2

FOCUS November 2001

FOCUS is published by theMathematical Association of America inJanuary, February, March, April, May/June,August/September, October, November, andDecember.

Editor: Fernando Gouvêa, Colby College;fqgouvea@colby.edu

Managing Editor: Carol Baxter, MAAcbaxter@maa.org

Senior Writer: Harry Waldman, MAAhwaldman@maa.org

Please address advertising inquiries to:Carol Baxter, MAA; cbaxter@maa.org

President: Ann E. Watkins, California StateUniversity, Northridge

First Vice-President: Barbara L. Osofsky,Second Vice-President: Frank Morgan,Secretary: Martha J. Siegel, AssociateSecretary: James J. Tattersall, Treasurer:Gerald J. Porter

Executive Director: Tina H. Straley

Associate Executive Director and Directorof Publications and Electronic Services:Donald J. Albers

FOCUS Editorial Board: GeraldAlexanderson; Donna Beers; J. KevinColligan; Ed Dubinsky; Bill Hawkins; DanKalman; Peter Renz; Annie Selden; Jon Scott;Ravi Vakil.

Letters to the editor should be addressed toFernando Gouvêa, Colby College, Dept. ofMathematics, Waterville, ME 04901, or byemail to fqgouvea@colby.edu.

Subscription and membership questionsshould be directed to the MAA CustomerService Center, 800-331-1622; e-mail:maahq@maa.org; (301) 617-7800 (outsideU.S. and Canada); fax: (301) 206-9789.

Copyright © 2001 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, thenewsletter of the Mathematical Associationof America (Incorporated).”

Periodicals postage paid at Washington, DCand additional mailing offices. Postmaster:Send address changes to FOCUS,Mathematical Association of America, P.O.Box 90973, Washington, DC 20090-0973.

ISSN: 0731-2040; Printed in the United Statesof America.

FOCUSNovember 2001

Volume 21, Number 8

Inside

4 The Hat Problem and Hamming CodesBy Mira Bernstein

8 An Experiment that Worked: Revising the Calculus CurriculumBy William W. Johnston, Alex M. McAllister, John H. Wilson

10 An Interdisciplinary Educational Issues Seminar Emanating from theMathematics DepartmentBy Jeremy Case and Matthew DeLong

12 The Math Forum Has a New Home

13 Happy Abstract Algebra ClassesBy John Fraleigh

14 Sectional and National Awards for Distinguished Teaching

15 Mathematical Sciences Reports Suggest Improved Job Market

16 My MathFestBy Fernando Gouvêa

17 Mathematicians on Michael Feldman’s “Whad’ya Know?”By Frank Morgan, Paul Holt, Joe Corneli, Nick Leger, and Eric Schoenfeld

20 Secretary’s Report

22 Treasurer’s Report

Cover artist John Johnson imagines a game show about mathematicians andcolored hats. See page 4.

January February MarchEditorial Copy November 15 December 14 January 15Display Ads November 26 December 21 January 29Employment Ads November 12 December 10 January 15

FOCUS Deadlines

FOCUS

3

November 2001

Norway has established a new interna-tional prize in mathematics, named afterNiels Henrik Abel (1802-1829), the mostfamous Norwegian mathematician of the19th century. Jens Stoltenberg, the Nor-wegian Prime Minister, announced thatthe Norwegian government will set up a$22 million fund to endow the prize in2002, the two hundredth anniversary ofAbel’s birth. The Abel Prize will beawarded annually beginning in 2003. Itwill include a monetary award worthabout $500,000. An independent inter-national committee of mathematicianswill choose the laureates.

“We need to strengthen mathematics andthe sciences,” Stoltenberg said. “NielsHenrik Abel was an internationallyknown Norwegian mathematician whonearly 200 years ago made a lasting im-pact in the world of science. An interna-tional prize in mathematics dedicated tohis name, is an expression of the impor-tance of mathematics, and is intended toencourage students and researchers.”

The Norwegian Government hopes thatthe Abel Prize will increase public aware-ness of research in mathematics and thatit will also reinforce Norway’s image as acountry that values knowledge and learn-ing. The International MathematicalUnion and European mathematical so-cieties have expressed support for theprize.

An Abel Prize was first proposed in 1902,by Oscar II, then king of Sweden andNorway. When the union between thetwo countries was dissolved in 1905, theplan was abandoned. As a result, math-ematics has never had an internationalprize of the same value and importanceas the Nobel Prize. The hope of the Nor-wegian Government is that the Abel Prizewill come to play that role.

For more information, see the Septem-ber 7 issue of Science and the NorwegianGovernment web site at http://odin.dep.no/smk/engelsk/aktuelt/pressem/001001-990157/index-dok000-b-n-a.html.

Norway Establishes Abel Prize

in Mathematics

The American Association for the Ad-vancement of Science will be holding itsAnnual Meeting and Science InnovationExposition in Boston, MA on February14–19, 2002. The program, as usual, in-cludes several items of interest to math-ematicians.

Several of the scheduled symposia con-cern the teaching of science and math-ematics, including sessions on quantita-tive literacy, articulation between schooland college, advanced mathematics andscience in U.S. high schools, and genderissues. Other program items that havesignificant mathematical content include:a “Topical Lecture” by Carl Pomerance on“Hard Number Theory Problems andCryptology.” Further information, lastminute updates, and online registrationare located on the AAAS website: http://www.aaasmeeting.org.

Mathematics at the

AAAS, February 2002

As FOCUS goes to press, it is the earlyOctober, and we are just beginning to re-cover from the shock of September 11.We have only heard of one death directlyrelated to the mathematics community:Daniel Lewin of Akamai, a good friendof mathematics and of the MAA. See page14 for a short obituary.

The officers and staff of the MAA appre-ciates the notes and emails from mem-bers and friends expressing sorrow andconcern for us in light of the terrorist at-tacks on America. In particular, the MAAthanks the Greek Mathematical Societyand the Canadian Mathematical Societyfor their support and friendship.

This is a somber note with which to be-gin my third year as editor of FOCUS.This issue, which simultaneously looks

back to the Madison MathFest and for-ward to future activities of the MAA,emphasizes the continuity of our lives andof the activities of the Association. Pleasecontinue to send me news about anyevents that might be of interest to MAAmembers. In particular, I’m always look-ing for cover images, for short expositoryarticles about interesting new mathemat-ics, and for good photographs of MAApeople and events.

FOCUS is your newsletter. Please let meknow what you’d like to see us do! Youcan contact me by email (the best way!)at fqgouvea@colby.edu, or by mail at De-partment of Mathematics, Colby College,Mayflower Hill 5836, Waterville, ME04901.

Fernando Q. Gouvêa

From the Editor

The MAA’s new Journal of OnlineMathematics and its Applications(JOMA) is now a year old. We will cel-ebrate this event at the Joint Meetings inSan Diego with a panel session featuringJOMA authors discussing their JOMAmaterials and their work in general.Members of the panel include:

• Dennis DeTurck, University ofPennsylvania

• John Kiltinen, Northern MichiganUniversity

• Tom Leathrum, Jacksonville StateUniversity

The session will be introduced and mod-erated by JOMA editor David Smith(Duke University). The session will beheld on Monday, January 7, 9:00 a.m.-10:30 a.m. The JOMA web site is http://www.maa-joma.org. This session has beenorganized by Lang Moore, Executive Edi-tor of the Mathematical Sciences DigitalLibrary (MathDL).

JOMA Panel Session

in San Diego

4

FOCUS November 2001

By Mira Bernstein

Consider the following two problems–one, an entertaining new puzzle, theother, an important practical question:

Problem 1: At a mathematical game showwith n players, the host blindfolds thecontestants and puts colored hats on theirheads. The color of each person’s hat– redor blue–is determined by a coin toss, in-dependently of all the others. After theblindfolds come off, each player can seehis teammates’ hats, but not his own.When the host gives a signal, all playerssimultaneously either guess the color oftheir own hats or pass. If there are no in-correct guesses and at least one correctguess, the players share a $1,000,000 prize.There is no communication between theplayers during the show, but they are toldthe rules in advance and allowed to dis-cuss their strategy. What should they doto maximize their chances of success?

Problem 2: In storing or transmittingdigital data, there is always some risk ofdistortion: a 0 might accidentally flip toa 1 or vice versa. One way to deal withthis problem is to introduce some redun-dancy into the transmission–for instance,by sending each bit multiple times. How-ever, transmitting too many extra bits iscostly and ineffective. How can we pro-tect k bits of data against the possibilityof an error by using the minimal num-ber of additional “check bits”? Note thatour method must not only detect the er-ror, but also determine its precise loca-tion, so that the user can recover the origi-nal message every time.

On the surface, these two problems ap-pear quite different. Underlying them,however, are very similar questions in thegeometry of binary n-space. The prob-lems are, in fact, dual to one another: theyapproach the same goal from opposite di-rections. This theoretical optimum is notattainable for most values of n and k, butwhen it is, the two problems have solu-tions based on the same “perfect” struc-ture: the binary Hamming code.

Richard Hamming originally designed hiscode to tackle Problem 2, but let’s begin

our discussion with Problem 1, a recenthit in the mathematical community. Anarticle in last April’s Science Times [2]dates the puzzle’s first appearance to 1998and tells of it “spreading through net-works of mathematicians like a juicy bitof gossip”. Since the publication of thearticle, the gossip has spread ever fasterand further. While the “hat problem” it-self is just a restatement of an old ques-tion in coding theory, certain variationshave led to new constructions that are thesubject of current research. We discussthese briefly at the end of the article.

The Hat Problem

Many mathematicians, when they firstencounter the hat problem, reason as fol-lows. (a) Since the color of each hat is cho-sen independently, a player can obtain nouseful information from studying herteammates’ hats: whatever she sees andwhatever she does, her probability ofguessing correctly is always 1/2. (b) Sincea single correct guess is enough to win,and a single incorrect guess spoils thegame, a good strategy should always havemost of the players passing, to minimizethe risk. (c) Therefore, the optimal strat-egy is to have everyone pass, except forone designated player who (for example)always guesses “red”. The probability ofwinning under this strategy is 1/2, andyou can do no better.

The argument certainly sounds plausibleenough. The first statement is certainlytrue, which seems to imply that nothingcan be gained from cooperation amongthe players. In fact, however, cooperationallows the team to win with probabilitymuch higher than 1/2. Even with onlythree players, they can increase theirchances to 3/4, and as the number ofplayers grows, the probability of winning(under the correct strategy) rapidly ap-proaches 1.

To explain why, it is better to introducesome notation and terminology. For nplayers, there are 2n possible arrange-ments of hats; we refer to these as con-figurations. If we substitute 1 for “blue”and 0 for “red”, the set of all configura-tions can be identified with binary n-

space, F2n = {0,1}n. A strategy S is a com-

plete set of instructions for each player:if you observe X, do Y (we consider onlydeterministic strategies). Given S, eachpossible hat configuration will be either“winning” or “losing”. Denote the set ofwinning configurations by W

S and the set

of losing configurations by LS. Since all

configurations are equally likely, theteam’s probability of winning is |W

S| /2n.

The goal, therefore, is to find S such that|W

S| is maximal.

We might try to accomplish this goal bymaximizing the frequency of correctguesses. However, a glance at statement(a) shows that this is hopeless. All it takesto change a good guess into a bad one isto change the guesser’s hat. If the game isplayed 2n times–once with each possiblehat configuration–the total number ofguesses made by the players will varyfrom strategy to strategy. However, thenumbers of correct and incorrect guessesfor a given strategy S will always be equal.Thus the expected fraction of correctguesses, both for an individual player andfor the group as a whole, is 1/2, regard-less of S.

