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Home , Constance Reid, David Eisenbud, George Dantzig, Jerzy Neyman, Paul Halmos, Peter Lax

FOCUS August/September 2005

FOCUS is published by the Mathematical Association of America in January, February, March, April, May/June, FOCUS August/September, October, November, and Volume 25 Issue 6 December. Editor: Fernando Gouvêa, Colby College; [email protected] Inside

Managing Editor: Carol Baxter, MAA 4 , 1909-2005 [email protected] By John MacDonald Senior Writer: Harry Waldman, MAA [email protected] 5 Encountering Saunders Mac Lane By Please address advertising inquiries to: Rebecca Hall [email protected] 8George B. Dantzig 1914–2005 President: Carl C. Cowen By Don Albers First Vice-President: Barbara T. Faires, 11 Convergence: , History, and Teaching Second Vice-President: Jean Bee Chan, An Invitation and Call for Papers Secretary: Martha J. Siegel, Associate By Victor Katz Secretary: James J. Tattersall, Treasurer: John W. Kenelly 12 What I Learned From…Project NExT By Dave Perkins Executive Director: Tina H. Straley 14 The Preparation of Mathematics Teachers: A British View Part II Associate Executive Director and Director By Peter Ruane of Publications: Donald J. Albers FOCUS Editorial Board: Rob Bradley; J. 18 So You Want to be a Teacher Kevin Colligan; Sharon Cutler Ross; Joe By Jacqueline Brennon Giles Gallian; Jackie Giles; Maeve McCarthy; Colm 19 U.S.A. Mathematical Olympiad Winners Honored Mulcahy; Peter Renz; Annie Selden; Hortensia Soto-Johnson; . 20 Math Youth Days at the Ballpark Letters to the editor should be addressed to By Gene Abrams Fernando Gouvêa, Colby College, Dept. of 22 The Fundamental of ______Mathematics, Waterville, ME 04901, or by email to [email protected]. By Jeffrey Nunemacher Subscription and membership questions 24 2005 Award Winners for Distinguished Teaching should be directed to the MAA Customer 26 “I Like Change” An Interview with Tina Straley Service Center, 800-331-1622; e-mail: [email protected]; (301) 617-7800 (outside By Don Albers U.S. and Canada); fax: (301) 206-9789. MAA 32 Third Annual Mathematical Study Tour—Home of the Ancient Maya Headquarters: (202) 387-5200. 34 Archives of American Mathematics Spotlight: Copyright © 2005 by the Mathematical Association of America (Incorporated). The New Mathematical Library Records Educational institutions may reproduce By Robin Howard and Kristy Sorensen articles for their own use, but not for sale, 36 What I Learned About…Online Assignment Management provided that the following citation is used: “Reprinted with permission of FOCUS, the By Glenn Ledder newsletter of the Mathematical Association 38 Letters to the Editor of America (Incorporated).” 40 Finding Common Ground in K-12 Mathematics Education Periodicals postage paid at Washington, DC By Michael Pearson and additional mailing offices. Postmaster: Send address changes to FOCUS, 41 U.S. Team Survives Hurricane to Place 2nd in the Mathematical Association of America, P.O. International Mathematical Olympiad Box 90973, Washington, DC 20090-0973. By Steve Dunbar ISSN: 0731-2040; Printed in the of America. 42 The Missouri Collegiate Mathematics Competition By Alvin Tinsley and Curtis Cooper 44 Short Takes

2 August/September 2005 FOCUS

William McCallum and Ken Ono Named Distinguished Teaching Scholars

On June 21, the National Science William McCallum’s research work is in Ken Ono’s work has focused on modu- Foundation named seven Distinguished theory and arithmetic algebraic lar forms and their relations with ellip- Teaching Scholars, honoring scholars geometry, but he is best known as a leader tic curves. He has obtained remarkable who have achieved success in both re- in the calculus reform movement and results on congruence properties of the search and teaching, and who have suc- one of the main authors of the “Harvard” partition that are closely related cessfully integrated the two. Among the calculus textbook. He has also been in- to congruences discovered by Ra- recipients are two : Wil- volved with the Winter School manujan many years ago. His work with liam McCallum of the University of Ari- on Arithmetic , undergraduates has successfully involved zona (and a member of the MAA) and which has made this area accessible to them in research, resulting in papers Ken Ono of the University of Wisconsin countless graduate students and schol- published by students and in collabora- Madison. The awards, which are worth ars wishing to learn more about the sub- tion with students. He has also worked up to $300,000 over four years, are “NSF’s ject. NSF reports that “His new work will with K-12 students on projects related to recognition of accomplishments by sci- focus on better communication among . NSF reports that “His entists and engineers whose roles as edu- mathematicians, teachers and math edu- award will help fund summer institutes cators and mentors are considered as cation researchers in a systematic con- that will provide high-school and under- important as their ground-breaking re- tent analysis of problems in algebraic graduate students with a structured re- sults in research,” said NSF Director thinking that should lead to new instruc- search environment. It will also allow for Arden L. Bennet, Jr. The Distinguished tional materials for a broad range of stu- Ono to travel to conduct lectures and Teaching Scholars program has existed dents.” hands-on activities with middle- and since 2001, and has so far honored 34 high-school students alongside Nobel people. Prize and National Medal of Science win- ners.”

MAA Election Results

Joseph A. Gallian Deanna B. Haunsperger President-Elect (2006) First Vice-President (2006-07) Second Vice-President (2006-07)

The MAA’s national election concluded at the of May; 3759 votes were cast, about 35% of them electronically. Joseph Gallian was chosen as President-Elect for 2006, which means that he will be President of the Association in 2007–2008. Carl Pomerance and Deanna Haunsperger will be vice-presidents during 2006–2007.

3 FOCUS August/September 2005 Saunders Mac Lane, 1909-2005

By John MacDonald

Saunders Mac Lane was one of the most directly involved in the proofreading of influential mathematicians of the 20th each chapter as it was typed and used to century and was, together with Samuel delight in finding misprints because Eilenberg, a creator of Theory. Saunders would say I had an “eagle eye”. Details of this achievement together with He liked to have his students read much other information about his career original source material and had me read can be found in his new book Saunders early papers by Hopf, his papers with Mac Lane, A Mathematical Auto- Eilenberg on K(π,n) spaces, Lawvere’s biography, published by A.K. Peters, Ltd. thesis, as well as Freyd’s, at that time new, in 2005. However, in this article, I will book on abelian categories. present my personal point of view, since Saunders Mac Lane was my thesis At Oberwohlfach he climbed through the advisor, mentor, and lifetime friend. hills with his special walking stick. In the 1970s there were many category meetings I first met Saunders as a graduate student there and many impassioned discussions in Chicago in 1961 when I took a course amongst the participants, especially with him in . I probably about the new developments in first came to his attention around that theory. There were discussions during time, when I pointed out a slight error the day and in the evening, and in one of the exercises in his book with arguments too, often with both Eilenberg Birkhoff on Modern Algebra. Saunders Mac Lane and Mac Lane present. Freyd, Lawvere, MAA President 1951-52 Johnstone, Kock, Tholen, Rosicky, Isbell, Saunders’ influence on me did, in fact, suggest some other lemma or theorem Barr and Tierney and many others were extend much further back in time, to that needed proving. involved. Eilenberg’s students were September 1956, when I took a course in always part of the inner circle. Jon Beck’s algebra at Harvard from Andrew Gleason My most important interactions with work, in particular, continues to using the book just mentioned. This Saunders, in fact, took place this way command the highest respect. course resulted in my changing majors when I was a graduate student. He from to mathematics, since it insisted on weekly meetings as well as on I remember Saunders’ first wife, Dorothy, convinced me that mathematics had the a progress report from the previous week. quite vividly from the 1974 International richness and mystery that I wished to It was not enough to have read Conference in Vancouver when I had explore further. something — he wanted evidence of many categorists at my house. I met some thought applied to the research Saunders’ second wife Osa at MSRI in In his later years when he had passed problem at hand. In this way he was very 1993. I have a much treasured photo of from his role of advisor and mentor to serious and not nearly as light and Osa, Saunders and my wife standing friend I was constantly amazed by his easygoing as he seemed with some around my son Ian, then 18 months old. tenacity and independence. To give an visitors. I saw her later at notable events like the example, here is a story from when he celebration in Coimbra, Portugal, in was in Coimbra, Portugal, at the At this time in the early 60s, there was 1999, and kept in touch with her up and Category Theory meeting in 1999. After indeed a whirl of categorical ideas through the time of Saunders’ memorial the lectures one day, everyone was urging evolving with visits to Chicago from service at MSRI on May 4, 2005. him to take a cab back to the hotel Eilenberg, Freyd, Lawvere, Beck, and because of the steep walk from the Linton. Tom Hungerford was also Of course, these reflections are not a university to the town. But, no, he wanted writing a thesis with Mac Lane at the chronological or balanced view of to walk back and I ended up walking back same time and John Thompson and Saunders’ life but rather an im- with him. The trek was difficult for him, Jonathan Alperin could be heard in the pressionistic view of a person with a very but he was very determined. halls discussing exciting new rational and orderly view of the world. developments in theory. Please see the autobiography mentioned He showed the same tenacity, and at the and John Gray came up frequently for above for a more complete story. same time a great attention to detail, the Midwest Category Seminar. In the when acting as a thesis advisor. When I meantime Saunders kept producing all As he used to say at the time of someone’s had written something up and thought these books, neatly typed chapter after passing — Hail and Farewell! it was fine, Saunders would read it chapter. was the first one. I was through, make corrections and usually

4 August/September 2005 FOCUS Encountering Saunders Mac Lane By David Eisenbud

No man could so stimulate others unless, alongside an incisive intellect, he was possessed of enthusiasm and warmth, a deep interest in his fellow man, and a sympathy the more real for being unsentimental. Those who proudly call themselves his friends know these things: others will infer them in reading [his works]. — M. Kelly, in Saunders Mac Lane: Selected Papers, edited by I. Kaplansky

Saunders Mac Lane was my teacher, paying attention to his class, later told me mentor, and model almost from the he thought he could always see who was beginning of my mathematical life. It is following and who was not. In a moment a relationship I’ve cherished. He was a like a thunderclap, I looked up from my figure of great honesty and integrity, who seat and found him pointing directly at worked hard to advance research and to me from across the room. “You!” he said serve the mathematical community. His peremptorily, “you don’t believe this belief in the good, the right, and the proof, do you?” Belief and disbelief were rational, his care for the essence of equally beyond me; I sat petrified. He mathematical ideas, his powerful advanced toward me, and I don’t know enthusiasm, and his essential optimism what I imagined — that he would pick were and are deeply attractive to me. me up by the scruff of my neck and throw me from the room? He stopped, turned Nearly everything about Saunders in back to the board, and proceeded to action was colorful, starting with the red- explain the proof to satisfy me. Of course, and-green plaid sports coat (the Mac I still understood nothing — but I sat in Lane tartan, of course) and red pants that rapt attention. he would wear for important occasions. Perhaps a few anecdotes and reflections Fortunately the class ended soon, and as from my experience of him over 40 years students asking questions surrounded will help the reader appreciate this color. him, it was easy for me to slip out. I didn’t tell Saunders this story until many years Saunders Mac Lane, c. 1931 First Encounter afterwards, when I had the privilege of Norwalk, Connecticut re-enacting it (from the other side) in a I first met Mac Lane — in a sense I’ll lecture at the conference in honor of his make precise — in 1963. He was one of mathematics. It was the beginning of my seventieth birthday. Needless to say, the the most important figures in the second quarter, and I was scheduled to event hadn’t left a trace in his memory, Mathematics start a basic class that though it remains sharp for me to this Department, or indeed in American morning. I happened to arrive a little day. mathematics: His first student, Irving early, settled down in the first row of the Kaplansky, was Chair of the department, class, and sank peacefully into a Saunders and Tolerance and two other students were on the daydream. Being so new, I wasn’t faculty — one, John Thompson, a Fields surprised not to know the other students Saunders believed strongly in principles, Medalist. Mac Lane was an inventor of who settled in around me, and I didn’t in the rightness of right positions. I never group , a founder of know the teacher that I’d have. In due once saw him personally intolerant, but and category theory, course, Mac Lane walked in and began he could sometimes be direct and candid known for the Eilenberg-Mac Lane lecturing. His style was lively and to the point of offending. People whose spaces in topology. He was past President colorful, and I was immediately judgment I respect have felt injured by of the Mathematical Association of interested — but almost at once aware what he said, and sometimes by the America, and he would soon be Vice that I’d made a big mistake: this was not bluntness of his expression. In some way President of the National Academy of an undergraduate linear algebra course, perhaps he didn’t appreciate the Sciences, member of the Board but an advanced graduate course on magnitude of his position in governing the National Science Category Theory. I’d come an hour early. mathematics, or the seriousness with Foundation, and President of the which people took him. In a lesser American Mathematical Society, as well. I understood nothing whatever after a personage some of his extreme positions few moments, but was far too might have been regarded as charming I knew none of this. I was sixteen, an early embarrassed to get up and leave — eccentricities. But given Saunders’ entrant to the University, an uneven instead I sank into daydreams, glassy- stature, they could injure, and he might student with a great enthusiasm for eyed. Mac Lane, who prided himself on have been more cautious.

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An event from late in Saunders’ life may and grandfather were Congregational Kegel), it was finally time for me, by now give a bit of the flavor. It was a special ministers, seemed to feel that since his a second-year graduate student, to settle session run by him and Richard Askey at view was right, his view would prevail. on an area for a PhD thesis. I obsessed the Joint Mathematics Meeting in 1999, Once he had stated it, all he could do was about how to make the choice. A close a session boldly entitled mathematical friend, “Mathematics Education Joe Neisendorfer, and Mistaken Philosophies explained to me an of Mathematics.” The : forget the audience was enormous. I topic, look around the found the title charming faculty for the person (and still find it so, even now you like the most. It as I become more involved didn’t take me long to with ideas in K-12 choose Saunders. education), and I imagine that Saunders meant it to be I wouldn’t say I ever controversial but playful. felt personal intimacy Predictably, it annoyed and with Saunders, but he needled some practitioners. did go out of his way Saunders began the session to make me and other with introductory remarks students feel welcome that I found fascinating: he in more than his said that he now considered office. Saunders and the extent of his own his late wife, Dorothy, emphasis on category theory had a small but as a tool for learning and comfortable cottage in teaching mathematics to the Indiana Dunes, a have been too extreme. This beautiful area on the humbleness may have helped shore of Lake soften the critical tone of the Michigan about an session. hour south of Chicago, and they Saunders and Sammy occasionally invited students to spend an One of Saunders’ great afternoon there. mathematical friendships Saunders was an and collaborations was with enthusiastic sailor, (widely and I can report, from known as Sammy, or even as a ride in a small S2P2: “Smart Sammy the sailboat on rough Polish Prodigy”). I got to see water, that he was them in action together only Mac Lane lecturing abroad ready to provide once, at the AMS Summer needed instruction Research Institute on Category Theory bang his fist. The devious and not only in mathematics but also on how at Bowdoin College, in 1969. They had sophisticated European versus the to handle the absence of a toilet — or special status at this three-week innocent but honest American? That’s any privacy — in that difficult situation. conference, not only as the senior how it seemed to me at the time. Maybe members, but also as the very founders I was a little innocent myself. A loyal If you look at the list of Saunders’ 39 of the subject. So, when they began student, I was rooting from the students, you’ll see that , discussing its origins one evening after beginning for Saunders’ point of view, who worked on theory of dinner, everyone gathered around to but I came away feeling that he was fields, came first; I’m near the end, with listen. trounced in the contest. a thesis on noncommutative rings. Along the way are such people as John I dearly wish I could recall the substance Being Saunders’ Student Thompson (finite groups), of their debate, but I don’t; only my sense (logic and computation), and Robert of the contrast in the two men’s styles After flirting a while with operator Szczarba (). How did stays with me. Sammy drew Saunders out theory ( and this variety come about? and egged him on, always slightly evasive were my teachers) and group theory and mocking; Saunders, whose father (learned from Jon Alperin and Otto

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Perhaps the answer lies in Saunders’ people I called on for advice and blessing, worked his way through the thesis until hospitality to these many ideas. He and he gave both. he’d compiled a list of exactly 25 wanted to learn finite groups, and substantive suggestions. Then he taught a course on them. By the end of Returning to the more fundamental stopped, and returned the document to the course he’d decided that he’d never matter of being Saunders’ mathematical me for an overhaul. When I had finished really understand the subject, but in student: I tried for a while, dutifully, to making the corrections he’d flagged and Thompson he found a fabulously find a thesis topic in Category Theory, all their analogues, I gave it back to him, strong student. eager to be Saunders might done. But… have tried to turn after a week or such a student so I got a toward interests second list of close to his own, exactly 25 more but I think he suggestions. would not, on The third list principle: he was was a bit happy to en- shorter, and courage his Saunders al- students to do lowed the what excited process to con- them. verge before I got too frus- Saunders fol- trated. lowed an inter- esting, curving It must be clear trajectory by now: over through mathe- these forty matics, from log- years I learned ic and founda- many lessons tions to from Saunders. theory and the I’m deeply beginnings of Just sailing along on his own off the coast of Maine. grateful to him. homological algebra, through topology to category Saunders’ passion in that part of his life. David Eisenbud is Director of the theory, with smaller diversions along But I failed; somehow, the things I read Research Institute the way into Hamiltonian mechanics, and learned in that domain just didn’t in Berkeley, CA and President of the finite groups, and many other subjects. inspire me. When I developed an interest American Mathematical Society. This Perhaps his students, or many of them, instead in a problem on non-com- article is an edited version of the preface could be described as coming off on the mutative rings posed by a visitor of he wrote for Saunders MacLane: A tangents to this path, a kind of devel- Herstein, the young Chris Robson, Mathematical Autobiography, published opable surface reaching broadly across Saunders could easily have washed his by A K Peters, Ltd., Wellesley, MA. It is mathematics. hands of the project. He did not: though reprinted here with the publishers’ kind it was far from his current area of interest, permission. Some other aspects of Saunders are also he welcomed what I had done, and reflected in his students: Saunders was painstakingly read draft after draft of my always active on behalf of the thesis. ICM 2006 in community, whether as Chair working to build the department at the University Saunders’ mode of instruction in thesis Madrid, Spain of Chicago or, near the end of his career, writing bears mention. I had written a as a member of the National Science couple of papers, jointly with Robson, of All information currently available Board or as manager of the elaborate which my thesis results were partially an about the International Congress of reports for the National extract. Robson cared a lot about of Mathematicians (ICM) 2006 Academy of Sciences. Many of his exposition, and so (learning from program, organization, and regis- students and grand-students have Saunders among others) did I. We’d gone tration procedure can be found on followed him into this willingness for through many drafts, and I thought the the ICM 06 website at: public service. When I was worrying writing pretty polished. Saunders did about whether to move to my current not. He began at the beginning and http://www.icm2006.org. position at MSRI, he was one of the first

