is the range of g?

Selling b = -1 reflects thegraph of f in they-axis.Change thevalues of a, b, c, and d to obtain theother basic types of transformations. State whichtypeof transformation youhave obtained.

Graphof ffxl andglxl

Define f(x).

Specify constants for g(x). (You caneditthisline.b must be nonzero.)

Define g(x)asthetransformation of f(x).

Specify the input values for f{x).

V :=;f(b>O. 2·lcl + 2, -~)

Specify the input values for g(x).

o11,12

-2

c:=O d:=O

, - /'-" /-.

[(xl)

g(1t.2)

g(x) := a.f(ll-x +c) +d

xl :=0,0.125.. 2·1c1 + 2

L :=if(b>O.-~. -lcl'2 - 2)

x2:= L,L+0.125.. V

FOCUS

Figure 2 In this exploration. the user is invited to "play" with the values ofa. b, c, and d. As each ofthese constants is changed, the graph immediately updates. The goal of the exploration is for the user todiscover how each constant affects the graph of the function!

tion on a word processor-"discover calculus for yourself." In this case thecomputer's interaction consists of keeping a record of the user's voyage. At thehigh end, the computer could play the roleof an all-knowing tutor who would guidethe user on his or her voyage. Each step upthe interactive ladder involves more timeand money for the developer of the text.

The Mathcad screen shown in Figure I isan example of interactivity. The user isable to edit the definition off, the choiceof x I, and the x-range of the graph. Aftereach change, the entire screen updates toreflect the new input. Similar versions ofthis exploration can be written with symbolic languages such as Derive, Maple,and Mathematica.

This "exploration" invites the user to findx-values graphically at which the derivative is zero. Because the tangent line isredrawn with each new choice of x I, theuser can discover that the appropriate values are close to ±O.816. This explorationcan also be used to confirm that the exact(or analytic) values are ±-fiT3 .The learning opportunities provided byinteractive explorations like those shownin Figures I and 2 are very different fromthose provided by a printed textbook. Foran author, the challenge in writing suchexplorations is to move out of thestorytelling mode and into the "discov-

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June 1994 FOCUS

t!u -3.0 -2.0 -1.0 -0.5 -0.1 0.0 0.1 0.5 1.0 2.0 3.0

f(4 + il.x)

T(4 +t!u)

are indicating that interactive learningsituations have resulted in increased retention. This appears to be occurringbecause interactivity can provide immediate feedback to the user.

Roland E. Larson is a professor ofmathematicsat Penn State University at Erie. Thefifth editionof his calculus text. Calculus, with D. C. Heathand Company. is available in printed and CDROM versions.

- Increased Motivation The dynamic interaction between user and computer hasbeen shown to focus a user's attention.Users tend to respond more to the activesetting of a CD-ROM textbook than thepassive setting of a printed textbook.

In a May 1993 article in Newsweek titled"An Interactive Life," Barbara Kantrowitzand Joshua Cooper Ramo discuss thepromises that interactivity hold for communication, entertainment, and education.They claim that the products that are beginning to appear are just the "firstgeneration." What can we expect withinthe next decade? The answer appears tobe: no one knows. It seems clear, however, that we can expect change.

The second article in this series will focuson two other advantages of a CD-ROMcalculus textbook-use ofmultimedia andnonlinear organization.

h f, U3ethe "zoom in" feature to obtainthe graph in the neighborhood of the point

times. the graph should appear almost linear,

-

,.. Maple@

III: MATHCAD@ (4)'* Mathematica@ .ing through the given point!.-._----,.,....-__---..,.-:---..,.---,=:!ffi the graphs off and T on the same set of

coordinate axes, Note that T is a "good" approximation off when x is"close to" 4, What happens to the accuracy of the approximation as youmove farther away from the point of tangency?

d. Demonstrate the conclusion of part (c) by completing the table.

"mainstream" courses.

- Privacy When used in aone-on-one learning environment, an interactive bookallows students the opportunity to "ask"questions without riskingembarrassment.

«lncreasedRetention Preliminary studies

Figure 5 An interactive calculus text can be programmed to interface with a variety ofexisting products.For instance. the tools button of this interactive textbook allows users to open a symbolic language window and use the language to solve the exercises provided in the textbook. Similar interfaces can beprogrammed to include word processors. graphing calculator simulators, curve-fitting software, testingsoftware, and spreadsheets.

9. f(x) = 2x2 +x-l

f'(x) = Urn f(x +il.x) - f(x)II X""0 il.x

= Urn 12(X +il.x)2 + (x + il.x) - 1l=J2x2 + x - 11llx ....O il.x

= Urn (2x 2 +4xil.x +2 (il.x)2 + x +il.x -1) - (2x2 +x -1)llx....O il.x

= Urn 4xil.x + 2(il.x)2 + il.xllx ....O il.x

= Urn (4x +2il.x + 1)llx ....O

=4x +1

In EXercises 5-16, use the definition of the derivative to findf' (x),

III f(x) = 3 6. f(x) = 3x +2

• f(x) = -5x 8. f(x) = 9-}X

III f(x) = 2:? +x -1 10. f(x) = 1 - x2

Screen design and interface are importantconsiderations for developers of interactive textbooks. For instance, using largetype with an open format is not only moreengaging for individual users, it also allows the textbook to double as a classroomlecturing device.

Figures 4 and 5 illustrate two ways thatinteractivity can be incorporated into theexercise sets of an interactive calculustextbook. At the push of a button, workedout solutions can provide additionalexamples for the user, as shown in Figure4, or problem-solving tools can be provided, as shown in Figure 5.

Rockley L. Miller, editor of Multimedia& Videodisc Monitor, lists the followinglearning benefits of interactive textbooks.

- Reduced Learning Time Several studieshave found that interactive technologiesreduce learning time. Although most ofthese studies have concerned remedial ortechnical subjects, one could expect similar results in calculus and other

Figure 4 Because of its vast ability to store data. a CD-ROM calculus textbook can provide many moreexamples for users. Including the worked-out solutions of the odd-numbered exercises ofa typicalprinted calculus textbook would require several hundred pages of text. With a CD-ROM textbook. prose.art. explorations. animations. videos. examples. exercises. and worked-out solutions can all be stored onthe same disc.

various approaches to finding the slope ofthe tangent line. The last screen asks theuser to try using one or more of the approaches to find the slope of the tangentline at a different point on the graph offix) = x?-.

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FOCUS June 1994

Student Attitudes in First-Semester CalculusDavid Bressoud

One ofthe large lecture sections' offirstsemester calculus at Penn State wassurveyed to determine student attitudestoward calculus and the learning of calculus. This is a summary of the picturethat emerged. The student described hereis not entirely typical: only halfofthe students were in class that day. 91% ofthosewho answered this survey reported thatthey regularly attend class. This is therefore a picture ofthe conscientious studentwho comes regularly to class. It says nothing about the student who does not cometo class. I have chosen to name this conscientious student Chris.

Chris is a science or engineering major inthe first semester of the freshman year andis, by definition, a good student who didwell in high school. Chris has always likedmath,' feels confident about the ability tohandle it,' and believes that it will play animportant role in the chosen major.' Especially appealing is the precision found inmathematics and the sense of satisfactionthat comes from getting an exact solutionto a problem.' Chris took an AdvancedPlacement calculus course in high schoollast year," but entered Penn State feelinginsecure about the ability to use calculus.Expecting that it will provide an important foundation for later courses, Chrisbegan this class looking for a better understanding of the topics that were studiedthe previous year.'

Having had an AP calculus class the yearbefore, Chris has found all of the topicscovered in first-semester calculus, andsome of those in second-semester calculus, are familiar. Nevertheless, Chriscomes to all or almost all of the lectures."The professor spends the first half of theperiod talking about the theory involvedin the topic at hand, and then moves intoexamples of problems that are based onthis theory." The material is projected ontoa screen with the use of transparencies,many of which are prepared beforehand.Being conscientious, Chris strives to writedown everything that is projected. Becausethe concentration is focused on gettingeverything written down, there is little timeto listen or try to make sense of what isbeing said. 10Chris will not ask questions; II

the large lecture is simply too intimidating.'? Not all of the other students areworking as hard. Many in the back aretalking, and some are sleeping. I)

Later that same day," it is time to studycalculus. This means doing the homeworkproblems." Chris is conscientious, doingevery problem that is assigned, 16 but feeling frustrated by the even numberedproblems for which there are no answersin the back of the book.'? The homeworkproblems are worked alone, but sometimesChris will discuss the most difficult oneswith friends who are taking, or have taken,this course." When doing the homework,Chris initially relies on what is remembered from last year. The next recourse isusually the notes from that day which areused to find examples of similar problems.Such examples are often helpful, but thenotes from the theory part of the day'slecture make little or no sense. '9 The thirdrecourse is the book, which is searched forworked-out examples." Chris will spendabout an hour working on the assignedproblems that relate to that day's lecture. 21

Once a week, there is a lab session thatuses graphing calculators. ToChris, this isa mechanical exercise that is required, butseems to bear no relationship to the rest ofthe course." It would be much more helpful to go over the homework problems."

There are two midterm examinations givenon Wednesday evenings. Serious studybegins the weekend before the exam.24 Thismeans putting in two to three hours a nightuntil the day of the test" redoing homework problems, working the practiceexams, going over notes, and sometimeslooking through the textbook. 26 The theorythat was explained at the beginning of thelectures is unimportant because it neverappears on the exams." There is a gooddeal of frustration after the test becausethe exam questions are not quite like thehomework questions," and it does notseem that the instructor has tested whatChris has studied. The fact that the examsare multiple choice with no partial creditand that each exam counts for such a majorportion of the grade reinforces a sense thatthe purpose ofthe exam is to trap and fail

as many students as possible."

Chris is becoming increasingly cynicalabout the importance of calculus. Therehas been no sign that it is actually usefulfor anything." Maybe calculus really isjust a "weedout course."!' Having arrivedat Penn State with the hope of beginningto understand calculus, Chris thinks it hasnow become more confusing than ever.There seemed to be more emphasis onconcepts and understanding in high schoolthan is being found at the university."

M. Kathleen Heid and Barbara Edwardssuggested many ofthe questions that wentinto this survey. A list ofthe questions anda more detailed summary ofthe responsescan be obtained by anonymous ftp underpublbressoud/survey.dvi at math.psu.edu.

Notes

'Initial enrollment was 350 students.

20n a scale of 1-10, 87% of respondentsrated how much they like math as 7 orhigher, 63% rated it as 8 or higher.

)58% of respondents felt that they coulddiscover some mathematics on their own.Only 33% were certain that they couldnot.

"Among the most common responses to"How is calculus used?" were engineering, physics, science, and highermathematics.

"This was the most common response tothe question "What do you like aboutcalculus?" Typical answers were "I likecalculus because, as in all math, it hasdefinitive answers," and "I like calculusbecause it is like a puzzle. If you knowhow to play or figure it out, you win." Ina related vein, many students like the algorithmic nature of mathematics: "I likehow it is very methodical and tedious."

689% of respondents had taken calculuspreviously; ofthese, 53% took AdvancedPlacement calculus in high school.

"The most commonly given answer to thequestion "What do you expect to get outof class?" was "better understanding ofcalculus."

8As stated above, 9 I% of respondentscome regularly to class. Many of theseemphasized that they have never misseda class.

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June 1994 FOCUS

30The second most common response to

What Interactive Calculus Is.

A DYNAMIC calculustext - the ideal toolfor presenting themathematics ofchange and motion.

125 Spring StreetLexington, MA 02173

David Bressoud recently left Penn State forMacalester College in St. Paul, Minnesota, notout ofdiscontent. but attraction to a student-centered environment.

the question "Whatdon't you like [aboutcalculus]?" was it is not useful for anything. The most common response wasto describe one or more specific topicssuch as volumes of solids of revolution.

3\A term used by several students.

32When asked to describe the differencebetween high school and universitymathematics, 32 respondents talkedabout the quality of explanations, theemphasis on concepts, the familiarity ofthe instructor with the material, and therequirement ofmore thought and deeperunderstanding. 81% felt they got moreof this in high school; 19% said that theygot more of it in college.

DCIIeaIIIA.........Ca1lJEr'rf

Interactive Calculus CD-ROM for Windows isCalculus, Fifth Edition, by Larson, Hostetler,andEdwards in an expanded multimedia format. It'sthe only calculus text that interfaces withMathematica, Mathcad, Maple, and Derive togiveyou "live"editable math and on-screen problem-solving. Interactive Calculus isaudio, video,animated art, and many other features not foundin anyprinted text. Interactive Calculus is powerful problem-solving, exploration, and real-lifeapplications in an exciting, easy-to-use format.Need more information? Call 1-800-235-3565.

o

25Before an exam, students average anextra nine hours of studying.

26As reported above, 90% said that whenthey study they do homework or practice exam problems; 22% go over theirnotes; 17% go over the book. Few students distinguished between "studying"and "studying for a test."

27"1 try to take notes and pay attention, butit is very hard to [do] so because the material is not always necessary to do wellon the test."

28Several students complained that the testsbore no relationship to the homeworkand seemed to be based on memorizedtricks rather than on understanding.

29'fhese were common complaints aboutthe course.