The key to solving the hat problem is torealize that the guesses do not have to bedistributed uniformly over the space ofconfigurations. Our goal, after all, is tomaximize not the number of correctguesses, but the size of W

S. Suppose that,

under strategy S, the total number ofguesses made by all players in all configu-rations is 2w–of which, necessarily, w arecorrect and w are incorrect. Since a win-ning configuration requires at least onecorrect guess, |W

S| cannot exceed the

number of correct guesses, w. |WS| will

be equal to w only if in each winning con-figuration, exactly one person guesses andthe rest of the players pass. On the otherhand, the minimal number of configu-rations in L

S is w/n. This can happen only

if in every configuration not in WS, all of

the players guess incorrectly. (In particu-lar, there should be no configurationswhere all players pass.) Intuitively, wewant to spread out the correct guesses aswidely as possible and to concentrate allthe incorrect ones on just a few badpoints. (Contrast this with statement (b),above.)

A strategy that achieves such a distribu-tion of correct and incorrect guesses is

The Hat Problem and Hamming Codes

FOCUS

5

November 2001

called perfect. At this stage, we do notknow whether perfect strategies exist, forany n. However, we do know that theprobability of winning under a perfectstrategy is

This gives an upper bound for what ourteam of n players can aspire to.

In fact, it is not hard to see that perfectstrategies can exist only for very specialvalues of n. We must have

Since | LS | = w/n is an integer, we con-

clude that n + 1 | 2n, which is only pos-sible if n = 2m – 1 for some m. Remark-ably, for all such n, perfect strategies doexist. Before going on, the reader maywant to try her hand at finding such astrategy for the simplest case, n = 3.

Coverings

As we have seen, any strategy S partitions

F2n into two disjoint subsets, L

S and W

S.

For a perfect strategy, these subsets havespecified sizes, but as yet we know littleabout their other properties. A natural

question is: can every subset of F2n func-

tion as LS for some strategy S? If not, what

are the necessary and sufficient condi-tions?

The conditions, it turns out, are geomet-

ric. F2n has a natural metric space struc-

ture, the Hamming metric, in which thedistance d (x,y) between two vectors x, y

in F2n is defined to be the number of en-

tries where x and y differ. (In terms of ourgame, the distance between two hat con-figurations is the number of playerswhose hats would have to change color.)The ball of radius r around a point y is

defined as usual: Br(y) = { x in F2

n : d (x,

y) ≤ r}.

Fix a strategy S, and consider a point x inW

S. Since x is a winning configuration,

the instructions in S must lead at least oneplayer to guess his own hat color correctly.If we secretly change his hat, his instruc-

tions will still be the same–but in the newconfiguration his guess will be wrong, andthe team will lose. Thus for every win-ning configuration x, some configurationat a distance 1 from x must be losing.Equivalently, each x in W

S is contained

in a ball of radius 1 around some point yin L

S. Thus, together, the unit balls around

the points of LS cover the entire space of

configurations. Formally, the union of the

balls B1(y) for y in L

S is all of F2

n .

Conversely, given any set L with this prop-erty, we can construct a strategy S suchthat L

S = L. In this strategy, the instruc-

tions to Player i read as follows: “You donot know what configuration of hats yourteam has been given. However, from look-ing at your teammates’ hats, you can de-termine that it is one of two configurations,x and x', which differ only in the i-th coor-dinate (corresponding to the color of yourown hat). Guessing your hat color is thesame as guessing whether the actual con-figuration is x or x'. If exactly one of theseconfigurations is in L, guess the other. Ifneither or both are in L, pass.” The role ofthe covering condition is to ensure thatin every configuration outside of L, atleast one player will make a guess.

Readers familiar with coding theory ordiscrete geometry will recognize the defi-nition of a covering. For any metric spaceX, we say that a subset C of X is an r-cov-ering of X if the union of all B

r (y) for y in

C is equal to X. Coverings are easy to vi-sualize in Euclidean space: just imaginefilling a region with balls of radius r, pos-sibly overlapping, in such a way that ev-ery point lies in (“is covered by”) at leastone ball.

What we have shown is that strategies forthe n-player hat problem are exactly

equivalent to 1-coverings of F2n . The cen-

ters of the balls are precisely the points ofL

S–the losing configurations. The chal-

lenge, therefore, is to minimize the num-ber of balls, by “spreading them out” asevenly as possible and avoiding overlaps.

Ideally, we would like to cover all of F2n

without any of the balls overlapping atall. This is called a perfect covering, andcorresponds, not surprisingly, to a per-fect strategy. In Euclidean space, perfect

coverings are impossible. In F2n , they

cannot exist unless the size of a unit ball(n +1 points) exactly divides the size ofthe whole space (2n points). This leadsto the same conclusion as before: n = 2m

– 1 for some m. We have thus reducedthe hat problem to a classic question incoding theory: finding maximally effi-cient (if possible, perfect) 1-coverings in

F2n .

Packings

Let us now go back and look at Problem2. Here our task is to transmit a k-bitmessage in such a way that the originalcan be recovered even if one bit in thetransmission gets distorted. Suppose weuse a total of n bits to transmit the mes-sage. In our original formulation, the goalwas to minimize n given k, but we canalso invert the problem and ask how wecan maximize k given n.

To each of the 2k possible messages, we

assign a vector in F2n , which is just the

sequence of 0’s and 1’s that we intend totransmit. The resulting set of 2k desig-

nated points in F2n is called a code; its

elements are called codewords. Now sup-pose an error occurs in transmission: onebit is changed, and the addressee receivesa vector at distance one from the intendedcodeword. If the received vector is itself acodeword, the addressee will not evennotice the error. If the received vector isnot in the code, the addressee will realizethat something went wrong and will lookfor the closest codeword. However, if hefinds multiple codewords at distance onefrom the received vector, he cannot besure which one was intended. In eithercase, he will not be able to recover themessage correctly. Thus we must deviseour code in such a way that any vectorcontained in a unit ball around acodeword has distance at least 2 from anyother codeword. In other words, the unitballs around the codewords must be dis-joint.

There is a name for this condition: our

code must be a 1-packing of F2n . In gen-

eral, an r-packing of a metric space X is asubset C such that the balls of radius rcentered at the points of C are disjoint.Recall that for coverings, the challenge

W

W L

w

ww

n

n

nS

S S+=

+=

+1

21n

s sW L ww

n

w n

n= + = + =

+( )

6

FOCUS November 2001

was to use as few balls as possible and stillhave them fill the entire space. Forpackings, on the other hand, the challengeis to squeeze as many balls into the spaceas possible. A perfect covering is, at thesame time, a perfect packing–an idealequilibrium arrangement that solves bothproblems simultaneously. For r > 1, thereis only a single instance of such a perfectarrangement (aside from some trivialcases): the Golay code, with n = 23, r = 3[1]. But when r = 1, for every value of nwhere such an arrangement can exist, itdoes exist. These are the Hamming code.

Hamming Codes

Recall that perfect 1-coverings or 1-packings can exist only when n = 2m – 1for some m. Let P be the m x n matrixwhose i-th column is the binary expan-sion of the number i, padded with extra0’s at the beginning if necessary. For in-stance, for m = 3,

As usual, P defines a linear map from F2n

to F 2m . The binary Hamming code H is

the kernel of this map. Since P is always

full-rank, H is a linear subspace of F2n of

dimension n – m. Thus H contains 2n – m

= 2n/(n + 1) points, exactly the rightnumber for a perfect 1-covering/1-pack-ing.

To see that H really is both a packing anda covering, let’s check that it gives a solu-tion to both of the problems we have beendiscussing. (We could also have checkedthe packing and covering conditions di-rectly, but it is more interesting and in-structive to see H at work.)

Transmitting a message of k = n – m bits:It does not matter how we assign pointsof H to the 2k possible messages. Whatmatters is that when the addressee re-

ceives a vector x in F2n , she should be able

to determine which vector from H wasactually transmitted. To do this, she sim-ply computes Px. If the result is 0, then xwas the intended transmission and noerror has occurred. Otherwise, Px is thebinary expansion of some integer i be-

tween 1 and m. To recover the originaltransmission, the addressee must changethe i-th entry of x: this is the entry thatgot distorted in transmission.

The perfect strategy for n players: Be-fore the game, each player is assigned anumber from 1 to n. Once the hats are inplace, each player mentally constructs thevector x corresponding to the hat con-figuration that he observes, with a 0 inhis own entry. He then computes Px. Ifthe result is 0, he guesses “1”. If the resultis the binary expansion of his own num-ber, he guesses “0”. Otherwise, he passes.

As an exercise, the reader can try to pre-dict the actions of a team of seven play-ers whose hats are Red, Blue, Red, Red,Blue, Blue, and Red. Who will guess andwho passes if “1” = Blue? What about if“1” = Red? Is the answer surprising?

Open Problems: Imperfect Strategiesand Multicolored Hats

When the number of players is not 2m –1, the team has to settle for a 1-covering

of F2n that is not perfect. A good mea-

sure of the “imperfection” of a 1-cover-ing (or 1-packing) C is its density, definedby d = | C |(n + 1)/2n. For a packing, d is

the fraction of the space F2n that is cov-

ered by the unit balls: d ≤ 1. For a cover-ing, d is the average number of balls that

contain each point of F2n , so d ≥ 1. In the

perfect case, of course, d = 1.

Asymptotically, it is known that the den-sities of the best coverings and packingsfor all values of n come arbitrarily closeto 1 (see [1]). However, these packingsand coverings do not have a nice algebraicstructure like the Hamming codes, andno general method is known for con-structing them. For now, the best generaladvice we can offer a team of n players isto have 2m – 1 of them play according to aHamming code strategy, while the restpass. This is clearly not optimal (as n goesto infinity, d can get arbitrarily close to2), but it is not too bad either: the prob-ability of winning is still at least n/(n +2).

The problem of finding optimal 1-cov-

erings in F2n is hard, but it is not new. In

contrast, if we try to play the game withmore than two hat colors, we enter intouncharted territory. Let Q be a set of qcolors, q ≥ 2. In the configuration spaceQ n, we can define L

S, W

S, and the Ham-

ming metric exactly as before. We caneven talk about perfect strategies –strat-egies in which, for each hat configuration,either one player guesses correctly or allguess incorrectly. But although there is awell-developed theory of coverings for q≥ 2 (for instance, the definition of a Ham-ming code can be extended to work overany finite field), a moment’s thoughtshows that none of these results have anyrelevance to the multicolored hat prob-lem. For q ≥ 2, the condition on the set L

S

is no longer the same as the 1-coveringcondition. It leads to a new sort of “code”that has never been studied before.

Recently, a few coding theorists have be-gun to tackle the multicolored hat prob-lem. For instance, Hendrik Lenstra andGadiel Seroussi have shown that no per-fect strategies exist for q ≥ 2. On the otherhand, for any q, they can construct strat-egies which, as n approaches infinity, al-low the team to win with probability ap-proaching 1. (Remarkably, these strate-gies use binary Hamming codes for all q.)It is not clear what practical applicationsthese constructions might have, but cod-ing theory is full of unexpected connec-tions. The original hat problem, once itsgeometric structure was exposed, seemedlike the most natural instance of 1-cov-erings imaginable–yet no one hadthought of it until a few years ago. Per-haps in a few more years, the codes thatarise from the multicolored hat problemwill also turn out to be useful in an en-tirely different–maybe even practical–context.

Acknowledgments: I am grateful toGadiel Seroussi for sharing the slides fromhis recent talk, “On Hats and Other Cov-ers”.

References

[1] Cohen, G., Honkala, I., Litsyn, S., Lobstein,A., Covering Codes. North-Holland, 1997.[2] Robinson, S., “Why Mathematicians NowCare About Their Hat Color.” The New YorkTimes, April 10, 2001. Online at http://w w w. m s r i . o r g / a c t i v i t i e s / j i r / s a r a r /010410NYTArticle.html.

P =

0001111

0110011

1010101

FOCUS

7

November 2001

In January of 2002, Jerry Porter willcomplete ten years of service as MAATreasurer, which followed eight years asan elected member of the Finance Com-mittee. During that period, the MAA hasbeen put in a much stronger financialposition and its procedures have beenmodernized. But Porter has not only beenthe guardian of the MAA’s finances. Fromthe time he was first elected to the Boardas the Governors from the EPADEL Sec-tion in 1980, he has repeatedly acted asthe Association’s conscience on humanrights issues as well.