7 FOCUS August/September 2005 George B. Dantzig 1914-2005 By Don Albers

George Bernard Dantzig, the “fa- About six weeks later, on a Sunday ther of ,” died at morning, he and his wife were home in Palo Alto, CA on May 13. His awakened by someone pounding father , named him on the door. It was Neyman. He after the famous writer George Ber- rushed in with papers in hand all nard Shaw in the hopes that he would excited: “I’ve written an introduc- become a writer. Fortunately for the tion to one of your papers. Read it world of mathematics, he became a so I can send it out right away for . He was the inventor publication.” It turned out that of the simplex method, which has what Dantzig thought were home- powerful applications in several work problems were in fact two fa- fields. Many feel that he should have mous unsolved problems in statis- received the Nobel Prize in Econom- tics. The story of these problems ics for his work. has proven to be inspirational to several ministers who have incor- His achievements in mathematics are porated it into their sermons. particularly striking in view of the fact that he had trouble with algebra In 1941, before finishing his doc- in junior high school. In a 1984 in- torate, Dantzig, wanting to contrib- terview with the author of this obitu- ute to World War II, took a posi- ary, he was disarmingly modest about tion with Air Force Statistical Con- his trouble with algebra: “To be pre- trol. In that position he became cise, I was flunking. I remember walk- expert at programming planning ing home one day, furious with my- methods and set up a reporting sys- self. How is it, I asked myself, that I, a tem for combat units on the num- son of a mathematician, do poorly George B. Dantzig ber of sorties flown, aircraft lost while all the other kids in class do so and damaged, bombs dropped, and much better? I was angry with my- targets attacked. In those days the self. After that I sailed through algebra.” Michigan. He and Anne Shmuner, who only “computing machines” they had From then on, he got top marks in math- he had married the previous summer, were people using hand-operated desk ematics. then moved to Washington, DC where calculators. he got a job as a statistical clerk in the Dantzig attributed the influence of his Bureau of Labor . While there In 1946 he returned to Washington as the father to the development of his analyti- he reviewed a paper on double sampling civilian head of the Air Force’s Headquar- cal power: “My father taught me by giv- by the famous statistician . ters Statistical Control. Two of his col- ing me problems to solve. He gave me Dantzig was very excited by the paper leagues challenged him to mechanize the thousands of geometry problems while and soon wrote to Neyman saying that planning process; that is, to find a more I was still in high school. I would say over he would like to finish his Ph.D. under rapid way to compute a time-staged de- ten thousand. But it was I who asked for him at the University of , Ber- ployment, training, and logistical supply problems. After he gave me one, he would keley. program. Mechanization in those days say, ‘Well, I’ll give you another one.’ It meant using analog devices or punch- seemed as if he had an infinite storehouse During his first year at Berkeley, he ar- card equipment. He set out to formulate of them.... The mental exercise required rived late one day to one of Neyman’s a model, and soon became fascinated by to solve them was the great gift from my classes. On the blackboard were two ’s Input-Output Model father.” problems that Dantzig assumed had been of the American Economy. assigned as homework. A few days later He completed his bachelor’s degree in he apologized to Neyman for taking so By late 1946, he had formulated a linear 1936 at the University of Maryland, long to do the homework, saying that the programming model, but did not yet where he did not recall seeing a single problems seemed to be a little harder have an algorithm. Over the next year he application of mathematics in any of his than usual. Neyman told him to throw succeeded in producing the simplex mathematics courses. In 1937, he earned the homework on his desk. method, an algorithm for solving linear a master’s degree from the University of programming problems, and his group

8 August/September 2005 FOCUS at the Air Force started experimenting number of alternative courses of action Center of the University of California, with it. He continued to look for a better to find one that is optimal. Berkeley, and in 1966 professor of opera- algorithm, but in June of 1948, his group tions research and at asked him why he continued to look else- In 1952, he became a research mathema- , where he remained where when the simplex method was tician at the RAND Corporation where until his retirement. working so well on the test problems. he began implementing linear program- His pioneering book Lin- The rest, as they say, is ear Programming and Ex- history. The simplex al- tensions was published in gorithm has proven to 1963 and revised in 1997 be of great utility in a and 2003. wide range of applica- tions. Airlines use it to Dantzig was the recipient schedule crews and of many prizes and eight make fleet assignments, honorary degrees. He was shipping companies use a member of the National it to determine how Academy of Engineering, many planes they need the National Academy of and where their delivery Sciences, and the Ameri- trucks need to be de- can Academy of Arts and ployed. It is used in re- Sciences. In 1975 he was finery planning, manu- awarded the National facturing, telecommuni- Medal of Science. cations, architecture, revenue management, Don Albers is Associate Ex- circuit design, advertis- ecutive Director and Direc- ing, and many other ar- Receiving the Medal of Science from President Ford in 1975. tor of Publications at the eas. It has provided Mathematical Association managers with a powerful tool for mod- ming on computers. In 1960, he was of America. eling problems and comparing a large named chair of the

An Earth Filled with Computers*

By

In my Linear Programming and Exten- steady-state model and what was needed program with a staircase struc- sions you will notice that I pay great trib- was a highly dynamic model, one that ture. Initially there was no objective func- ute to Leontief. It was Leontief who could change over time. In his model tion; in other words, no explicit goal. around 1932 first formulated the Inter- there was a one-to-one correspondence Such goals did not exist in any practical industry Model of the American between the production processes and sense because planners simply had no Economy, organized the collection of the items produced by these processes. way to implement them. data during the Great Depression, and What was needed was a model with many finally tried to convince policy makers to alternative activities. Moreover, the ap- A simple example illustrates the funda- use the output from the analysis. All of plication had to be large scale — with mental difficulty of formulating a plan- these things are necessary steps for suc- hundreds, perhaps thousands of activi- ning program using such an activity- cessful applications, and Leontief took ties and items. Finally, it had to be com- analysis approach. Consider the problem them all. That is why in my book he is a putable. In other words, once the model of assigning 70 men to 70 jobs. An “ac- hero. was formulated, there had to be a practi- tivity” consists of assigning the i-th man cal way to compute what quantities of to the j-th job. The restrictions are (a) Leontief’s model had a matrix structure these activities to engage in so as to be that there are 70 men, each of whom which was simple enough in concept compatible with their input-output char- must be assigned, and (b) that all of the with sufficient detail that it could be use- acteristics and given resources. The jobs, also 70, must be filled. The level of ful for practical planning. I soon saw that model I formulated would be described an activity is either 1, meaning it will be it had to be generalized. Leontief’s was a today as a time-staged dynamic linear

9 FOCUS August/September 2005 used, or 0, meaning it will not. Thus there which is best. So, invariably, man has al- are 2 x 70, or 140, restrictions and 70 x ways had to turn to a leader whose “ex- 70, or 4900, activities with 4900 corre- perience” and “mature judgment” would sponding zero-one decision variables. guide the way. The leader’s guidance usu- Unfortunately there are also 70 factorial ally consisted in the issuance of a series permutations, or ways to make the as- of edicts or ground rules to those devel- signments. The problem is to compare oping the programs. Although such 70 factorial ways and to select the one methods are still widely used, the world which is optimal, or “best” by some cri- today is far too complex for such sim- terion. plistic methods to work, and they don’t.

Now in this example 70 factorial is a very In late 1946, before we knew that high- big number. To get some idea of how big, speed electronic computers were soon to suppose we had had an IBM main-frame exist, I had formulated a mathematical computer available at the time of the Big model that satisfactorily represented the Bang fifteen million years ago. Would it– technological relations usually encoun- between then and now—have been able tered in practice. However, in place of any to examine all the possible solutions? No! explicitly stated goal, or function to be But suppose that an even more powerful extremized, there were a large number of computer had been available, one that ad hoc ground rules issued by those in could have examined one billion assign- authority to aid in the selection of the ments per second. The answer would still George Dantzig as a graduate student. solution. Without these it would have be no. Even if the Earth were filled with been impossible to choose from the as- nanosecond-speed computers, all work- tronomical number of feasible solutions. ing in parallel, the answer would still be This example illustrates why, up to 1947 no. If, however, there were ten Earths, all and for the most part up to this day, a *Excerpted from George Dantzig in More filled with nanosecond-speed computers, great gulf exists between man’s aspira- Mathematical People edited by Albers, all programmed in parallel from the time tions and his actions. Man may wish to G.L. Alexanderson, and , of the Big Bang until the sun grows cold, state his wants in terms of an objective Harcourt Brace Jovanovich, Inc, Boston, then perhaps the answer would be yes. to be extremized, but there are so many 1990. The remarkable thing is that the simplex ways to go about doing the job, each with method with the aid of a modern com- its advantages and disadvantages, that it puter can solve this problem in a split has been impossible to compare them In Memoriam second. and to choose among them that one Ronald C. Biggers, age 59, died on April 23, 2005 after a stroke. He held the dis- tinct honor of being the first African Study Shows Gains for Women American to earn a Ph.D. in pure math- ematics from the University of Califonia in Mathematics at Irvine. His area of emphasis was alge- braic geometry and combinatorial group theory. He taught at many institutions recently-concluded study (spon- A The study used the ranking of graduate before joining Kennesaw State University sored by the MAA together with the mathematics departments by the National in Kennesaw, GA, in 1989. Biggers is sur- American Mathematical Society, the Research Council to investigate the status vived by Celo Biggers, his wife of more American Statistical Association, and the of women in the top 48 mathematics de- than 30 years, and his two daughters. He Institute of ) partments in the US. Women received 25% was an MAA member for 32 years. shows that women are participating in of the doctorates at these institutions, up mathematics in increasing . The from 21% the previous year. Also noted Edward N. Mosley, age 66, passed away study, which was published in the August were the large percentage of women among on June 12 after an extended battle with issue of the Notices of the AMS, shows undergraduate mathematics majors, the cancer. He taught at Lyon College for 35 that about one third of all doctorates in the increased visibility of women in math- years and was a member of MAA for mathematical sciences during 2003-2004 ematics competitions such as the Math- 43 years. Mosley also served as Governor went to women. This continues a long- ematical Olympiad and the Putnam Com- of the Oklahoma-Arkansas Section. He term trend of increasing participation by petition. For the details, see the August is- is survived by his wife, Mary Eleanor, a women that has persisted since the 1980s, sue of the Notices or visit http:// son, John Mosley, and his brother Dr. when gender records began to be kept. www.ams.org/notices/200507/survey.pdf. James Mosley.

10 August/September 2005 FOCUS Convergence: Mathematics, History, and Teaching An Invitation and Call for Papers By Victor Katz

Convergence: Where Mathematics, His- tory, and Teaching Interact, is the MAA’s new online magazine about the history of mathematics and its use in teaching. Part of MathDL, the mathematics digi- tal library, Convergence is aimed at teach- ers of mathematics, be they secondary teachers, two- or four-year college teach- providing information on the his- fee will be charged beginning later this ers, or college teachers preparing second- tory of mathematics. year. ary teachers. (We describe the range of topics as “grade 9-14 mathematics”: al- • Classroom suggestions. These may Currently, we have a limited supply of gebra, synthetic and analytic geometry, be self-contained articles showing articles in our pipeline. Because our goal trigonometry, probability and statistics, how to use history in the teaching is to bring out new material on a regular elementary functions, calculus, linear al- of a particular topic or they may be , we need a continual flow of articles gebra, and differential equations.) The materials closely related to a main and classroom suggestions. We therefore editors, Victor J. Katz, from the Univer- article, showing in some detail how welcome your ideas for articles as well as sity of the District of Columbia, and to use the article in a classroom set- your completed manuscripts. In particu- Frank Swetz, from Penn State University, ting. lar, we welcome short classroom sugges- Harrisburg, welcome all members to log tions that can immediately be imple- in to the Convergence website at http:// • Historical problems. These prob- mented by teachers. convergence.mathdl.org and see what the lems will appear in a section entitled magazine has to offer. “Problems from another time,” with Materials should be sent both in new problems appearing frequently. hardcopy and electronically. Hardcopy Among the types of material appearing should be sent to Victor Katz, Conver- in the magazine are the following: • What Happened Today in History? gence, Mathematical Association of Each day, there will be a listing of 2- America, 1529 18th St. N.W., Washington, • Expository articles dealing with the 3 “mathematical events” which hap- DC 20036. Electronic files should be sent history of various topics in math- pened on that date in history. to Victor Katz at [email protected]. We can ematics curriculum. These may con- take articles in Word or TeX, but please tain interactive components and • Quotation of the day. A new and include illustrations (in jpg format), color graphics, to take advantage of interesting quotation about math- applets, etc. as separate files, and give the capabilities of the Web. Each ar- ematics from a historical figure will explicit instructions for both internal and ticle will have a discussion group appear in this section each day. The external hyperlinks. If there are many il- attached, where readers can share reader will also be able to search our lustrations or applets, it is best to send suggestions as to how the material of quotations to find addi- the electronic version on a CD to the can be used in the classroom and tional ones. MAA address. point out strong points and possible pitfalls. • An up-to-date guide to what is hap- If you have an idea for an article, but do pening around the world in the his- not know how to produce applets for it, • Translations of original sources, tory of mathematics and its use in we suggest that you contact an expert on generally accompanied by commen- teaching. The magazine will report your own campus for help. If necessary, tary showing the context of the on past meetings and give notice of however, we can provide help in the edi- works. The goal of these translations future meetings. Where abstracts torial office, provided you give us very is always to show teachers how ideas are available for a particular meet- explicit instructions as to what you need. were developed in various cultures ing, these will be included. We may and how knowledge of this develop- also include copies of handouts for If you are interested in writing reviews ment is useful to teaching the same easy access, as well as links to the for Convergence, send your name and ideas to today’s students. author’s webpage, if available. some indication of the kinds of material in which you are interested to Frank • Reviews of current and past books, The magazine is currently free to all, due Swetz at [email protected]. We also articles, and teaching aids on the his- to the support of the National Science welcome interesting quotations as well tory of mathematics of use to teach- Foundation, but registration is required as information on mathematical dates to ers, as well as reviews of websites to access the site. A small subscription add to our database. 11 FOCUS August/September 2005 What I Learned from… Project NExT

By Dave Perkins

Project NExT is an MAA professional ish the problem myself.” If a student zling out the answer on the whiteboard. development program for undergradu- made a mistake, I replied in the shared There is still one proof in chapter 8 that ate professors who have recently acquired folder, and expected a response. At the none of us can explain! Once again, the a Ph.D. You can recognize a Project NExT end of the year, the students’ evaluations student evaluations were all positive, and fellow at any MAA national meeting by of the course included only positive re- one bright physics student points to that the colored dot on his or her nametag. actions to this approach. experience as the reason why he is ap- When I learned in spring of last year that plying to graduate school next year not I had been accepted as a NExT fellow, I This past semester, I team-taught a in physics, but in mathematics. speculated that the award might be little course in advanced logic that met once a more than a feather in my cap (and an week. My colleague from philosophy and These are by no means the only peda- orange dot on my nametag). It is no ex- I chose an article each week (e.g., “Com- gogical impacts that Project NExT had aggeration, however, to state that my ex- puting Machinery and Intelligence” by on me. My attitudes toward both under- periences as an orange dot fundamen- Alan Turing) and asked the students to graduate research and my own research, tally changed my approach to teaching. I read it at least once during the next two for example, have also changed greatly. am grateful for the chance to tell the sto- days. While reading, they were to formu- Consider this my thank you to an under- ries of two professors who spoke to us late a list of questions that were raised taking that has affected me profoundly, and inspired me to change. by the material, then choose one and post and an encouragement to you to let your it to a shared folder. Later, after having new colleagues know about Project Thomas Banchoff of Brown University read the article more carefully and dis- NExT. You can find out more at http:// demonstrated the online system he uses cussing it in class, they each were to archives.math.utk.edu/projnext/. to collect and respond to homework. choose another student’s question and Programmed by his undergraduate stu- respond to it with an essay. All of this Dave Perkins is an assistant professor of dents, this system informs Dr. Banchoff discourse was public to the rest of the mathematics at Houghton College in west- when a student has posted the answer to class; a general shared folder collected ern . He is particularly interested a homework problem, allows him to re- whatever discussion spilled over — fur- in discrete mathematics. His email address ply to the student’s work, notifies the stu- ther questions, links to websites, and so is [email protected]. dent, and so on. Once a certain deadline on. has passed, the system releases these dis- cussions for public view. Like many pro- The second professor, sarah-marie fessors, I have on occasion allowed stu- belcastro of Xavier University, surprised dents to rewrite their homeworks, hop- the orange dot crowd with her approach ing that the dialogue thus created will to teaching, an approach I like so much benefit the students in a way that the that I plan to teach over half of my classes simple “here is your paper with my com- next year in her style. During her pre- ments” model cannot. Dr. Banchoff’s sentation she said, “If all we want is to approach, however, takes the dialogue convey information to our students, we approach even further by moving it into might as well just read a text to them, and the public forum. in that case, they might as well just read the text themselves.” I think that state- Since hearing his presentation a year ago, ment is perfectly well put, and I could I have experimented with Dr. Banchoff’s not agree more. Correction idea in two classes. In a special topics class on the constants φ, π, e and i, my stu- A month after hearing her speak, I taught A note in the May/June issue of FO- dents posted their homework answers in a real analysis class with her approach. CUS referred to “ICM 2006, a shared folder, accessible to all. Because We read Elements of Real Analysis by Barcelona, Spain.” That, of course, is I did not have software that would un- David Sprecher page by page; at the start a mistake: as indicated on page 7 of lock the answers at a given time, the stu- of each class, I asked who had written this issue, ICM 2006 will be held in dents were able to view answers that were questions into their reading journal, and Madrid. We apologize for the error. posted before the deadline. Thus, I asked what the questions were. Usually, other them to give each other credit every time students answered these questions, and credit was due — e.g., “I was stuck, but occasionally I suggested my own answers. saw how Shawna factored the polynomial Every now and then, none of us knew to get started, so I used that idea to fin- what to say, and we spent the class puz-

12 August/September 2005 FOCUS

You do the math.