23There was no question on thesurvey that addressed this issue, but itwas one ofthe most common complaints.

2448% start studying at least five daysbefore the exam, another 26% start theweekend before the exam.

work problems," and "I alwaystake notes! I read them but theymake no sense."

22Asked to describe a typical labsession, 66% of the respondents said things such as, "I dothe lab, then I leave." 29% weremore negative: "We getworksheets that make no senseand put down answers that wedon't understand."

20"[1 use the text] when my notesfail to make sense. The textdoesn't provide too much helpeither, it's written so technically that it's almost likereading a foreign language,"and "You must filter out whatis important and what is just toconfuse you."

2\55% of the students normallyspend two to four hours a weekstudying calculus. Half of theremainder spend less than twohours, half over four hours.Students who have not hadcalculus before will spendabout twice as much time.

\9Some of the comments were "I try to use[the notes] to figure out homework, buta lot of the notes seem like useless information," "I rarely read the notes, unlessI have a problem with one of the home-

\2'Veryrarely [ask questions in class]. Theatmosphere is not conducive to interaction between lecturer and student. It'sjust too big."

"These were common responses to "Whatdo other students do?"

1455% of the students will do the homework relevant to a given lecture eitherbefore or on the same day as the lecture.An additional 24% will do the homework problems within one week of thelecture.

1590% said that when they study they dohomework or practice exam problems;22% go over their notes; 17% go overthe book.

\666% ofrespondents do all ofthe assignedproblems. Homework is not collected orgraded.

17A common complaint is expressed in thiscomment: "On the syllabus, there arequite a bit of even problems assigned.This is pointless, especially when theanswers to such problems are posted oneweek later. You need to know if you'redoing the homework right as you do it,not one week later. Either assign moreodd problems or give us the answers tothe even problems sooner."

1856% study alone, although many ofthesewill sometimes get help from or help others. An additional 27% will sometimesstudy alone and sometimes study withothers.

"This summary of what happens in theclassroom is based on the respondents'descriptions.

1095% of the respondents said that theytake notes in class; 43% said that theylisten, pay attention, watch, or try to understand. A typical response was thestudent who said, "The prof gives notesand does a few examples. Most studentspay attention and try to take notes, but hemoves quickly. I usually end up behindand start doodling."

"85% of the students never ask questionsin class.

FOCUS June 1994

freshman calculus one semestereach year,because, as he told me, "Teaching calculus is the most important thing thedepartment does, and so as chair I felt Ishould be leading that effort." When I wasyoung, I knew I wanted to be a math professor like my father so I could have theimmense satisfaction that I saw him getfrom this profession.

I believe my father passed on to me a sensehe got from his father that teachers had .aspecial responsibility to look out for theirstudents' interests (a bit like ministerstowards their parishioners). One of myfather's stories conveyed that message instark terms. While a graduate student atPrinceton, my father taught a section ofanalytic geometry. The professor inchargetaught a specially chosen class oftalentedstudents. He expected all sections of thecourse to move at the fast pace of his special section. My father felt that pace wastoo fast for his own class and he asked theprofessor for a meeting to discuss the syllabus. The professor began the meeting byinsisting that my father agree in advanceto adhere to the syllabus they would decide on. My father refused, and theprofessor went to the mathemati.cs ~hair,

demanding that my father be dismissedfor insubordination. Fortunately, a fellowTAsought help from another professor (S.Lefschetz, who respected people whostood up to intimidating professors), andmy father was transferred to anothercourse. I asked my father where he got thecourage to stand up to the professor, risking his career. He told me he had no ~hoi.ce,

the welfare of his students was hIS firstpriority as long as he was their instructor.

Alan Tucker is at the State University of NewYork at Stony Brook.

question, and probably most other s~ude~tswould benefit from hearing that basic pomtreiterated. An occasional, truly lazy question is a small price to pay for having aclassroom environment in which askingquestions is recognized by stude?ts andinstructor as a welcome and essential contribution to the learning process.

I should add another miracle to my list ofmiracles: the power of mathematics. Working in mathematics, we take for gra~ted

how amazingly dependable mathematicaltruths are. Compare mathematics to biology or economics. While biology hasaccumulated vast knowledge about theprinciples of how organisms ~ork at themacro and molecular levels, If one performs some subtle biochemical test on thecontents of some flask today and then doesit tomorrow or in another lab, there is agood chance that the results :-vill dil!er.The same applies to a fiscal stimulus Imposed on two different countries orcompanies. But in mathematics, when students at my institution or anotherinstitution ten years from now integratesin(5x) from 0 to 1t, they will always getthe same answer. Layer upon layer of techniques and concepts in mathematics arepredicated on this consistency and reproducibility. Facility in such complex,multi-layered reasoning is great preparation for performing effectively in any sortof career, whether mathematically-basedor not.

My appreciation for teaching is a familyaffair. My grandfather taught math beforebecoming a Methodist minister, and myfather, A. W. Tucker, an eminent mathematician at Princeton, always spoke ofhimself as a teacher. He said that his university paid him to teach and gave himfree time to do research. He had alwayswanted to be a teacher, originally a highschool math teacher like his father, but hedid not want to stop studying after a B.S.or an M.S. and ended up on a path leadingto university teaching. At home, he talkedconstantly about his students and his joysin watching them grow and enter satisfying careers. When he was chair ofPrinceton's math department in the I950s,he never had a reduction in his twocourses-per-semester teaching load-ifanything suffered, it was his research.Further, as chair, my father appointed himself as the course coordinator teacher in

Reflections from page J

brain, and strenuous mental exercise is theonly way that growth will occur. Co~ching can include some standard lectunng,but such lectures are meant to provokequestions that show me where studentshave problems. I often remind myself ofthe Chinese proverb, "I tell you, you forget. I show you, you remember. I involveyou, you understand."

One way of trying to stay excited aboutbeing a mathematics teacher is by reminding myself constantly of how hard it is ~o

learn. There are two miracles that I WIllnever understand, given the very imperfect creatures we humans are: one, thatlarge, complex, human organizations function at all (I am constantly amazed whenthe lights are on every morning when Icome into work!); the other miracle is thatsemester after semester, most students inmy classes are learning lots of completelynew ideas. As I age, I appreciate more andmore how hard it is to learn new things.Perhaps teachers focus too much on whatstudents have not learned, not adequatelycelebrating how much they have learned.Several times each semester, I stop duringclass and ask my students to look back athow much their knowledge has advancedsince entering college or since eighthgrade.

It annoys me when a student, upon solving a problem that had stumped him orher, says to me, "That was dumb of me notto see how to do it." In reality, such students usually figured out for themselves,with a little help from me, some problemthat required reasoning skills they werejust starting to acquire. The stud~nt shouldhave said, "I think I'm developing a goodunderstanding of how to solve these problems." Another instance is the term "dumbquestion." If a student has not been following a presentation and asks a "dumbquestion," it usually means that he or s?eis now trying hard to follow the class discussion while realizing the need to knowsome important fact that did not registeras being important earlier. Even if thestudent's mind was wandering earlier orthe student dozed off briefly (tired frombeing up late with a part-time job and acomputer project, students cannot be expected to be alert constantl~), the ~tudent

tried to do the right thing m askmg the

@l)------------

June 1994

Technical Work Force from page 1

telecommunications, manufacturing, andagriculture. As educators have developedstandards for mathematics and other subjects, so now industry associations aredeveloping occupational standards in"skills clusters" to provide a portable national credential that will fit the workplacebetter than the traditional high school orcollege diploma.

At the school level, many of these occupational training programs fall under thegeneral title of "tech-prep"; at the collegelevel-most often at two-year collegesthe moniker of a new NSF program,Advanced Technical Education (ATE),provides a convenient descriptor. Manyof these programs combine the last twoyears in high school with the first two yearsof post-secondary education into "2+2"programs.

In a recent study titled Preparing for theWorkplace, the National Research Council reports that only one in five adults hasa four-year college degree. "The other80%" enter the world of work either directly out of high school, or aftercompleting some non-baccalaureate formof post-secondary education in institutionsranging from community colleges to proprietary vocational institutes, fromemployer-sponsored training programs tocommunity-based organizations. Overall,50% of recent high school graduates takepost-secondary education in contexts thatare not leading to a four-year bachelor'sdegree.

No matter the institution or the program,mathematics plays a critical role in all suchtechnical education programs: virtuallyevery course of study requires mathematical preparation, often of a type that is quitedifferent from the traditional "school-tocollege" preparation that leads fromalgebra to calculus. That's where the other"T" comes in: the growth of school-towork programs, valuable as they may beto many students, may well lead to a newand potentially invidious form of tracking.

The historical form of tracking that NCTMargued against in their Standards relegatedlarge numbers of students, disproportionately minorities, to dead-end courses inwhich students received little challenge

and even less education. The new careerbased tracking will be moreseductive-who can resist the allure ofhigh-tech occupational preparation-butequally capable of perpetuating socio-economic class distinctions. Decisions madein early high school years-especiallydecisions about mathematics coursescan program students into a tech-prep orcollege-prep curriculum with little opportunity for switching. Yet students'dispositions for change between ages 15and 20 virtually guarantee that no inflexible system of early tracking can bepersonally or educationally sound.

ATE programs employ distinctive examples in which mathematics is oftenembedded in occupational contexts ratherthan being presented within a traditionaldisciplinary framework. That such programs differ from the traditionalpre-college track does not necessarilymake them worse, weaker, or "watereddown." But the presence of two paralleland quite different tracks, possibly beginning in grades 9 and 10, may prematurelyforeclose future options at a time whenneither students, parents, nor teachers canreliably predict a student's educationaltrajectory.

This poses a special challenge to the mathematical community: how to support theclear national need for tech-prep and ATEprograms without undermining students'options for a traditional four-yearbachelor'sdegree program. Educators alsoshould be concerned that some studentswho are on the fast track for traditionalcollege programs-sometimes under parental or peer pressure that may not reflecttheir true interests or abilities-may finish high school ignorant of options forpost-secondary technical or vocationalpreparation that may suit them better thantraditional college programs.

One resolution of this dilemma would bea common mathematics program thatcould serve equally well as preparationfor college and as preparation for skilledwork. All students could benefit from thebroadening effects of such a high schoolpreparation, yet there are currently fewgood models of curricula that serve bothmasters. The challenge of a common program opens up questions on many fronts:

• What would it take to convince high

FOCUS

school and college teachers of mathematics that there is a common good insuch a common curriculum?

• How can mathematicians contributetheir understandings about the practiceof mathematics to the tech-prep andATEmovements, and learn from these movements insights that may benefit theirteaching and research?

• Can one develop performance expectations for high school mathematics thatsuccessfully reflect the goals of both theschool-to-work movement and theschool-to-college traditions?

• At what grade should a common curriculum give way to tracks that lead indistinctly different directions?

Mathematicians who think about curricular issues typically focus primarily on the"academic track" that leads to scientific,engineering, and mathematics courses atthe university level. Those who developcurricula for the technical and vocationalprograms rarely work with mathematicians or mathematics educators to ensureconsistency with theexpectations and standards of mathematics education. Despitethe current schism between these two communities, the increasing political pressurefor educational alternatives such as techprep and ATE provide mathematicianswith a marvelous opportunity to demonstrate the universal applicability of theirdiscipline-to show that it is good forsomething other than preparing more university professors.

It is far too early to say with assurancehow the many challenges of school-towork programs will play out in the realworld of education and politics. Wecan becertain, however, that they will occupy anincreasing shareof attention and resources,perhaps even becoming the dominant curriculum in the schools. Mathematicianshave much to contribute to this discussion; we ignore it at our peril. Now is thetime to get involved, to learn what the issues really are, and to use the manyresources of our discipline to work withother constituencies to frame an effectiveresponse to this clear national need.

The authors are at the Mathematical SciencesEducation Board, ofwhich Steen is the Director.

FOCUS

Civilized MathematicsDan Alexander

While attending Kenneth Ribet's marvelous lecture on the gap in Andrew Wiles'proof of Fermat's Last Theorem at therecent Cincinnati meetings, I was struckwith how civilized the discussions aboutWiles' accomplishments have been. Noneof the principals in the recent developments leading up to Wiles' work have usedthe gap as opportunity to toot their ownhorns at the expense of Wiles. Indeed, although Ribet's talk directly addressed thegap in Wiles' proof, I was left with theclear impression that he was genuinelyrooting for Wiles to close it.

Things don't always work in such a civilized manner. One need only look back afew years to the controversy which exploded in the pages of the Mathematicallntelligencerover who deserved credit forthe discovery of the Mandelbrot set in thelate 1970s. Perhaps an even betterreminderthat it is human beings who make mathematics is a dispute that broke out in Franceat the end of 1917 just as the First WorldWar was entering its final year.

This dispute involved two of France's bestmathematicians, Pierre Fatou and GastonJulia, and at issue was the priority for several fundamental developments in thestudy of the iteration of complex rationalfunctions (which, incidentally, providedthe foundation for the discovery of theMandelbrot set some 60 years later).