MAA looks for a New Treasurer

Call for Nominations: MAA Treasurer

The Committee charged with the re-sponsibility of recommending candidatesfor the position of Treasurer of the MAAseeks input. Members of this TreasurerSearch Committee are Barbara Faires,Chair, Ron Graham, and BarbaraOsofsky. If you would like to suggest acandidate or would like to be consideredyourself, please contact Barbara Faires bye-mail (faires@westminster.edu) as soonas possible.

The term of the treasurer position is fiveyears. According to the MAA Bylaws,

• The Treasurer shall have the usual du-

MAA President Ann Watkins has createda search committee charged with recom-mending a new Treasurer to the Board.The committee members are:

• Barbara Faires (chair), former chair ofthe Budget Committee;

• Ron Graham, member of the Invest-ment Committee and soon to be Presi-dent-Elect;

• Barbara Osofsky, First Vice-President.

The committee has issued the call fornominations printed below.

ties pertaining to the office including thecollection of dues and the supervisionand safekeeping of the funds of the Asso-ciation.• The Treasurer shall be responsible forthe control and administration of all in-vestment funds; endowment, trust, andgift funds; and such other funds as theBoard may designate.

The Search Committee makes its recom-mendations for candidates for the posi-tion of Treasurer to the Executive Com-mittee. The Executive Committee willpresent the nominations to the Board,which will elect the MAA Treasurer.

Copies of the Conference Board of theMathematical Sciences report on TheMathematical Education of Teachers weresent to mathematics departments acrossthe nation in September. The report at-tempts to distill the essence of currentthinking on curriculum and policy issuesand to apply these ideas to programs forthe preparation of future teachers. Theauthors hope that the report will moti-vate the mathematical community to getmore seriously involved with teacherpreparation. One of the major themes ofthe report is that school mathematics hasintellectual substance and depth, and thatthis should guide the efforts to reformteacher preparation programs.

The report is available in two formats.Part I only, published by the MAA, canbe obtained free of charge from CBMS(1529 Eighteenth St. NW, Washington,DC 20036, or send email tokolbe@math.georgetown.edu). The com-plete report (Parts I and II) was publishedby the American Mathematical Society.Both parts are also available online athttp://www.maa.org/cbms.

Report on the Math-

ematical Education of

Teachers Released

In addition, the Clay Mathematics Insti-tute has produced a video recording ofthe musical. It is available in VHS andDVD formats and comes with a bookletdiscussing the production and the historyof Fermat’s Last Theorem and its proof.For information about obtaining thevideo, please visit the Clay MathematicsInstitute web site at http://www.claymath.org.

Fermat’s Last Tango Available on CD and Video

Both the cast album and a video record-ing of the York Theatre Company’s pro-duction of Fermat’s Last Tango are nowavailable. As described in the November,2000 issue of FOCUS, Fermat’s LastTango is a musical play by JoshuaRosenblum and Joanne Sydney Lessnerbased on Andrew Wiles’ proof of Fermat’sLast Theorem. The musical was describedby The New York Times as “rollicking,whimsical, catchy and clever”, and com-

pleted a successful run at the York in De-cember 2000.

The CD, which will be distributed on theOriginal Cast Records label, contains allthe musical numbers from the show. Theaccompanying booklet features the com-plete libretto, production photos, and es-says by the authors and by Andrew Wiles.Visit http://www.fermatslasttango.com formore information about the cast record-ing and the play.

8

FOCUS November 2001

By William W. Johnston, Alex M.McAllister, John H. Wilson

In 1998, at Centre College , we launcheda significant curricular experiment withour introductory precalculus-calculus se-quence. The Precalculus course was elimi-nated and the Calculus I course was reor-ganized with an emphasis on makingfunctions, rather than the calculus tools,the primary objects of study. The out-comes of this revision have been quitepositive: most significantly, they have ledto an increase in both the number of stu-dents completing Calculus II and thenumber of students majoring in math-ematics at Centre.

The Centre College academic calendarconsists of three terms: a twelve week FallTerm, a six week Winter Term, and atwelve week Spring Term; every collegecourse meets for 36 hours. Until the1998–99 academic year, Precalculus–Cal-culus I–Calculus II was a standard Fall–Winter–Spring course sequence for fresh-men, allowing those students with weakermathematical backgrounds to completeCalculus II by the end of their first yearand be on track for a mathematics or sci-ence major. While this plan seemedpromising in theory, the results were quitedisappointing.

During the 1990’s, we found that moreand more of our entering freshmen re-ported having some exposure to calculusin their high school preparation for col-lege. In spite of this fact, we continued toplace many students in the precalculuscourse. As a placement tool we use a testthat measures knowledge of algebraic,trigonometric, exponential and logarith-mic functions. Those students who hadtaken calculus in high school but did notperform well on the placement test be-lieved they had taken a step backward intheir mathematics education when theywere enrolled in the precalculus course.

Even students who had not taken calcu-lus often viewed the precalculus courseas a repeat of a high school precalculuscourse they had failed to understand or

see the point of in the first place. The ar-gument that precalculus prepares stu-dents to study calculus was not persua-sive even for those students who had beenexposed to calculus in high school. Theseprecalculus students often became verydiscouraged and most dropped out ofmathematics altogether after one or twocourses. In the fall terms from AY 92–93to AY 97–98, 308 freshmen took precal-culus, but only eight of those completedCalculus II in the spring, a retention rateof 2.6%.

Many students entering at the precalcu-lus level were extremely bright and justas capable at understanding new conceptsas those entering at the calculus level; theywere usually “just” weaker in their alge-braic and function skills. Surely we couldfind a way to shore up their weaknesses,while also furthering their forwardprogress in the mathematics curriculum.

In AY 97–98, we began making plans toeliminate the traditional precalculuscourse from the curriculum. We wantedto take advantage of two particular op-portunities: our 12–6–12 academic cal-endar and the increasing number offreshmen exposed to calculus in highschool. We read about curricular experi-ments mixing precalculus and calculus inthe same course and the use of “just intime algebra”. Instead of scheduling thetraditional Precalculus in the fall followedby a compressed Calculus I in the winter,we realized that we could extend the Cal-culus I topics over a two-term fall/wintersequence and use the extra class meetingsto reinforce precalculus concepts. Stu-dents following this track would be readyto enroll in Calculus II in the Spring Term.

As we began developing the syllabus fora new precalculus-calculus sequence, wedecided we should do more than justteach the usual calculus course with someextra days of precalculus review. We de-vised a calculus course with a new anddifferent perspective. The focus of our

traditional Calculus I had been to definethe three calculus tools (limit, derivative,and integral) and apply these tools to“any” known function. This approachworked well for those students who hada good understanding of the many dif-ferent types of functions. Unfortunately,this traditional approach seemed to over-whelm students who did not have a work-ing knowledge of the different types offunctions to which we were applying thecalculus tools.

We decided to organize the new precal-culus-calculus sequence around threeclasses of functions – algebraic, exponen-tial, and trigonometric. The primary ob-jects of study are the functions, not thecalculus tools used to investigate the func-tions; indeed, the heart of this revision isthis paradigm shift from a calculus coursedesigned around calculus tools to a se-quence of calculus courses designedaround functions. We named this two-course sequence Math Functions I and II,and Table 1 provides a general courseoutline.

About two-thirds of the first course (inthe Fall Term) studies differential calcu-lus for algebraic functions. We begin witha brief precalculus review with an empha-sis on improving algebra skills. After that,a first exposure to limits, the definitionof the derivative, rules of differentiation,and applications of the derivative all oc-cur in the context of only algebraic func-tions. Students learn and understand howthese tools of the calculus provide infor-mation about the behavior of the “safeand easy” algebraic functions. We spendthe last weeks of the first course studyingexponential and logarithmic functionsand both limits and derivatives for thesenew functions. Thus, students learn aboutexponential and logarithmic functions inthe context of an extended review of thedifferential calculus.

The second course (in the Winter Term)begins with the Mean Value Theorem and

An Experiment that Worked: Revising the Calculus Curriculum

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some graph sketching for algebraic, ex-ponential, and logarithmic functions. Wethen take up the study of trigonometricfunctions. We start with the unit circledefinitions and some basic trigonomet-ric identities and then continue to inves-tigate trigonometric functions using thetools of the differential calculus. The lastthird of the course takes up the study ofintegral calculus, particularly the Funda-mental Theorem, for all of these differ-ent types of functions.

We have been very pleasantly surprisedand encouraged by the initial success ofthis curriculum revision as measured byboth the enrollment figures for ourcourses and the numbers of majors. Thegains in retention from course to courseare quite impressive.

• Winter Term calculus enrollment in-creased from fewer than 10 freshmento more than 40;

• During the first three years of thisnew curriculum, an average of thirteenfreshmen per year completed the in-troductory sequence MathematicalFunctions I and II and Calculus II,compared to an average of one fresh-man per year completing the Precal-culus–Calculus I–Calculus II duringthe previous six years;

• During the first two years of this newcurriculum, six students who matricu-

lated at the precalculus level later de-clared a mathematics major, com-pared to a total of only two studentsin the previous six years.

Anecdotal reaction to the new sequenceof courses has also been quite positive.Students commented on course evalua-tions at the end of the two courses; hereare some excerpts:

• “The [new courses] gave those of uswho hadn’t had calculus a chance tostill major in math if we wanted to andthose of us who already hadcalculus…a nice review.”

• “Combining [Math Functions I] with[Math Functions II] is an excellent wayto introduce calculus to students withlittle or no experience. Trust me onthis.”

• “Together, these two classes appro-priately teach students the fundamen-tals of calculus.”

• “I really liked the mixture of calculusand precalculus. It gave a great reviewand extra clarification of the subject.”

• “I like [the way students taking ei-ther [Math Functions I] or…CalculusI…can still be at the same level in thespring.”

General Syllabus for Math Functions II – Winter 2001 (36 class hours)

Days 1 – 4 Applications of the Derivative (Graph Sketching, Differentials)Days 5 – 8 TrigonometryDays 9 – 13 Differential Calculus of Trigonometric FunctionsDays 14 – 21 Riemann Sums and the Fundamental Theorem of CalculusDays 22 – 24 Applications of the Integral

Table 1:

General Syllabus for Math Functions I – Fall 2000 (36 class hours)

Days 1 – 6 Algebraic FunctionsDays 7 – 13 Limits of Algebraic FunctionsDays 14 – 21 Derivatives of Algebraic FunctionsDays 22 – 29 Applications of Derivatives (Related Rates, Optimization)Days 30 – 36 Exponential and Logarithmic Functions and the Differential Calculus

At a time when we sense that a higherpercentage of students may be enteringcollege with an inadequate understand-ing of functions, mathematics programsacross the country might do well to re-visit course offerings with an eye onboosting enrollment in courses at the cal-culus level and beyond. There are manyrich opportunities that come with the“problem” of students with weaker math-ematical backgrounds, and we can enablethese bright young minds to both under-stand and appreciate mathematics in allits beauty and power.

Any questions or comments about thiscurriculum revision and its outcomes canbe addressed to William W. Johnston(johnston@centre.edu), Alex M.McAllister (alexmcal@centre.edu), andJohn H. Wilson (wilson@centre.edu).

Bill, Alex, and John teach mathematics atCentre College. Somehow, in the midst ofrevising their calculus curriculum, theyfind time to train for their annual croquetmatch with Centre’s chemists.

10

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By Jeremy Case and Matthew DeLong

Considerable effort has been made inrecent years by the mathematics commu-nity to improve the teaching and learn-ing of mathematics. Similar efforts havebeen made in some of the sciences. Yet,we often are not aware of what the “otherside” is doing.

For the past three years we have helpedorganize an Educational Issues Seminarfor the faculty in the Science Division atTaylor University, a Christian liberalarts college with about thirty facultymembers in the division. Such a semi-nar is one way to encourage these im-provement efforts at a local level, andto pool resources from across the dis-ciplines.

The seminar at Taylor is held monthly.The format of the seminar is for theattendees to read a short article or twoon a teaching topic prior to the meet-ing, and then to use the reading as aninstigator for a one-hour informal dis-cussion on the topic. A lunch is pro-vided for the participants to encour-age attendance.