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The infinite possibilities of mathematical literacy.

13 FOCUS August/September 2005 The Preparation of Mathematics Teachers: A British View Part 2 By Peter N. Ruane

What should teachers know about mathematics in order to All well and good, you might say, but how were such principles be able to present the subject most effectively in the primary put into practice? The specialist modules included one on Pre- and secondary classroom? Calculus, another on Proof, Logic and , and a module on Geometry, but the one described below was devoted For primary teachers, the problem isn’t so simple; they have to to Number; here’s a compressed outline of its contents. teach a range of subjects and can’t possibly be expert in all of them. But English primary schools have subject coordinators ______who assume leadership roles for particular aspects of the pri- mary curriculum. Mathematics coordinators provide guidance NUMBER in curriculum matters and take the lead for in-school staff de- velopment. They should be able to shed light upon the under- Children’s formation of number concept and counting moti- lying complexities of primary mathematics and have under- vated discussion of cardinality (the Russell definition of natu- standing of its continuation in the secondary school. So, what ral number), ordinality and a formal description of bijective follows is a brief description of the specialist maths compo- equivalence of sets (finite or infinite). (For secondary B.Ed. stu- nents of a primary B.Ed. degree that took a different approach dents, the Peano Axioms would be included). towards extending the subject knowledge of those primary teachers who were likely to become maths coordinators. Classification of : figurate numbers (square, rectangle, triangle etc), prime numbers and the fundamental Prior to choosing maths as a specialist study, students would theorem of arithmetic and other aspects, such as additive par- have completed several basic curriculum modules (Number, titioning and digital roots etc. Shape & Space, Data Handling and Algebra), which had a strictly pedagogic emphasis and prompted students to reflect Initial discussion usually focussed upon primary school activi- upon their previous learning. But the later specialist modules ties and then became more analytical, as exemplified by the were founded upon the following principles: following assignment given to final year primary B.Ed students as part of their coursework: 1. Mathematical topics were considered from various perspectives, so complex numbers, for instance, would Assignment be viewed traditionally as extended solutions of qua- dratics; they would be introduced in the form of 2x2 (a) By applying the Fundamental Theorem of Arithmetic, find matrices and, finally, as a of (R, +, x). a natural number, x, such that: Geometric and wider algebraic relevance would also be explored. x4 – 1296 =0. 2. Teachers at one level (infant, primary, secondary, higher education) should have insight into the nature Hence show that, if x is a triangle number, it is possible for x2 of the mathematics that is taught at subsequent and to be a triangle number. Is a generalisation possible in this re- preceding levels. spect? For the value of x specified by the above equation, de- 3. Excursions into ‘advanced’ topics, wherever possible, termine the triangle number that is nearest in value to x3 and emanated from analysis of a relevant aspect of school that which is nearest in value to x4 . mathematics. 4. Equal emphasis was placed upon mathematics as a (b) Again by use of the FTA, decide which of the numbers 1225 body of knowledge (content objectives) and as a way and 1485 is a square number. Prove that both are triangle num- of knowing (process objectives), and coursework bers and determine how many other triangle numbers and how therefore included mathematical investigations and many square numbers lie between them. special studies with an historical emphasis. (c) If the difference between consecutive triangle numbers,

14 August/September 2005 FOCUS

tm and tm+1 , is 1000, find, for the same value of m, the differ- lum modules, but the approach now became more serious. For ence between the consecutive square numbers sm and sm+1 . example, whilst it is true that the vast majority of students/ teachers may accept that 13= 03. , too many are unable to ex- Is there any evidence to suggest that triangle numbers occur more or less frequently than square numbers? Is there any va- plain why this is so. But statements like 1 = 09. always gener- lidity in the claim that there are fewer of one of these two types ate widespread disbelief, illustrating the sort of intuitive and of number than the other? Justify your answers to both ques- mathematical leeway to be made up when teaching this aspect tions of number. Nonetheless, the connection between rational num- bers and finite or recurring decimals has to be established and (d) A group of children are building cuboids amongst which formally proved. Naturally, this involves intuitive ideas of lim- are one with a volume of 2079cm3 and one of volume 2464cm3 its and questions of infinite processes (e.g. every rational num- but they find that these two particular cuboids have bases of ber is the sum to infinity of a GS). equal area. Calculate the dimensions of both cuboids given that they are all specified in terms of natural numbers (greater than Many teachers (trainee or otherwise) have a poor grasp of the 1). structural properties of number and tend to confine their insights to algorithmic procedures. So, when distinguish- (e) For two natural numbers, a, b, the concepts of highest com- ing between Z and Q, say, it comes as a surprise to realise that mon factor (hcf) and lowest common multiple (lcm) can be a concept such as consecutiveness has no meaning in Q. And, defined as follows: apart from the algebraic difference between the two systems, there is also the anti-intuitive revelation that Q is dense whilst (i) x = hcf(a,b) if natural numbers m,n can be found such also being countable. that a = mx and b = nx where m and n are co-prime. (ii) y = lcm(a,b) if a|y and b|y and if, also, a|X and b|X⇒ y|X The Euclidean geometric derivation of rational numbers would also be considered, together with many surprising results like ,

Illustrate these definitions by means of one specific example ∞∞1 1 1 3 for each. ∑∑nn== 114 10 3 (f) Prove that, for any natural numbers a,b; which arose from the analysis of a primary school activity of dividing a square into thirds by repeatedly taking quarters. Fi- a x b = lcm(a,b) x hcf(a,b) nally, the (ordered) field Q of rational numbers would be con- structed from the (well-ordered) domain Z of all integers (re: Note: Part (d) of this assignment is based upon a hypothetical equivalent fractions). classroom situation due to the policy of assessing subject knowl- edge in the context of classroom activities. Irrationals: Students are strictly averse to irrationals, and sym- bols such as 2 aren’t readily accepted as representing ‘proper Integers: The starting point would be to see how directed num- numbers’. Despite evidence as to the precise geometric exist- bers are introduced in various school textbooks. One such ap- ence of such things, they can’t resist replacing them by some proach portrays them as 1-dimensional vectors and distin- rational approximation; so π has to be 22/7 and 2 has to be guishes between use of the symbols +, – to denote operations something like 1.414. But, having accepted rational numbers as opposed to their use as labels for directed numbers (e.g. the as repeating decimals, one has to take account of the set Q' of difference between 7-3 and 7+-3). Practical and theoretical infinite non-repeating decimal numbers. Algebraically, this is importance of the integers would also be considered and the shown to be different to Q and is not so easily comprehended topic would culminate with construction of the system of in- as a whole (e.g. there is no countable array). But students grudg- tegers (Z,⊕,⊗) from equivalence classes of ordered pairs of ingly accept it as the lesser-known partner in the marriage natural numbers, where ⊕ and ⊗ are the binary operations Q ∪ Q' = R. on Z induced from (N, +, x). So, for the first time in their lives, students would be able to prove that -1 ⊗ -1=+1. Yes, they found Cantor’s diagonal argument introduces the notion of this challenging, but the point was to expose the complexities uncountability, and showing that Q has zero estab- lying just below the surface of school mathematics lishes the fact that Q' is really the senior marriage partner. Informal ways of showing this include the generation of an Rational numbers: Fractions, equivalent fractions and deci- mal representation would have been treated in earlier curricu- 15 FOCUS August/September 2005 infinite decimal number on a random digit-by-digit basis. Next Question 1. on the scene is the probabilistic trauma that, if a real number is randomly selected, then P(irrational) = 1 and P(rational) = (a) Using the laws of the algebra of propositions (provided), 0. A similar conclusion is achieved with respect to algebraic show that the proposition and transcendental numbers. ~(p v q) v (~p ∧ q) The Reals. For older primary school children, there are vari- ous calculator activities devised to enhance estimation skills. is logically equivalent to a one-variable statement. Specify which For example, to calculate the edge-length, L, of a cube whose of the laws is being used for each step of the simplifying proce- volume is, say, 10 = L x L x L , children each produce a list of dure. converging ‘guesstimates’. In the absence of silly errors, each list will form a finite rational Cauchy sequence approximating (b) Two groups of children each design a simplest possible elec- 3 10 . At this juncture, our student-teachers would be intro- trical circuit for the simulation of the scoring system for each duced to the definition of real numbers as equivalence classes of the games they are playing. Each game is based upon the of rational Cauchy sequences with binary operations induced simultaneous tossing of three coins (1p, 2p and 5p) and the from those on Q. Other aspects of (R, +, x) were considered, circuits are to be as follows: such as completeness and its algebraic and practical versatility. For game 1, a bulb lights up if the three coins are either all Complex numbers. This theme was introduced in accordance heads or all tails. with principle 3 given above, whilst the treatment was in line For game 2, a bulb lights up if the three coins show either ex- with that described in 1. In terms of equation solving, the com- actly 2 heads or exactly 2 tails. plex numbers would be revealed as the most versatile of all systems hitherto discussed and the topic would conclude with (i) Using the symbols p, q, r, respectively, to denote the state- the fundamental theorem of algebra. ments for which the 1p and the 2p and the 5p coins land as ______‘heads’, write down a compound proposition for each of the circuits to indicate when the bulbs are lit. Sketch the two cir- That concludes a compressed description of the module on cuits. Number but, as in all modules, the exploration of ideas was rooted in some aspect of school mathematics. For example, in (ii) What is the simplest compound statement logically equiva- the pre-calculus module, the main theme was ‘growth’ and, from lent to the negation of that which represents the circuit for a textbook intended for 12 year-olds, there was an exercise based game 2? Give reasons. upon the notion of repeated doubling for which the book showed the graph of y = 2x as being a continuous curve. How- (iii) What would be the effect of arranging the two circuits in ever, although students (and most teachers) could readily as- parallel? sign values to 2x when x is an integer, many are at a loss regard- ing the meaning of 2x for non-integral values of x (say, x = 2 ). Question 2 Yet, whatever their knowledge of the laws of indices, few teach- For each of the parts (i) to (iv) below, you may select methods ers seem able to apply it to make links of the sort: of proof from: counterexample, exhaustion, contradiction, de- ductive proof or mathematical induction.

1414 221.414 ==1000 1000 21414 ⇒≈ 22 1000 21414 If you employ these (or any other) methods of proof, you should specify the exact point in your argument where this is so. The module on Proof, Logic and Boolean Algebra began with analysis of the range of early school activities described in my (i) State, with proof, the number of interior angles in a penta- article on “What I learned by Teaching Logic and Proof” (FO- gon that may be right angles. Illustrate your proof by means of CUS, May/June 2004). Proof would be discussed in its widest clear diagrams. pedagogical context and the logical structure of various forms of proof would be revealed. Applications of logic to switching (ii) Without reference to decimal representation of real num- circuits has particular relevance to the practical work on cir- bers, prove that, if 5 is irrational, then is 51+ also irra- cuitry that appears in the primary and secondary science cur- tional. 2 riculum, so here’s a couple of illustrative examination ques- tions used for this module.

16 August/September 2005 FOCUS

(iii) Prove, or disprove, that, for all 2 x 2 real matrices A, B, strongly connect school mathematics with its pedagogy and (A+B)2 = A2 +2AB +B2 its historical development. Pursuit of additional mathematical expertise could then be a matter of subsequent in-service pro- (iv) Prove, or disprove, that 7n + 4n + 1 is divisible by 6 for any fessional development. natural number n. Note: Geometry There were a few transcription errors in part 1 of this article (in the May/June 2005 issue). Two of them were mathematical and On entry to higher education, most students have a very frag- have caused some consternation among some readers. Firstly, the mented knowledge of geometry — even those with good A- equation level (high school) grades. So the approach here began with 2 4 2 5 many of the practical, visual and investigative activities that ÷=× 3 5 3 4 characterise the geometry of primary and early secondary years. Lack of space debars adequate description of the work cov- appeared with an addition sign on the left-hand side instead of ered, but isometries, shear, stretch, enlargement would be en- the division sign, which obviously made no sense. Secondly, the countered within the context of classroom activities and then 1 formally defined and then applied to the proof of geometric fraction 3 should have been shown equal to the infinite repeat- (e.g., Pythagoras’ theorem, or a rotational proof of ing decimal 0.333… and not the terminating decimal 0.3333. the fact that, in a circle, angles on the same chord are equal). Work on three-dimensional geometry was carried out in the Peter Ruane ([email protected]) is retired from univer- curriculum modules. sity teaching, where his interests lay predominantly within the field of mathematics education. The first part of this article ap- Conclusion peared in the May/June issue of FOCUS.

The course described above is not offered as any sort of exem- plar to which others might aspire, but merely to indicate how the mathematical preparation of teachers can commence with analysis of school mathematics and be built outwards from We are deeply grateful for the there. As for the students themselves, they were a very mixed generosity of the following individuals, bunch in terms of prior mathematical background, but the who have made bequests to the approach of relating more abstract ideas to school mathemat- Mathematical Association of America. ics was hugely motivating. So the pay-off is that teacher sub- Every bequest is a powerful ject knowledge pertains more strongly to that which they are expression of their loyalty, called upon to teach, and yet there is no implied upper their lifetime involvement, and regarding subsequent mathematical learning. their faith in the future of the MAA. We remember each of them fondly Experience with secondary in-service courses revealed many and with deep personal and gaps in the basic subject knowledge of graduate maths teach- professional respect. ers. Most of the contents of the above Number module would be alien to the majority of them and, with respect to calculus, Barbara C. Beechler very few appreciative of the ambiguities in Leibniz notation Member for 53 years (for example). As for something like circular and hyperbolic functions, there would be appreciation of their behavioural Charles F. Hicks similarities but no awareness of the strong geometric analogy Member for 45 years between the two, and little idea of their historical development. Murray S. Klamkin This, then, is the basis of the claim that the mathematical edu- Member for 56 years cation of teachers should commence with a thorough explora- b tion of school mathematics, together with an introduction to Kenneth E. Kloss the history of mathematics. The best route for the achievement Member for 43 years of such a goal is via appropriately written B.Ed. degrees, which

17 FOCUS August/September 2005

So You Want To Be A Teacher By Jacqueline Brannon Giles

In a two-year college, it may be that or woman” needs to be attended to. De- many of the teachers did not plan to be veloping the climate for students to learn one when they were in college. Over the often requires the attentive heart of a years, they may have been business mother, and the protective demeanor of people, engineers, social workers or in a father. other professions. No doubt their need to connect and communicate with Why do I say these things? Well, in the people is what drove them to the teach- two-year college I have encountered stu- ing profession. Or, as in my case, in 1968 dents who were abused, misused, and a colleague (Dr. Eugene DeLoatch, who confused. Of course, there are those stu- later founded the School of Engineering dents who are gifted, healthy, focused and at Morgan State University) looked at my mature, but we cannot ignore the pres- personality and attributes, and decided ence of the others who also need to be to encourage me to become a mathemat- thoroughly “taught.” As students in the ics instructor. two-year college, they are seeking a bet- Jacqueline Brannon Giles ter way, and another chance to make it perienced hardships and barriers beyond There were great educators in my fam- in life. They are looking for new ways our comprehension deserve an opportu- ily: an aunt taught for more than 40 years and new paths to reach their goals — nity to be taught and to be in the pres- in the Houston Independent School Dis- goals that are often unclear but certainly ence of someone who points the way to trict; and another aunt worked at Texas different from experiences in their past. access and to a greater degree of func- Southern University for 30 years, and We, the teachers, stand before them with tionality in society. founded the Texas Southern University a tremendous amount of influence. In Gospel Choir for students that was led some ways, their lives are in our hands A conclusion is that our job as math- by the outstanding musician V. Michael for each 50 minute period, and their view ematics teachers and as educators is not McKay, and had members such as Jenni- of themselves and their potential can be an easy one; it is a challenging one that fer Holliday, Finnessa White, and other re-shaped, released or distorted and op- we do and learn to love to do it well. We great singers. So I have always been aware pressed, depending on what we teachers realize that our totality of perceptions of the role of teachers, since both aunts do. and experiences are simply an example and my mother nurtured and directed and that the next generation can build me during my formative years. Even if we did not want to be teachers on our core beliefs and extend those be- during our early career, our presence in liefs for their generations. My first teacher was my mother (now 81 the two-year college classroom is confir- years young) who was in the first calcu- mation of our “call” to be there. And, Granted, each generation is dependent lus class at Jack Yates High School in with each call in life comes a responsi- on the previous one, yet if taught well, Houston, Texas. (One of her classmates bility — in this case, for perpetual prepa- will extend and improve, and move to- was the late Dr. Robert Terry, who was a ration and dedication because the stu- ward a more excellent way for all of man- president of Texas Southern University dents sitting before us may have never kind. If this vision is set before us, it at at the peak of his career.) She patiently experienced the supportive, instructive, least directs us toward a more excellent played games and puzzles with me when corrective and even therapy-type envi- way, even if our reality is muttered by I was a child. She corrected my speech, ronment we can create for them. life’s tendency to drift toward entropy: posture, logic and overall presentation in “a state of disorder and chaos.” Never- life, including my choice of clothing and So you want to be a teacher! Yes, because theless, we must aim toward the good, colors. She was a role model and coach. you and I want to do something to assist the just, and the excellent and learn to Her opinion was readily given when I had our nation remain strong and secure. tap into those things that promise true a need, and even when I did not recog- Our role as a mathematics teacher is a excellence in this professional/academic nize my need. Her keen eye and steady role in the “army” at home in the United journey. gaze would scan me up and down, and States. To fail to teach and to learn more in and out to identify that I had attained about teaching this generation translates Jacqueline Brannon Giles is on the edito- the level of excellence that she expected into failure to be an effective citizen in rial board of FOCUS. She teaches at the in all of her children. the United States. Our mindset should Houston Community College System — be that all students deserve a chance (and Central College. She wrote this article I claim that this level of attentiveness is even a second or third chance) to develop thinking of some of her students and needed in classroom teachers. We are into fully functional citizens. We have friends who had expressed the desire to be beginning to realize that the “whole man to believe that even those who have ex- teachers. 18 August/September 2005 FOCUS

U.S.A. Mathematical Olympiad Winners Honored

The 34th U.S.A. Mathematical Olympiad Awards Ceremonies took place in Washington DC on Sunday and Monday, June 26 and 27. This event honors the twelve top winners of the annual USA Mathematical Olympiad exam, the premier high- school level mathematical problem solving competition in the United States. The two day celebration began with a Sponsors’ Reception at the MAA Headquarters. Representatives of the sponsoring organizations of the American Mathematics Competitions along with members of the MAA Ex- ecutive Committee, were there to meet and greet the winners and their families. On Monday morning, the winners toured the Cryptologic Mu- seum at the National Security Agency and enjoyed a talk from one of the on site mathematicians on classical cryp- tography.