At the time of the dispute, Fatou was almost 40 years old. He worked at the ParisObservatory as an astronomer, havingoriginally taken a position there becauseacademic jobs in Paris had been scarcewhen he entered the job market after receiving his dissertation in 1906. For thefirst few years of his employment at theobservatory, he evidently had insufficienttime to pursue mathematics. This wasunfortunate, since Fatou's early mathematical work revealed him to be amathematician of great potential. Not onlydid his doctorate contain the importantmeasure-theoretic result known as Fatou'sLemma, but he had also published a resultin 1906 regarding the iteration of complexrational functions which represented amajor breakthrough in that field of study.

Nevertheless, Fatou published relativelyfew mathematical works until 1917, whenhe again took up the study of iteration.

Julia was 24 at the end of 1917, muchyoungerthan Fatou. He had been a prodigyin his youth and by 1917 had already established a strong mathematicalreputation. Julia was also a war hero. In1915, during a furious German attack, heexhibited uncommon bravery and wasawarded the French Legion of Honor. Theprice of Julia's valor was extreme: he suffered a terrible wound to his face and wasleft with a disfigured nose which he customarily covered with a patch. Accordingto his medal citation, his wounds left himalmost blinded and unable to speak, yet heissued a written directive that he was notto be taken from the battlefield until afterthe German attack was repulsed. Julia'sstanding as a mathematician is indicatedby the fact that, during the recuperationfrom his wounds, he was visited from timeto time by two members of the FrenchAcademy of Sciences, Emile Picard andGeorge Humbert.

As far as the public record is concerned,the dispute between Fatou and Julia waswaged exclusively by Julia. At the centerof this dispute was the most prestigious ofthe Academy's mathematical prizes, theGrand Prix des Sciences mathematiques.It was announced in 1915 that the 1918Grand Prix would be given to the bestpaper the Academy received regarding theiteration of functions. The Academy suggested as well that the entrants confinethemselves to the study of rational functions of one complex variable, which bothFatou and Julia did.

As was the custom, preliminary resultswere announced in the public record ofthe Academy's weekly sessions, theComptes rendus hebdomadaires desSeances de l'Academie des Sciences. Thefirst to do so was Pierre Fatou, who, onDecember 17, 1917, announced severalfundamental results regarding the iteration of rational complex functions. Thisannouncement apparently raised the ire ofJulia, because two weeks later Julia announced that prior to the appearance of

June 1994

Fatou's note, he had established identicalresults. He offered proof of the matter inthe form of the following letter, whichappeared in the Compte rendu of December31,19l7:

I read with interest the note of Fatoupublished in the Compte rendu of December 17, 1917. The essential resultswhich it contained, I had previouslyentered in a series of four sealed letterswhich I have deposited with the Secretary of the [Academy] and which wereregistered with numbers 8401, 8431,8438,8466, dated, respectively, June 4,1917, August 17, 1917, September 17,1917 [and] December 10, 1917. Theacademy canjudge, upon opening theseletters, whether they contain the resultsgiven by Fatou ... The Academy canassess, at the same time, whose methods and whose results ought to be givenpriority.

The Academy heeded Julia's request andimmediately following Julia's letter was aresponse titled "On a Communication ofGaston Julia," written by Humbert. Hereported that in response to "questionsraised [by Julia] concerning a question ofpriority," he had opened Julia's sealed letters and discovered that "the assertions ofJulia are founded" since the letters indeedcontained "all the results for which he hasclaimed priority."

In the folk history of mathematics thisincident has grown to epic proportions,and it is sometimes said thatJulia publiclyaccused Fatou of stealing his results. Although Julia made no such claims in hisletter, he did refer to the similarity ofFatou's approach as a "curious coincidence." And although it would be anexaggeration to term this note a personalattack on Fatou, Julia did take a rathersuperior tone in observing, perhaps withsome justification, that his approach was

@----------------

June 1994 FOCUS

Involving Undergraduates in the Professionboth more precise and more general.

Surely discouraged by the Academy'sverdict, Fatou never formally entered thecontest, and the fact that he had publishedresults regarding the iteration of complexfunctions in 1906-when Julia was only13-must have eaten at him.

I do not mean to suggest that the Academydisplayed any favoritism towards Julia.Despite the fact that there is little substantive difference between the initial resultsof Fatou and Julia, it is important to keepin mind-and difficult to disagree witha characterization ofJulia's work from theAcademy's announcement that it hadawarded Julia the Grand Prix: "Julia'smemoir is marked by a mathematical spiritof the highest order."

Nonetheless, it is interesting to speculatewhether patriotic sentiments or personalcontacts would have had any influence onthe prize deliberations had Fatou decidedto enter his paper. After all, Julia was a warhero-while Fatou did not go to battleand was visited during his recovery fromhis wounds by both Humbert and Picard,both members of the Academy, the latteran ardent nationalist.

Dan Alexander is an assistant professorofmathematics at Drake University in Iowa. Hespecializes in the history of mathematics. Hisbook, A History of Complex Dynamics fromSchroder to Fatou and Julia, has just been published by Vieweg Publishing, in their Aspects ofMathematics series.

Most Successful YearYet for Greater MAAFundThe Greater MAA Fund concluded 1993with its best record in the Fund's twelveyears.

A total of 1,168 donors contributed$73,173. These numbers represent a 31%increase in number of donors, and a 55%increase in gift totals from 1992 figures.

The fund will assist such MAA projectsas Electronic Services, student activities,and Math Horizons.

Thanks and appreciation to everyone whohelped make 1993 a banner year for theFund!

Joseph A. Gallian andJohn Greever

Historically, scientists have involved undergraduates in their research and in theirprofession much more deeply than mathematicians. Recently, however, there hasbeen a major effort by many mathematicians to increase the role thatundergraduates play in the professionalcommunity. Examples abound: the MAAStudent Chapters; Math Horizons; REUprograms; the Mills program; a mathematical and computer sciences divisionin the Council on Undergraduate Research(CUR); the AWM Alice Schafer Prize.Moreover, the NSA, IDA, AT&T,Bellcore, and the NSF Regional Geometry Centers have expanded or createdundergraduate research programs in thepast few years. Pi Mu Epsilon and theMAA have provided travel support andsponsored paper sessions and activitiesfor undergraduates at the annual summermeetings in recent years. Both PME andthe MAA award prizes for the best presentation at the student paper sessions.

Perhaps one good way to gauge thischange in the mathematics culture is tocompare the annual joint AMS-MAAwinter meeting in 1989 with the 1994meeting in Cincinnati. The 1989 meetingdid not have a single organized activityfor undergraduates and it is doubtful ifmore than a handful of undergraduatesattended. Moreover, committees of theprofessional organizations spent little orno time discussing ways to engage undergraduates in research or the professionalcommunity at the 1989 meeting. In contrast, at the 1994 meeting:

• the AMS sponsored a special session onresearch by undergraduates in which 38students gave fifteen-minute presentations of papers written by 69 authors;

• the MAA Subcommittee on Undergraduate Research in Mathematics andthe Mathematical and Computer Sciences Division of CUR cosponsored aposter session on research by undergraduates that featured the works of 19students;

• the MAA Subcommittee on Undergraduate Research in Mathematics

formulated proposals for the provisionof publication outlets for undergraduate students;

• the MAA Committee on Student Chapters sponsored the Student Workshopand the Student Lecture followed by a"make your own sundae" social;

• the MAA Committee on Student Chapters sponsored a paper session for facultyon topics pertaining to activities for undergraduates;

• the AMS and the MAA held a social forfirst-time attendees;

• the MAA Committee on Student Chapters sponsored a hospitality/informationcenter;

• both the Mills program and the Study inBudapest program had informationaltables;

• the MAA offered a minicourse on organizing an undergraduate researchprogram which had 43 participants from25 states;

• an MAA committee passed a motion toestablish a joint AMS-MAA $1000prize for research by an undergraduate;

• an AMS committee discussed ways toincrease undergraduate research.

So, injustafew years we went from wherethe annual meetings had little significanceregarding the involvementof undergraduates in our profession to where thisinvolvement is now one focal point of themeetings. Most gratifying, undergraduates now make a rich contribution to themeetings.

We wish to express our appreciation tothose who made these changes possibleand encouraged even greater participation by undergraduates in the life of ourcommunity.

Joseph Gallian is at the University of Minnesota-Duluth; John Greever is at Harvey MuddCollege in Claremont. California. Both arehighly active in nationwide initiatives to involveundergraduate students in research activities.

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FOCUS

Table 1

Yes, the effective tax rate can be 5 percent

Given solution Proposed alternative

Personal O~_im_·o_n _Can Students Solve Math Problems?

June 1994

Table 2, Column 1 shows the suggestedsolution to the second question. The ruleseems to be to remove parentheses firstoff, willy nilly. This requires a calculation, and can complicate the algebra ratherthan simplify it. Ironically, in this case itgives the solution sooner than the writersrealized-since any number minus 600 isless than the number itself. The third lineis unneeded and unwanted. (Perhaps thecontradiction in the second line was hard

An extended-response problemfor Grade 12asks why thesquare of a number ending in 5must end in 25.Only 2% earned4 or 5 on thisone. The problem penalizes students who do know howto expand (IOn+5)2yet slip, writing 100n2

+ 25, apparently solving the problem andthus lessening the incentive to go back andcheck. The analysis in the report showsthat at most 2% of the students could havebeen affected in this way; but the principleshould be kept in mind.

The following extended-response problemis from Grade 12: Under a proposed income tax, you pay nothing on income upto $10,000, then 6% on any excess over$10,000. The effective tax rate is the percent of total income that you pay in tax.Can this be 5 percent? Can it be 6 percent?

Only 3% of the students earned a score of4 or 5-2% earning 4 for answering thefirst question correctly and I% earning 5for answering both correctly.

I think this abysmal performance is due inpart to the plodding way students are taughtto handle equations. Such simple tacticsas doing the steps in a different order canbrighten things up considerably. Table 1,Column 1 shows the suggested solutionto the first question. My own principle isto clear clutter as soon as possible. Express income in units of $1,000, therebysidestepping a welter of zeros. Cleardecimal clutter right away instead of carryingit doggedly to the very end. The result,shown in Column 2, saves time, inhibitscopying errors, and is light in spirit, assimilable almost at a glance, and very easyto check.

of three other days, and nothing on Sunday.This is a problem in simple arithmetic,but the solution requires more than onestep. Only 22% in Grade 4 and 59% inGrade 8 answered it correctly.

The outstanding feature of this year's testsis the inclusion of questions requiringextended responses in which students explain their reasoning either directly(perhaps by a diagram) or by examples.(The usual multiple-choice and short-answer formats are there, too.) The reportquotes three extended-response questionsin each grade level, along with examplesof students' answers, detailed commentson them, and models for possible solutions. These questions were graded on ascale of 0 (Blank), I (Incorrect), 2 (Minimal), 3 (Partial), 4 (Satisfactory), 5(Extended). Ofthe students in Grade 12currently sitting in your freshman calculusclass-only 9% earned 4 or 5. (In Grade 8it was 8%; in Grade 4, 16%.)

In any enterprise as vast as this one, thereare bound to be some rough spots, thoughI found none that affect the results in anysignificant way; however, I think it will beuseful to point them out.

A short-answer problem for Grade 8 asksfor two numbers that make the sentence54 < 3 X true. Only 49% gave an acceptable response. Apparently, students aretrained to solve first, think later-and halfwere unable to solve. The suggested solution is 19, 20. Why not 54, 55-requiringno computation? Or 1000,2000?

Let x be the incomein units of $10,000.

O.06(x - 10) =O.05x

6(x- 10) = 5x

6x-60= 5x

x=60

0.06(x - 10,(00) = O.05x

O.06x - 600 = O.05x

O.OIx= 600

x= 60,000

Let x be the incomein dollars.

In this article I will comment on the reportCan Students Do Mathematical ProblemSolving? Results/rom the 1992 Mathematics Assessment, by John A. Dossey, Ina V.S. Mullis, and Chancey O. Jones, NationalCenter for Education Statistics, U.S. Department of Education, 1993, 226 pp.Copies of the report can be obtained fromEducation Information Branch, Office ofEducational Research, Department ofEducation, 555 New Jersey Ave, Washington, DC 20208-5641; (800) 424-1616.

The report is required reading for all whotake mathematical education seriously.The tests under discussion were administered to 250,000 students in grades 4, 8,and 12, from 10,000 schools in 41 states.They were designed with extreme care,and graded in painstaking detail. The results are analyzed by region (Northeast,Southeast, Central, West), state, ethnicbackground, sex, societal environment(advantaged urban, disadvantaged urban,extreme rural, other), and type of school(public, private). The comparisons arewhat one would guess, with one exception: there was no significant difference inperformance between the girls and theboys, a fact mysteriously not included inthe list of major findings.

I expected the mathematical performanceoverall to be dismal, but was not preparedfor the actual numbers. "Jill's Class Trip"asks how many weeks Jill must work toearn $45 for the trip if every week sheearns $2 on each of three days, $3 on each

Leonard Gillman

tt------------

June 1994 FOCUS

Income

Tax

x

•

-a contradiction.

~~ocm~,~m800-446-9323 (Visa or M/C)

x- 10

NOW TO ,IT fJVlR " AN' ,ITONW"N YOIIR tlill

x = Income

10...

-a contradiction.

Table 2

No, the effective tax rate cannot be 6 percent.Given solution Proposed alternative

Letx be the income in dollars. Letx be the income in units of $10,000.