The purposes of the seminar are to fosterconversations about teaching among thedivision members, to encourage the ex-change of ideas on teaching philosophyand practices among the division mem-bers, to introduce faculty to books andarticles on teaching issues, to connect newfaculty with established faculty, and toconnect faculty from different depart-ments within the division. Ultimately, thegoal is to make teaching a communityproduct rather than an individual enter-prise.

In this article we describe the history ofthe seminar, provide more details con-cerning its implementation, share someoutcomes and disappointments. We alsoprovide suggestions for establishing simi-lar seminars.

History

As a graduate student at the Universityof Michigan, Matt was on the initial or-ganizing committee for an EducationalIssues Seminar for the graduate studentsand faculty of the Mathematics Depart-ment. That seminar was similar in for-mat and purposes to the one describedabove.

Based on that experience, Matt organizeda planning committee, which includedJeremy, for a similar seminar at Taylor. Be-

cause the Mathematics Department atTaylor is not large enough to sustain arobust seminar such as the one at Michi-gan, we decided to expand the scope toinclude the entire Science Division, con-sisting of the Biology, Chemistry, Com-puter Science, Environmental Science,Mathematics, and Physics Departments.

During the Spring semester of 1999, apilot version of the seminar was held withan average attendance of six. Feelingslightly disappointed at the turnout buthoping that we had a solution, we askedour dean to pay for lunch for the attend-ees the next year. During the 1999-2000academic year, the seminar had an aver-age attendance of nearly thirteen. Appar-ently, free food is a larger motivator to acollege professor than an opportunity todiscuss teaching! During 2000-2001, the

seminar had an average attendance ofnearly fifteen, which is 50% of the divi-sion faculty.

Implementation

To implement the seminar, the planningcommittee meets before the beginning ofthe school year to identify the seminartopics. Then the committee divides up theresponsibilities for each seminar. Occa-sionally a guest speaker is invited, al-though the primary format of the semi-nar is for active discussion among the at-tendees.

Each month during the week precedingthe seminar, the committee sends outemail invitations to the entire division ex-

plaining the seminar topic and askingfor an RSVP. The committee identifiesthe readings for the seminar, and thensends copies two days in advance of theseminar to those planning to attend.The committee orders and picks up thefood. During the seminar, one of thecommittee members leads the discus-sion, although this is done rather in-formally. Finally, one of the committeemembers takes notes of the discussion,which are then emailed to the entiredivision.

The seminar topics have included stu-dent learning goals, assessment, moti-vating students, using questions to im-prove understanding, using course

portfolios to document teaching, gettingstudents to read the textbook, under-graduate research, cooperative learning,advising students, constructivism, teach-ing with technology, encouraging womenand minorities in mathematics and thesciences, service learning, students’ per-ceptions of what makes a good professor,faith and learning integration, and assign-ing grades.

Because the planning committee ismostly mathematics and physics profes-sors, the suggested readings have beenprimarily drawn from literature in thosedisciplines. For example, many of the ar-ticles have been taken from the MAANotes series and the journals PRIMUSand Physics Today. However, we make aneffort to include readings of interest tothe entire division.

An Interdisciplinary Educational Issues Seminar

Emanating from the Mathematics Department

Professors from different disciplines get togetherat Taylor University for the Educational IssuesSeminar.

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November 2001

Outcomes

When asked for feedback on the seminar,attendees said that the greatest benefit hasbeen the building of friendships withinthe division. Even at a small school, timeconstraints can make it difficult to buildrelationships across departments or agegroups. The seminar has been a fairly easyway to encourage such professional ca-maraderie. For example, our divisiontended to “bond” during the discussionof grades, when it was generally agreedthat math and science students workedharder yet faced tougher grading stan-dards than students in some of the otherdisciplines.

Some particular benefits to the Math-ematics Department have come from theinteraction among disciplines. For ex-ample, during a recent review of Taylor’score curriculum, the Mathematics De-partment discovered that faculty mem-bers across campus had stereotypical anduninformed views about mathematics.The seminar discussions showed thateven some science faculty were unawareof the calculus reform movement or thecurrent practices of mathematics teach-ing at Taylor. The seminar enabled the de-partment to tell its story to its most vitalclient disciplines.

At the same time, the seminar has giventhe Mathematics Department a better un-derstanding of the objectives of client dis-ciplines so as to serve them better. It hasalso been illuminating for us to see howthey try to meet these objectives in dif-ferent settings.

Finally, we note that while several mem-bers have done some incremental experi-mentation due to the seminar, partici-pants have not made wholesale changesin their teaching philosophies or prac-tices. Rather, the seminar has encouragedawareness, personal examination, reflec-tion, and dialogue on issues surroundingteaching.

Disappointments

Informal discussions can easily get “onthe wrong track.” As an example, we sharean anecdote that occurred in a seminaron student expectations. This story also

gives evidence to our previous observa-tion that change is slow.

Instead of an introductory reading for theseminar on student expectations, we hada video for the attendees to watch. Jer-emy took a video camera to the galleriaof Taylor’s library, which is a commonstudent study hangout, and asked severalstudents “What makes a good professor?”Contrary to our cynical preconceptions,most of the student responses dovetailedwith many of the previous seminar issuesthat we had discussed—motivation, ac-tive learning, technology, etc.

We were excited by the fact that the stu-dents’ ideas of good teaching practicecoincided with many of the practices thatwe were exploring in our seminar. Un-fortunately the discussion that followeddidn’t really recognize that fact. Insteadthe discussion focused on the fact that fewof the students interviewed were sciencemajors, and hence somehow “soft,” andthat this generation of students in gen-eral has a need to be entertained. Al-though those statements may in fact betrue, we were disappointed that the largerpoint was missed.

Suggestions

Based on our experiences, we suggest thefollowing advice for mathematics facultywho may be interested in starting a simi-lar seminar at their home institutions.

First, organize a planning committee. Al-though one person could easily organizethis kind of seminar, the multiple ideasand the extra hands of a planning com-mittee will make the seminar more viable,more valuable, and more interesting to avariety of people. A planning committeewill also give multiple individuals own-ership in the seminar, guaranteeing a coregroup of supporters.

Second, invite different “types” of peopleto the seminar. In other words, if theseminar is organized in a large mathemat-ics department, include everyone frombeginning graduate students to full pro-fessors. If the seminar is organized in asmaller college, include faculty from sev-eral different disciplines. We have foundthat having a wide variety of viewpoints

makes the seminar discussions more ro-bust and enlightening. It is also impor-tant that the make-up of the planningcommittee be similarly diverse.

Finally, get funds for free food! As ourattendance figures show, this is vital. Aprofessor’s life is hectic, and it can be dif-ficult to give up one hour even for some-thing as worthwhile as a collegial discus-sion. However, most professors also lovea free lunch, and this may easily offset theextra hour.

Conclusion

We have found that the seminar has beena good public relations move for theMathematics Department. Our adminis-tration has given the seminar very posi-tive feedback, commending our activitiesto the Board of Trustees and asking us topropagate the seminar to other disci-plines. We have a better understanding ofour client disciplines and their needswhile having an avenue to dispel stereo-types of an uninvolved, irrelevant, andaloof discipline. Rather, as the motivat-ing force behind the seminar, the depart-ment can project itself as a leader in stu-dent-centered instruction that takesteaching and collaboration seriously.

Jeremy Case (jrcase@tayloru.edu) is Asso-ciate Professor and Chair of the Mathemat-ics Department at Taylor University. MattDeLong (mtdelong@tayloru.edu) is Assis-tant Professor of Mathematics at Taylor.They are both Project NExT graduates. Inaddition to common interests in math-ematics and teaching, Jeremy and Matt areboth passionate about music, theater, andsports.

The MAA is seeking Visiting Math-ematicians for one or both terms ofacademic year 2002-03. Areas of ex-pertise that are of particular interestto the MAA are teacher preparation,public policy, public awareness, andon-line publishing. Contact ExecutiveDirector, Tina H. Straley, for more in-formation at tstraley@maa.org.

Visiting Mathematicians

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FOCUS November 2001

The Math Forum, one of the pioneermathematics websites, began life as theGeometry Forum through an NSF granton the Swarthmore College campus. Un-der founding director Gene Klotz, its fo-cus in its early days was on K-12 math-ematics, and it soon enjoyed a large fol-lowing. Since then it has grown into oneof the largest mathematics sites on theweb. The Math Forum has now been ac-quired by Drexel University and is nowknown as the Math Forum @ Drexel. Thenew address of the site is http://mathforum.org/. Pointers from the previ-ous addresses will seamlessly redirect youto the new location.

Spearheaded by President ConstantinePapadakis and under the direction of Dr.Harvill Eaton, Drexel University’s seniorvice president for research and graduatestudies, the University will maintain theMath Forum @ Drexel site as a leadingcenter for mathematics and mathemat-ics education on the Internet. The site willcontinue to offer previous Math Forumservices, and will introduce new featuresand communities to ensure the site re-mains on the cutting edge.

The Math Forum has a New Home

Drexel brings many strengths to the MathForum, including expertise in mathemat-ics, information science and technology,and education. See http://www.drexel.edu/univrel/aboutdu/ for more about the uni-versity. Drexel is Philadelphia’s techno-logical university. In 1983, it was the firstuniversity in the nation to require all stu-dents to have a personal computer. In2000, Drexel became the first major uni-versity to operate a fully wirelessCyberCampus.

The Math Forum staff believe that this istheir right home for many reasons. Drexelis interested in online learning, technol-ogy in teaching, digital libraries, math-ematics education, and similar issues.Moreover, Drexel brings an exciting com-bination of the practical and the theoreti-cal into education. The Math Forum isvery comfortable returning to the aca-demic fold, certainly wiser for its dot.comexperience. The staff is grateful to WebCTfor its generous support and for the con-cern they displayed for the Math ForumWeb site and their continuing researchinitiatives.

Although all of the Math Forum serviceswill go forward, the Problems of the Week(PoWs) will initially provide services at amuch lower level since they are so labor-intensive. The Math Forum will also goforward in new directions, and, for ex-ample, hopes to develop tools and tech-niques to train mentors for the PoWs sothat they can be built up in a cost-effec-tive way. Another avenue they hope toexplore is using the Math Forum andother digital libraries to study studentlearning. Now that their energies are nolonger focused on reorganization, theMath Forum staff is ready to pursue manyother collaborations as well.

The Math Forum @ Drexel will be underthe direction of Harvill Eaton. KristinaLasher is the Forum director, and GeneKlotz (klotz@mathforum.org) will con-tinue as the Math Forum’s senior advi-sor. Bookmarks and links to the MathForum should be changed to point tohttp://mathforum.org/.

Daniel M Lewin, co-founder and ChiefTechnology Officer of Akamai Technolo-gies, was on board one of the planes thatwere hijacked and crashed on September11, 2001. Lewin was a brilliant computerscientist and a good friend of the math-ematics community and in particular ofthe MAA. Akamai Technologies has beena valuable supporter of the AmericanMathematics Competitions/USA Math-ematical Olympiad program of the As-sociation. The MAA was deeply saddenedby the news.

Lewin founded Akamai Technologies inSeptember 1998, together with TomLeighton and a group of computer sci-

In Memoriam

Daniel M. Lewin

entists from MIT. As Chief TechnologyOfficer, he was responsible for Akamai’sresearch and development strategy andfor many of the innovative ideas that ledto Akamai’s success. Lewin was 31 yearsold and is survived by his wife and twosons. The family has asked that contri-butions in Danny’s memory be made tothe Daniel Lewin Science ScholarshipFund, dedicated to providing scholar-ships to students pursuing careers in sci-ence.

More information on Lewin and on theLewin Science Scholarship Fund can befound on Akamai’s web site at http://www.akamai.com.