The 2005 USAMO winners (in alpha- USAMO winners from left to right: Zhou Fan, Yi Sun, Robert Cordwell, Hyun Soo Kim, betical order) are Robert Cordwell of Brian Lawrence, Eric Price, Rishi Gupta, Peng Shi, Yufei Zhao, Sherry Gong and seated on Albuquerque, , Zhou Fan of foot: Albert Ni. Standing to the left: Tina Straley, MAA Executive Director and to the right: Parsippany, New Jersey, Sherry Gong of Carl Cowen, MAA President. Photograph taken at the Einstein Memorial on the National San Juan, Puerto Rico, Rishi Gupta of Academy of Sciences grounds, Washington, DC, Sculptor Robert Berks. Photograph cour- Cupertino, California, Hyun Soo Kim of tesy of Robert Allen Strawn. River Edge, New Jersey, Brian Lawrence of Kensington, Maryland, Albert Ni of named in honor of Gerhard C. in the world of mathematics competi- Carmel, Indiana, Natee Pitiwan of North Arenstorff, twice a winner of the tions. Andover, Massachusetts, Eric Price of USAMO and a member of the first USA Falls Church, Virginia, Peng Shi of team in the International Mathematical The highlight of the evening came when Toronto, Canada, Yi Sun of San Jose, Olympiad. the Akamai Foundation Scholarships California, and Yufei Zhao of Toronto, were presented to the 1st place winner, Canada. Dr. Olsen, in her pre-dinner address, read Brian Lawrence, 2nd place winner, Eric a letter with laudatory greetings and con- Price, and the tying 3rd place winners, Dr. Kathie Olsen, Associate Director for gratulations from President Bush. Peng Shi and Yufei Zhao. These scholar- Science at the Office of Science and Tech- ships are in the amounts of $20,000, nology Policy in the Executive Office of After dinner, Brian Lawrence received the $15,000, and $10,000 (divided) respec- the President, was our host in Dr. John Samuel L. Greitzer/Murray S. Klamkin tively. By awarding these scholarships, the Marburger’s absence, at the celebratory Award for his superior achievement in Akamai Foundation hopes to encourage reception and dinner at the U.S. Depart- the Olympiad exam. Dr. James Carlson, these and other students to continue ment of State. The formal awards cer- President of the Clay Mathematics Insti- their pursuit of mathematics education. emony, presided over by MAA President tute, designated Sherry Gong as the sev- Carl Cowen, took place at the National enth CMI Mathematics Olympiad On June 12th the students traveled to Academy of Sciences. Dr. Donald G. Scholar. Sherry best fulfilled the prize’s Lincoln, Nebraska to participate in the Saari, professor of mathematics and eco- criteria of elegance, beauty, imagination, Mathematical Olympiad Summer Pro- nomics at the University of California- and depth of insight. The newest prize is gram (MOSP) for advanced training for Irvine, delighted the winners and the the Robert P. Balles Distinguished Math- the International Mathematical Olym- audience with his USAMO Address ematics Student Award, given to each of piad (IMO). This program is funded in titled “Mathematics is Everywhere.”The the twelve winners, in an effort to recog- part with a grant from the Akamai Foun- winners received the USAMO Medal, nize and reward their high achievement dation.

19 FOCUS August/September 2005 Math Youth Days at the Ballpark By Gene Abrams

About 100 times bigger, expressed by ERA is: if the and about 100 times louder! pitcher pitches a lot of com- That’s how I’d compare the plete 9 inning games, how environment at the four re- many runs would we expect cent Sky Sox Math Youth Days him to allow each game? to the environment of my About 3.] typical math classrooms. The contest works as follows. For the fourth consecutive Starting with the break be- year, the math department at tween the first and second in- the University of Colorado at nings, and continuing for the Colorado Springs has next five half-inning breaks, partnered with the Colorado one of our UCCS math stu- Springs Sky Sox baseball team dents stands on top of the Sky and Agilent Technologies to Sox dugout with the wireless provide mathematics-based microphone and reads a activities for kids at the question. The question is ballpark. The general idea of Students enjoying a day of baseball and mathematics. clearly heard by everyone in Math Youth Days is this: the stadium. Simultaneously, throughout the academic year, students given below. Why “Mound Master”? Be- an abbreviated version of the question in grades 4 through 8 from schools cause the student who wins the contest appears on the scoreboard. The school- throughout the Pikes Peak region com- gets to throw out the ceremonial first children are encouraged to work to- plete various activities which connect pitch of the ballgame!! gether, work with their teachers, use cal- mathematics to baseball (samples given culators if necessary, in order to come up below). The culmination of those activi- The second activity is called the Stat Star with the correct answer. Here are some ties is that the kids (and their teachers) competition. As students enter the sta- sample Stat Star questions: get to spend a whole school day at a real dium they are given an 8 1/2 x 14 inch baseball game! (The games have an un- sheet which contains information about 1. Last season my favorite Sky Sox pitcher usual 10:35 AM start time to accommo- that day’s game, and about baseball in gave up 90 earned runs in 180 innings date the school schedule.) general. One section contains up-to-date pitched. What was his Earned Run Aver- player statistics, including such quanti- age last season? There are two math activities that we ties as batting average, home runs, runs deliver at the ballpark. The first is called batted in (for hitters), and innings 2. Henry Aaron holds the major league the Mound Master Competition. About 30 pitched, strikeouts, earned run average record for career home runs with 755. In minutes before the start of the game, (for pitchers). Player names and uniform actually circling the bases during those 755 eight kids from eight different schools are numbers are listed so that kids can keep home runs, how many miles did he run? randomly chosen to come down on the track of who is on the field during the (Give your answer to the nearest mile.) field to participate in this single-elimi- game. A second section of the Stat Star nation contest. Each pair of students is Sheet contains a handful of information 3. Sky Sox pitcher Jason Young’s curveball asked a question; the first one to raise items regarding how some of these sta- travels sixty feet to home plate in half a her/his hand and give the correct answer tistics are computed. They might specify, second. How many miles per hour is that? moves on to the next round. (If both kids for example, that the distance between (Give your answer to the nearest m.p.h.) answer incorrectly then the one whose bases is 90 feet. Or they might be more answer is closest to the correct answer complex: As you can see, students need to bring moves on. Kids must answer using ‘men- some of their own knowledge to the tal math’; no calculators allowed here!) A pitcher’s Earned Run Average (ERA) question (e.g. number of feet in a mile), The quizmaster has use of a wireless mi- is: (R / IP) x 9 where R = number of as well as some knowledge of baseball crophone, so questions can be heard runs allowed and IP = number of innings (which they get from the information clearly by everyone in the stadium. Stu- pitched. Earned Run Average is expressed sheets). Once all five questions have been dents in the stands are encouraged to play to two places of accuracy. So, for instance, announced, students are given an inning along (without shouting out answers), if a pitcher has allowed 50 runs, and has to complete their answers and write them and to root for their classmates! Some pitched 150 innings, then his Earned Run in the space provided at the bottom of sample Mound Master questions are Average is (50 / 150) x 9 = 3.00. [The idea the sheet. They then write their names

20 August/September 2005 FOCUS next to the answers, tear these off from the chance to experience, in yet another powerful impression on kids when they the bottom of the sheet, and put them in venue, how mathematics really is all find out that the reader is “…a captain boxes which are circulated through the around us. in the Air Force and a master’s degree stands by the UCCS students. Our UCCS student in Applied Math at UCCS…”, or students then sort through these answers, “… a young woman with a 3.9 GPA. as a distill from them those which mathematics major, who have all five answers correct, is attending her first ever and place correct answers in baseball game today …”, a box. The winner of the Stat or “… a high school stu- Star competition is chosen at dent who just this morn- random from among the cor- ing took a final exam in rect responses. The winner’s calculus, and will attend name is announced, and s/he college next year as a math gets to come on top of the major …”, or, perhaps dugout and receive prizes most compellingly, “… a from both the Sky Sox and second year math major from UCCS. More impor- and mother of two young tantly, the winner is show- students who are in the ered with wild cheers from ballpark today …”! her/his classmates! After the game, the Stat Star questions Gene Abrams, is Professor (and answers) are posted at of Mathematics at the Uni- the Sky Sox website at http:// versity of Colorado at Colo- www.skysox.com for stu- rado Springs. He has been dents, teachers, and parents a faculty member at UCCS to use as a part of follow-up since 1983. He is the author activities. The Sky Sox mascot looks on as winners from the Stat Star contest are of more than two dozen re- announced. search articles in math- All in all, the day is fun and ematics. Along with his col- exciting for the school kids (more than Over and above all of these positive as- league Jeremy Haefner, Abrams co-devel- 12,000 this year!), their teachers, and the pects of the day, in my opinion perhaps oped the MathOnline program at UCCS volunteers from UCCS. Both the Sky Sox the most important thing the kids take in 1998. In 1988 he was named the UCCS and UCCS have gotten significant posi- away from the day is the following. In campuswide Teacher of the Year. In 1996 tive feedback regarding all of the activi- large part due to our desire to help de- he earned lifelong designation as a Uni- ties. The UCCS volunteers provide a stroy harmful stereotypes regarding versity of Colorado systemwide President’s valuable community service in an enjoy- mathematics ability, prior to each of the Teaching Scholar. In 2002 he received the able environment. The school teachers Stat Star questions a Sky Sox representa- annual Burton W. Jones Outstanding not only have a good time, but also are tive gives a brief introduction of the Teaching Award from the Rocky Mountain able to provide valuable mathematics les- UCCS math student who will read that Section of the Mathematical Association sons for their kids. Of course the kids get question. I believe it makes an extremely of America.

Sample yearlong class Sample Mound Master Questions or individual projects If a batter hits a home run, he runs 360 feet around the Calculate the total distance traveled by the team for the bases. If he only hits a double (and runs to second base), away games played during last year’s Sky Sox season. how far does he run?

Research the salaries of major league baseball players The Sky Sox expect 5,000 fans at the ballgame today. If today, including the Colorado Rockies. Compare the the Sky Sox play 70 home games, and they draw this salaries of early players (e.g. Babe Ruth) to those of many fans to each game, how many fans will have at- today’s players. Make a chart of graph showing salary tended a Sky Sox game by the end of the season? increases or decreases. The Sky Sox play at the AAA level, one step below the Measure and calculate the average distance your class- major leagues. Over the years, 75% of Sky Sox players mates can throw a baseball. (Three throws each.) Make have gone on to the majors. If the Sky Sox currently a list of the average distances. have 24 men on their roster, how many of the current players would we expect to go on to the majors?

21 FOCUS August/September 2005 The Fundamental Theorem of ______

By Jeffrey Nunemacher

In 2003 I needed to design a written cal algebra, single-variable calculus, pro- Cauchy’s Theorem, which leads directly Honor’s Exam in mathematics as part of jective geometry, and Galois theory. All to the Cauchy Integral Formula and the my duties as the outside examiner for the these areas of mathematics have a named Residue Theorem. Actually when I teach honor students at a sister institution. As fundamental theorem. One could also complex analysis, I prefer instead a a final question, I wanted something that include the theory of Abelian groups, broader theorem, which includes would touch the whole mathematics cur- even though it is a rather narrow sub- Cauchy’s result. I call it the Grand riculum and leave some room for indi- area within . In case you Equivalence: the class of analytic func- viduality and opinion. I decided to ask have never studied projective geometry tions can be defined on an open disk in the students to state what they consid- (it used to be the most common sort of the complex plane by any one of the fol- ered to be the fundamental theorem in a geometry to encounter in the under- lowing conditions: complex differentia- variety of named areas of mathematics graduate curriculum), the Fundamental bility (differential calculus); Cauchy’s and briefly to defend their choices. Re- Theorem of Projective Geometry asserts Theorem (integral calculus); the Cauchy- action to this question was sufficiently that given two four-tuples of non-col- Riemann equations (partial differential positive — from both students and fac- linear points in the projective plane, there equations); power series (infinite series) ulty — that it seems worth posing this is a unique projective transformation ; and conformality (geometry), at least question to a broader audience. Think- mapping one to the other. As an under- away from points where the derivative is ing about this issue is a good way to so- graduate at Oberlin in the late sixties, I zero. lidify one’s understanding of a math- well recall learning each of these funda- ematical subject. mental theorems and trying to under- For classical statistics the fundamental stand why they were deemed fundamen- theorem is surely the Central Limit What should be the characteristics of the tal. Theorem, which relates the Gaussian dis- fundamental theorem of a branch of tribution to a well behaved sum or aver- mathematics? Probably we all agree that Into category (2) I place multivariable age of a large number of independent such a theorem should have far-reach- calculus, (ordinary) differential equa- random variables. It is this result which ing applications within the branch, tions, complex analysis, classical statis- sheds light on the behavior of large should not be too trivial, should not be tics, and perhaps Euclidean geometry. In samples and allows the construction of too esoteric, and should somehow cap- each case I think there is a definite best the most common kinds of confidence ture the essence of the subject. In addi- choice (with which you may certainly intervals and significance tests. tion, it would be nice if the fundamental disagree). For multivariable calculus it is theorem had a proof founded on basic the general Stokes’s Theorem, which re- Finally, for Euclidean geometry the natu- principles and with some aesthetic ap- lates integration over a set to behavior ral candidate is the analog of the projec- peal. These attributes apply, for instance, of some kind of antiderivative on the tive fundamental theorem. My candidate to the Fundamental Theorem of Arith- boundary of the set. In dimension 1 this is the assertion that given any two three- metic, which asserts the existence and result reduces to the Fundamental Theo- tuples of non-collinear points in the Eu- uniqueness (up to order) of the prime rem of Calculus, and in dimensions 2 and clidean plane, there exists a Euclidean factorization of any positive integer 3 it yields Green’s Theorem, the Diver- transformation, which maps one to the greater than one. gence Theorem, and the classical Stokes’ other, if and only if the respective dis- Theorem. tances are preserved, and such an isom- With this in mind, we might divide un- etry is unique whenever the three dis- dergraduate mathematics into four cat- For ordinary differential equations the tances are distinct. If one takes seriously egories: (1) those for which there is a gen- fundamental theorem is the basic exist- Klein’s view that geometry is the study erally agreed upon fundamental theorem ence and uniqueness theorem, which of those properties of a space which are that is called “the fundamental theorem guarantees that every smooth system of preserved under the action of a group, of ______”; (2) those for which a best ODEs has a smooth local solution (the then this assertion is the basic result upon choice can be made and defended; (3) flow). Since we now typically use soft- which all of the rest of Euclidean geom- those for which there are several good ware to sketch these solutions so that we etry depends (even though a typical high candidates to be called the fundamental can study the geometric behavior of the school student never encounters it!). theorem; and (4) those areas for which flow, it is clearer than ever that this theo- it is not possible to identify a single most rem, which asserts the existence and Let us now consider Category (3), con- fundamental theorem. Let us briefly con- uniqueness of the flow, is the central re- sisting of areas for which there are sev- sider each of these categories. sult of an ODE course. For complex eral reasonable alternative choices. Into analysis most mathematicians would this grouping I place linear algebra, ba- Into category (1) fall arithmetic, classi- declare that the fundamental theorem is sic real analysis, ,

22 August/September 2005 FOCUS group theory, and advanced real analy- These two results are related, but they are ness of the spaces L2 and L1 is a more sis (measure theory and Lebesgue inte- not the same theorem. Which is more fundamental result. This fact of com- gration). fundamental? Notice that unless we con- pleteness establishes that the extension struct the reals from the rationals, we to these natural classes of Lebesgue inte- Here are three possible choices for the cannot select the completeness of R as grable functions has filled all the holes fundamental theorem of linear algebra our fundamental theorem, since we de- left by consideration of only continuous with a one- sentence reason for each. sire a theorem and not a postulate. or Riemann integrable functions.