O.06(x - 10,(00) =0.06x O.06(x - 10) =0.06x

O.06x- 600 =O.06x x - to =x

0=-600

oo

Tax

1

Tax agaln8t Income

(1 unit=$1000)

=rJ. ~c0r4&JL~~ :&Py_~<'2? @g~~ CLAUDIA ZASLAVSKY

(Q)f15:(' CB(f) "Claudia Zaslavsky is one ofthe finest and mostsensitive math teachers I know!"

--educational reformer Herbert Kohl, author of 36 ChildrenZaslavsky gives a host of reassuring methods, drawn from many cultures, for tackling real-world math problems. Sheexplodes the myth that women and minorities are not good at math.

250 pp. 100 b & w illus., cartoons, sidebars.Paper, $14.95; Hardcover, $37.00

Leonard Gillman is at the University ofTexas atAustin. and is a former president of the MAA.

Can the effective tax rate be 5%? Let b bea line from the origin that moves up to theright. If its slope is 5%, which is less than6%, then b is less steep than a and willintersect it at some point P. The coordinates of P represent an income x and thetax on it. The slope of b--tax divided byx-is the effective tax rate. This showsthat the effective tax rate can be 5%.

We don't expect today's students to comeup with this proof, but the report would dowell to point it out. It uses no computation. The only special fact about thenumber 5 is that it lies between 0 and 6;hence, the proof shows with no additionaleffort that the effective tax rate can be anypositive number less than 6%. Moreover,this conclusion is derived from a diagramconsisting of not much more than twostraight lines. The reader can now seesomething ofthe spirit and power ofmathematics, as well as its elegance, beauty,and charm.

The instructions explicitly invite usingwords or diagrams. Here are some words:Can the effecti ve tax rate be 6%?Ofcoursenot! The tax rate of 6% is the tax dividedby part of the income. The effective rate isthe same tax divided by the total incomeso must be smaller. (Thus, the secondquestion, which gave the students so muchtrouble, actually requires only trivial arithmetic.)

Not all high school seniors have studiedcoordinate geometry; but let's try a diagram. Line a in Figure 1 shows the tax forincomes above 10. (For readability, we usedifferent horizontal and vertical scales.)The slope of this line-"rise over run"is equal to tax divided by x -1 O. That's thetax rate, 6%.

to spot amid the clutter.) But the wholematter of parentheses succumbs to theprinciple of clearing clutter: first cancelthe 0.06, as shown in Column 2.

The authors feel that the concept of limitsomehow plays a role in the solution, butI don't see that limits have anything to dowith it.

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FOCUS June 1994

MAA Contributed Papers in San FranciscoThe Mathematical Association of America and the American Mathematical Society willhold their annual joint meetings from Wednesday, January 4, 1995 through Saturday,January 7, 1995 in San Francisco, California. The complete meetings program willappear in the October 1994 issues of FOCUS and the AMS Notices. This preliminaryannouncement is designed to alert participants about the MAA's contributed paperssessions and their deadlines.

Please note that the days scheduled for these sessions remain tentative. The organizerslisted below solicit contributed papers pertinent to their sessions; proposals should bedirected to the organizer whose name is followed by an asterisk (*). For additionalinstructions, see the Submissions Procedures box on page 16.

Sessions generally must limit presentations to ten minutes, but selected participants mayextend their contributions up to 20 minutes. Each session room contains an overheadprojector and screen; blackboards will not be available. You may request one additionaloverhead projector, a 35mm slide projector, or a 1/2 inch or 3/4 inch VHS VCR with onecolor monitor. Persons needing additional equipment should contact, as soon as possible, but prior to October 27, 1994: Donovan H. VanOsdol, Department of Mathematics,University of New Hampshire, Durham, NH 03824, e-mail: [email protected].

Experiences with Modeling inElementary DifferentialEquations

Thursday afternoon and Saturdaymorning

Professor Robert Borrelli *Department of MathematicsHarvey Mudd CollegeClaremont, CA 91711-5990phone: (909) 621-8896FAX: (909) 621-8366e-mail: [email protected] ColemanHarvey Mudd College

Modeling is playing an increasingly important role in elementary differentialequations courses mostly because of theready availability of inexpensive ordinarydifferential equations solver platforms.These platforms open the door for studentsto consider an incredibly rich variety ofapplications and models which moreclosely fit the needs of client disciplines.Papers are invited on how modeling hasbeen integrated into the teaching of differential equations, whether by case studies,cooperative projects with industry, or alaboratory experience.

MakingStatistics Come Alive

Thursday afternoon and Saturdayafternoon

Professor Robert W. Hayden *20 River StreetPlymouth State CollegeAshland, NH.03217-9702phone: (603) 968-9914e-mail: [email protected] R. ParkerAustin Community College, Texas

The MAA has a joint committee with theAmerican Statistical Association. One ofits goals is to support mathematiciansteaching statistics, and this session is partof that effort. We invite papers related tostatistics in any undergraduate course.Possible areas include specific curriculumtopics and examples, the use of real data,student projects, writing, computers, innovative approaches to grading, usingstudent feedback to improve teaching, andcurriculum organization issues.

The First Two Years

Thursday morning and Saturdaymorning

Professor William Davis *Department of MathematicsThe Ohio State UniversityColumbus, OH 43210-1328phone: (614) 292-0365FAX: (614) 436-7919e-mail: [email protected] B. SmallU.S. Military Academy

As various reform efforts are implemented,

the character of courses in the first twoyears ofcollege mathematics is changed.Certain aspects of differential equations,linear algebra, and discrete mathematicshave been incorporated into most of thecalculus courses, and the several variablesofferings thus lookquite different than theyused to. This session will look at variousefforts toward building a coherent mathematical sciences program for the first twoyears of college.

Chaosand Dynamics

Wednesday afternoon and Fridayafternoon

Professor Denny Gulick *Department of MathematicsUniversity of MarylandCollege Park, MD 20742-0001phone: (301)405-5157FAX: (301) 314-0827e-mail: [email protected] ScottMontgomery College

To commemorate the 20th anniversary ofthe scientific terms "chaos" and "fractals,"a special session on chaotic dynamics anddynamic geometry is being offered. Thesession invites papers on topics related toeither chaotic dynamics or fractal geometry.The papers need to have anexpositoryflavor.

Teaching with Original Sources

Thursday afternoon and Fridayafternoon

Professor David Pengelley *Mathematical Sciences DepartmentNew Mexico State UniversityLas Cruces, NM 88003-000 Iphone: (505) 646-3901FAX: (505) 646-6218e-mail: [email protected] LaubenbacherNew Mexico State University

Teaching with original sources is becoming increasingly widespread. This sessionis a forum for exchanging experiencesusing original sources from all time periods in teaching at any level. Papers areinvited which address the incorporationof specific sources in a particular instructional setting.

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June 1994

Dynamic Geometry

Thursday morning and Saturdaymorning

Professor James R. King *Department of Mathematics GN-50University of WashingtonSeattle, WA 98195phone: (206)543-1915FAX: (206) 543-0397Doris SchattschneiderMoravian College

The availability of "dynamic" geometrysoftware such as CARRI Geometre andGeometer sSketchpad has opened thedoorto many changes in the teaching of geometry. This session invites papers thatpresent instructive examples of the use ofdynamic geometry software in the classroom or for research, and also invitespapers that discuss the implications of dynamic geometry software for whatmathematics we teach and how we teachit. Note: if you will need computer equipment for your presentation, please includewith your submission a clear statement ofyour needs.

Laboratory Approaches toTeaching Mathematics

Wednesday morning and Thursdayevening

Dr. Jon Wilkin *6725 North 27th StreetNorthern Virginia Community CollegeAlexandria, VA 22213-1204phone: (703) 536-5290FAX: (703) 450-2536e-mail: [email protected] MaysNorth Lake College

The use of laboratory techniques and approaches in teaching mathematics isgradually permeating all undergraduatemathematics courses. This session willpresent papers that deal with laboratoryapproaches to teaching mathematicscourses other than the calculus. We areinterested in papers which present courseswhich have laboratory experimentationand/or exercises as their primary mode ofteaching and learning. Presentations mayinclude methodology,equipment used, andsample laboratory exercises, as well as adiscussion of the merits of the particularapproach.

Innovations in Teaching LinearAlgebra

Wednesday afternoon, Thursdayevening, and Friday afternoon

Professor Donald R. LaTorre *Department of Mathematical SciencesClemson UniversityClemson, SC 29634-1907phone: (803) 656-3437FAX: (803) 656-5230e-mail: [email protected] J. Leon (ATLAST)Universityof Massachusetts at DartmouthDavid C. Lay (LACSG)University of Maryland

The teaching oflinear algebra is undergoing substantial change. This session invitespapers on personal experiences with innovations in teaching linear algebra,including: (I) the creative use of computeralgebra systems, supercalculators, or computer software; (2) experiences with theNSF-funded ATLAST summer workshops; (3) experiences with the corecurriculum recommended by the LinearAlgebra Curriculum Study Group(LACSG); (4) "gems" of exposition inlinear algebra; and (5) other innovativeteaching or curriculum initiatives in undergraduate linear algebra.

Recruitmentand Retention ofFemale Faculty

Thursday morning and Fridaymorning

Professor Marcelle Bessman *Frostburg State University644 Geneva PlaceTampa, FL 33606phone: (813) 253-6584e-mail: [email protected]. Miriam CooneyUniversity of Notre DameGerald PorterUniversity of Pennsylvania

Women are obtaining doctoral degrees inmathematics in increasing numbers. Mathematics departments are seeking qualifiedwomen faculty. This session, sponsoredby the MAA Committee on Participationof Women, seeks papers on strategies thatare being used, or could be used, to recruitand retain female faculty. Papers shouldtake approximately twenty minutes topresent.

FOCUS

New Directions in StudentAssessment

Wednesday morning and Thursdayevening

Professor Rose Hamm *Honors ProgramCollege of CharlestonCharleston, SC 29424-000 Iphone: (803)953-7154FAX: (803) 953-7135e-mail: [email protected] VanderveldeHope College

Assessing students' performances withrespect to critical thinking, mathematicalcommunications skills and the use of technology demands assessment techniquesdramatically different from those manyinstitutions and instructors have traditionally relied upon. We invite contributedtalks from persons regarding new (and old)evaluation techniques which are effectivein this new environment.

Preparing Teachers toImplement Change(3 models)

Wednesday afternoon and Fridaymorning

Dr. Bettye Clark *Department of Mathematics, Box 171Clark Atlanta UniversityAtlanta, GA 30314phone: (404) 880-8188FAX: (404) 880-8250Robert BixUniversity of Michigan-FlintM. Kathleen HeidPennsylvania State University

Sponsored by the Committee on the Mathematical Preparation of Teachers(COMET), this session solicits papersdescribing model programs, creative partnerships, research, and position papers onhow colleges and universities are changing attitudes and curricula in themathematics and mathematics educationdepartments to implement change in teaching mathematics in all collegiatemathematics and mathematics educationcourses.

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Research in UndergraduateMathematics Education

Friday evening and Saturdayafternoon

Professor Daniel L. Goroff *Department of MathematicsHarvard UniversityCambridge, MA 02138phone: (617) 495-4869FAX: (617) 495-3739e-mail: [email protected] Ferrini-MundyUniversity of New Hampshire

We solicit research papers which addressquestions about teaching and learning ofundergraduate mathematics. We are especially interested in including reports fromcompleted research studies. Areas of particular interest are: visualization, the roleof technology in learning undergraduatemathematics, student learning in reformedcurricular settings, student learning ofadvanced undergraduate mathematicaltopics, and the preparation of teachers.

Student ChaptersSessionThe MAA Student Chapters Sessionserves as a forum for the exchangeofideasamong advisors to individual chapters andSection coordinators. Each fifteen-minutetalk in this Special Session will focus onone or several activities implemented bya campus chapter or by a Section, or onactivities supported by the Exxon grants.Ifyou are interested in presenting a paperon your Student Chapter activities, pleasesubmit the title and an abstract of the proposed talk, including your name,affiliation, address, telephone number, email address, and whether you are achapter advisor or a Section coordinator.Send to Karen Schroeder, MathematicalSciences Department, Bentley College,175 Forest Street, Waltham MA 021544705; phone (617) 891-2267; FAX (617)891-2457; e-mail [email protected];so as to arrive by September 16, 1994.

June 1994

CorrectionThe Young Mathematicians' Network article (February 1994, page23) incorrectly stated that CharlesYeomans was an assistant professor at the University ofKentucky.We regret the error.

DO NOT FORWARD COMPLETED ABSTRACTS TO THE MAA ORTO THE AMS. THEY ARE TO BE SENT TO THE SESSION ORGANIZER.

If you wish to obtain an abstract form in advance, please contact: TheExecutive Department, MAA, 1529 Eighteenth Street, NW, Washington,DC 20036; e-mail: [email protected].

Submission Procedures for Contributed PapersAfter you have selected a session to which you wish to contribute a paper,forward the following directly to the organizer (indicated with an (*»:

• the name(s) and address(es) of the author(s); and

• a one-page summary of your paper.