MATHEMATICAL SOFTWEAR FORYOUR HEAD

KLEIN BOTTLE CAPS ANDMOEBIUS EARBANDS

“ONE SIDE FITS ALL”

www.mathhatter.com

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November 2001

By John Fraleigh

Every teacher of abstract algebra fromwhom I have heard says that undergradu-ates find abstract algebra to be a difficultsubject. At the University of Rhode Island,we have only a one-semester course thatis not required for all math majors, prob-ably because the majors found it difficult.When I started teaching it, Ihad the usual experience: theA and B students of calculuswere dismayed with theirfirst test grades in the 60’s or70’s, and the C studentsdidn’t like their grades in the40’s. There were dropoutsbefore the first test. Somestudents appeared sullen.One student accused me ofdoing nonstandard math-ematics.

I am ashamed that it was notuntil about the last ten yearsof my teaching career that Isolved this problem, hadhappy abstract algebraclasses, and was a happy andrelaxed instructor. My solu-tion may horrify some, andthere might be grumblings of relaxedstandards and watered down courses, butthey really weren’t and it works. I am con-vinced that the happy students learnedmore than others did before and had agreater appreciation of the beauty of thesubject, precisely because they were re-laxed.

My course grades were based on dailyhomework (2/7), three hour-long tests (3/7) and a final exam (2/7). The daily home-work handed in consisted of about threeof the problems suggested for that lesson,marked with an asterisk on the syllabus.I was often asked about them in class be-fore they were passed in, and I obliged,trying of course to get students in the classto give solutions. Furthermore, I told thestudents that this homework was part oftheir notes and that I had no objection totheir correcting it in class before theypassed it in. Consequently, everyone hada pretty good homework average.

A week or more before each test and thefinal exam, I handed out a preparation,The students were told that the defini-tions and proofs requested on this prepa-ration would be exactly the same on theactual test, but that data and structureswould be altered in other questions. Thusif a preparation question asked if twogroups were isomorphic, and why, thestudent could expect different groups on

the test. If the preparation asked for theirreducible polynomial over the rationalfield for some algebraic number, thenumber would be different on the test.Finally and most important, each prepa-ration stated clearly that students couldwork together and get help from anyone(student, graduate student, faculty) otherthan their instructor in working thepreparation. I also told them in class thatthey should write it all out once with nonotes, timing themselves so they could besure that they could do it with no notesin the available 50 minutes (3 hours forthe final).

Here is what happened. I was able to givetests that would have been unreasonableto expect students to do without thepreparation and in the time allowed. Evenso, most students finished the 50-minutetests in 40 minutes or less, and the finalexam in less than 2 hours. The A studentsof calculus had semester grades in the

90’s, the B’s in the 80’s, and so on. Thenumber of students electing the classmore than doubled. There were hardlyany drops before Test 1. As I indicatedabove, the class was happy and relaxed,and so was I. In addition to a betterknowledge and appreciation of the sub-ject, some students also learned a verypractical lesson. If you need help withsome problem, be sure you ask a compe-

tent person, and even then,do your best to check thatthe information is valid.

This method might not besuitable for honor studentsat schools like Harvard, al-though the occasional stu-dent of that calibre that Ihad never seemed to feel in-sulted.

Anyone who wishes to seemy syllabus, the prepara-tion final exam and the ac-tual final exam for Spring2000 can find them at http:// w w w. m a t h . u r i . e d u /~fraleigh.

John Fraleigh received hisMA in mathematics from

Princeton University in 1956. He taught atDartmouth College until 1962 when hejoined the mathematics department at theUniversity of Rhode Island. He was hap-pily enjoying his December, 2000 retire-ment until Addison-Wesley pushed for a7th edition of his text, A First Course inAbstract Algebra, first published in 1967.

How do you teach Abstract Algebra?

FOCUS would be interested in hearingcomments on this article and other in-novative teaching ideas and stories aboutthe teaching of Abstract Algebra at theundergraduate level. We’ll run a reporton what our readers have come up within a couple of months. Send your com-ments to fqgouvea@colby.edu.

Happy Abstract Algebra Classes

John Fraleigh

14

FOCUS November 2001

Sectional and National Awards for Distinguished Teaching

Twenty-four mathematicians receivedthis year’s Section Distinguished Teach-ing Awards, conferred at the Spring meet-ings of each of the Sections of the MAA.The list of all the winners appears on page15. The twenty-four section award win-ners represent the very highest level ofmathematics teaching. Dedicated, caring,inspiring, and innovative, they all richlydeserve the honors accorded them bytheir sections. They are all winners

The three winners of the nationalDeborah and Franklin Tepper HaimoAward for Distinguished College or Uni-versity Teaching of Mathematics are cho-sen from among the last two years’ sec-tional winners. This year’s winners areDennis DeTurck of the University ofPennsylvania, Paul J. Sally, Jr. of the Uni-versity of Chicago, and Edward L.Spitznagel, Jr. of Washington Universityin St. Louis. The three recipients of theHaimo Award will each give a talk at theJanuary Joint Mathematics Meetings inSan Diego. The talks will be on Tuesday,January 8, from3:30 to 5:00 pm.

People reach forwords like “cre-ative” and “charis-matic” when theywant to describeDennis DeTurck.In addition to be-ing a brilliantclassroom teacherwho inspires students at all levels to learnand to love the subject, DeTurck has cre-ated a variety of programs to enhance theteaching and learning of mathematicsand science. He founded the Middle At-lantic Consortium for Mathematics andits Applications Across the Curriculum,which creates and disseminates interdis-ciplinary course materials. He serves asthe faculty coordinator for AmericaCounts at the University of Pennsylvania,which sends some 40 undergraduates tospend 8 to 10 hours a week as tutors inWest Philadelphia schools. He also playsan active role in the Penn Summer Sci-ence Academy, a pre-freshman program

emphasizing Mathematics, Physics, andChemistry. DeTurck is a creative user oftechnology. He teaches a web-basedcourse on Ideas in Mathematics and wasinstrumental in introducing substantialuse of computers in the calculus classesat Penn. His many-faceted work adds upto a distinguished career dedicated to theteaching of mathematics and science.

Paul Sally is a respected research math-ematician working in representationtheory, but he is also committed to edu-cational excellence at all levels. His superbclassroom teaching and his long-rangeeducational programs have impactedthousands of students and teachers. Sallyis Director of Undergraduate Studies inthe Mathematics Department of the Uni-versity of Chicago. In addition to his ex-cellent classroom teaching, he hascoached the Putnam team, helped to es-tablish a math club, and has overseen cal-culus courses at all levels. He has hadmany Ph.D. students who have gone onto distinguished careers of their own.

But Sally also has had an impact on schoolmathematics. In 1983, he became direc-tor of the University of Chicago Math-ematics Project, a leading education re-form effort. Within this project he workedwith teachers from school districts allover the country.

In 1992, Sally founded SESAME (Semi-nars for Elementary Specialists and Math-ematics Educators), a professional devel-opment program for elementary schoolteachers in the Chicago public schools. Heco-founded a Young Scholars Programfor mathematically talented students,providing personal mentoring and en-couragement. The broad impact of hiswork marks him as an exceptional edu-cator.

Edward Spitznagel is described by mem-bers of the Mathematics Department atWashington University in St. Louis astheir “preeminent teacher, preeminentstatistics guru, and preeminent computerjock.” His lectures on statistics are packedwith real-world applications, often drawn

from his own cur-rent research work.His courses makeeffective use ofcomputers, withstudents typicallyusing a computeron the second dayof class. Hisbreadth of scholar-ship and his col-laborations with investigators in medi-cine, pharmacology, marketing, engineer-ing, and psychology allow him to fill hisexamples from the real world. It is nowonder that students regularly oversub-scribe his courses.

Spitznagel’s creativity is not limited to sta-tistics courses. When his department de-cided to create a calculus sequence forpre-med students, Spitznagel devised acourse based on research in pharmaco-kinetics that introduces students to bothstatistical and calculus techniques inmedicine. It has been enthusiastically re-ceived both by the students and by theirpre-med advisors. Such creativity anddedication are typical of Spitznagel. Heis a truly great teacher who has had ex-traordinary success in applying his vastpractical experience and his great enthu-siasm to the classroom.

Since teaching is one of the main con-cerns of the MAA, it is fitting that thesesectional and national awards serve toidentify, honor, and reward exceptionalcollege and university teaching. J. J. Price,chair of the Committee on the HaimoAwards, noted that not all sections chosewinners. “The national committee urgesall sections to continue to nominate andrecognize their outstanding teachers andencourages all members of the Associa-tion to nominate worthy candidates,” hesaid. The participation of more MAAmembers in this process can only makethe awards even more valuable as a trib-ute to talented teachers who have a deepimpact on many student lives.

Edward Spitznagel

Dennis DeTurck

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November 2001

2001 Distinguished Teaching Award Winners

EPADEL SectionDennis DeTurckUniversity of Pennsylvania

Florida SectionScott HochwaldUniversity of North Florida

Illinois SectionPaul J. SallyThe University of Chicago

Indiana SectionFrancis L. JonesHuntington College

Iowa SectionCathy GoriniMaharishi University of Management

Kansas SectionCharles J. HimmelbergUniversity of Kansas

Kentucky SectionJ. Brauch FugateUniversity of Kentucky

LA-MS SectionJeffrey L. StuartUniversity of Southern Mississippi

MD-DC-VA SectionElizabeth MayfieldHood College

Metro NY SectionSheldon GordonSUNY at Farmingdale

Missouri SectionLouis J. GrimmUniversity of Missouri, Rolla

NE-SE South Dakota SectionAlan PetersonUniversity of Nebraska, Lincoln

North Central SectionTed VesseySt. Olaf College

Northeastern SectionPaul BlanchardBoston University

Northern California SectionWade EllisWest Valley College

Ohio SectionAlan StickneyWittenberg University

Oklahoma-Arkansas SectionStanley EliasonUniversity of Oklahoma

Mathematical Science Reports Suggest Improved Job Market

Pacific Northwest SectionBruce Lind & Ron VanEnkevortUniversity of Puget Sound

Rocky Mountain SectionJim LoatsMetropolitan State College of Denver

Seaway SectionDavid E. ManeSUNY College at Oneonta

Southeastern SectionJohnny HendersonAuburn University

Southwestern SectionWilliam D. StoneNew Mexico Tech

Southern CA SectionJennifer QuinnOccidental College

Texas SectionRobert NorthcuttSouthwest Texas State University

Wisconsin SectionRanjan RoyBeloit College

The Second and Third Reports from theAnnual Survey of the Mathematical Sci-ences were released in July and Septem-ber. Together, they show that job open-ings for mathematics PhDs increased by22% in 1999-2000. The growth seems tobe caused primarily by faculty retire-ments. After many years of difficult jobmarkets, the report suggests that thingsmay finally be improving. The AnnualSurvey is produced jointly by the Ameri-can Mathematical Society, the American

Statistical Association, the Institute ofMathematical Statistics, and the MAA.The reports were published in the Noticesof the American Mathematical Societyand can also be found online at http://www.ams.org/notices/.

The Chronicle of Higher Education pub-lished a story in its September 6 issuedescribing the results of the survey andincluding anecdotal evidence that themarket is indeed heating up, with several

departments reporting that they are hir-ing more people than they have in a longtime. The article also notes that manymathematicians and statisticians are be-ing hired outside academia, with the mar-ket for statisticians being particularly hotat the moment.

16

FOCUS November 2001

It was very hot in Madison this August.That much was clear from the moment Igot off the plane. But the hotel was nicelysituated, close to the State Capitol build-ing and to a pleasant street with shops andeating places. I found an espresso shopand a bookstore specializing in sciencefiction and mysteries, so life was good.Plus, there was a shuttle to/from the placewhere most of the meeting would actu-ally happen, so I wouldn’t have to walkin the sun.

The meeting happened at the MononaTerrace Community and ConventionCenter, a beautiful building overlook-ing a beautiful lake. It wasn’t too hardto get around, and it was pleasant to gointo the book exhibit area and be ableto look out over the lake.

The high points of this meeting were theplenary talks. Ingrid Daubechies gave agreat set of Hedrick Lectures entitledWavelets in Action. She managed to hitjust the right level of technicality to keepthe talks serious and interesting at thesame time. The result was a model ofwhat expository mathematical talks canbe like.