(1) Every nonzero has a ba- In mathematical logic there are two pos- What is left to fall into Category (4) sis. If linear algebra is the study of vector sible choices, both due to Gödel. The (those with no obvious choice for a fun- spaces, then the space is completely de- Completeness Theorem guarantees that damental theorem)? Here is a short list: scribed once a basis is known. in any first order theory anything which (point set) topology, combinatorics, is true (in all models of the theory) is also , graph theory, nu- (2) Strang’s Fundamental Theorem (see provable. This result shows the strength merical analysis, and number theory. his Linear Algebra and Its Applications): of the concept of first order logic. But Some would argue that the fundamental Given any matrix there are associated the Incompleteness Theorem guarantees theorem of combinatorics is Newton’s four fundamental subspaces, which have that in any (simple) theory strong binomial theorem (perhaps, with its gen- dimensions determined in terms of the enough to encompass arithmetic there eralization for non-integral exponents). rank and certain orthogonality behavior. exist true statements which are not prov- While this result underlies the solution If linear algebra is the study of matrices able. This result focuses on the limita- of many different counting problems, I or linear mappings, then the geometric tions of classical logic and is the single remain unconvinced that it deserves the action of a rectangular matrix on Euclid- result of logic most important to non- title of the fundamental theorem. ean space is fully described by this theo- logicians. rem. If number theory is merely the natural Here are two candidates for the funda- continuation of arithmetic, perhaps we (3) Every square matrix is similar to a mental theorem of group theory of very should regard the Fundamental Theorem matrix in Jordan canonical form. This different levels of sophistication. The of Arithmetic as the correct result as well statement includes diagonalization First Isomorphism Theorem decom- for number theory. That choice, however, theory and is the key result in many ma- poses any group in seems too mundane when we contem- trix calculations in linear algebra, phys- terms of its kernel and, in particular, de- plate the Theorem or ics, and elsewhere within mathematics. I termines all homomorphic images of a Wiles’s proof of Fermat’s Last Theorem. find it hard to choose any one of these group. This result focuses on the nature I leave you to ponder the appropriate over the others. Of course, some main- of mappings between groups. The very fundamental theorem in these areas. tain that linear algebra is a subject with- difficult classification theorem identifies out any theorems, but that position is too all finite simple groups. Thus it estab- Jeff Nunemacher enjoys teaching a wide extreme for most of us. lishes the building block from which all range of mathematics courses and some finite groups are made. computer science at Ohio Wesleyan Uni- For basic single-variable real variables, if versity, where he has been chair for twelve we have in mind the structure of subsets Finally, here are two candidates for the years. He was educated at Oberlin and of the real line, then a reasonable choice fundamental theorem of advanced real Yale and has previously taught at UT is the compactness and connectedness of analysis. The Dominated Convergence Austin, Oberlin, and Kenyon. (email: a closed bounded interval. But if we fo- Theorem provides a useful and simple [email protected]) cus on functions rather than on sets, we condition under which the Lebesgue in- might select the fact that a real-valued tegral behaves nicely with respect to con- Editor’s note: Respnses to this article are continuous function on a closed inter- vergence. It is likely to be the most used encouraged! val attains its maximum and minimum result from this theory in applications. as well as all values in between them. But it can be argued that the complete-

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23 FOCUS August/September 2005 2005 Award Winners for PACIFIC-NORTHWEST

NORTH CENTRAL

IOWA

Chris Meyer Pacific Lutheran University

NORTHERN CALIFORNIA, NEVADA, HAWAII Ivy Knoshaug Bemidji State University

ROCKY MOUNTAIN Keith Stroyan University of Iowa

INTERMOUNTAIN

Russell Merris California State OKLAHOMA-ARKANSAS University, Hayward

SOUTHERN CALIFORNIA- Bryan Shader NEVADA University of Wyoming

Afshin Ghoreishi

John Wolfe Oklahoma State University SOUTHWESTERN Jacqueline M. Dewar Loyola Marymount University TEXAS

Art Duval University of Texas, El Paso

John Quintanilla University of North Texas

24 August/September 2005 FOCUS Distinguished Teaching NORTHEASTERN

SEAWAY

WISCONSIN MICHIGAN

David L. Abrahamson Rhode Island College

Jim Conklin Ithaca College Michael Slattery Professor Ted Sundstrom Marquette University ALLEGHENY MOUNTAIN METRO NEW YORK Grand Valley State University

OHIO

Jim Reynolds Professor Melvin Hausner Clarion University of Pennsylvania David Singer MD-DC-VA Case Western EPADEL Reserve University

KENTUCKY

Professor George Nakos Elizabeth McMahon US Naval Academy Lafayette College

Dora Ahmadi SOUTHEASTERN Morehead State University

FLORIDA LOUISIANA-MISSISSIPPI

David Stone Georgia Southern University

Talmage James Reid Patrick McDonald The University New College of Florida of Mississippi

25 FOCUS August/September 2005 “I Like Change” An Interview with Tina Straley By Don Albers

Tina Straley was named Executive Di- DA: Were your parents also born in the rector of the MAA in January 2000. Her U.S.? path to Washington and the position of Executive Director had its start in Brook- TS: Yes, both in New York. My grand- lyn with stops in Florida, Alabama, and parents all emigrated from Eastern Eu- Georgia. It's unlikely that there is a stan- rope in the big wave of immigration from dard path to becoming an executive di- 1890 to 1910. rector, but in the interview that follows we examine her trajectory and learn what DA: Did your mother work outside of attracted her to the job. During her first the home? five-year term, she was at the center of growth and many changes, expecially in TS: When I was young, she was a stay- the area of professional development. at-home mom. When I turned 12, she Now in her second term, she is excited went to work. She was a saleswoman in about further growth of professional de- women’s clothing. velopment programs and starting the new conference center. For someone at- DA: And your father, what stands out in tracted to initiating new things and your memory about him? change, the MAA has proved to be a good match for her. TS: He was a good guy. Everybody loved him. In Florida, he was a food sales man- The following interview was done in Dr. ager. His first product was beer, later soft Straley's office in March of 2005. drinks, and then a whole line of pack- Young Tina minus green paint. aged food. He traveled extensively Don Albers: Where were you born? throughout South Florida. He knew ev- erybody because not only would he meet Tina Straley: I was born in Brooklyn, Somebody got paint and brushes. I was the people working in the stores, but the New York. We lived in a six story apart- very small, and I was also one of the people who came to shop as well. He al- ment building of mostly young families. youngest. My mother dressed me that ways carried candy because he had a real I remember my friends there, and I re- day in brand new blue jeans, and remem- sweet tooth. When I was small, he’d al- member the neighborhood park. My ber I said I was the smallest. We only had ways give me little candies. To my daugh- aunt and uncle and two cousins and my green paint. All the big boys painted the ter Jessica, I think that stands out most. grandparents also lived there. But then roof and the paint dripped everywhere. Whenever she saw him, Grandpa Al al- we all moved to Florida when I was seven. I went home covered from head to toe in ways had some little hard candy for her. green paint. DA: This had a real nuclear family aspect DA: You had an early interest in draw- to it. Do you remember things that you DA: How did your mother react to a ing and still take art classes. Did you have particularly enjoyed as a little girl before green Tina? an early interest in reading, too? going to Florida? TS: That day she was not pleased, but TS: It was not until I was in school that TS: I always enjoyed drawing. My sister she never punished me for anything. My I became interested in reading. I first re- Carol, who is older than my cousins and mother was the kind of person that you member reading books on my own in I, used to play school with us. She was could confide in. She was supportive, third grade. I read the Bobsey Twins and the teacher. We used to write stories, tell and she had a wonderful sense of humor. Gulliver’s Travels. I also read one Nancy stories, and do drawings. Although this She loved laughing. If everybody went Drew after another. We moved a lot and was in Brooklyn, behind the apartment to a restaurant and laughed, then for her books were hard to move. We had a set building was a big empty lot. We used to it was a good time and she thought the of encyclopedias that had been bought catch butterflies and lightning bugs there. food was excellent. If there was no laugh- for my sister and some assorted books To us the lot was a field. There was a little ing, then the food couldn’t have been my mother must have liked. In addition shack on the field with nothing in it. The very good either. to the ones I mentioned, there were short kids decided to make it a clubhouse. stories by Guy de Maupassant, plays of

26 August/September 2005 FOCUS

Shakespeare, a set of encyclopedias, and know. I considered chemistry, but never TS: When I went to Georgia State, I just a few other books. One was a book called really pursued it. I always wanted to do continued along in calculus and beyond. Nobody’s Boy, which was very sad. By the something in art, so I think it may have However I had to start over in the sopho- time I was in the fourth grade I had read been the first or second semester that I more English sequence. Therefore, I was all the children and teen books we had. ahead in math, but I was no longer ahead In fourth, fifth, and sixth grades I read in English. the encyclopedia, the plays of Shakespeare, and the short stories of de DA: So that helped you make the choice. Maupassant, not all, of course, but a good number of them. By sixth grade, I was TS: I got to interesting courses in math- an avid reader. ematics faster than in English. And truth- fully, I liked calculus a lot more than any Prison Windows other course. I loved the ideas of infinity and infinitesimals. DA: Was school fun at that stage of life? At the end of my sophomore year, I had TS: I didn’t like going to school. I liked to declare a major. I thought, “I bet if I doing well in school. But I didn’t like major in math, I could always get a job.” being in school. I remember feeling the There’s actually more to the story. I had windows looked like prison windows. I a scholarship from the State of Florida didn’t like school until activities became to study actuarial science at Georgia available. But I still didn’t like the classes. State. That was part of my reason for I liked all the activities. That’s why I went leaving Florida and going to Georgia to school. And yet I was a good student. State. As long as I was studying applica- I worked hard, and I got good grades, and Math major Tina ready for the next tions of calculus, which comprised the I didn’t realize it at the time, but I look challenge. first actuarial exam, I liked it. At the be- back now and I think I was always pretty ginning of calculus class, everyday, Mr. competitive for grades and positions. decided I would try architecture because Eason, my instructor, walked across the I thought it would combine mathemat- front of the room, he’d look at me and DA: How did you choose a college? ics and art. The first course in the archi- ask, “Are you a math major yet?” I’d say, tecture program was on building mate- “No.” Then he’d shake his head and start TS: I went to a public college because I rials. Talk about a bad way to start. It class. had assumed that that was all we could was about cement, and I decided then I afford. I went to a good one–The Uni- didn’t want to be an architect. I dropped Classes were five days a week. He’d come versity of Florida. the course, the only course I ever in the next day, and he’d start class the dropped in all of college. same way. “Are you a math major yet?” DA: What was it like? “No.” Shake his head, and start class. I continued to double major in English TS: Big and intimidating. Adjusting to a and in mathematics. In the summer be- I had been taking actuarial science for a big state university having come from tween my freshman and sophomore couple of quarters. I met with the actu- Miami Beach (which was a very close knit years, I visited my sister who had recently arial science advisor to plan my sched- community) was like going from a small moved to Atlanta. ule. For the junior year and beyond, the town to a big city. My best friend from program was all business courses. I asked, high school was my roommate, and the DA: So when she invited you to Atlanta, “When do I take history and languages? other students from Miami Beach you went up right away. When do I take philosophy, if I have to formed a community. In that sense it was take all of these business courses?” He certainly not going alone. I had a ready TS: I was looking for a job for the sum- said, “When you’re a math major.” He made support system. I had a good time, mer, and I was having trouble finding a was being sarcastic. I looked at him and and I did well. job in Miami. I went up to Atlanta, and said, “bye.” I went directly to the dean’s I did get a job. I had a fabulous time, so I office and changed my major to math- Are You a Math Major Yet? decided to just stay in Atlanta. A friend ematics. In the calculus class the next of ours, the Dean of the Business School, day, Mr. Eason asked, “Are you a math DA: As a freshman, did you have an idea encouraged me to study actuarial science major yet?” And I said, “Yes.” And he what you were going to major in? at Georgia State University. said, “Good.”

TS: I started out as a double major in DA: At that point were you still major- DA: Was Mr. Eason a good teacher? English and math, because I really didn’t ing in English and math?

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TS: Excellent. He was a country guy. He TS: No, not a lot, but not what I would Auburn was the place that Wilt wanted used to come to class with a straw in his call few either. The students were excel- to go, plus I had an NSF traineeship there, mouth. Maybe it was because he couldn’t lent. The faculty was small and collegial so we went to Auburn. When I got there smoke in class. and very supportive. The President and I was very surprised to find that the al- the faculty encouraged the students to gebra was almost totally lacking. I started The next course I took at the end of my pursue Ph.D.’s and return to Spelman as working in topology. sophomore year, the first upper division faculty and role models. course, was real analysis. The next was Lo and behold Auburn hired Curt , which I loved. Lindner from Emory in combinatorics and universal algebra. Lindner had just DA: You eventually ended up doing gotten his degree under Trevor Evans, a universal algebra. leader in universal algebra. Evans had given a wonderful talk at Auburn. It just TS: Right. I liked set theory, linear al- made me wish that I was in a place gebra, and modern algebra. I minored where I could be studying that topic. in philosophy because I loved logic. And then Curt gave a talk as part of his When I was going into my senior year, I interview on similar topics. And I just started getting nervous about what I kept my fingers crossed that they would would do after graduation. I went to hire Curt and he would take the job, and school that summer and took the four he did. I started studying with him right education courses I needed to get certi- away. The next year, I finished my de- fied to teach high school mathematics. gree. I practice-taught one of the terms of my senior year because by this time I actu- DA: That’s fast. ally had gotten ahead in credits. I gradu- ated at the end of winter quarter. TS: Yes. I finished in three years. Curt led a seminar of advanced students DA: Had you been thinking graduate reading research papers. I read a paper school? by Jean Doyen. Because it was in French, I had to translate it; so, therefore, I had TS: When I graduated, Georgia State to think about every word. Curt said, nominated me for a Woodrow Wilson “Gee, that would be such a great result fellowship and they had just started a Tina with her daughter Jessica, who recently if it could be generalized,” and that’s new masters degree program in math- completed her Ph.D. in English. what I did. That was the first chapter of ematics. I was dating my future hus- my dissertation and my first published band. He was in the masters degree pro- paper in the Journal of Combinatorial gram at Georgia State. I realized if I ap- DA: So you were there for one year. Theory. plied for the Woodrow Wilson, I’d have to go someplace else. So I never applied. TS: Right. And then Wilt and I went to DA: Did you and Wilt finish at the same Auburn to graduate school. time? The head of the department at Georgia State asked me if I would like an assis- Auburn TS: No. I didn’t have to teach, and I got tantship there, so I said yes. My onto this problem very early. It’s easier bachelor’s degree was okay, but not great. TS: Then my husband Wilt and I started when you have a well-defined problem However, Georgia State had hired three graduate school at Auburn. Wilt wanted that you can solve; once you solve it, new Ph.D.s just out of graduate school to work in topology, and Auburn had an you’re pretty much done. I went on and to offer this masters program. They were excellent faculty in topology. did other generalizations as part of the all gung-ho. Once I finished the masters dissertation, but the first generalization program, I had a really strong back- DA: Many of the Moore refugees went was the big result. Wilt had heavy teach- ground. I went back home to Florida, and to Auburn from UT-Austin after R.L. ing obligations at some points. He took I taught in the high school I had attended Moore’s forced retirement from the Uni- five years, an average length of time. for one year. Then I got married and went versity of Texas–Austin. back to Atlanta, where I taught for a year DA: So you had two years to do other at Spelman. TS: Auburn was one of the principal things while he was finishing. places the Moore folks went. I was inter- DA: In that period, the mid-60’s, were ested in algebra, and I very naively TS: Right. And again, Auburn was won- there many white faculty at Spelman? thought every place will have algebra. derful to us. They gave me a full-time instructorship for two years.