The summary should enable the organizer(s) to evaluate the appropriateness ofyour paperfor the selected session. Consequently, you should includeas much detailed information as possible within the one-page limitation.

Your summary must reach the designated organizer by Friday, September2,1994.

The organizer will acknowledge receipt of all paper summaries. If theorganizer accepts your paper, you will receive a standardized abstract form.Use this form to prepare a brief abstract. Please return the completed formto the organizerby Friday, September 16, 1994.Abstracts received after thedeadline will not be published in the 1995Joint Annual Meetings AbstractsJournal. This abstracts journal will be available in the meetings registrationarea during the conference in San Francisco.

Mathematical Sciences,Technology, and EconomicCompetitiveness

Friday morning and Saturdayafternoon

Dr. S. Brent Morris *5088 Lake Circle WestColumbia, MD 21044-1442phone: (301) 688-0332FAX: (30 I) 688-0289but call voice phone before sending FAXe-mail: [email protected] Dale McCraySearle & Co.

A 1991 National Research Council report,Mathematical Sciences, Technology, andEconomic Competitiveness, emphasizedthe central importance of mathematics tomodem industry. This session invites papers that describe case studies ofmathematics in use in industry. A casestudy should explain what mathematicalproblems were involved, what solutionswere generated, and how the solutionswere implemented. It should also describeproject sponsorship-how the companydefined, approved, and managed theproject, the working relations between themembers of the project team, and the reasons for success or failure.

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June 1994

1994 Summer Geometry WorkshopExperiencing Geometry presented by David Henderson

Geometry and the Visual World presented by Maria Terrell

June 20-25, 1994 at Cornell University, Ithaca, New York

As curricular reform includes more informal, intuitive, and applied geometry, collegefaculty need opportunities to explore new options for college geometry courses.

Henderson's Experiencing Geometry is revolutionary in its approach to teaching. Hisapproach provides a rare opportunity to experience using materials designed to encourage both student and instructor to construct their own understanding of basic conceptsin plane and spherical geometry.

Terrell's Geometry and the Visual World uses linear perspective and geometrical opticsto motivate geometric models in Euclidean and projective geometry. Topics include thehistorical development of geometrical optics, linear perspective and projective geometry from Euclid to Desargues, a hands-on approach to applied geometry, and a crashcourse in analytic projective geometry.

Room, board, and a modest stipend will be provided by an NSF grant. Workshop islimited to 25 participants. For further information, contact David Henderson([email protected]) or Maria Terrell ([email protected]), Department of Mathematics, Cornell University, Ithaca, NY 14853-7901.

FOCUS

The MathematicsArchivesThe Mathematics Archives, making useof information provided by mathematicians in the Division of UndergraduateEducation at the National Science Foundation, has established "gopher links" totheabstracts on the National Science Foundation gopher of funded proposals inmathematics and statistics education. Using a gopher client, one can locate theseabstracts by accessing the MathematicsArchives at archives.math.utk.edu (port70) and choosing the menu items:

Teaching Materials and Other Information

NSF Awards in Mathematics EducationAbstracts.

The abstracts are arranged first by program and then by the institution of theprincipal investigator.

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FOCUS June 1994

The MAA Gopher is Here!Wearepleasedtoannouncethatthefirst phaseofMAA'sElectronicServices, theMAAGopher, willbeavailablefor testingbyMAAmembers,as ofJune 15. On theMAAGopher, you willfind information about theMAAandits structure, how to become a member, the MAA Publications Catalog (including ordering information) acompletecalendarofmeetingsand workshops,and sectionand studentchapternews. In addition,a pilotforCelebratingProgress in CollegiateMathematics, an electronicdatabaseofprogramsand projects in undergraduate mathematics, is availablefor your perusal.

You can access the MAA Gopher several different ways:

• Ifyour computer runs a gopherclient,oryou can logintoa computerwith a gopherclient,you can access the MAA gopher server by typing "gopher gophenmaa.org",

• The MAA gopher server is also listed in the appropriate menus under "Other Gopherand Information Servers"at the University ofMinnesota mastergopherserver, and maybe accessed that way as well.

• You can also connect to MAA on the World Wide Web at URL ''http://www.maa.org!''.

The MAA Gopher is in a test stage. We are in the process of refining, improving, and expanding the gopher.This is the timefor you to sendus youropinionsand commentsso that wecan make the gophera usefulservicefor you. You will fmd a Suggestion Box on the MAAGopher. Please use it.

Art and MathematicsConferenceThe third annual interdisciplinary conference relating art and mathematics will beheld June 25-29, 1994 at SUNY in Albany, New York. For information, contact:Nat Friedman, Department of Mathematics, SUNY, Albany, NY 12222; (518)442-4621; FAX: (518) 442-4731; e-mail:[email protected].

Women in Mathematics and Computer ScienceOctober 21, 1994

Kean College of New Jersey

This one-day conference will bring together mathematics and computer science professors and high school teachers concerned with increasing the number of women studentsin these disciplines. Practitioners from industry and academia will address the problemsand explore various intervention techniques. The conference will feature a keynoteaddress by Prof. Patricia Kenschaft of Montclair State College, followed by a paneldiscussion, lunch, and breakout groups. Ifyou are interested in leading a breakout groupor would like further information, contact Prof. Barry J. Arnow, Chairperson, Mathematics and Computer Science Department, Kean College of New Jersey, Union, NJ 07083;(908) 527-2104; e-mail: [email protected].

The MAA Gopher is Here!See the top of this page!

@------------------

June 1994 FOCUS

New Mathematical Sciences Curriculum Initiative at the NSF: AMS andMAA Team Up to Offer Workshop to Spark Discussion and Focus IdeasAllynJackson

The National Science Foundation (NSF)has launched a major new initiative thatshould be of interest to every mathematical sciences department in the country. Theinitiative, entitled Mathematical Sciencesand Their Applications Throughout theCurriculum, will fund large-scale interdisciplinary educational projects at theundergraduate level. The basic goal of theprogram is to increase students' understanding of mathematics and their abilityto apply mathematics in other disciplines.This initiative differs from the NSF's calcuIus reform efforts in that projects fundedunder this initiative should aim at comprehensive, coordinated change involvingmore than one course (most likely severalcourses) and should have strong participation by faculty from other disciplines.The NSF is insisting upon serious dissemination plans to insure nationwide impactso that the projects will have influence farbeyond the confines of the institutionsreceiving the grants.

The NSF expects to fund a few projects atthe level of about $1 million per year forthree to five years. Formal proposals aredue in February 1995; proposals for$50,000 planning grants are due June 6,1994. Instructions about how to obtainprogram announcements and additional information are at the end of this article.

In order to inform the mathematical sciences community about this initiative, theAMS and the MAA teamed up to organizea workshop entitled Changing CollegiateEducation: Mathematical Sciences andtheir Uses in Other Disciplines. HeldMarch 27-28, 1994 in Washington, DC,the workshop brought together about 130participants who listened to presentationsabout how such large-scale projects canbe organized, and asked questions aboutthe initiative. What follows is a reportabout the conference.

What the NSF is Looking For

The new initiative is a cooperative effortof NSF's Course and Curriculum Development (CCD) Program, which is part of

the Division of Undergraduate Education(DUE) of Directorate for Education andHuman Resources, and the NSF's Division of Mathematical Sciences. Anoutgrowth of the NSF calculus reformprogram, CCD funds projects in all disciplines to improve undergraduate scienceand engineering courses. The new initiative is an attempt to build on theseimprovements through coordinated, comprehensive reform of a large number ofcourses that use the mathematical sciences.CCD has launched an analogous initiativein chemistry that is further along than themathematical sciences initiative: Out of120 proposals for planning grants for thechemistry initiative, CCD recently madefourteen awards.

Robert F. Watson, director of DUE, notedthat the initiative is not a replacement forcalculus reform projects, but rather is anatural sequel. One of the motivationsbehind the initiative is the common complaint that students do not possess enoughmathematical and quantitative skills tohandle problems in the disciplines in whichthey study.Watson noted that the NSF maybe least interested in mathematics majorsas part of this initiative-the aim is to reachout to other areas. The initiative shouldnot just strengthen a few departments or afew institutions, he noted, but rather shouldengender widespread change. The "endproduct" should be students who can utilize mathematical and quantitative skillsin whatever they do and who can functioneffectively in our increasingly technological society. Watson emphasized thatproposals must be multi-disciplinary andmust involve partnerships among facultyin different disciplines. One or two faculty focusing on a small number of coursesis not sufficient. There must be a largerinterdisciplinary group looking at severaldifferent courses or sequences of courses.The focus should be on improving studentlearning through new courses, new activities in existing courses, new teachingmethods, new software, etc.

Watson also noted that strong support from

•"beyond the departmental level"-meaning from deans and provosts-is likely tobe a key factor. In response to a questionfrom the audience, Watson said that he didnot mean that one must have a dean as aco-principal investigator, but rather thatlaunching a large-scale project such as thiswould naturally need the support of higheradministration. In addition, the NSF wantsto support projects that do not die out whenthe NSF funding dries up; unless the institution makes a commitment to the projectfrom the beginning, the project's futurecan be uncertain. Matching funds from theinstitution are not required, but, as Watsonput it, "it never hurts."

James H. Lightboume, an NSF programdirector working on the initiative, said thatthe NSF hopes to build on successes withearlier reform efforts, such as with calculus, but that it wants to bring on boardmore faculty, both from the mathematicalsciences and from other disciplines. Planning grant proposals will be limited to fivedouble-spaced pages, he noted; within thatspace, proposers must set forth a vision ofthe project, tell which courses will be involved, and describe the interdisciplinaryteam. He stressed that planning grant proposals should make it clear thatinterdisciplinary communication has already begun; confining planning withinthe mathematics department in the hopethat other departments will join in later isprobably not a winning strategy. Otherfaculty may join in as the project evolves,but there must be significantcross-departmental communication and collaborationfrom the outset.

Another NSF program officer working onthe initiative, William Haver, noted thatalthough the calculus reform effort hasbeen successful in many ways, it has notreally reached a large percentage of students nationwide. The NSF is hoping that,through this initiative, a small number ofinstitutions can attack the larger problemsof linking the mathematical sciences andother disciplines to improve mathematical sciences instruction and bring about

-----------------@

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pervasive changes that reach large numbers of students.

One participant from the audience wondered why the NSF was making so fewgrants of such a large size instead of seeding grassroots efforts through a largenumber of smallergrants. Haver acknowledged that the mix oflarge and small grantsis a serious question for the NSF; he alsonoted that DUE will continue to fundsmaller projects through its other programs. In addition, Haver pointed out thatthe initiative will focus on projects thatwill be useful to institutions all over thenation. Therefore, institutions not receiving these grants could indirectly benefitfrom the intiative. Tina Straley, anotherNSF program officer working on the initiative, noted that there is no restriction onthe type of institution that can apply forthe initiative grants, and coalitions of institutions areespecially encouraged. Haveralso stressed that the NSF remains committed to supporting calculus renewalefforts. In addition, it is expected that thenew initiative will strengthen calculusrenewal efforts by supporting someprojects which will build upon the successof the calculus program and link the newlydeveloped curricula and approaches toother courses.

Large-scale Projects RequireNew Thinking

Denice Denton is on the electrical engineering and the chemistry faculties at theUniversity of Wisconsin at Madison. Shewas part of a group who submitted a successful planning grant proposal to CCDfor the chemistry initiative. In addition,she has been involved in a couple oflargescale curricular reform projects. Dentonsaid that flexibility is crucial in suchprojects. There will be a natural ebb andflow-some faculty will join the projectearly on and later drop out; some willobserve at first and join only later. Onemust be flexible enough to allow theproject to evolve and change.

Also, she noted that although $1 millionsounds like a lot of money, it really isn't:given the high overhead rates at some institutions one might be left with aroundhalf that amount. She suggested talking tothe grants and contracts office at one'suniversity to get a more realistic picture of

20l .

what one will have to spend. In addition,if n institutions are involved in the project,and the money is divided by n, the fundsare stretched yet further. Denton also suggested getting institutional fundingcommitments in writing.

One needs to get the word out on campus,not only to bring in those who want toparticipate, but also to get feedback to helpguide the project. In preparation for theirwork on the chemistry initiative, Dentonand her colleagues sent out a survey toother departments asking what they expected their students to get out of chemistrycourses, what they liked, what they didn'tlike, etc. The survey form also describedthe NSF initiative and asked for those interested in participating to contact theproject directors.

Getting students to contribute their ideasis another important factor.

As for writing up the proposal, Dentonnoted that many proposals come into DUEwithout references; she suggested writingthese proposals like research proposals,with appropriate supporting references tothe literature. She also stressed the importance of the dissemination and evaluationcomponent. Ronald Douglas of the StateUniversity of New York at Stony Brook,one of the organizers of the workshop,echoed this point, saying that dissemination and evaluation cannot be add-ons butmust be integral parts of the project. Veryfew mathematical sciences faculty havesufficient expertise in this area, he said,and most will need to bring in someonefrom outside of mathematics for this component of the project.

Harvey Keynes of the University of Minnesota stressed that, with large-scaleprojects like the ones the NSF hopes tofund, there has to be a clear mission inmind. "Reform" does not just mean a newtextbook and new activities, he noted; ithas more to do with looking at the goals ofthe courses and formulating ways of meeting those goals. In particular, the NSFinitiative requires a substantial componentfor evaluation, and one's goals must beclear in order to evaluate whether or notthey have been achieved.