There were lots of other good talks. I en-joyed Judy Grabiner’s portrayal ofNewtonianism in Action in her lecture onColin Maclaurin, Robert Witte’s explana-

tion of What I Have Learned From theMathematics Community, and MichaelStarbird’s discussion of The Other Lessons:What Students Keep for Life. RhondaHatcher’s student lecture on methods forranking football teams was interestingeven for someone who cares not a whitfor football (i.e., someone like me). Thosewere, of course, only some of the manytalks. There was no way I could go to themall, so I’m sure I’m leaving out somethingreally interesting that I just happened tomiss.

In addition to the plenary talks, therewere special sessions and mini-coursesabout all sorts of things, plus the usualbook exhibit. I attended only a few of thetalks given in special sessions (mostly theones about The Use of History in theTeaching of Mathematics), but the ones Idid go to were interesting and useful. And,of course, I spent time at the book ex-hibit, paging through new books, askingpublishers to send review copies to MAAOnline, and chatting with people.

Since I am editor of FOCUS and of MAAOnline, I have to attend committee meet-ings whenever I go to national meetingsof the MAA. This time I was at the Boardof Governors meeting (see the report be-low, written by the student visitors to thatmeeting), the Committee on Publica-tions, and the Committee on ElectronicServices. Committee meetings are neverreally fun, but these were at least produc-tive.

My MathFest

By Fernando Gouvêa

There were also several social events. TheMC for the opening banquet was EdwardBurger of Williams College, who had funshowing us several examples of “math inthe news,” ranging from the discussion of“fuzzy math” during the presidentialcampaign to the notorious “I’d rather goto math camp” ad. He then introducedEzra Brown of Virginia Polytechnic whogave a light-hearted but serious talk onnumber theory and cryptography.

At the other end of the meeting, the ban-quet honoring long-time members ofthe MAA was also a remarkable event.Many people who have been membersof the Association for 25 years or morewere present, and all were recognized byname. A special tribute was paid toWalter and Mary Ellen Rudin of the Uni-versity of Wisconsin at Madison. Friendsand students told stories about theRudins, but they were all upstaged byMary Ellen herself, who had (of course)the best stories to tell. The banquet con-cluded with a talk by Sylvia Wiegand

about the mathematicians in her family,from William Young and Grace ChisolmYoung to the present day.

In all, MathFest was a pleasant and pro-ductive meeting, without the hustle andbustle of the winter meetings but with lotsof interesting mathematics, interestingpeople, and good things to do. I met newideas and interesting people, boughtsome books, and enjoyed myself. I thinkmost of those who attended felt the same.

MAA members at the 25 Year Member Banquet.

Sylvia Wiegand

Walter and Mary Ellen Rudin

Photographs courtesy of Gerald J. Porter

FOCUS

17

November 2001

By Frank Morgan, Paul Holt, Joe Corneli,Nick Leger, and Eric Schoenfeld

“Where’s the guy who’s in KNOTtheory?” asks Michael Feldman on hisMadison PRI show “Whad’Ya Know?”during the MathFest.

Colin Adams waves from the back, andFeldman turns to others in the first row:

“And you guys are mostly, what, inbubbles?”

“Bubble math, that’s right,” answersFrank Morgan.

“So that’s why you were blowing bubblesearlier.”

“It was purely professional.”

“So you make a living studying bubbles?”

“Absolutely. First we got one bubble, andthen a double bubble, and then we’ll workon the triple bubble, and… it’s a wholecareer.”

“Very exciting, very exciting for you. Butyou’re studying bubbles and you call itmathematics.”

“It’s math. Everything’s math!”

“Oh, don’t give me that. [laughter] If ev-erything was math, wouldn’t I be mak-ing more money?”

“Aren’t you?”

“Yes! Perhaps. Perhaps. We’re not goinginto that math right now.” He turns to thestudent beside Morgan. “Are you inbubbles, too?”

“Yeah, I’m working with him for the sum-mer,” responds Eric Schoenfeld.

“Is that long enough? Were you in bubblesbefore that?”

“No, I’m just an undergraduate.”

Feldman moves on to the next student.“Okay, and you’re in bubbles?”

“No, I do algebra,” answers Sonja Mapes.

“You do algebra.”

“Yeah.”

“Bubbles don’t interest you at all?”

“No, I don’t like bubbles.” [laughter]

“Don’t care for bubbles?”

“I’m not good at bubbles. Bubbles arehard.”

“I would think so.” Feldman turns to thenext student. “And then, are you inbubbles, or knots?”

“Yeah, I’m a part of the bubble group,too,” answers Nick Leger.

Feldman turns to the woman beside him.“And then are you bubbles?”

“No, I’m an analyst —but the mathemati-cal kind,” answers Janine Wittwer.

Feldman moves on to the next student.“Okay, and what are you in?”

“What can I say – it’s all about thebubbles,” answers Paul Holt.

Feldman looks at the next student.“Bubbles.”

“Ergodic theory,” responds Kate Gruher.

“What?” [laughter]

“Ergodic theory.”

“Oh, I have those theories. But what arethey?”

“We study dynamical systems – like theplanets going around the sun, as an ex-ample.”

“Why don’t you study something impor-tant, like bubbles? The planets goingaround the sun are all well and good, butit’s not bubbles.”

“We think it’s kind of important.”

Feldman turns to the last student. “Okay,and you are in…”

“More bubbles,” says Joe Corneli.

“More bubbles, okay. And what is the lureof bubbles? I mean, what can you tellfrom a bubble?”

“It’s got an interesting geometry.”

“It’s round.” [laughter]

“Yeah, but try sticking a bunch of themtogether – it gets complicated.”

“Well, everything does, that you stick to-gether, you know…but that’s true.”Feldman returns to Morgan. “And then,so this has application? Can we make abetter—let’s say a better bomb—or aweapon of destruction—out of bubbles?

“Bomb, no…well, a better economy,maybe.”

“Out of bubbles. Okay, fine, whatever,uh… [laughter] and are you paid for outof grants that come out of the taxpayers’pocket? [loud laughter] Speaking ofbubbles, speaking of living on bubbles…”

“Yes.” [laughter]

“Well, that’s good, that’s good. Is this theuniversity—where are you from?”

“Williams College.”

“Williams College, which is where?”

“Massachusetts—northwestern Mass.”

“Massachusetts, let’s hear it for WilliamsCollege, yes! Did you go there? It’s thebubble capital of America–they’re doingall the research—the important research—on the fundamental use of bubbles.”

How did mathematicians end up on“Whad’Ya Know?” Charlotte Chell ofCarthage College got some twenty ticketsmonths before the MathFest. The Williamscontingent, including several undergradu-ate research students, arrived early and oc-cupied the front row. This show, of August4, 2001, may be heard at the website http://www.notmuch.com. The mathematiciansappear at the end of the third segment.

Mathematicians on Michael Feldman’s “Whad’ya Know?”

18

FOCUS November 2001

By Joseph Corneli (New College), Paul Holt(Williams College), Nicholas Leger (UT-Austin), and Eric Schoenfeld (WilliamsCollege)

We would like to share with the FO-CUS readership our reflections on thesummer meeting of the Board of Gover-nors of the MAA, held in Madison justbefore the start of this August’s MathFest.As special guests, we had the opportunityto meet and talk to many governors andofficers. Student reporting on these meet-ings has become something of a tradition,one that we are pleased to continue here.

A good place to begin is with the obser-vation that, for such a large meeting,things were really very well run. Thespeakers dealt respectfully with one an-other, especially when it came to settlingopposing positions. On the other hand,the Board of Governors is not a body ofprofessional bureaucrats, so for all theformality of the meeting, it was clear thatat least a few people were learning someof Robert’s Rules of Order as they wentalong. President Ann Watkins was an ex-cellent facilitator.

Although the technical details evaded us,it was interesting to realize that the MAAhas finances. Happily, they appear to behandled very well, through a mixture ofboth conservative and some less conser-vative investments.

One of the most important sources ofrevenue for the MAA is their publications.At one point in the meeting, Associate Di-rector Don Albers brought in several tallstacks of books, the newest volumes. Thiswas a fairly exciting moment, and copieswere passed around for everyone to lookat. Albers said, “If you ever want to judgea book by its cover, you should take a lookat this one.” And he produced a very beau-tiful volume, Symmetry, by Hans Walser.

The most emotional part of the meetingwas the report by Titu Andreescu on theMathematical Olympiad. Andreescu elo-quently conveyed to the audience aglimpse of the feelings of excitement, lovefor mathematics, and camaraderie sharedby the Olympiad competitors.

Lunch, when it came, was welcomed en-thusiastically by everyone. The hotel staff,who were otherwise entirely punctilious,had neglected to number the tables. Twoof us ended up sitting at table number 1(after the numbering had been imposedby executive order), with President AnnWatkins, past President GeraldAlexanderson (one of the editors ofMathematical People), Second Vice Presi-dent Frank Morgan, and John Watson,the Governor of the Oklahoma-Arkan-sas section. The other two of us sat withExecutive Director Tina Straley, amongothers. The conversations we had at lunchwere most enjoyable.

Student Section Members at Board of

Governors Meeting

MAA President Ann Watkins

Jim Daniel, MAA ExecutiveCommittee Member

Barbara Osofsky MAA First Vice President

After lunch, out in the hall, we got in-volved in an interesting conversationabout careers in mathematics with Straleyand Chris Stevens, the director of ProjectNExT. Each described her own very ex-ceptional career. Chris told us aboutsomething that we never heard the likeof before, namely how as a recent gradu-ate she had put her mathematics degreeto use as an aide in the House of Repre-sentatives. We also compared notes onNational Science Foundation ResearchExperiences for Undergraduates (REUs)with Joseph Gallian from the Universityof Minnesota at Duluth.

A highlight of the afternoon session wasthe characteristically carefully delibera-tion over a new prize for young faculty. It

was necessary to decide, for example, howexactly one would qualify for the prize.

We enjoyed all of the interactions we hadwith so many distinguished mathemati-cians and supporters of the mathemati-cal community. We also had a great timeat MathFest, but as they say in the newsbusiness, that’s another story.

Photographs courtesy of Gerald J. Porter

FOCUS

19

November 2001

By Charles Dimminie

On Thursday and Friday, August 2–3,eight MAA Student Paper Sessions wereheld at the Madison MathFest. Sessions1–4 were held from 1 to 5 P.M. on Thurs-day, and Sessions 5–8 took place from 1to 5 P.M. on Friday.

The program included 46 talks involving49 students from 32 colleges and univer-sities. There were 27 student chaptermembers, of whom 16 qualified for travelgrants. Three students had also spokenat previous MathFests. There were 23 stu-dents from six REU Programs (17 stu-dents from Williams College, 2 fromRutgers University, and 1 each from In-diana University, Mount Holyoke Col-lege, Lafayette College, and the Univer-sity of Idaho).

Cash awards of $150.00 from the MAAwere presented to the following studentsfor outstanding presentations:

1. Aliyah Ali–Rutgers University (RutgersUniversity REU Program) —“Graphsand matrices.”

2. Eric Katerman–Williams College (Wil-

liams College REU Program)—“Knotcomplements: the hyperbolic alter ego ofour twisted friends.”

3. Cody Patterson–Texas A&M University—“Distinct element vectors over finitegroups.”

4. John Meth–Indiana University—“Idempotent cocycles on cyclic groups.”5. Eva Kashat and Daniela Silva (jointly)

Report on the MAA Student Paper Sessions

at the Madison MathFest

Winners of the student poster sessions with Charles Dimminie.

–Wayne State University—“Geometricapplications of a system of congruences.”

6. Ellen Panofsky–Millersville University—“Geometric analysis of distance mini-mizing paths crossing the same rim of acircular can twice.”7. John Bryk–Williams College (WilliamsCollege REU Program)—“Completionsof unique factorization domains.”