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DA: Then he was done. What was the DA: And you had a young daughter to about five of us, with Chris Schaufele as next stop? care for. department chair. I was the first faculty member to chair the college-wide pro- Kennesaw TS: Right. I was divorced and a single motion and tenure committee. That was mother, so it was hard to leave a place before I was a chair. In the winter of 1987 TS: Well, it was 1973, and the job mar- that treated me really well. The roots of we split the department between math- ket was awful. When I had finished two my career were strange, but at that point ematics and computer science. I became years earlier in 1971, I actually got let- it was actually on a pretty nice trajectory. acting chair, and then after a search, I be- ters from mathematics departments at Kennesaw was growing, and I was grow- came chair. major research universities asking me to ing with it. The thing that I had to give apply. No one was writing letters to any- up was research. At first I wrote papers DA: Then later you went to NSF. body to apply in 1973. We needed two jobs in TS: I went to NSF in the same location and 1993. we had a new baby. We went back to Atlanta Earth Algebra and took positions not at all where we thought DA: Before coming to we would have ended NSF, what stands out as up. I went to Kennesaw your biggest accom- Junior College, just tem- plishment? porarily, on a one-year appointment. And Wilt TS: It’s hard to say just took a job in the busi- one thing. But it was ness school at Georgia clear that college alge- State under the condi- bra was a disaster. In a tion that he would study department meeting, business courses. the curriculum com- Tina with family: left to right: Tina, Father, Mother, Carol, Jessica, and Wilt. mittee chair presented DA: Wow. That’s quite suggestions, cosmetic a condition. with Curt and on my own. I had to do changes, for revising college algebra. I my work on weekends and during vaca- thought, “I’ve been in this same discus- TS: And he did. He actually completed tions because the demands of the job at sion hundreds of times. I had read the two masters, one in finance and one in Kennesaw were very great. At a place that NCTM Standards; this was 1989 or 1990. international business, and he changed is growing and changing, in addition to I told the department, “I want to throw his career to the quantitative areas of the normal academic committees, we al- out the course and start over.” And I re- business. I took a job at a two-year col- ways had new responsibilities, courses, peated the recommendations from the lege, thinking for just one year, and then policies and procedures, and programs Standards. I said, “I want group work. I I’d look for a real job. to develop. want real world problems. I want writ- ing. And I want technology.” Well, I had a baby, a nice group of col- DA: As the institution grew, you said you “Now, who’s willing to create a new leagues, and pretty good students. I were getting promoted. course?” Chris Schaufele wanted to re- stayed there a second year and a third vise the business calculus course that fol- year. Kennesaw Junior College converted TS: Most of my years at Kennesaw I was lowed college algebra. I said, “first you’ve to a four-year college after just a few the ranking female on the faculty. And got to change college algebra.” He and years. We had a faculty of young Ph.D.s, there weren’t that many men promoted Nancy Zumoff headed up the project. and we created a whole program from to full professor before me. Their challenge was how to make college scratch. We could do anything we algebra interesting. wanted. It was fun. Kennesaw grew at DA: As you said, you were getting pro- six to ten percent a year. We kept adding moted. When did more serious leader- Nancy and Chris came up with the idea students, adding programs, and I kept ship roles begin to develop for you at of the environment being the context of getting promoted. There were so many Kennesaw? the courses. That was brilliant. They cre- opportunities to be creative and to do ated Earth Algebra. I had contacts at new things. What I thought was just a TS: In 1987, I became department chair. Georgia Power Company through the stop-gap position for one-year turned But when you say leadership, it didn’t just state mathematics coalition. They gave out to be a great place for a career. Again start there. As we transitioned from di- us a $6,000 grant to pay part timers. I lucked out. vision to department, there was a lead- Nancy and Chris each had a two-course ership group in the math department of reduction to develop the course in the

29 FOCUS August/September 2005 spring, followed by summer stipends nity to do something new, to do some- So I went back to Kennesaw, but in a to- from Kennesaw. I worked with them on thing exciting, to make a difference. tally different position. the first grant proposal and some on sub- sequent proposals. It’s a long story. DA: That sounds right for some- one who likes change. The project received both FIPSE and NSF funding. In fact, when NSF fund- TS: That’s right. I like change. The ing came through, we already had fund- portfolio that I was given was a bit ing from FIPSE. We received additional ambitious. I was overseeing tech- funding from NSF/DUE. Nancy and nology, the library, sponsored pro- Chris had funding from both agencies grams, assessment, space assign- for the next 10 years for Earth Algebra ments. There were some things in and other courses with similar design. there I told the VP I didn’t want, but I got them anyway. The next NSF year, I also became Dean of Gradu- ate Studies. In four years I was able DA: When you went to NSF in 1993, to restructure the position into a what were you thinking? What was mo- really great job and worked on tivating you? Were you just looking for building up scholarship, external something new? Were you getting funding, and graduate programs. bored at Kennesaw? The MAA Calls TS: I had been department chair for six and a half years, and that is really DA: Well, it wasn’t too much later long enough. that you became aware that Marcia Sward was going to retire as Execu- DA: Your daughter Jessica was settled tive Director of the MAA. at Cornell. TS: Right. That was the beginning TS: She was going into her junior year. of 1999. I went to NSF and she went off to Ox- ford University, England. DA: How did that develop? You The Executive Director does not always wear evening had been very active in the MAA DA: She would have been about 20 in clothes to the office. at the section level prior to that. 1993. So she was fairly independent. DA: This was 1993-1995. You were there TS: I also was active at the national level. TS: Yes. It was a good time for me per- two years as a rotator and then you re- I was the chair of the local organizing sonally. The department was in good turned to Kennesaw. committee for the MAA’s first solo shape, and it was time to make a change. MathFest in Atlanta, and I was chair of One person shouldn’t be department TS: Right, but not to what I was doing. the program committee for the UCLA chair forever. You can’t go home again. You’re differ- MathFest. I was active both at the na- ent, and they’re different. Kennesaw was tional and the section level. I had been DA: But why NSF? keeping my position for me, but I real- Newsletter Editor and Chair of the ized after two years, I couldn’t go back in Southeastern Section. I was also work- TS: Because of Earth Algebra, and the the department and say, hey, I’m back. ing on the MAA books program as Notes fact that I had reviewed at NSF several That’s not fair to the department; they Editor. times. As department chair I had brought had moved on and I had, too. teacher preparation in secondary math- DA: You may not have had much of an ematics into the department. NSF was I told my Dean, Vice President, and Presi- idea of what an executive director did. looking for someone with that interest. dent that if I came back to Kennesaw I My plan in life has always been when an would want to do something different TS: Very little, except that I thought a opportunity comes along, I take it. I’ve that would capitalize on what I had lot of Marcia. When I heard that she was never planned out my career. But there’s learned. I had started a third year at NSF retiring from this position, I told her how something worthwhile about being ready when I got a call from Kennesaw and, as sorry I was that she was leaving. I remem- to capitalize on new opportunities. In they say, got an offer that I couldn’t refuse ber telling her how much MAA meant the case of NSF, here was an opportu- to become associate vice president. And to me. She thanked me for that. That was again, it seemed like a great opportunity. at the Joint Mathematics Meetings in San Antonio.

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DA: So did you begin thinking about the DA: What was the biggest attraction of better. But we had virtually nothing in position then? the job? the area of professional development programs. We had national and sectional TS: No, not at all. In San Antonio Wade TS: The biggest attraction of the job was meetings with mini-courses and short Ellis stopped me and said, “You know, the MAA. This is the association that I courses. But in the whole area of ongo- Marcia Sward is retiring as Executive had been involved with for a long time, ing professional development activities, Director.” He said that he was on the and in which I have felt very much at we had little to offer. Some individuals, search committee for Marcia’s successor. home. To me, the MAA was my profes- on their own, had obtained grants for the I thought he was going to ask me if I had sional family, my professional support MAA to deliver workshop programs; and some good ideas for people to nominate network. I have very strong feelings I, with Brian Winkel, was one of those for Executive Director. Instead he told about this association and the members. people. me the search committee wondered if I Coming to work for an association that would be interested in applying. I was to- I already cared that much about was the I came here with an interest in deliver- tally blown away. I would never on my top attraction. ing programs. We did not have a direc- own have applied for this position if tor of programs. We had a very small people didn’t suggest it to me and ask me DA: You have outlined your biggest fear programs department of one full-time to think about it. So I did. I applied. of the job. In practice, what have you had person and two half-time people. That the most fun with? department supported committees, DA: Has it turned out to be different placement tests, sections, student chap- from what you expected? TS: I love creating new things. I like ters, and liaisons. But they were not in- change. I like coming up with ideas and volved in delivering programs. TS: Of course I had no idea of what the then working with a group on those job would be like, seeing it from the out- ideas; the ideas get refined, and they get We’ve come a long way. We have a full side and not having worked here. I re- better. I love setting new things in mo- spectrum of programs now. But it’s very member talking to you about it during tion, going in new directions. That’s highly dependent on National Science the interview. For example, I realized that what I still like best about the job. Sur- Foundation funding. I would like to get in being the executive director I would prisingly, the running of the business has it to the point of being a self-sustaining have a level of responsibility that I didn’t turned out to be something I like as well. program integral to the MAA. I think have within a state university system, we’re headed in that direction. What where there was always someone I was DA: What’s been the toughest part of Michael Pearson, as Director of Pro- reporting to. As MAA’s Executive Direc- your job? grams, has done is fantastic, and he has tor, I wouldn’t have that kind of support. wonderful ideas for how to grow the pro- TS: The toughest part is personnel. gram and how to bring it to self-suste- I knew that I would have to deal with We’re very lucky that we have an excel- nance. everything from budgets to personnel, lent staff of very dedicated people. But that the entire operation would report there’s been some change to get to that, DA: You’ll soon have the new conference to me. I never thought of myself as a busi- and some hires that didn’t work out, and center. That should help out a bit, too. ness person. I never thought of myself some people who were here who have as anything but an academician in some left. Most of the staff have been, even TS: That will be a great step forward. In cog of the academic wheel. I was con- from the very beginning, just excellent, fact, I hoped that I would be here for two cerned about how I would handle these highly qualified, dedicated people who five-year terms, and I’m delighted that new responsibilities. Raising money are wonderful to work with. Firing some- I’ve been invited to stay on a second term. through a grant and worrying about the body is the worst thing I have had to do. It would really hurt to leave at this point bottom line of an organization’s finances when we’re just getting the conference are two different things. I never had to Big Goals center off the ground. This is something worry about how to pay the bills. And that I’ve dreamed of, and I’m, again, very that’s just an example of the feeling that DA: You’ve now completed your first five fortunate to have the kind of support that you’re out there on a limb in this posi- years, and you recently signed a contract I got from you, Don, and from Jerry tion, responsible for the people who for five more years. What do you hope Alexanderson, and the generous finan- work for you, and responsible to all the to accomplish in the next five? What are cial support of Paul and Virginia Halmos members. I never had that sense of re- your big goals? to make this dream come true. What an sponsibility before. opportunity! And it’s my dream to see it TS: My number one priority has been come to fruition. I knew that those things would be part and remains professional development. of the job, and I was truly concerned Five years ago we had a well established Don Albers is Associate Executive Direc- about my being able to do that job. and highly appreciated publications pro- tor and Director of Publications at the gram. We still do. In fact, it’s bigger and Mathematical Association of America.

31 FOCUS August/September 2005 Third Annual Mathematical Study Tour—Home of the Ancient Maya

The Maya tour was our second tour with the MAA and we plan on participating in others. We enjoy the mathematics history strand that is a core of these tours. It becomes clear that mathematics has a significant impact on the development of cultures. The excellent leadership has developed outstanding tours that are both educational and fun, and the tours have left enough free time to enjoy relaxing in the areas visited. We have also enjoyed making new friends in the mathematics community through these tours.

Gregory Dotseth, Professor Emeritus, Mathematics Department, University of Northern Iowa and Carol Dotseth Gregory

As a participant in all three MAA study tours, I was struck by the focus of this trip. In Greece, the topics covered a time span from Pythagoras through issues in mathematics education in Greece today. In England, our study ranged from the construction of Stonehenge to decoding messages during World War II. This year in Mexico we im- mersed ourselves deeply into the contributions of a single culture, with special emphasis on the classic era of Mayan civilization. The sites were both beautiful and extensive and the guides were superb, offering insights only practicing archaeologists can give. Addi- tionally, what a treat it was to be able to experience new flavors in the regional cooking!

Joel Haack, Interim Dean, College of Natural Sciences Professor, Department of Mathematics, University of Northern Iowa and Linda Haack Linda

I was excited by this trip because I have always had a fascination with ancient civilizations and how they made sense out of their world. It’s great fun to travel with a mathematics group. These individuals frame questions particularly well and organize fragmented information into coherence. I returned with new respect for the Mayan civilization and new appreciation for the archeologists who unearth the remains and interpret their findings. An unanticipated bonus was gaining some insight into contempo- rary issues in Mexican society and especially issues that affect the more than 7 million Mayans in Mexico now.

Joan R. Leitzel President Emerita, University of New Hampshire

The Maya trip was an exciting adventure for this budding historian. Our three guides gave us the most current insights into Maya cosmology. Numerologist Christopher Powell described the construction of prevailing ratios found in Maya architecture. Archeo-astronomer Alonzo Menendes demonstrated how building alignments provided sight lines for celestial events. Finally, Alfonso Morales led us through the complex structure of Maya writing. I’ll be reporting results at the Mediterranean Studies Confer- ence this July as well as the undergraduate Mathematics Colloquium at Dowling College in the fall.

Sandra Monteferrante Professor, Department of Mathematics Dowling College and Steve Frohock

32 August/September 2005 FOCUS

Where do I begin to talk about this trip? I went because I knew that I would probably never be adventurous enough to do something like this by myself. What did I learn? The list is long, but I think what I will take away the most is a much greater appreciation for the Mayan people. Their science and mathematics were much more advanced than I ever imagined. My fellow travelers were awesome! It was a grueling trek some days and I never heard a single complaint from anyone. In addition, our guides (Chris, Alfonso & Alonso) were so knowledgeable and helpful. My personal thanks go out to everyone.

Herb Kasube Professor, Department of Mathematics Bradley University

I teach the History of Math course both at the undergraduate and graduate level, and I always include a section on the Mayan numeration system. So I felt that this MAA Study tour would enhance my knowledge on the subject as well as get to see first hand some of the great Mayan archaeological sites. Well, both expectations were exceeded!!! The expertise of Alfonso, Alonzo, and Christopher is indeed admirable. It was also a distinct pleasure to meet all the wonderful people on the trip. What a great bunch!!! I’m looking forward to future study tours.

Philip Scalisi Professor and Chair, Department of Mathematics Bridgewater State College

I embarked on the tour to learn about Mayan mathematics which I believed I’d be able to incorpo- rate into an inservice course for teachers. But I was also looking to fellowship with a group of mathematicians with similar interests. Needless to say, both missions were realized. As to the trip itself, of course there were the wonderful sites at Chichen Itsa and Palenque that I’d been reading about, but there were some lovely gems of surprises along the way, such as the bumpy ride through the jungles to Bonampak to view the beautiful, still vibrantly colored stelae depicting Mayan court life. I was so impressed by our guides and teachers — Alfonso, Alonso, and Chris. Their love for their work and these enthusiasm for sharing what they knew with us was a joy. As with all great teachers, they inspired me to learn more!

Carole Lacampagne, Retired, U.S. Department of Education

It was extraordinary to watch Alfonso read through a long heiroglyph text almost like reading the newspaper. Robert Bumcrot Retired, Department of Mathematics Hofstra University

Name glyphs drawn by Cristin Cash of the Maya Exploration Center

33 FOCUS August/September 2005 Archives of American Mathematics Spotlight: The New Mathematical Library Records By Robin Howard and Kristy Sorensen

The Archives of American Mathemat- ics is pleased to make an online inven- tory of the New Mathematical Library Records available to researchers. This collection documents the work of editor Anneli Lax to bring engaging mathemati- cal texts to young students of mathemat- ics, and provides an inside look at the work of mathematical publishing.

The New Mathematical Library (NML) is a series of monographs on various mathematical topics. They are not text- books, but are meant as supplements for the interested high school or early col- lege student. The monographs are writ- ten by individual mathematicians, and at the NML’s beginning, most of the au- thors had not written for the high school level prior to their work in the series.

The first monographs appeared in 1961 and were originally published for the School Mathematics Study Group Monograph Project, begun in 1958 to remedy the perceived shortage of well- written mathematical materials for young people. Initially published by Ran- dom House and the L.W. Singer Com- pany in conjunction with , the Mathematical Association of America took over publication in 1975.

The NML was intended as a temporary project, set to come to an end after the publication of approximately thirty monographs or after commercial Ameri- can publishers began to produce similar books for high school students. Instead, books are still being published in the NML series as of 2003, though at a slower A letter from Ivan Niven to Anneli Lax regarding the publication of his book, Numbers: pace than during its height in the 1960s. Rational and Irrational, the first book in the New Mathematical Library series. From the New Mathematical Library Records, ca. 1929, 1957-1997, Archives of American Anneli Lax, the NML’s technical editor Mathematics, Center for American History, The University of Texas at Austin. for almost forty years, was born Anneli Cahn on February 23, 1922, in the Kattowitz, part of Germany at the time, against , moved to Paris, Palestine, in New York University’s Aeronautics but part of soon after. Her fam- and finally, in 1935, to the United States. Department and joined NYU’s Courant ily left Kattowitz for Berlin in 1929 to She studied mathematics at Adelphi Col- Institute as a graduate student in 1943. escape discrimination against Germans, lege and, following her graduation in but in 1933, to escape discrimination 1942, she became an assistant researcher

34 August/September 2005 FOCUS

A handwritten note by Anneli Lax describing a September 18, 1974 meeting at which the transfer of the New Mathematical Library from Random House to the Mathematical Association of America was discussed. From the New Mathematical Library Records, ca. 1929, 1957-1997, Archives of American Mathematics, Center for American History, The University of Texas at Austin.

While at NYU, Lax met and married This collection reflects the progress of the The Archives of American Mathematics mathematician . In 1955 Lax New Mathematical Library, specifically is located at the Research and Collections received her PhD and in 1961 NYU ap- under the editorship of Anneli Lax. It division of the Center for American His- pointed her to the faculty of the Depart- includes correspondence with authors tory on the University of Texas at Austin ment of Mathematics, where she stayed and publishers, outlines and drafts of campus. Persons interested in conduct- until her retirement in 1992. Dr. Lax ac- monographs, and various production ing research or donating materials or cepted the position of technical editor of records. who have general questions about the the New Mathematical Library series at Archives of American Mathematics its inception in 1958, and remained edi- The online inventory for the New Math- should contact Kristy Sorensen, Archi- tor until her death in 1999. The MAA ematical Library Records is available here: vist, [email protected], (512) renamed the series the Anneli Lax New 495-4539. Mathematical Library in her honor in the http://www.lib.utexas.edu/taro/utcah/ year 2000. 00387/cah-00387.html Web page: http://www.cah.utexas.edu/ collectioncomponents/math.html

35 FOCUS August/September 2005

What I Learned… about Online students who do not pass must repeat all of the questions on Assignment Management their next attempt. Mastery protocols do not work well for exams, but they are ideal for short assignments.

By Glenn Ledder 8. Choose the right material for online instruction

Online instruction is fine for routine computation and ideas nline assignment-management (OAM) systems allow O that can be developed through examples, but it is no substitute instructors to create assignments or exams that are posted on for a live class meeting for the teaching of nuanced ideas, the web, taken by students on their own time, and computer- problem-solving strategies, or techniques in which individual graded to provide instant feedback for students and a record steps need a separate presentation. The typical homework for instructors. Interest in these systems can be measured, problem from a textbook is too complicated to be of perhaps, by the number of talks at mathematics meetings on instructional value as an all-or-nothing computer-graded issues related to their use. problem.