Fostering Collaborations

Keynes mentioned the idea of collaborat-

June 1994

ing with two-year colleges, which alsocame up in later discussions during theworkshop. Many students start out at twoyear colleges and transfer to four-yearcolleges or universities, so it makes sensefor the two types of institutions to coordinate the mathematics students learn. Oneparticipant in the audience wondered howreceptive two-year colleges would be tosuch collaboration. One of the speakers,Deborah Hughes Hallett of the Universityof Arizona, responded that, in her experience, it is often the two-year colleges thatbring four-year institutions on board whenit comes to curricular innovation.

On the subject of collaboration with otherinstitutions, Hughes Hallett noted that herwork with calculus reform involved arange of different institutions, fromHarvard University to the University ofSouthern Mississippi to Haverford College. Although involving a wide range ofinstitutions can slow down the process ofgetting started, she said, ultimately thedifferent viewpoints are very helpful, because each institution brings its individualstrength into the group. In addition, dissemination is easier because differentkinds of institutions have already tried outthe ideas.

Hughes Hallett noted that initiating collaborations with other fields is a difficultchallenge for mathematics, and one thatshould be addressed even were it not forthe NSF initiative. She had some practicaladvice for getting started: invite a colleague from another department to lunch.Large meetings with people from different disciplines are often much moredifficult, because different groups tend tostake out their own "territories". What isneeded, she said, is a good, friendly relationship, not a political alliance. HughesHallett also warned against deciding beforehand what to do and then asking otherdepartments to join in. Talking, listening,and looking for common ground will insure that all sides have a vested interest inmaking the project work.

It almost goes without saying that proposals for the NSF initiative should includesomething about the impact ofthe projectson the achievement of groups traditionally underrepresented in science,mathematics, and engineering. UriTreisman of the University of Texas at

June 1994

Austin described some of his experiencein working with underrepresented groups.He noted that it is important to try to builda sense of "community" centered on intellectual interests. In response toquestions from the audience, he suggestedthat if one wanted to improve minorityachievement in mathematics, one should"start small" with students who havestrong backgrounds and are already interested in mathematics. Once one has"existence proofs", it is much easier toreach out to more students. In addition, hesuggested getting mainstream, senior faculty involved in such programs, not justyoung faculty.

Building on Experience

Although this initiative is a first for themathematical sciences community (at leastat the undergraduate level), the community has built some experience incross-disciplinary curricularprojects. Twoof the presentations at the workshop described successful ongoing projects. FrankGiordano and Fletcher Lamkin of theUnited States Military Academy at WestPoint described an interdisciplinary mathematical sciences curriculum they havedeveloped, which brings in applicationsranging from pollution along a river topredator-prey problems to flight strategiesfor aircraft range. Louis J. Gross of theUniversity of Tennessee at Knoxville described a program he designed to enhancethe quantitative skills of students in thelife sciences.

Such programs can succeed ifthe departments involved see improved studentlearning and understanding as the primarygoal. Gross said that he told his colleaguesin biology that they are ultimately responsible for what life sciences students learnor do not learn, so biology faculty mustalso make changes in their own courses ifthey want to increase their students' quantitative skills. He suggested working withother departments to prioritize what quantitative skills they want to emphasize andto assist them in including a larger quantitative component in their courses.

Arnold Ostebee ofSt. Olaf College broughtout some key curricular issues thatproposers should consider. What are themost important things for students to retain six weeks, six months, or six years

after the course? Are communication andoral skills important? What level and typeof problem solving skills are needed? Howdo mathematics courses play into othercourses the students are taking? In the areaofmathematical content, he said that mostsuccessful curricular reform efforts don'task, "What should we delete?" but ratherstart from ground zero and ask, "What dowe put in?" Bringing in applications canbe accomplished in many ways-fromsetting the entire course around a few specific applications to introducing thempiecemeal. Which applications are themost important? How does one get students to appreciate the mathematics in theapplications? What level of rigor shouldbe expected in courses taken by non-mathematics majors?

Some Common Questions

A number of questions about the NSF initiative were posed during the workshop asthe participants tried to get a handle onexactly whatthe initiative was about. Someof the commonly asked questions were:

Question: Can the projects include calculus or precalculus?

Answer: One thing to remember is thatthe NSF has formulated a vision for thisinitiative, but ultimately the shape it takeswill be determined by the proposals theFoundation receives. The NSF is not ruling out any courses, but the focus must beon a number of courses, not just one ortwo. The CCD program funds projects forcurricular development for individualcourses, whereas the initiative is an attempt to stimulate curricular reform in acoordinated way across many courses.

Question: What about non-science majors?

Answer: Projects should focus on the areas that fall under the NSF's mandate:science and engingeering, including social science and economics. Reformingcourses like "mathematics for humanities"majors could conceivably be worked intosuch a project, but again, if that is the solefocus, one would do better to make a proposal to the general CCD program thatfunds curricular reform for individualcourses.

FOCUS

Question: What about equipment?

Answer: Support for equipment is not ruledout, but the projects must be "people-oriented" rather than centered on equipment.

Question: What will be the composition ofthe review panel?

Answer: The NSF intends to have abouthalf mathematical scientists on the reviewpanel. The composition of the other halfwill be determined by what kind ofexpertise is needed based on the different areasemphasized in the proposals.

For More Information

E-mail addresses for NSF staff workingon this initiative are:

Lloyd Douglas, [email protected] Lockhart, [email protected] Lightbourne, [email protected] Straley, [email protected] Teles, [email protected] Levitan, [email protected]

These are Internet addresses; for Bitnet,replace @nsf.gov with @nsf. The telephone number is (703) 306-1669. Themailing address is:Division of Undergraduate Education,Directorate for Education and HumanResourcesNational Science Foundation4201 Wilson BoulevardArlington, VA 22230

The program announcement may be foundin Funding Information in the Mathematical Sciences in the current issue oftheAMSNotices. Copies are also available throughSTIS, NSF's online information service.For information on how to use STIS, sende-mail to [email protected] (Internet) orstis@nsf (Bitnet); or telephone 703-3060214.

Allyn Jackson is Associate Managing Editor, Notices of the AMS. This article also appears in thecurrent issue of the AMS Notices.

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AdoptaD entJune 1994

ror to another deserving institution

Honor your Alma Mater by donating a bulk subscription ofMath Horizons tothe Mathematics Department. Write a check that will bring mathematics to lifefor students.

Specifically geared for undergraduate students, Math Horizons delivers livelyarticles, interviews, career information, and problems designed to challengeand broaden the knowledge ofstudents interested in mathematics. A specialcard will be sent to the institution, informing them ofyour generous gift.

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June 1994

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FOCUS June 1994

200 pp., Paperbound, 1994

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List: $25.50 MAA Member: $19.50

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Liang-shin HahnThis book demonstrates that complexnumbers and geometry can be blendedtogether beautifully, resulting in easyproofs and natural generalizations of manytheorems in plane geometry-such as theNapoleon theorem, the Ptolemy-Eulertheorem, the Simson theorem, and theMorley theorem.

Beginning with a construction of complexnumbers, readers are taken on a guidedtour that includes something for everyone, even those with advanced degrees inmathematics. Yet, the entire book is accessible to a talented high-school student.

The book is self-contained-no background in complex numbers isassumed-and can be covered at a leisurely pace ina one-semestercourse. Over100 exercises are included. The bookwould be suitable as a text for a geometrycourse, or for a problem solving seminar,or as enrichment for the student who wantsto know more.

Complex ...Numbers anll'JiflGeometry

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This is the story of a century ofmathematics in America.

Browsing throughits pages, readerssee what haschanged (the kindsof mathematics infashion, for example) and whathas stayed the same(our concern aboutteaching and our .complaints aboutthe deplorable state of our students).

This is a book about history-a samplingofhistory, meant to be savored rather thanstudied. For one-hundred years, theMonthly has contained articles by some ofthe greatest mathematicians in the world,as well as articles by students and facultyfrom small midwestern colleges wherethose great names were barely known. Thisbook gives a glimpse of both worlds. Ittells a story rather than the details of history.

This is a book about mathematics-aboutteaching and research, applied and pure,elite universities and community colleges.

Century of MathematicsThrough the Eyes 01 the Monthly

John Ewing, EditorThis is the story of American mathematics during the past century. It containsarticles and excerpts from a century of theMonthly, giving the reader an opportunityto skim all one-hundred volumes withoutactually opening them. It samples mathematics year by year and decade bydecade. Along the way, readers canglimpse the mathematical community atthe tum of the century, and the divisionsbetween the mathematical communitiesof teachers and researchers. Read aboutthe struggle to prevent colleges from eliminating mathematics requirements in theI920s, the controversy about Einstein andrelativity, the debates about formalism inlogic, the immigration ofmathematiciansfrom Europe, and the frantic effort to organize as the war began. At the end of thewar, hear about new divisions betweenpure and applied mathematics, heroic efforts to deal with large numbers of newstudents in the universities, and the rise offederal funding for mathematics. In morerecent times, see the advent of computersand computer science, the problems facedby women and minorities, and some ofthe triumphs of modem research.

Exploring Mathematics With Your Computer -.--

You need a Pascal compiler in order to usethe 3.5" IBM-eompatible disk packaged withthis volume.

It is assumed thatreaders have a PCat their disposal.


ISBN 0-88385636-0

List:$38. 00MAAMember: $26.50

Catalog Number NML-35

teachers looking for fresh ideas. It is fullof diverse mathematical ideas requiringlittle background. It includes a largenumber of challenging problems, many ofwhich illustrate how numericalcomputation leads to conjectures whichcan then be proved by mathematicalreasoning.

You will find 65 interesting and substantial mathematical topics in this book, andover 360 problems. Each topic is illustrated with examples and correspondingprograms. The major goal ofthe book is touse the computer to collect data and formulate conjectures suggested by the data.

This is a mathematics book, not aprogramming book, although it explainsPascal to beginners. It is aimed at highschool students and undergraduates witha strong interest in mathematics, and

Arthur EngelToday's personal computer gives itsowner tremendous power which can beusedfor experimental investigationsand simulations ofunprecedentedscope, leading to mini-research. Thisbook is a first step intothis excitingfield.

<fj}---------------------------

June 1994

You're The •.Professor, // .What Next?Ideas and Resources lor Preparing CollegeTeachersBettye Anne Case, Editor

You're the Professor, What Next? is avaluable guide for doctoral mathematicalsciences departments wishing to preparetheir advanced graduate students andpostdoctoral instructors for collegiateteaching and related academic responsibilities. The book also will be useful tofaculty mentors of new assistant professors and as personal reading for many,especially inexperienced, members ofmathematics faculties.

Users will find discussion of a wide rangeof pedagogical issues, extensive referencesto other sources of information, and numerous practical suggestions. Forty essays,published for the first time, and a hundredreprinted articles provide a varietyof viewsand reflections from the mathematicalcommunity. The centerpiece of the book isa collection of reports from eight graduatemathematics programs which piloted special seminars in teaching andprofessionalism for students about to receive Ph.D. degrees. Doctoral departmentsin other subjects may find this section helpful as they seek to establish disciplinebased programs to enhance readiness forthe professoriate.


ISBN 0-88385-091-5

List: $24.00

Catalog Number NTE-35

David M. Bressoud

This book is an undergraduate introduction to real analysis. It can be used as atextbook, as a resource for the instructorwho prefers to teach a traditional course,or as a recourse for the student who hasbeen through a traditional course yet stilldoes not understand what real analysis isabout and why it was created.

This course of analysis is radical; it returns to the roots ofthe subject, but it is nota history of analysis. It is designed to be afirst encounter with real analysis, layingout its context and motivation in terms ofthe transition from power series to thosethat are less predictable, especially Fourier series.

The book begins with Fourier's introduction of trigonometric series and theproblems they created for the mathematicians of the early nineteenth century. Itfollows Cauchy's attempts to establish afirm foundation for calculus, and considers his failures as well as his successes. Itculminates with Dirichlet's proof of thevalidity of the Fourier series expansionand explores some ofthe counterintuitiveresults Riemann and Weierstrass were ledto as a result of Dirichlet's proof.

To facilitate graphical and numerical investigations, Mathematica commands andprograms have been included in the exercises. Mathematica is powerful andconvenient, but any mathematical tool withgraphing capabilities-including thegraphing calculator--can be substituted.

336. pp., Paperbound, 1994

ISBN 0-88385-701-4


Catalog Number RAN

FOCUS

Game Theor,.and Strategy"'"Philip D. Straffin, Jr.This important addition to the New Mathematical Library series pays carefulattention to applications of game theory ina wide variety of disciplines. The applications are treated in considerable depth. Thebook assumes only high school algebra,yet gently builds to mathematical thinkingof some sophistication. Game Theory andStrategy might serve as an introduction toboth axiomatic mathematical thinking andthe fundamental process of mathematicalmodelling. It gives insight into both thenature of pure mathematics, and the way inwhich mathematics can be applied to realproblems.