8. Jarod Alper–Brown University(Lafayette College REU Program)—“Thenumber theory of the composition alge-bra.”

9. Nicholas Leger –University of Texas atAustin (Williams College REU Program)—“Double bubbles on flat two-dimen-sional tori –Part I.”

10. D. Jacob Wildstrom –MassachusettsInstitute of Technology (University ofIdaho REU Program)— “On pairs ofmonochromatic and zero-sum hookedsets.”

Also, a special research prize of $150.00from the Council on Undergraduate Re-search was awarded to: Paul Holt – Will-iams College (Williams College REU Pro-gram)— “Double bubbles on flat two-di-mensional tori – Part II.”

Thanks are due to Katarina Briedova andJim Tattersall for their assistance through-out the process of setting up the sessionsand to Ron Barnes for chairing some ofthe sessions and evaluating the speakers.

The MAA makes it easy to changeyour address. Please inform the MAAService Center about your change ofaddress by using the electronic com-bined membership list at MAAOnline (www.maa.org) or call (800)331-1622, fax (301) 206-9789, email:maaservice@maa.org, or mail toMAA, PO Box 90973, Washington,DC 20090.

Have You Moved?

20

FOCUS November 2001

By Martha Siegel, Secretary of the MAA

First, let me remind you of the resultsof the national elections. Dr. Ronald L.Graham was elected President-Elect. Hewill be President-Elect in 2002 and be-come President for a two-year term inJanuary 2003. Newly elected 1st and 2ndVice Presidents are Carl C. Cowen and Jo-seph A. Gallian, respectively.

In January, the Board elected Daniel P.Maki to the Budget and Audit Commit-tees and the Executive Committee. He fillsthe unexpired term of Barbara T. Faires,who asked to be replaced.

At the Board meeting in Madison, Trea-surer, Gerald J. Porter, announced his in-tention to retire from that position. Jerryhas served the MAA in so many ways andover so many years that it is difficult toimagine his absence from ExecutiveCommittee and Board of Governorsmeetings. He has served as Governor ofthe EPADEL Section, member of the Au-dit and Budget Committees, member andchair of the Investment Committee, andas chair of the Committee on ProfessionalDevelopment. He was the PI on severalsuccessful and innovative MAA projects,including the Interactive MathematicsText Project, and he has helped to estab-lish the Journal of Online Mathematicsand Its Applications and the Digital Li-brary. His commitment to the MAA in-volves both financial support and anenormous dedication of his time. Asearch is being conducted for a new Trea-surer; the Search Committee, appointed

by President Ann Watkins is chaired byBarbara Faires. Other members areRonald L. Graham and Barbara L.Osofsky.

Also at the Board meeting, UnderwoodDudley was elected as the editor to serveon the Executive Committee after De-cember 31, 2001. Roger Horn will stepdown from the Executive Committee atthat point, since he will be leaving hisposition as Editor of the Monthly. BrucePalka has already been working as Edi-tor-Elect. I want to thank Roger for allhe has contributed to the MAA, for hisincredible dedication to the editorial po-sition, especially during his long recov-

ery after a bad skiing accident. TheBoard also approved the recommenda-tions of the Haimo Awards Committeeand we are pleased to announce that theawards for Distinguished College orUniversity Teaching of Mathematics willbe presented to Dennis DeTurck, Paul J.Sally, and Edward Spitznagel at the Janu-ary Prize Session in San Diego. Mem-bers should be sure to attend their talksat the San Diego meetings.

The Board also elected Jim Lewis of theUniversity of Nebraska as Leitzel Lec-turer for MathFest 2002 and established

a standing Committee on Graduate Stu-dents. The Florida, Kansas, Michigan,Northeastern, Rocky Mountain, andTexas Sections presented their nomineesfor Certificates of Meritorious Service.The Board enthusiastically approved.Certificates will be presented in San Di-ego.

We now have three SIGMAAs: theSIGMAA for Research in UndergraduateMathematics Education, the SIGMAA forMathematicians in Business, Industry,and Government, and the SIGMAA forStatistics Education. We are thrilled withthe progress of the SIGMAA programand we now have a standing Committeeon SIGMAAs. I want to publicly acknowl-edge the hard work of its first chair, EdDubinsky. Stephan Carlson follows Ed aschair of the committee. He and his com-mittee are reviewing applications of threemore groups seeking to becomeSIGMAAs.

Our JPBM activity has been limited thisyear by the new structure of the group.The MAA Science Policy Committee,chaired by Lida Barrett, has been work-

ing diligently to keep the MAA informedand involved in national policy. AlBuccino has been the man behind thescenes in creating a terrific web page dedi-cated to science policy issues. I urge youto get into MAA Online and click on thescience policy page on a regular basis.

The MathFest in Madison was extremelysuccessful. Attendance was among thebest we have ever had at a MathFest, thesite was beautiful, and the program wassuperb. Hedrick Lecturer, IngridDaubechies, and Leitzel Lecturer, BobWitte, as well as all those delivering In-vited Addresses afforded all who attendedan opportunity to enjoy mathematics ina relaxed and hospitable place. Thanks toJim Tattersall, Associate Secretary, and allthose involved in arranging the scientificand social programs.

Once again, I am struck by the caring anddedication of our members to the Asso-ciation and to each other. I left Madisonearly because of a family emergency.Many members checked to be sure I wasokay. To our friends in the Metro NewYork and the MD-DC-VA Sections andthose in neighboring sections who mayhave had family or friends in the disasterarea on September 11, I send my bestwishes and my hopes that they will knowpeace in the days ahead. May we alwaysrecognize the importance of every humanlife.

Secretary’s Report

Jim Tattersall, Associate Secretary

Lida Barrett

Photographs courtesy of Gerald J. Porter

FOCUS

21

November 2001

Online Tutorials and Mathematics Students

Mathematics students increasinglyhave access to online sources of tutorialassistance, whether from textbook pub-lishers, educational institutions, or inde-pendent companies. Having recently beeninvolved in a beta test of a new online tu-torial site for calculus and precalculus(http://www.hotmath.com), I am wonder-ing about the impact of these resourceson student learning. Hotmath suggests onits site that the benefits of their tutorialhomework assistance outweigh the risksof abuse by students. I am very curiousto know if this is true in practice.

Letter to the Editor

Call for Organizers:

MAA Contributed

Paper Sessions,

Baltimore 2003

The MAA Committee on Minicoursesis soliciting proposals for minicourses tobe given at the Joint Mathematics Meet-ing in Baltimore, Maryland, January 15–18, 2003. Most minicourses are related toundergraduate curriculum, although anytopic of interest to the MAA membershipwill be considered.

To find more information on how to sub-mit a proposal, see http://www.maa.org/meetings/miniguide.htm. The deadline forsubmissions the Baltimore Joint Math-ematics Meeting is December 1, 2001.

Call for Organizers:

MAA Minicourses,

Baltimore 2003

As a full-time faculty member at Ameri-can River College (Sacramento, CA) anda math education graduate student at theUniversity of California at Davis, I wouldwelcome information from my teachingcolleagues on any observations theymight have made concerning how stu-dents use online tutorial sites, which onesthey use, and what effect, for good or ill,the sites appear to have. Surely this is anarea meriting further investigation.

I can be contacted via email atabarcellos@ucdavis.edu.

Anthony BarcellosDepartment of MathematicsAmerican River College& Graduate Group in EducationUniversity of California, Davis

The MAA Committee on Sessions ofContributed Papers selects the topics andorganizers for the contributed paper ses-sions at Mathfest and at the winter JointMeeting. The committee would be de-lighted to hear from MAA members whoare interested in organizing sessions orwho have suggestions for topics.

Planning is now underway for the AMS-MAA-SIAM Joint Meeting in Baltimore,Maryland, January 15–18, 2003. Thedeadline for submissions for the Balti-more Joint Mathematics Meeting it isDecember 31, 2001.

Send (preferably by e-mail) proposal title,name(s) and address(es) of theorganizer(s), and a one-page summary tothe chair of the committee, Howard Penn.

E-mail: hlp@usna.eduAddress: Department of MathematicsU.S. Naval Academy, Annapolis, MD21402Phone: (410) 293 6702Fax: (410) 293 4883

The Mathematical Association of America(MAA) is seeking a highly qualified person forthe position of Director of Programs and Ser-vices. Candidates should have a doctorate or theequivalent in a mathematical science or math-ematics education and at least six years of expe-rience as a collegiate faculty member. A candi-date should have successful experiences in all ormost of the following areas: grant proposal writ-ing and project management; administration;improvements in teaching and learning; andMAA committees, sections or programs. Ap-pointments may be made for two or three years,with the option of renewal for multiple years.

The Director oversee programs and member ser-vices which include professional developmentactivities; program development; support ofmember run activities, including those of com-mittees, sections, and special interest groups;grant management and support; preparation andsubmission of proposals to foundations and gov-ernment agencies. The Director reports to theExecutive Director. He/she is a key member inthe MAA’s staff leadership team, and will workclosely with the Executive Director and otherstaff members, national officers, section officers,committee chairs, and others in strategic plan-ning and program development.

The MAA, with 30,000 members, is the largestassociation in the world devoted to collegiatemathematics. Membership includes college anduniversity faculty and students, high schoolteachers, individuals from business, industry andgovernment, and others who enjoy mathemat-ics. The Director of Programs and Services hasresponsibilities that encompass all major com-

ponents and activities of the MAA. These includetwo annual national conferences, the summerMAA MathFest, and the winter Joint Mathemat-ics Meeting organized with the American Math-ematical Society; twenty-nine Sectional organi-zations that hold annual meetings and conductprofessional development activities; publicationof three scholarly journals, two magazines, anewsletter, and 15-20 books annually; over 100committees and councils that are responsible formuch of the work of the Association; and exter-nally funded projects and programs.

The deadline for submission of applications isJanuary 21, 2002. Interviews will be held duringthe months of January and February. It is ex-pected that the new Director will begin work byJuly 2002, earlier if possible. The position is lo-cated at the national headquarters of the MAAin Washington, DC. Salary will be based uponthe candidate’s credentials or current salary fora reassignment position. The MAA offers a gen-erous benefits package.

Candidates should send a resume and letter ofinterest to:Ms. Julie KramanMathematical Association of America1529 18th Street, NWWashington, DC 20036.Applications may be submitted electronically tojkraman@maa.org. References will be requestedafter review of applications. Applications fromindividuals from underrepresented groups areencouraged. Additional information about theMAA and its programs and services may befound on MAA’s website: http://www.maa.org.AA/EOE.

MATHEMATICAL ASSOCIATION OF AMERICADIRECTOR OF PROGRAMS AND SERVICES

22

FOCUS November 2001

Treasurer’s Report—2000 Financial Year

The chart below gives an overview of the performance of theMAA Operating Funds (excluding extraordinary transfers fromthe Investment Fund) during the last five years. This year weinclude the American Mathematics Competitions (AMC) in theOperating Funds report. In the past the AMC was included inthis report as an off-site project. The decision to integrate fullythe AMC into the MAA budget and financial reports was madein recognition of the increasing role the AMC plays both in theAssociation’s programmatic and financial activities. During thistransition year we display below both the results without AMC(labeled General Fund) and with the AMC (labeled “All Funds”).“All Funds” includes the journal and book programs, meetings,the American Mathematics Competitions, governance, andmember services. It does not include grant funded programs orthe operation of the MAA headquarters buildings.

In 2000 there was a surplus of $46,627 in All Funds. This re-sulted from a surplus of $53,983 in the AMC and a deficit of$7,356 in the General Fund.

The Board of Governors had approved a budget for 2000 thathad a $20,300 surplus in the General Fund and a $52,400 sur-plus for the AMC. The AMC results were in line with the bud-get; however, the General Fund results were $27,656 under bud-get. While this is a shortfall of only 0.5% of the total budget, itremains true that the Association must monitor its expendi-tures carefully or find a way to increase revenues significantly.