The Department of Mathematics at the University of Nebraska- 7. Write good questions Lincoln uses on OAM system called EDU, from Brownstone Learning; however, most of these principles apply equally well Many of my students try to learn the material only when they to other systems. (EDU, as well as Maple T. A. and several have exhausted all other ways to pass an exam. It is good publisher-hosted products associated with textbooks, is instructional practice to help students reach this point quickly. founded on common code architecture developed by Professor Randomized parameters allow for an enormous variety of John Orr of our department.) answers to problems that look almost identical, eliminating any hope of passing an exam by memorizing the answers. Grouping I have been using OAM long enough to compile an impressive problems by category has a more subtle effect. Many students list of mistakes and bad choices as well as a smaller list of good who fall short of a passing mark on an exam will quickly repeat practices. In the hope that others might profit from my errors the exam, hoping to get an easier set of questions. The more without having to repeat them, I have collected here what I closely different settings of an exam are related, the more likely believe are the ten most important principles for OAM use. students will come to appreciate that trying to get an easy Some of the items in the list are specifically for exams or for version is hopeless. This desirable uniformity of appearance daily assignments, while others apply equally to both. Here, and difficulty comes from having an exam constructed by then, are the top ten things I’ve learned about Online random selection of one problem each from several Assignment Management: standardized categories. 10. Use matched sets of questions 6. Set high standards and allow retakes

When several different cases occur in a problem class, it is good Our standard educational system is built around one-shot to require students to work a problem from each case. An exams. These allow us to categorize students by level of example would be two problems of the form “Find the (exact) achievement, but they encourage students to work towards x coordinate of the global minimum of f(x) = 3x3 + bx2 + cx on performance rather than learning (so that the goal is to [-1,1],” where b and c are chosen so that the global minimum complete assignments and get good grades rather than to learn occurs at a critical point for one of the questions and at an the material), and may let them move on without mastering endpoint for the other one. fundamental material. An alternative plan is to set high performance standards and require students to repeat an exam 9. Use a mastery protocol until they achieve the standards. This plan does not distinguish the stronger students from the weaker, but (see point #8) it is In a mastery protocol, students are required to successfully work not necessary to do that on material that is well suited to online one problem from each of several categories. This means that testing. Another objection is that allowing retakes can students repeat only those problems they have not yet worked encourage students not to study for an exam. I solve this correctly, and it also allows the instructor to set up question problem by basing the students’ grades on the date at which hierarchies (see item #5). This is qualitatively different from they pass the exam and limiting them to one attempt per day. the case where a student must get a certain number correct to Students have three days in which they can receive the full 30 pass and has multiple attempts at passing. In the latter case,

36 August/September 2005 FOCUS points for passing my exam. After that, the number of points learned the background material from the previous class they get decreases by one each day. This policy is very effective meeting. Online instruction has the potential to ameliorate at motivating students to pass the exam quickly. this difficulty, provided assignments are timed so that each is due before the next class meeting. There is no harm in 5. Use a question hierarchy extending the short time window as needed, but the standard practice should be to require online work to be done in time to Students learn best from success that build on previous success. prepare the students for the next class. Success rates can be improved by using a hierarchy of questions that allow students to progress from easy questions to hard 2. Minimize instructor commitment ones. For example, consider the problem of finding the 32–cosx No matter how useful OAM is, most faculty are not going to derivative of . Experienced students should find this 47+ sin2 x use it if they perceive it as a large time commitment. I set up problem to be straightforward; however, it does require students an administrative structure that utilizes EDU’s facility to have to successfully combine the quotient rule, the chain rule, and class folders created as copies of existing folders, with the the formulas for the derivatives of the trig functions. Students daughter folders inheriting any materials kept in the parent who have just learned one of these components are not likely folder. I have a master course folder that contains the question to be able to do this problem successfully. I use a hierarchy of banks, the single online examination used by all sections, and the non-credit assignments used by students to practice their 2x problems in which students must first differentiate , skills or study for exams. I also have an enhanced master course 54– x folder that is a copy of the master folder with the addition of a 3x 14+ cos x , and before tackling the desired problem. set of assignments-for-credit used by some of the course 34+ sin x 62+ sin x instructors. These course folders can be reused each semester The average student will probably have to work seven problems just by changing the assignment dates. Each individual class to get through the four-question hierarchichal assignment. The has its own folder that is copied from the appropriate master same student would probably also need seven tries to succeed folder. I find that a one-hour training session is sufficient to at the corresponding single-question assignment, but she will teach instructors how to do the only tasks that they must have got only one question right in the process instead of four. actually perform themselves: changing assignment dates, observing and re-grading student work, and downloading the 4. Encourage students to rework missed problems before gradebook. trying an exam or assignment again 1. Give minimal credit Students who take the time to analyze their mistakes show much greater improvement in their next attempt than students who Few of my students will take the time to do an assignment that don’t; however, few students use this seemingly obvious method does not count for credit, no matter how interesting or valuable of study. I suspect that the problem is the all-too-common the assignment. Yet the same students will do an inordinate performance orientation. To students whose goal is to complete amount of work for minimal credit. Last semester, I gave two the assignment rather than to learn the material, time spent points for each of my 30 web assignments, out of 600 total studying is time not spent on the task of completing the points for the course; my students completed 75% of the total assignment. Even students who have a learning orientation number of assignments possible in the course. My colleague often have not developed good study techniques. It is worth offered the same assignments for no credit, and his students spending a little bit of class time teaching students how to learn completed only 2% of possible assignments. The credit I gave mathematics. for the assignments had only a minimal direct impact on student grades; not a single student passed because of the 3. Use a short time window availability of a few easy points. The indirect impact was much larger: my students were better prepared for class and were able Mathematics lectures generally build on previous material. The to get more out of the limited class time available for the course. standard calculus course includes daily lectures, a small amount of daily or weekly homework, weekly quizzes, and monthly Glenn Ledder teaches at the University of Nebraska-Lincoln. He exams. Most students fall behind during the first week and can be reached by email at [email protected]. only catch up when studying for an exam. This means that most lecture material is delivered to students who have not

37 FOCUS August/September 2005 Letters to the Editor

Another Mac Lane Story pushed a that has ruled Las For me, a science and technology teacher, Vegas out as a conference site! I was it was a very special event to have the Since the next issue of FOCUS will shocked to discover it when my interest opportunity of seeing first hand some of contain an extended obit of Saunders in the Meetings grew in 1980’s. A couple these original items, and thereby to Mac Lane, I thought your readers might of math bigwigs essentially told me, “You experience a degree or closeness to enjoy the following anecdote. forget Las Vegas as long as Mac Lane is Einstein himself. Later on, I was able to alive!” Once I wrote him a very persuasive share this experience with my students. In the Fall of 1959, he was teaching an letter on this issue. He graciously replied algebra course to undergraduates at the in his long hand, but refused to change I subsequently decided to write about I University of Chicago, using (of course!) his mind. had seen, as well as my reactions. These Birkhoff-Mac Lane’s A Survey of Modern thoughts, together with a selection of Algebra. A student asked him why some One may talk of his legacy both archival photographs were published in of the exercises were marked with an mathematical and non-mathematical. an article entitled, “’s asterisk. He replied that those were the They are two separate things. One does Personal Papers: A Physics Teaching ones Birkhoff couldn’t do! not have to study category theory to Resource”. This appeared in the British know its impact on many branches of journal, Physics Education, January 2000 Jack Driscoll mathematics including computer (volume 35, No.1, page 69). science. His influence on mathematics organizations is deep for the number of Samuel Derman Las Vegas and Saunders Mac Lane people who know him directly (his College of Technology colleagues and PhD students at Level 1) Today’s early morning e-mail from my and indirectly (his students’ students, More on Baley Price neighbor wanted me to click on an Level 2, 3, 4, and 5!). To give an idea of internet link of an article from the his hold over math organizations, I never Like Steve Carlson, I obtained three Boston Globe. Its heading read: got a reply from any officer of the AMS degrees from the University of Kansas, “Saunders Mac Lane, developed key or MAA of my written suggestions for all in mathematics. So I was pleased to Algebraic theory; at 95.” I said, my god, holding a Joint Meeting in Las Vegas! It see the article honoring G. Baley Price in this man is doing fundamental research was essentially a one man crusade that I the May/June 2005 issue of FOCUS. But in mathematics at 95! But this reality gave up five years ago. I wish to note that G. Baley chaired the lasted only for a second. In his death on KU Mathematics Department for a April 15, 2005, Mac Lane may have set a Yes, it was five years ago when I last saw longer period than 1959–1970 as record as the longest living and active Mac Lane at a Joint Meeting. He spoke indicated in Steve Carlson’s article. Dr. mathematician of the 20th century! at a panel discussion on Philosophy of Price also chaired the department Mathematics. People were standing wall throughout 1954–1959 during which While replying my Bostonian neighbor, to wall in a big hall to hear him! He had time I completed my BA and MA I said, “He is the man responsible for not notes written and rolled up in yellow degrees. I first met him in the fall letting the Joint Annual Meetings of the sheets. Most of the time, he rambled on. semester of 1954 when my College AMS and MAA take place in Las Vegas! In fairness, that was a great mathematical Algebra and Trig instructor (William Very stubborn man to the end.” performance at 90! Hartnett, later of SUNY Plattsburgh) encouraged me to ask Dr. Price about I continued to muse over my comments. Satish C. Bhatnagar opportunities in mathematics. The result How can you crack a long standing University of Nevada Las Vegas of that conversation was that I enrolled mathematics problem or develop a new in Analytic Geometry the next semester, concept without persistence and A Visit to Einstein’s Papers and then proceeded to major in perseverance? Never! One’s being mathematics. I had never given a thought stubborn in dealing with colleagues may May I add a footnote to Herbert Kasube’s to becoming a math major before this. not be a positive quality. However, in interesting suggestion, “Why You Should certain areas of “pure” mathematics, take a Mathematical Study Tour” Unfortunately I don’t know when G. there may be a correlation between (FOCUS, Dec 2004,Vol 24, Page 9)? Baley first served as Chair of the KU originality and stubbornness. It is not a Math Department. I have tried to find litmus test, though in my sample, it Back in 1999, fulfilling a long-standing that information in the 788 page “History worked with 95% accuracy! desire, I visited the Einstein archives at of the Department of Mathematics, the Hebrew University in Jerusalem. University of Kansas, 1866–1970” which During the first and only Joint Meetings Einstein, in his will, specified that his Dr. Price published in 1976. I may have held in Las Vegas in Jan 1971, Mac Lane papers be kept there. 38 August/September 2005 FOCUS missed it, but it appears to me that G. to do this mapping on page 129 of his subject of the FOCUS piece, and Andre Baley did not consider the precise dates book, but I had not fully understood on Weil (at the time he wrote the letter), I of his chairmanship to merit publication, my very first quick read through the am an incarcerated mathematician. (To though the fact that William Scott (later book. I understood much better after be precise, Mr. White only completed a of the University of Utah) served as thinking about the claim and discussing minor in mathematics, but in light of the Acting Chair in 1959–1961 while Dr. the key ideas of the book with others. fact that he’s incarcerated, that’s close Price was fully occupied with CBMS is enough for me). I’d like to share with recorded. So, here’s a way to do the mapping in my FOCUS’s readership and the Notices’ own words. readership that there is interest in math- W. M. Greenlee, ematics in prison. To be fair, Mr. White, Professor Emeritus 1. Convert each real number base 10 Professor Weil and I are in .a very small University of Arizona between 0 and 1 inclusive to its base 2 minority. The vast majority of prison equivalent number. inmates, at least in the United States, have Report on a Meeting of the no interest in mathematics and only Northeast Section 2. Convert each 2 in the denominator of slightly less appreciation of math than the equivalent number to a 3 and convert the general public. A large number of I greatly enjoyed the June 17, 2005 each 1 in the numerator of the equivalent prison inmates have not graduated high meeting of the Northeast Section of the number to a 2. school. Nonetheless, the number of in- MAA at Bates College in Lewiston, mates who enjoy mathematics is a posi- Maine. For example, tive number; indeed, it is currently at least 2! The number of inmates who are The college campus was lovely. It was at least curious about mathematics and 1 1 0 0 1  fun to see and hear presentations by 12345+++++. . . are at the high school algebra level or recent college seniors on the 2 2 2 2 2 above is greater. Those of us who actu- isoperimetric problem in Gauss space ally do problems for fun or for and prion disease as well as presentations maps to coursework (i.e., Mr. White and I, hope- by graduate students on internet search fully among others) know this from per- and by professional 2 2 0 0 2 sonal experience. In prison, there is very mathematicians on subjects as varied as +++++. . . little privacy, and most inmates have at 3123453 3 3 3 transcendental numbers, the logistic least a passing curiosity about anything equation, paying employees fairly, another does which is outside the norm. bubbles, code breaking, Pascal’s triangle and vice versa. applied to trigonometry, and the history Spring is coming, and with that, for many of mathematics. I’ve probably forgotten I returned home having met some of you, spring cleaning, office reorgani- something or someone, but I found all interesting people, enjoyed many fine zation, and so on. As you find books, the presentations and conversations to be conversations and presentations, and papers, or publications you don’t need, of interest. with a much deeper understanding of my give some thought to donating them to reading of a very interesting explanation a prison library in your state. Books at Part of the fun of these conferences are of the and Banach- the high school algebra level (or lower) the “hallway” or “unplanned” Tarski Theorem. would be most useful, to a prison’s gen- discussions. I enjoyed discussing the eral population, especially those prepar- book “The Pea and the Sun” by Leonard Randall J. Covill ing for the GED. I personally have a fond- M. Wapner A.K. Peters, Wellesley, MA [email protected] ness and high opinion of the traditional 2005 with a math book editor and an high school geometry course (with em- industrial mathematician. The book Helping Incarcerated Mathematicians phasis on proofs) and would love to see explains the axiom of choice and a proof one or two geometry textbooks in my li- of the Banach-Tarski Theorem. Editor’s Note: The letter that follows was brary. I also lament the absence of this addressed simultaneously to the editors of course in the curriculums of junior col- I remembered there being a claim in this FOCUS and of the Notices of the Ameri- leges and other institutions offering re- book that Cantor dust can be placed in a can Mathematical Society. medial mathematics courses, but that is one to one correspondence with all the another letter. I think pre-calculus and real numbers between 0 and 1 inclusive. Dear Editors, calculus texts would be welcome in a sur- This surprised me until I thought about prisingly large number of prison librar- it because Cantor dust is sparse and I read with much interest the March 2005 ies. And post-calculus mathematics texts consists of all the real numbers between FOCUS Short Take “Math Set Him Free” (even graduate-level texts), while not 0 and 1 inclusive that can be represented and the March 2005 Notices article on likely to be checked out by a significant as ternary fractions without any 1s in the Andre Weil’s 1940 letter on Analogy in portion of a prison’s population, has a numerators. The author explains how Mathematics. Like William White, the chance of being a godsend to that one

39 FOCUS August/September 2005 person in the population looking for a ematician in their population, they are pleasure that most mathematicians en- book on analysis. Or number theory. Or likely to know him. joy. In prison, where intellectual chal- groups. Allowing for a moment of self- lenge is in short supply and monotony is ishness, I’d be delighted to see a text or As a closing thought, I’d like to encour- the norm, the MAA journals would be two on differential equations, number age the MAA to offer Mr. White a com- more of a welcome relief than you could theory, and “second-level” linear algebra plimentary one-year membership as a imagine. (canonical forms and the classical means of congratulating him on his re- groups). If you have a mind to donate cent achievement of earning a degree. Calvin A. Curtindolph old texts or materials, I strongly recom- Like most of us (mathematicians), his New Correctional Institute mend contacting the librarian or the di- interest in mathematics likely stems from New Lisbon, WI rector of education of the prison you in- the satisfaction of facing the challenges tend to donate to. They can guide you a good problem provides. The Problems on procedures and on what materials sections and articles of any of the MAA would be most useful. If there’s a math- journals would provide the challenge and

Finding Common Ground in K-12 Mathematics Education

By Michael Pearson

The MAA hopes to help encourage and facilitate constructive discourse between mathematicians and mathematics educators in order to seek common ground in their mutual efforts to improve K-12 mathematics teaching and learning. The success of two pilot meetings (one at NSF in December 2004 and a second at the MAA offices in June 2005) with two mathematicians (R. James Milgram and Wilfried Schmid), three mathematics educators (Deborah Loewenberg Ball, Joan Ferrini-Mundy and Jeremy Kilpatrick) and a moderator from the business community (Richard Schaar) demonstrated that such common ground does exist among individuals who are thought to be strongly aligned with different sides in what has come to be known as the ‘Math Wars.’

These meetings resulted in a document designed to serve as a starting point for future conversations. Agreeing that “All students must have a solid grounding in mathematics to function effectively in today’s world,” the group started with three fundamental premises:

1. Basic skills with numbers continue to be vitally important for a variety of everyday uses.

2. Mathematics requires careful reasoning about precisely defined objects and concepts.

3. Students must be able to formulate and solve problems.

From there, the group explored a number of topics, including the importance of automatic recall of basic facts, the use of calculators in lower grades, instructional methods and teacher knowledge, and found significant points of agreement. The full text of the document is available on the MAA website at www.maa.org/common-ground.

Other groups have met with similar intentions of focusing serious effort on what is essential in the K-12 mathematics curricu- lum and how best to achieve some level of consensus between various constituencies. We expect further articles and reports will become available that help the mathematical community participate more effectively in guiding our schools towards providing students with the skills they need to succeed, both in higher education and the workplace. Watch these pages for future commu- nications from the MAA regarding these efforts.

Michael Pearson is Associate Executive Director and the Director of Programs and Services at the Mathematical Association of America.