Since its creation by John von Neumannand Oskar Morgenstern in 1944, gametheory has contributed new insights tobusiness, politics, economics, social psychology, philosophy, and evolutionarybiology. Inthis book, the fundamental ideasof game theory share the stage with applications of the theory. How mightstrategic business decisions depend oninformation about a rival company,and how much would such information be worth? When is itadvantageous to vote for a candidate who is not your favorite?What can we learn about theproblem of "free will" byimagining playing a gamewith an Omnipotent Being? Game theory givesinsight into all of thesequestions.

200 pp.,Paperbound, 1993ISBN 0-88385637-9

List: $27.50M1I1Member:$22.00

FOCUS June 1994

IQ"._,.-.._VI:lilt ............: 817.•C._-.CRTII

Catalog Number TRI

Newly Revised •

160pp" Paperbound, 1993

ISBN 0-88385-700-6

List: $27.50 MAAMember: $22.00

Catalog Number PWW

Proofs without words are generally pictures or diagrams that help the reader seewhy a particular mathematical statementmay be true, and how one could begin togo about proving it. While in some proofswithout words an equation or two mayappear to help guide that process, theemphasis is clearly on providing visualclues to stimulate mathematical thought.

Proofs without words have a long history.In this collection you will find modernrenditions of proofs from ancient China.classical Greece, twelfth-century Indiaeven one based on a published proof by aformer president of the United States!However. most of the proofs are morerecent creations, and many are taken fromthe pages of MAA journals.

The proofs in this collection are arrangedby topic into six chapters: Geometry andAlgebra; Trigonometry, Calculus andAnalytic Geometry; Inequalities; IntegerSums; Sequences and Series; and Miscellaneous. Teachers will find that many ofthe proofs in this collection are well suitedfor classroom discussion and for helpingstudents to think visually in mathematics.

The readers of this collection will findenjoyment in discovering or rediscovering some elegant visual demonstrationsof certain mathematical ideas that teachers will want to share with their students.Readers may even be encouraged to create new "proofs without words."

Roger B. Nelsen

Exercises in VisualThinking

ProofsWithoutWords

The TrisectorsFormerlyentitledABudget 01 T,.isections

Underwood DudleyAs a companion to Mathematical Cranksthe MAA is happy to offer its readersUnderwood Dudley's The Trisectors. Thisbook is also about mathematical cranksangle trisectors.

It is impossible to trisect angles withstraightedge and compass alone, but manypeople try and think they have succeeded.This book is about angle trisections andthe people who attempt them. Its purposesare to collect many trisections in one place,inform about trisectors, amuse the reader,and, perhaps most importantly, to reducethe number of trisectors. The Trisectorsincludes detailed information about thepersonalities of trisectors and their constructions.

According to Dudley, "hardly any mathematical training is necessary to read thisbook. There is a little trigonometry hereand there. but it may be safely skipped.There are hardly any equations. There areno exercises at the end of the sections, andthere will be no final examination. Theworst victim of mathematics anxiety canread this book with profit and dry palms.It is quite suitable to give as a present."


ISBN 0-88385-514-3



ISBN 0-88385-507-0


This is a truly unique book, written withwit and style. Kenneth O. May calls thework of mathematical cranks part of folkmathematics that should not pass unrecorded.

Catalog Number CRANKS

Underwood DudleyMathematical Cranks is about peoplewho think that they have done somethingimpossible. like trisecting the angle,squaring the circle, duplicating the cube,or proving Euclid's parallel postulate;people who think they have done something that they have not, like provingFermat's Last Theorem, verifyingGoldbach's Conjecture, or finding asimple proof of the Four Color Theorem;people who have eccentric views, frommild (thinking we should count by 12sinstead of lOs) to bizarre (thinking thatsecond-order differential equations willsolve all problems ofeconomics, politics,and philosophy); people who pray in matrices; people who find the AmericanRevolution ruled by the number 57; peoplewho have in common something to dowith mathematics and something odd, peculiar, or bizarre.

MathematicalCranks

<t4;>------------------

June 1994 FOCUS

Pack... PrIc&--Ba1h B_1lilt: 848.00; Member: 838.00

Gatalollllmber: .....

The Words ofMathematics explains theorigins of over 1500 mathematical termsused in English. While other dictionariesof mathematics define technical terms.this book concentrates on where thoseterms came from and what their literalmeanings are.

As in any dictionary, the entries themselves are arranged alphabetically. Thewords are drawn from arithmetic. algebra, geometry, trigonometry, calculus.number theory, topology. statistics, graphtheory, logic, recreational mathematics,and other areas.

An Etymological D1cUonary or Ma emadcal Terms Used In English

Steven Schwartzman

CatalogNumberWORDS

This dictionary is an indispensable reference for every library that serves teachersand students of mathematics. It is a natural source of information for courses inthe history of mathematics and formathematics courses intended for liberal arts students. At the individuallevel, whether you are a teacher ora student of mathematics, a loverof words, or hopefully both;whether you are a veteranmathematician or a novice,you will find material in thisbook appropriate to yourlevel of language andmathematics.

ISBN 0-88385-511-9

262 pp.•Paperbound,1994

List: $27.00MAAMember:$21.00

The WordsMathematic

Rosemary Schmalz

Outofthe Mouths ofMathematicians: AQuotation BookforPhilomaths is a compilation of 727 quotations from 292contributors, almost all ofwhom are twentieth century mathematicians. Some ofthesubject categories include: The Development of Mathematics, Exhortations toAspiring Mathematicians, Pure and Applied Mathematics,About Mathematicians(by name), Anecdotes and MiscellaneousHumor, Particular Disciplines in Mathematics, Moments of MathematicalInsight, Mathematics and the Arts, ... andmuch more.

Out 01 theMouths 01MathematiciansAQuotation Book lorPhilomaths

Published as a companion volume to Robert Edouard Moritz's MemorabiliaMathematica, Rosemary Schmalz's Outof the Mouths ofMathematicians picksup where Moritz left off. Her work willgive you a sense of the "story" oftwentieth century mathematics. You willencounter the mathematicians, their collaborations and disputes, the movementfrom abstraction to application, the emergence ofnew areas of research, the impactof computers on mathematics, the challenges in mathematics education, andmore.

This book will be particularly useful toteachers of mathematics at all levels, toencourage, motivate, and amuse their students.


ISBN 0-88385-509-7

List: $29. 00 MAA Member: $23. 00

Catalog Number OMMA

The more than eleven-hundred fully annotated selections in this book, gatheredfrom the works of three-hundred authors.cover a vast range of subjects pertainingto mathematics. Grouped in twenty-onechapters, they deal with such topics as thedefinitions and objects of mathematics;the teaching of mathematics; mathematics as a language or as a fine art; therelationship of mathematics to philosophy.to logic, or to science; the nature of mathematics, and the value of mathematics.Other sections contain passages referringto specific subjects in the field such asarithmetic, algebra, geometry, calculus,and modem mathematics.

When Robert Edouard Moritz compiledhis book of quotations, MemorabiliaMathematica, which appeared in 1914,he stated that his primary objective was toseek out the exact statement of and exactreferences for famous passages aboutmathematics. His sources ranged from theworks of Plato to the writings of Hilbertand Whitehead. His second objective wasto produce a volume that would be a sourceof pleasure. encouragement. and inspiration to both mathematicians andnon-mathematicians alike.

MemorabiliaMathematicaThe Philomath'sQuotation BookRobert Edouard Moritz

To mathematicians the book will be a greatsource of pleasure, inspiration, and encouragement. To teachers of mathematics andwriters about mathematics, it will remainof inestimable value as a source of quotations and ideas. To the layperson, it will bea revelation.

440pp., Paperbound, 1993

ISBN 0-88385-321-3

List: $24. OOMAAMember: $19.00

Catalog Number MEMO

FOCUS

CryptologyAlbrecht Beutelspacher

FR BRX XQFHUVWCQG WKLV?I oucan't decipher this coded message, youmust read this book!

How can messages be transmitted secretly? How can one guarantee that themessage arrives safely in the right handsexactly as it was transmitted?Cryptology-the art and science of "secret writing"-provides ideal methods tosolve these problems of data security.

Technology advances have stimulated interest in the study of cryptology. Ofcourse,computers can break cryptosystems muchmore efficiently than humans can. Computers allow complex and sophisticatedmathematical techniques which achieve adegree of security undreamt of by previous generations. Today the applications

of cryptology range from the encryptionof television programs sent via satellite,to user authentication of computers, to newforms of electronic payment systems using smart cards.

The first half of the book studies and analyzes classical cryptosystems. Here wefind Caesar's cipher, the Spartan scytale,the Vigenere cipher, and more. The theoryof cipher systems is presented, includinga description of the best possible cipher,the one-time pad. An introduction to linear shift registers, which serve as buildingblocks for most presently used ciphers, isalso given.

The second half of the book looks at theexciting new directions of public-keycryptology, which since its invention in1976,has revolutionized data security.Theauthor also looks at the famous RSA-algorithm, algorithms based on "discrete

June 1994

logarithms," the so-called zero-knowledgealgorithms, and the smart cards that bringcryptographic services to the person-onthe-street.

Although the mathematics covered is nontrivial, the book is fun to read, and theauthor presents the material clearly andsimply. Many exercises and referencesaccompany each chapter. The book willappeal to a wide audience including teachers, students, and the interested layperson.

Cryptology was originally published inGerman by Vieweg. This edition has beenextensively revised.


ISBN 0-88385-504-6


Catalog Number CRYPT

FREE BOOIIS!! see page 30

• • .And the Winner Is. . .Knot TheoryCharles Livingston

Winner-Best New Book inMathematics for 1993 .. .Professional and Scholarly PublishingDivision of the Association ofAmerican Publishers

Knot Theory, a lively exposition of themathematics of knotting, will appeal to adiverse audience from the undergraduateseeking experience outside the traditionalrange of studies to mathematicians wanting a leisurely introduction to the subject.Graduate students beginning a programof advanced study will find a worthwhileoverview, and the reader will need no training beyond linear algebra to understandthe mathematics presented.

Over the last century, knot theory has progressed from a study based largely onintuition and conjecture into one of themost active areas of mathematical inves-

tigation. Knot Theoryillustrates the foundations of knotting aswell as the remarkablebreadth of techniquesitemploys--eombinatorial, algebraic, andgeometric.

The development ofknot theory has taken place within thecontext of the growth of topology. KnotTheory illuminates that context. The classification of surfaces, one of the majorachievements in topology, is described,and then applied to prove the existence ofprime decomposition of knots.

The interplay between topology and algebra, known as algebraic topology, arisesearly in the book, when tools from linearalgebra and from basic group theory areintroduced to study the properties of knots,including the unknotting number, the braidindex, and the bridge number. Livingstonguides you through a general survey ofthe topic, showing how to use the techniques of linear algebra to address some

sophisticated problems, including one ofmathematics' most beautiful topics, symmetry. The book closes with a discussionof high-dimensional knot theory and a presentation of some of the recent advancesin the subject-the Conway, Jones, andKauffman polynomials. A supplementarysection presents the fundamental group,which is a centerpiece of algebraic topology.

An extensive collection of exercises isincluded. Some problems focus on detailsof the subject matter; others introduce newexamples and topics illustrating both thewide range of knot theory and the presentborders of our understanding of knotting.All are designed to offer the reader theexperience and pleasure of working in thisfascinating area.


ISBN 0-88385-027-3

List:$31.50 MAAMember: $25.00

Catalog Number CAM-24

<~)------------------

June 1994

James J. Kaput and Ed Dubinsky, Editors

Research Issues in UndergraduateMathematics LearningPreliminary Analyses and Reports

.,.


FOCUS


ISBN 0-88385-090-7

List:$24.00

Cooperative learning does help the student. That is the conclusion ofBonsangue,who investigates how two carefullymatched classes of students in a statisticscourse perform on exams. How studentslearn to write proofs in group theory is thesubject considered by Hart. Rosamondbreaks new ground by comparing howemotions vary in their effect on the problem solving ability of novices and experts.

All college faculty should read this bookto find how they can help their studentslearn mathematics.

culties.

220 pp., 1994, Paperbound

ISBN-0-88385-088-5

List: $24.00


A survey of the literature begins the volume. Becker and Pence have broughttogether an unusually complete list ofreferences on research in collegiate mathematics. Their comments will guide thoseattempting to begin or to continue a program of research in student learning.

The sad fact that even good calculusstudents stumble over nonroutine problems is the theme of Selden, Selden, andMason. Their conclusions point to significant shortcomings in the curriculum.This study of student difficulties is continued by Ferrini-Mundy and Grahamwho investigate a single student's interactions with the fundamental concepts ofthe calculus. Baxter studies a group ofstudents to learn how they acquire theconcept of set, while Cuoco does thesame for the concept of function.

The conference was unique. Conferenceparticipants included pre-college mathematics teachers, community college anduniversity teachers, and research mathematicians. Papers were delivered insessions devoted to the classroom teacher,to the history of mathematics, and to pedagogical and research interests in geometry.Many lectures combined these subjects.This book presents some of those lectures.Anyone involved with teaching orproducing mathematics can find something in thisvolume that will be interesting to them.

Also included in this volume is a penetrating interview with Eves.