Income was $156,000 over budget while expenses were $184,000over budget. Significant differences from budget include thefollowing:

income over budget: dues $88,000; subscriptions $92,000;advertising $27,000; and interest income $30,000.income under budget: book sales $115,000; discontinuanceof placement test sales $28,000; and Greater MAA Fund$83,000.expenses over budget: journal printing $50,000; marketing$100,000; fulfillment $70,000 and travel $40,000.expenses under budget: salaries and benefits $40,000; scaledback JPBM $22,000.

The MAA Endowment Fund1 decreased in value by 2% afterthe normal transfer of $48,017 to the Operating Fund for pro-grams. At year end the value of the Endowment Fund was$2,413,904. This includes both restricted and unrestricted funds.

We discuss the operating budget, grant activity, the headquar-ters building fund, the American Mathematics Competitionsand the investment fund individually below. Last but not least,I am glad to acknowledge the assistance of our Director of Fi-nance, Neil Beskin, in preparing this report.

The Operating Budget

Where the money came from

Total income for 2000 was $6,069,181 up from $5,713,511 in1999. This was derived as follows:

• Dues includes member dues, institutional dues, corporatedues, and a payment from the Life Membership Fund forlife members.

• Contributions include the Greater MAA Fund, the duessupplement and other contributions to the Operating Fundbut does not include contributions to the endowment or tothe Building Fund.

• Journals include non-member subscriptions, sales of backissues, advertising, and royalties received. It does not includethe portion of dues allocated to journals (see footnote2 ).

• Publications income includes sales of MAA books and re-ports, placement tests, and video tapes.

• Investments and Trusts are funds that are transferred fromInvestment Funds and Trusts to support specified prizes andother activities.

• American Math Competitions and Olympiad The MAAmanages two high school and a junior high school nationalmathematics competition. These activities are managed fromour office in Lincoln, Nebraska. Students who perform wellon the high school examination are invited to compete forparticipation on the U.S. Math Olympiad team. This com-petition takes place through two additional exams, the AIMEand the USAMO. Income comes from sales of the exams,old exams and contributions including an in-kind contri-

Gerald J. Porter, Treasurer

1996

$600,000

$500,000

$400,000

$300,000

$200,000

$100,000

$0

($100,000)

($200,000)

($300,000)

1997 1998 1999 2000

General Fund BalanceAll Funds Balance

Dues

Contributions

Journals (other than member subscriptions)2

Publication other than journals

American Math Competitions/Olympiad

Transfer from Investments and Trusts3

Funded Programs

Indirect Costs on Grants

Meetings/Minicourses/Short Courses

Building Management Fee

Miscellaneous

TOTAL

1999

$2,008,578

$154,956

$921,875

$1,025,461

$987,444

$71,582

$111,933

$295,580

$25,000

$111,102

$5,713,511

2000

$2,096,376

$115,598

$981,490

$1,020,676

$1,174,325

$84,062

$28,849

$137,585

$287,993

$35,000

$107,227

$6,069,181

Where the money came from

FOCUS

23

November 2001

bution by the University of Nebraska at Lincoln. This con-tribution was $84,297 in 1999 and $87,668 in 2000. Prior tothis report the income and expense of AMC were not in-cluded in this report. We have adjusted the 1999 report sothat AMC is included.

• Funded Programs are programs to which a funding sourcehas been dedicated. Currently there are two such programs.Student Chapters and Women and Mathematics. StudentChapters is funded from revenue from the MBNA MAAcredit card program while Women and Mathematics isfunded from royalities on the book, She Does Math.

• Indirect Costs on Grants is income on externally fundedactivities that support MAA administrative activities. Notall grantors pay indirect costs.

• Meetings and Courses are registration fees from minicourses,shortcourses, and the online courses, net income from theJoint Meeting and all income from the summer MathFest.

• Building Management Fee is a transfer from the BuildingFund to the General Fund for management services.

• Miscellaneous includes various fees that we receive for man-aging activities (e.g., CBMS).

What happened in 2000

• Dues income increased by 4%, approximately the amountof the dues increase for 2000.

• Contributions to the Greater MAA Fund decreased by$40,000. This is a troubling trend that has continued for sev-eral years. The MAA is dependent upon the generosity of itsmembers and we need to do a better job soliciting contribu-tions.

• Journal subscriptions were up by more than the increase insubscription rates.

• Indirect cost recovery on grants increased by $26,000. Thistrend should continue in 2001 as the PREP and MathDLgrants have a full year of funding.

Where the money went

Expenses4 totaled $6,022,555 in 2000 compared to $5,490,046in 1999.

• Journals/Electronic Services include the cost of publishingand distributing the Monthly, Mathematics Magazine, theCMJ, FOCUS, Math Horizons, and our electronic newslet-ter, MAA Online.

• Publications is the cost of our book and video publicationprogram.

• General Programs and Services includes the cost of awards,minicourses, MAA portions of the Joint Meeting, the sum-mer MathFest, section support, SUMMA, student chapters,project support, and our participation in joint projects andactivities such as JPBM.

• American Math Competitions and Olympiad The 2000AMC expenses include support for the Math Olympiad Din-ner and the summer training program which were not in-cluded in the 1999 expenses. Prior to this report the incomeand expense of AMC were not included in this report.

• Administration is the cost of operating the Executive, Fi-

nance, Human Resources and Computer Service Depart-ments. These costs are not allocated to other activities.

• Governance includes the costs related to the Board of Gover-nors, section officers, executive and finance committees, andthe officers.

• Membership Processing is the cost of membership recruit-ment and fulfillment.

• Development includes the cost of the Development Depart-ment as well as costs related to the Greater MAA Fund. Thisis an investment in future gifts as well as present contribu-tions.

• Miscellaneous included telephone, copying, postage and of-fice supply expenses in 1999. In 2000 we have allocated mostof these expenses to the various departments. The $100,000decrease in expenses is artificial and corresponds to expenseincreases in administration, publications, membership pro-cessing and programs and services.

What happened in 2000

• It is difficult to make a direct comparison between 1999 and2000 expenses since we continue to improve the way thatexpenses are assigned to activity. In particular, as we noteabove, telephone, copying and office supply expenses are nowallocated to the departments.

• It was our goal to increase activity and staff in the memberservices department and that increase has resulted, as ex-pected, in increased expenses.

• The increased expense in publication and the decreased ex-pense in membership processing is in large part related tomore accurate allocation of fulfillment expenses.

• We note that both meeting expense and income increased in2000. This was due in large part to the very successful Wash-ington, DC meeting. As usual we lost money on the summermeeting and made money on the winter meeting. As a re-sult we broke even on meetings for the year.

• Both income and expense for AMC increased during 2000.The 2000 expenses include various Olympiad expenses notincluded in 1999. When one adjusts for these expenses thenet income in 2000 was essentially the same as it was in 1999.

Externally Funded Projects

During 2000 the MAA received external project support of$1,069,600. This was up from $864,340 received in 1999. Indi-

Journals/Electronic Services

Publications

Inventory Allowance

General Programs, Services and Projects

American Math Competitions/Olympiad

Administration

Governance

Membership Processing

Development

Miscellaneous

TOTAL

1999 2000

$1,810,724

$540,379

$17,981

$574,066

$843,908

$971,149

$175,140

$275,707

$157,710

$123,282

$5,490,046

$1,810,973

$768,346

$16,064

$686,362

$1,071,905

$1,027,881

$164,591

$189,591

$264,760

$22,082

$6,022,555

Where the money went

24

FOCUS November 2001

Supporting Materials

1. Expense and Income by Activity http://www.math.upenn.edu/~gjporter/maa/report20002000%20INCOME%20AND%20EXPENSE%20BY%20ACTIVITY.htm.

2. Statement of Financial Position http://www.math.upenn.edu/~gjporter/maa/report2000/STATEMENT%20OF%20FINANCIAL%20POSITION.htm.

3. Investment Fund Accounts http://www.math.upenn.edu/~gjporter/maa/report2000/MAA%20INVESTMENT%20FUND.htm.

4. Building Fund http://www.math.upenn.edu/~gjporter/maa/re-port2000/MAA%20BUILDING%20FUND.htm.

5. Grant Supported Programs http://www.math.upenn.edu/~gjporter/maa/report2000/GRANT%20SUPPORTED%20PROGRAMS.htm

rect cost recovery of administrative expenses was $137,585 upfrom $111,933 in 1999. We expect that this will continue to in-crease in 2001 because of grants such as MathDL and PREP.

Building Fund

The Association owns two adjoining townhouses and a “car-riage house” at 1527 and 1529 Eighteenth Street NW, Washing-ton, DC. The MAA Washington office occupies 1529, most ofthe “carriage house,” and a small amount of 1527. The remain-der of 1527 is rented to other mathematical organizations in-cluding the AMS and CBMS. In 2000 we “charged” ourselves$225,000 for the space we occupied. That amount is includedin Building Fund income.

In 2000, depreciation on the building and renovations was$82,006. In 2000, the Building Fund received contributions of$26,180. This includes a gift of $25,000 from the Dolciani Foun-dation. Building rental income has decreased in recent yearsbecause of the loss of JPBM as a tenant. It is essential for thelong term financial health of the buildings that we develop ad-ditional revenue streams to support the building expenses.

Endowment Fund

The MAA Endowment Fund includes both restricted and un-restricted funds. At the end of 2000 the Endowment was valuedat $2,413,904, a 2% decrease from the end of 1999. During 2000,$48,017 was transferred from these funds to support prizes andother activities designated by the original donors to the MAA.An additional, $32,046 was transferred from the Sliffe Trust. Agift of $1,074 was added to the endowment of the Benefactor’sFund.

The MAA Endowment funds are, according to accounting stan-dards, divided into Unrestricted, Temporarily Restricted andPermanently Restricted. The values of these funds at the end of1999 and 2000 are as follows:

The last will and testament of Edith May Sliffe established afund (The Sliffe Fund) to fund awards to selected teachers whose

teams qualified in the American Mathematics Competitions.The MAA was selected as the Trustee of this fund. On Decem-ber 31, 2000 the Sliffe Fund had a value of $612,374. The MAAalso is the trustee of a trust established by Clinton B. Ford inmemory of Walter B. Ford. This trust had a value of $100,908on December 31, 2000.

The MAA is also the beneficiary of two Charitable RemainderUnitrusts. At the end of 2000 these were carried on our balancesheet at a value of $336,511 in conformance with IRS rules.

General Fund Balance

The General Fund Balance is the cumulative sum of yearly bal-ances in the General Fund. It is a measure of how the Associa-tion has done over time. This balance decreased last year by$7,356, which was the deficit in the General Fund for the year.

This is my last annual report as Treasurer. I will leave that posi-tion at the conclusion of the annual meeting in January 2002. Ithas been my good fortune to work with exceptionally dedicatedindividuals, both MAA volunteers and staff. Because of theirefforts the MAA programs and financial strength has grown.The greatest asset that the MAA has is not its buildings or itsfinances; it is its members who are the true backbone of theAssociation. Because of their efforts the MAA will continue toprosper.

December 31, 1999 December 31, 2000

Unrestricted Board Designated

Temporarily Restricted

Permanently Restricted

$1,851,620

$492,909

$118,210

$2,462,739

$1,820,382

$416,614

$118,210

$2,413,904

Endowment Fund

Building Fund Income $335,709

Building Fund Expense $326,282

December 31, 1999 December 31, 2000

$218,240 $210,884

General Fund Balance

[1.] We use the name “Endowment Fund” to distinguish these assets from the MAA Investment Fund which consists of Endowment assets plus two trusts that the MAA administers.

[2.]Dues income allocated to journals was as follows: American Mathematical Monthly $424,178, Mathematics Magazine $197,479, College Math Journal $190,066, FOCUS $159,912.

[3.] Investment and Trust Income in 2000 were allocated as follows: awards $22,851, joint meetings $2,001, MathFest $2,265, Sections $10,895, Finance Dept $5,273, Sliffe Awards$27,278, Public Awareness $5,000, Publications $8,500.

[4.] Expenses include direct expenses, allocated building expense, and allocated direct service expense for the publications, marketing, and member services department. costs attrib-- utable to Governance, the Executive and Finance Departments and the Development Department are not allocated. They appear as Administrative expenses.