40 August/September 2005 FOCUS

U.S. Team Survives Hurricane to Place 2nd in International Mathematical Olympiad

By Steve Dunbar

Merida, Mexico - July 18, 2006 - The 2005 International Mathematical Olym- piad (IMO), 46th in the annual series of mathematical competitions for high school-age students, announced the medal winners today. At this year’s IMO, 513 of the best young mathematicians from 93 countries, making it the largest IMO ever, competed in solving 6 prob- lems posed in a grueling nine-hour test administered over two days (July 13 and 14). The competition which poses six math questions, each worth a total of 7 points, would challenge even the finest professional mathematician.

The U.S. finished 2nd overall with a total of 213 points out of a possible 252 points. came in 1st with 235 points, Rus- sia was 3rd with 212, Iran placed 4th with 201 points and Korea placed 5th with 200.

Upon word of their victory, Steve Dunbar Pictured left to right: Vivian, Brian, Scott, and Jim Lawrence at the USAMO Awards exclaimed, "This is an extraordinarily Ceremony in June.Brian Lawrence had a perfect paper at the International Mathematical strong performance by the U.S. team, Olympiad. Photograph courtesy of Robert Allen Strawn. since this is the first time that these six team members have represented the U.S. Graduated from Thomas Jefferson High at the International Mathematical Olym- The U.S. Team is sponsored by the Math- School of Science and Technology, Alex- piad. Congratulations to Team Leader ematical Association of America ( with andria, Virginia. Zuming Feng and Deputy Leader Melanie support by other mathematical societies, Wood for preparing the team and pre- the University of Nebraska and the Robert Cordwell, 36 points, senting their solutions to the judges." Akamai Foundation. Transportation is Gold Medalist. provided through a grant from the Army Members and competitions results of this Research Office. Additional contributions Graduated from Manzano High School, year’s team are: come from 19 organizations and compa- Albuquerque, New Mexico. nies in the mathematical sciences. The Brian Lawrence, Perfect Paper, team is chosen through a four-stage pro- Sherry Gong, 28 points, Silver Medalist. Gold Medalist. cess of mathematics testing by the MAA’s American Mathematics Competitions Attends Phillips Exeter Academy in Attends Montgomery Blair High School, Program Silver Spring, Maryland. Exeter, New Hampshire. Steve Dunbar is the Director of the MAA Hyun Soo Kim, 27 points, Eric Price, 41 points, Gold Medalist. American Mathematics Competitions. Silver Medalist. Graduated from Thomas Jefferson High School of Science and Technology, Alex- Graduated from the Academy for Ad- andria, Virginia. vancement of Science and Technology, River Edge, New Jersey. Thomas Mildorf, 39 Points, Gold Medalist.

41 FOCUS August/September 2005 The Missouri Collegiate Mathematics Competition

By Alvin Tinsley and Curtis Cooper

The tenth annual Missouri Collegiate point, say Q. Find the coordinates of P so for assessing all the solutions for a given Mathematics Competition was held in that the area bounded by the normal line problem number, and zero to 10 points conjunction with the spring meeting of and the parabola is a minimum. are assigned to each team. The results are the Missouri Section of the MAA on the reported to the scorer, and he only knows campus of Missouri Western State Col- The numbers ±±12,,..., ±2004 are writ- where the teams rank relative to each lege in April. The contest is sponsored other. The results of the two sessions are ten on a blackboard. You decide to pick two by the Missouri Section and is held at the combined to determine the top three numbers x and y at random, erase them, site of the spring section meeting. After teams and the ranking of all teams. and write their product, xy, on the board. ten years, the competition continues to You continue this process until only one be quite popular with students of Mis- The interest in the competition among number remains. Prove that the last num- souri colleges and universities, and, of institutions of all levels has been fairly ber is positive. course, the level of student participation constant over the past 10 years. Many in the state meeting has increased dra- four-year colleges, all the state universi- In addition, there is typically a number matically. ties, and the large research based univer- theory question on the test. The follow- sities participate annually. One would ing is an example: Ten years ago, when the contest was in anticipate that the later institutions incubation, it was reasoned that in order would dominate the awards, but that has Find all integer solutions (x,y) to the equa- to appeal to the interests of students and not always been the case. Often one very tion xy = 5x + 11y. institutions of higher learning at all lev- good student can carry his team to a high els the examination questions should ranking and even first place. The last question is typically a challeng- have a wide range of difficulty. The easi- ing analysis question. The following is an est should be at the level of typical cal- The expenses for an institution’s entry in example: culus and discrete mathematics prob- the competition and for the student’s lems, while the hardest would approach travel and boarding are borne by the in- Prove that in the MacLaurin series for the difficulty of those on the Putnam stitution. The contest entry fee is in- Exam. It was further decided that the tanθπ , – /22<< θπ / , every coefficient tended to cover the cost of materials, contest would consist of two sessions is non-negative. duplicating, and two meals for the stu- lasting two-and-one-half hour each, the dents. first to be held from 7:30 pm to 10:00 The contest is governed by a committee pm on the Thursday evening prior to the of seven college and university math- The team achieving the highest total first day of the spring meeting, and the ematics faculty members from around score for the two sessions receives a trav- second on Friday morning from 8:30 am the state. The committee members sub- eling trophy which their department dis- to 11:00 am. The competition would be mit questions to the chair and once per plays for the year prior to the next con- among teams of up to three students, and year they meet to finalize the exam. Tests test. In addition, each team member re- a college or university could enter two for the coming contest and the follow- ceives a plaque indicating that she/he was official teams to be accompanied by one ing year are prepared at the meeting. a member of the winning team for the or more sponsors from their institution. Questions need not be original, but they indicated year. Each participant in the At the request of a number of colleges very often are. Some may be found in competition receives a certificate of par- and universities, unofficial teams were journals and problem books, and some ticipation, and the awards are presented later allowed to enter, but they were in- are modifications of such problems. during the section meeting banquet on eligible for awards. Friday evening. The first two questions in each session The competition is held in a room which are of the easier type mentioned above will accommodate approximately 30 As mentioned above, snacks and food are and are intended to be solvable by all teams which participate annually. Each made available to the students during teams in the competition. The following team has its own table, each student re- their participation. To encourage the are examples of these questions: one is a ceives a copy of the exam, and scratch competitors to mingle and get ac- traditional parabola question and the paper is provided as needed. Snacks and quainted, a pizza party is given follow- other is a discrete mathematics question. drinks are available to the students dur- ing the second session. In addition, bowl- ing each session. At the end of each ses- ing lanes are reserved for friendly com- sion, the solutions which are secretly petition among teams that wish to par- Let P ≠ 00, be a point of the parabola () coded, are collected, separated by prob- ticipate after the banquet. A photogra- 2 y = x The normal line to the parabola at lem number and graded by the commit- pher takes pictures of the teams in com- P will intersect the parabola at another tee members. One grader is responsible petition, and a group photograph is shot

42 August/September 2005 FOCUS prior to the pizza party. The pictures are Found Math uploaded to the contest webpage before the end of the meeting. Great Moments in Public Education www.boston.com/news/odd/articles/2005/05/23/sports_fans_cry_foul_on_math_question/ For those who are interested, the contest rules and pictures are available at the fol- A state math exam for North Carolina Mildred Bazemore, chief of the state lowing web address: http://www.math- seventh-graders included a question Department of Public Instruction’s test cs.cmsu.edu/~curtisc/contest. on football that asked students “to cal- development section, said the question culate the average gain for a team on makes sense mathematically and was The popularity of the competition has the game’s first six plays.” But there was reviewed thoroughly. exceeded the expectations of the organiz- a problem: ers. A department chair at one of the par- “It has nothing to do with football,” ticipating universities has observed that The team opened with a 6-yard loss, Bazemore said. “It has to do with the the competition is the most significant a 3-yard gain and a 2-yard loss, mathematical concepts that you’re activity of the Missouri Section. which would have made it fourth studying.” down with 15 yards to go for a first Questions concerning the competition down. The team’s fourth play was just It seems mathematicians are out of touch with the real world — and even may be directed to Prof. Curtis Cooper a 7-yard gain, yet it maintained pos- who serves as chair of the contest com- session for a 12-yard gain and a 4- with the world of sports. mittee. His email address is yard gain on two additional plays. (From the Wall Street Journal’s “Best of [email protected]. the Web” online column, May 23, 2005; A state official defends the flawed ques- see http://www.opinionjournal.com/best; Alvin Tinsley and Curtis Cooper teach at tion: Central Missouri State University in reprinted with permission.) Warrensburg, MO.

the MAA’s 4th Annual Mathematical Study Tour

Journey to CHINA June 6 - June 21, 2006 Travel to the Land of Cathay and Explore Its Ancient and Modern Culture Contact Information: Lisa Kolbe Development Manager [email protected] 202-293-1170

Full details, itinerary, and registration form will be available September 1, 2005 on MAA Online www.maa.org

43 FOCUS August/September 2005 Short Takes Compiled by Fernando Q. Gouvêa

Seven Mathematical Scientists updated. The report, which can be found final round in this year’s competition Elected to the NAS online at http://www.nsf.gov/sbe/seind04, pitted the USA team against teams from contains data on elementary and second- Germany and Belarus to address the fol- Among the 72 new members elected by ary education, higher education, labor lowing problem: “Granular material is the National Academy of Science are force, research and development, and flowing out of a vessel through a funnel. seven mathematical scientists: Malcolm other aspects of Science and Engineer- Investigate if it is possible to increase the H. Chisolm of the Ohio State University, ing in the United States. The report is an outflow by putting an ‘obstacle’ above the Iain M. Johnstone of Stanford University, excellent source of raw data for those outlet pipe.” For more about the compe- Sergiu Klainerman and János Kollár of interested in Science Policy. tition and the USA team, visit the inter- , of national site at http://www.iypt.ch and the the University of California, Los Ange- The Chudnovsky Brothers USA site at http://www.usaypt.org. les, Margaret H. Wright of the Courant and the Tapestries Institute, and Adi Shamir of the Concern about Test Questions Continue Weizmann Institute of Science. Alas, not An article by Robert Preston in the April a one is a member of the MAA; we con- 11 issue of The New Yorker tells of how Given the increasing number of math- gratulate them nevertheless. the Chudnovsky brothers helped create ematics and science examinations and a digital image of the famous tapestries standardized tests, educators are facing New Journal to Focus on the History of from The Cloisters, a part of the Metro- the need to create more and more test Mathematics Education politan Museum of Art. The project, problems. An article in the 24 April 2005 which involved knitting together a large of highlighted the The first issue of the International Jour- number of digital images of portions of fact that a significant number of these nal for the History of Mathematics Teaching the tapestries, proved to be particularly problems are flawed in one way or an- is set to appear early in 2006. With a dis- challenging because of minute changes other: “ambiguous, imprecise, or just tinguished editorial board headed by Gert in the position of the fibers while the plain incorrect.” As a result of pressure Schubring and Alexander Karp, the new photographs were being taken. The ar- from critics, some states (Texas is one journal will be published twice a year by ticle can be found online at http:// that is mentioned in the article) are hir- the Teachers College at Columbia Univer- www.newyorker.com/fact/content/ar- ing university professors to review the sity. ticles/050411fa_fact. test questions. The article is available online at http://www.nytimes.com/2005/ IJHMT is an outgrowth of the success of USA Team Takes Second Place at Inter- 04/24/education/edlife/guernsey24.html, Topic Study Group 29, on The History of national Young Physicists’ Tournament but the Times requires payment for ac- Learning and Teaching Mathematics, at the cess. International Congress on Mathematics The USA team tied for 2nd place in this Education held in Copenhagen in 2004, year’s International Young Physicists Teaching Awards for the which suggested that there was a need for Tournament at the University of Zurich, MAA Financial Team “a permanent and stable international fo- placing among the top teams for the first rum for scholarly research in the history time in eighteen years. Five students rep- Two members of the MAA’s Budget and of mathematics teaching.” The journal is resented the United States: Phillip Audit Committees, Dan Maki and Jim calling for submissions of papers. For more Schwartz and Daniel Kerr of the Wild- Daniel, have received teaching awards information, contact Alexander Karp, wood School in California and Jonathan from their institutions. Dan Maki re- IJHMT, Program in Mathematics, Box Bohren, Robert Kirkham, and Divya ceived the 2004 Indiana University 210, Teachers College, Columbia Univer- Krishnan of the Rye County Day School President’s Award for Distinguished sity, 525 West 120th Street, New York, NY, in New York. Teaching. Jim Daniel was named to the 10027, or they can be reached by email University of Texas at Austin’s “Academy at [email protected]. The Tournament is a theoretical and of Distinguished Teachers” (see http:// practical competition for teams of high www.utexas.edu/faculty/academy). The Science and Engineering Indicators school students, who work on 17 phys- two, who make up the whole of the Au- ics problems and then present and de- dit Committee and two-thirds of the The NSF’s Science and Engineering Indi- fend their findings in “Physics Fights.” Budget Committee, are thus as distin- cators 2004 (a biennial report to Con- Rounds consist of three teams compet- guished as teachers as they are in their gress) was first published more than a ing against each other, each having a service to the Association. year ago, but the data have recently been chance to report, oppose, and review. The

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Mathematical Music, Anyone? There’s a famous fast-food restaurant you giving half or more of his annual income can go to, where you can order chicken to charity. Semmler’s largesse is made Visit http://www2.collegehumor.com/ nuggets. They come in boxes of various possible by his very simple lifestyle. “If I movies/149448/ for a delightful perfor- sizes. You can only buy them in a box of 6, didn’t do all of the things I was doing, I mance by The Klein Four Group, who a box of 9, or a box of 20. So if you’re really would probably have a new car every two sing A Finite Simple Group (of Order hungry you can buy 20, if you’re moder- years and I would have a huge house with Two), a piece M. Salomone based on ately hungry you can buy 9, and if there’s a huge pool,” Semmler told the Post, “but mathematical double-entendres. For more than one of you, maybe you buy 20 I would not do it that way. I want to do it more about the group, including more and you divide them up. this way.” According to the article, songs, lyrics, and the Klein Four Store, Semmler estimates that he has given away visit their web site at http:// Using these order sizes, you can order, for $770,0000 so far, and he intends to reach www.math.northwestern.edu/~matt/ example, 32 pieces of chicken if you one million before he retires. kleinfour/. wanted. You’d order a box of 20 and two boxes of 6. Here’s the question: What is the International Conference on Milken Family Foundation Focuses on largest number of chicken pieces that you Mathematics Teaching Teacher Quality cannot order? For example, if you wanted, say 37 of them, could you get 37? No. Is The 3rd International Conference on the The “No Child Left Behind” Act requires there a larger number of chicken nuggets Teaching of Mathematics will be held be- schools to meet specific goals with re- that you cannot get? And if there is, what tween June 30 and July 6, 2006, at spect to teacher quality. In its May 4 is- number is it? Instanbul, Turkey. Following on the suc- sue, Education Week reports that in or- cess of earlier conferences held in Samos, der to help schools meet these require- It’s probably too late to send in the an- Greece (1998) and Crete, Greece (2002), ments, the Milken Family Foundation swer and get a $26 gift certificate from the 2006 conference intends to focus on and the Broad Foundation have created their Shameless Commerce Division, but “new ways of teaching undergraduate a new foundation “to help urban schools it’s still a nice problem. mathematics.” The conference chairs are adopt proven strategies for improving Ignatios Vakalis of Capital University and the quality of their teaching staffs.” The Philosophia Mathematica Has Deborah Hughes-Hallett of the Univer- Milken and Broad Foundations have New Publisher sity of Arizona, and the conference will provided more than ten million dollars be co-sponsored by the MAA. For more to create the Teacher Advancement Pro- Philosophia Mathematica has been taken information, including information on gram Foundation. The foundation is ex- over by Oxford University Press, the how to submit a paper, visit http:// pected to continue the work of Milken’s world’s biggest and most significant pub- www.tmd.org.tr/ictm3. “Teacher Advancement Program” (TAP), lisher of philosophy. Philosophia is the which emphasizes “multiple career paths; sister journal of Historia Mathematica. It Sources: New NAS members: press re- instructionally focused accountability; was published over the last 12 years by lease. IJHMT: email communication, call ongoing, applied professional growth; the Canadian Society for the History of for papers. Science and Engineering In- and performance-based compensation.” Philosophy of Mathematics. As the dicators: email communication, NSF web The new foundation will be headed by names indicate, Historia (published by site. Chudnovsky Brothers: email com- Lewis C. Solmon. Elsevier) is a History of Mathematics munication, New Yorker web site. Inter- journal, while Philosophia deals with the national Young Physicists’ Tournament: The Education Week report is online at Philosophy of Mathematics. Congratu- press release. Test Questions: NASSMC http://www.edweek.org/ew/articles/2005/ lations to editor Robert Thomas for his Briefing Service, The New York Times. 05/04/34milken.h24.html, and the press success with the journal. For more about MAA Financial Team: email communi- release from the Milken Foundation is at Philosophia, including a searchable cation. Mathematical Music: email com- http://www.mff.org/newsroom/ archive, visit its web page at http:// munication. Milken Foundation: news.taf?page=447. www.philmat.oupjournals.org. NASSMC Briefing Service, Education Week, Milken web site. Mathematical A Mathematical Puzzler from Car Talk Puzzler: heard on air, email communi- Richard Semmler Featured in cation, Car Talk web site. Philosophia As the show’s many dedicated fans know, Washington Post Article Mathematica: email communication. NPR’s Car Talk regularly includes “puz- Richard Semmler: The Washington Post. zlers”, problems of various kinds that are An article in the June 11 issue of The ICTM: email announcement. sometimes about cars, sometimes just Washington Post profiles Richard tricky logic questions, and occasionally Semmler, who teaches mathematics at mathematics problems. On one of the Northern Virginia Community College. weeks during which FOCUS was being Describing Semmler as a professor who prepared, the puzzler was “The Chicken “finds fulfillment in emptying his pock- Nugget Conundrum”: ets,” the article lauds his generosity in

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