Joby Milo Anthony, Editor

Avery special volume lor aB 01 Eves'thousands 01 adnrers. nyour interest is history 01 mathematics, geometry, or pedagogy, then this book islor you.

Howard Eves celebrated his eightiethbirthday in 1991. To honor that occasion,the University of Central Florida sponsored a conference that focused on thelifelong interests of this prominentAmerican mathematician, namely, the historyof mathematics, the teaching of mathematics, and geometry. Eves iswell-known for his contributions to allthree areas.

This volume of research in undergraduatemathematics education informs us aboutthe nature of student learning in some ofthe most important topics in the undergraduate curriculum: sets, functions,calculus, statistics, abstract algebra, andproblem solving. Paying careful attentionto the trouble students have in learningmathematics will help us to work withstudents so they can deal with those diffi-

ESfClrCles

Research in undergraduate mathematicseducation is important for all college anduniversity mathematicians. Ifour studentsare to be more successful in understandingmathematics, then college faculty need tounderstand how mathematics is learned.This knowledge can guide us in curriculum reform and in improving our ownteaching. It can help us make mathematicsaccessible to all students and it can increase the number of graduate students inmathematics.

---------------------@

FOCUS June 1994

The Search lor E.T. Bellalso known as John TaineConstance ReidEric Temple Bell (1883-1960) was a distinguished mathematician and a bestselling popularizer of mathematics. HisMen of Mathematics, still in print afteralmost sixtyyears, inspired scores of youngreaders to become mathematicians. Under the name "John Taine," he alsopublished science fiction novels (amongthem The Time Stream, Before the Dawn,and The Crystal Horde) that served tobroaden the subject matter of that genreduring its early years.

In TheSearchforE.T. Bell,Constance Reidhas given us a compelling account of thiscomplicated, difficult man who never divulged to anyone, not even to his wife andson, the story of his early life and familybackground. Her book is thus more of a

mystery than atraditional biography. It beginswith the discovery of anunexpected inscription in an Englishchurchyard and a series of cryptic notations in a boy's schoolbook. Then comesan inadvertent revelation, by Bell himself,ina respected mathematical journal. ... Youwill have to read the book to learn the rest.

Originally agreeing to write only a profileof Bell, Mrs. Reid soon found herself involved in a full-length biography. Thediscoveries she made in the course of herfive years of research will necessitate afresh evaluation of his extensive mathematical work and his science fiction

Honorable Mention-Best NewBook in Mathematics for1993...Professionaland ScholarlyPublishing Division of the Association ofAmerican Publishers

novels as well as the revision of almostevery statement currently in print abouthis family background and early life.

Mrs. Reid is already well known as theauthor of acclaimed biographies of DavidHilbert, Richard Courant, and JerzyNeyman.

Includes a collection of over 75 photographs.


ISBN 0-88385-508-9

List:$35. 00 MAAMember:$28.00

Catalog Number BELL

~---------------------------------------I

IFREE MAA book with every order of $50 or more.IIIf your order comes to $50 or more, choose a FREE hardcover book from this list:

o Selected Papers on Precalculus MathematicsTom Apostol, Editor

o Selected Papers on AlgebraSusan Montgomery and Elizabeth Ralston, Editors

o Selected Papers on GeometryAnne Stehney and Tilla Milnor, Editors

o Mathematical Time ExposuresIsaac Schoenberg

o Studies in Applied MathematicsA.H. Taub, Editor

o Studies in Algebraic LogicAubert Daigneault, Editor

o Studies in Partial Differential EquationsWalter Littman, Editor

o The Chauvenet Papers, A Collection ofPrize WinningExpository Papers in Mathematics, Volume 1James Abbott, Editor

o The Chauvenet Papers, A Collection ofPrize WinningExpository Papers in Mathematics, Volume 2James Abbott, Editor

Membership Code _(Please(ojry the firs: six digits that appear 01/ the toj) liTIP of Jlmr mailing label)

Send this order form to:The MAA, 1529 Eighteenth Street, NWWashington, DC 200361-800-331-1622 FAX (202) 265-2384

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This offer is available to individuals only.Foreign Orders Please add $3.00 per item ordered to cover postage and

TOTAL handling fees. The order will be sent via surface mail. If you wan t your order sent

L byai~~i~e~~~send you~prof~~l::C~O~O~~e:.. _..J

@------------

June 1994 FOCUS

Cottey College of MissouriAssistant Professor, full-time, tenure-trackfaculty starting fall 1994; three person department. Curriculum now includes algebra,trigonometry, statistics, calculus and differential equations; four sections per semesterload. Ph.D. preferred ABD considered forinitial appointment, but continuation requiresdoctorate. Teaching oriented, small, private(non-church related), two-year, residential,liberal arts college for women. 10:1studentslfaculty ratio, students from 35-40 states, and18-20 countries. 80 miles from Kansas Citymetropolitan area, 40-140 miles from Ozarkrecreation areas. Our mission is the development of intellect, leadership andself-esteem in young women. Send c.v. andletter expressing interest to Vice PresidentforAcademicAffairs,CotteyCollege,Nevada,Missouri 64772.

El\IPLOY'IE~T OPPORIT:\ITIESRates for FOCUS Employment Advertisements are $65.00 per column inch, billed to thenearest 1/2 inch. FOCUS offers a 15% discount for the same advertisement in three or moreconsecutive issues. All invoices include a tear sheet. Advertising Copy Deadlines: The firstof the month, two months prior to publication. FOCUS is published in February,April, June,August, October, and December.

Advertisers should contact: Amy Stephenson, FOCUS Advertising Coordinator, The Mathematical Association ofAmerica, 1529Eighteenth Street, NW,Washington, DC 20036-1385(202) 387-5200; FAX: (202) 265-2384; e-mail: [email protected]

louisiana Tech University MathematicsDepartment of Mathematics and The Department of Mathematics anticipates

Statistics anopening for an assistantprofessor to teachA I" .. d f .. d the full range of mathematics courses from

pp ications ar~ Invite or an a~tlclpa~e algebra through calculus. Ph.D. in appropri-tenure-track assistant professorship special- ate discipline' experience and expertise inizing in an area o.f mathema~ics closely current class~oom pedagogy and technol-re.lated to ~omputatlonal analysis and mod- ogy including graphics calculator,ellng. Applicants must have a Ph.D. by the computer-assisted instruction reform of cal-time of the appointment (9/.94) and are ex- culus, etc. Resumes to: Dean Thomas M.~ectedtosho~strongpotentlalforexcellence Carroll, Human Resources, New York CityIn both teaching and research. A current re- Technical College (CUNY) 300 Jay Streetsume and three letters of recommendation Namm 321, Brookl n, NY 1; 201. AAlEOE/D

should be sent to: r--------y- - - - - - - - - - - - - - - - - - - - - - - - -

William McClungNebraska Wesleyan UniversityNumberTheory, Mathematica

Caren DiefenderferHollins CollegeDifferential Equations, Maple

Michigan University)MathematicalModeling, Mathcad

Margie HaleStetson UniversityDifferential Equations with Modeling,MathKit for Windows

Kyle SiegristUniversity ofAlabama in HuntsvilleProbability,MathKit for Windows

Charles HofmannLaSalle UniversityandRoseanne HofmannMontgomery County Comm. CollegePrecalculus and Calculus, MathKit for Windows

William SloughEastern Illinois UniversityA Brief Calculus for Non-science Students,Maple

William MuellerUniversity ofArizona (currently at Northern

The MAA's Interactive Mathematics Text Project has announced the selection of eightinteractive text developers. The developers were selected from the 138 participants in theIMTP workshops held during the summer of 1993. Each developer will receive an IBMPentium computer, printer, and software to be used in the creation of an interactive text.In addition, each developer will be supported to attend an IMTP advanced workshopduring this summer and the IMTP Developers'Conference to be held in October at theInstitute for Academic Technology at The University of North Carolina at Chapel Hill.

The Interactive Mathematics Text Project has as its goal the improvement ofmathematics learning through the use ofcomputer-based interactive texts. The project is supportedby the IBM Corporation, The National Science Foundation, Waterloo Maple, WolframResearch, MathSoft, and Eduquest. Twelve IMTP workshops are scheduled for thesummer of 1994. More information is available through the gopher: imtp.math.upenn.eduor by contacting the project director, Gerald J. Porter, [email protected].

The individuals chosen, the topics of their proposed texts, and the software platformsthey are using follows:

Ken DavisAlbion CollegeDiscrete Mathematics,Mathematica

Interactive Mathematics Text Project Chooses SecondRound of Developers

Please furnish curriculum vitae and threeletters of reference to Dr. S. Park, PersonnelCommittee, Department of Mathematics,Long Island University, University Plaza,Brooklyn, New York 11201. Long Island University is an AAlEO employer.

-----------------~.'~B:?

long Island UniversityBrooklyn Campus

Applications are invited for a tenure-trackposition at the Assistant Professor level beginning September 1, 1994. ReqUirementsinclude a Ph.D. in pure mathematics (preferably in geometric topology) and acommitment to teaching with an interest inresearch.

R.J. Greechie, HeadDepartment of Mathematics and StatisticsLouisiana Tech UniversityRuston, LA 71272

Thescreeningof applicantswill beginonJune1 and will continue until the position is filled.Louisiana Tech University is an Equal Opportunity/AffirmativeAction Employer.Weareinterested in receiving applications fromqualified women and minorities.

Oberlin CollegeA full-time, temporary position is available atOberlin College for the academic year 199495.Teachingload is5.c~s.Qualificationsinclude Pi!.D.i ~~expected by Fall1994, plu rat erest and potential excell c1!n~obeconsidered,send a letter icatlon, academic transcripts, curricu urnvitae and at least 3 lettersof reference to Michael Henle, Departmentof Mathematics,OberlinCollege, Oberlin, OH44074. Consideration of applicant files willbegin April 18, 1994.

National MAA MeetingsAugust 15-17, 1994 Sixty-ninth Annual

Joint Summer Meeting, MinneapolisMATHFEST 1994

January 4-7, 1995Seventy-eighthAnnualMeeting, San Francisco (Board ofGovernors, January 3, 1995)

Sectional MAA MeetingsALLEGHENY MOUNTAIN, Novem

ber 5, 1994, Montgomery CountyCommunity College, Blue Bell, PA

FLORIDA, March 5-6, 1995, Florida Institute of Technology, Melbourne, FL

NEW JERSEY, November 19, 1994,Georgian Court College, Lakewood, NJ

MinneapolisMathFest UpdatesBreakfast for MAA VisitingMathematicians.

There will be a breakfast on Wednesdaymorning, August 17, 7:00 a.m. to 8:20a.m., for past, present and potential futureMAA Visiting Mathematicians. These arepeople who arrange to work at the MAAheadquarters in Washington while on sabbatical, during retirement, etc. The pastand present Visiting Mathematicians willtell about their Washington experiencesand answer questions. Anyone interestedin being included should contact the MAAExecutive Director, Marcia P. Sward.

FOCUSThe Mathematical Association of America1529 Eighteenth Street, NWWashington, DC 20036-1385

CalendarNORTH CENTRAL, October 28-29,

1994, Minot State University, Minot,ND

NORTHEASTERN, November 18-19,1994, University ofHartford, Hartford,CT

NORTHERN CALIFORNIA, October21-22, 1995, Cal Polytech State University, San Luis Obispo, CA

OHIO, October 28-29, 1994, Universityof Findlay, Findlay, OH

SEAWAY,November 4-5, 1994, Rochester Institute of Technology, Rochester,NY

SOUTHERN CALIFORNIA, October21-22, 1995, Cal Polytech State University, San Luis Obispo, CA

TEXAS, March 30-April I, 1995, BaylorUniversity, Waco, TX

AWM Panel Discussion

Monday, August 15,3:30-5:00 p.m. Celebrating women's achievements inalgebra, analysis, cornbinatorics, and geometry: past, present, and future.

Panelists include: Jane Gilman (RutgersUniversity), Karen Saxe (Macalester College), Doris Schattschneider (MoravianCollege), Marie Vitulli (University ofOregon), and the Organizer and Moderator is Joan Hutchens (MacalesterCollege).

NAM Special Invited Lecture

The National Association of Mathematicians (NAM) is pleased to announce aSpecial Invited Lecture which will take

Other MeetingsOctober 13-15, 1994, Twenty-fourth Annual Conference of the InternationalSociety for Exploring Teaching Alternatives, Arizona State University, Tempe,AZ. For more information, contact: GloriaBalderrama, Society Registrar, Mathematics Department, Colorado State University,Fort Collins, CO 80523; (303)491-6452;FAX: (303) 491-2161; e-mail: [email protected].

place in Minneapolis and at all futureMathfests. The lecture, scheduled on Tuesday, I :00 p.m. to I :50 p.m., will be givenby David Blackwell (University of California, Berkeley), and is titled Largedeviations for martingales.

MAA-Mu Alpha Theta Lecture

Monday, August 15, I :40 p.m. Pamela J.Drummond (Kennesaw College), talk isentitled, Generating Enthusiasm whileAchieving Equity in Mathematics.

JUNE 1994Second class postage paid at

Washington. DC andadditional mailing offices

FOCUS_14_3.pdf - Mathematical Association of America

Documents

Transcript of FOCUS_14_3.pdf - Mathematical Association of America