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FOCUS February 2004

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Inside

4 “Great Internet Mersenne Prime Search” Locates Another One

5 Things I Learned in School Last YearBy Jim Loats

8 The 2004 IAS/Park City Mathematics Institute:The Undergraduate Faculty ProgramBy William Barker

9 Student Paper Contest in the History of Mathematics

10 Mathematics and Operations Research in IndustryBy Dennis E. Blumenfeld, Debra A. Elkins, and Jeffrey M. Alden

13 Prizes and Awards at the Phoenix Joint Mathematics Meetings

17 NSF BeatMentoring Through Critical Transition PointsBy Sharon Cutler Ross

17 Eileen Poiani Receives Several Awards

18 Octagon FeedbackBy Fernando Q. Gouvêa

20 Call for PapersContributed Paper Sessions at MathFest 2004

23 Employment Opportunities

On the cover: The prizewinner and the presidents. Left to right: AMS President DavidEisenbud, Melanie Wood, MAA President Ron Graham, and SIAM President James

Hyman. Photograph courtesy of Fernando Gouvêa.

Volume 24 Issue 2

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February 2004

Melanie Wood was named recipientof the 2003 The AMS-MAA-SIAM Frankand Brennie Morgan Prize for Outstand-ing Research in Mathematics by an Un-dergraduate Student at the January JointMathematics Meetings in Phoenix.Wood received a $1000 cash prize and acertificate at the meetings. Wood is thefirst woman to win the prize since its in-ception in 1995. The selection commit-tee cited Melanie’s honors thesis at Dukeon combinatorial structures associatedwith algebraic curves defined over num-ber fields, done under the supervision ofRichard Hain, and her work on algebraicnumber theory, done at the Duluth REUin 2000. Her work on algebraic curvesyields important insights into the actionof the absolute Galois group on funda-mental groups. This research has at-tracted the attention and admiration ofthe specialists working in this field.Wood’s results in number theory gener-alize the usual factorial function and ex-pand on ideas introduced by ManjulBhargava, for which he received the Mor-gan Prize in 1996. The Prize committeecalled her work “deep and original.”

Melanie is well known for her success inmath competitions. In 1998 she was thefirst woman ever to be named to theUnited States team for the InternationalMathematical Olympiad. In 2002 she wasthe second female to be named a PutnamFellow. In 2003 she was an instructor inthe Mathematical Olympiad Programand she accompanied the US team to theInternational Mathematical Olympiad inJapan.

Wood’s historic participation in the IMOcaused a media stir. The June 2000 issueof the science magazine Discover ran asix page profile about her and the IMO(see http://www.findarticles.com/m1511/6_21/62277745/p1/article.jhtml). Thetable of contents was printed over a fullpage photo of her doing a headstand infront of a blackboard. In 2002 she ap-peared on the television program To Tellthe Truth, in which four panelists and the

audience tried to determine which ofthree contestants was the real MelanieWood. (Only one panelist succeeded,perhaps because Melanie had taught theother contestants some “math talk.”)

Having received Fulbright and GatesScholarships, Wood is currently study-ing mathematics at Cambridge Univer-sity. In the fall of 2004 she will begingraduate school in mathematics atPrinceton.

Melanie also enjoys acting, especiallyclassical acting and voice work, direct-ing, dancing, and philosophy.

The Morgan Prize committee awardedHonorable Mention to Karen Yeats of theUniversity of Waterloo for a series ofthree papers making outstanding contri-butions on topics ranging fromasymptotics and number theory to math-ematical logic. Her work expands on re-sults of Schur and results of Dedekind.Karen was a member of the Waterloo2002 Putnam team that finished in thetop 10. She spent three summers as anNSERC (Natural Sciences and Engineer-ing Research Council of Canada) under-graduate research assistant at Waterloo.She graduated in 2003 with honors inPure Math and a Governor General’s Sil-

By Joe Gallian

Melanie Wood Wins Morgan Prize

Melanie Wood Karen Yeats.

Melanie Wood presenting her talk at the Joint Mathematics Meetings inPhoenix, AZ.

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FOCUS February 2004

The absolute Galois group, GQ, is the group of automorphisms of the algebraic

numbers which fix the rational numbers. Though this group is an importantobject in number theory, little is known about it. Grothendieck proposed study-ing G

Q by considering its faithful action on combinatorial structures called dessins

d’enfants (children’s drawings). A dessin is a bipartite graph with a fixed 2-color-ing and cyclic edge orderings at each vertex. The eventual goal of studying theaction of G

Q on dessins is a complete list of G

Q invariants computable from the

combinatorial structure of the dessins. Such a list would allow us to check whethertwo dessins are in the same G

Q-orbit by checking whether they agree on the list

of invariants. Some invariants are known, and the present work develops a methodfor constructing new invariants from old ones. We give an example of two dessinswhose orbits are distinguished by one of our new invariants but were previouslyundistinguished. The action of G

Q on dessins comes from an action on the fun-

damental group of the Riemann sphere minus three points, which gives an injec-tion of G

Q into Aut( F2

∧), the automorphisms of the profinite completion of the

free group on two generators. Our method for finding new dessin invariants ex-tends to a method for finding relations on the injection of G

Q into Aut( F2

∧

).

Abstract of Melanie Wood’s Morgan Prize Talk

“It was just a matter of time,” saidMichigan State University graduate stu-dent Michael Shafer, about identifyingthe largest prime number to date. In lateNovember, 2003, the number was found:It is 2 to the 20,996,011th power minus1. The number contains 6,320,430 digits— 2 million digits more than the previ-ous largest known prime number.

Mersenne primes are prime numbers ofthe form 2n – 1. It is easy to see that inorder for this number to be prime, nmust also be prime, but of course thiscondition is far from sufficient. It is con-jectured that there exist infinitely manysuch primes, so the search has the po-tential to go on forever. Mersenne primesare closely related to perfect numbers: if2n – 1 is prime, then 2n-1(2n – 1) is a per-fect number. (This result can be found

in Euclid’s Elements; Euler proved that infact any even perfect number is of thisform.) They are named after MarinMersenne (1588–1648), a French friarwho is best known for his correspon-dence with many scientists and math-ematicians. By passing along results,questions, and ideas, Mersenne hoped toget scholars to build on each other’swork. One of his areas of interest wasperfect numbers, which is why his namebecame attached to the primes involvedin the factorization of even perfect num-bers.

Shafer, who is 26 and studying chemicalengineering, helped find the number —using a Dell computer — as a volunteermember on the eight-year-old projectcalled the “Great Internet MersennePrime Search” (GIMPS). This is a world-

wide effort that involves thousands ofpeople and more than 200,000 comput-ers — in effect a supercomputer thatcould do 9 trillion calculations per sec-ond. As for his standing in the world ofmathematics, “I don’t think I’m going tobe recognized as I go down the street,”he said. “Somebody else could havefound the number… You install the pro-gram on the computer and it takes careof itself.” But, he continued, “I get thecredit, along with the people that devel-oped the software.”

The home page for GIMPS can be foundat http://www.mersenne.org, where onecan find more information aboutMersenne primes, the GIMPS projectand its latest discovery. One can evendownload software to join the search.

“Great Internet Mersenne Prime Search” Locates Another One

ver Medal. She is currently a graduatestudent in mathematics at Boston Uni-versity.

Karen is an accomplished recorder player,and also enjoys playing clarinet and sing-ing in choirs, as well as the occasionalforay into making teddy-animals andworking on free software.

Both Wood and Yeats gave talks on theirresearch at the Phoenix meeting. Al-though most of the recipients of theMorgan Prize have yet to finish a Ph.D.degree, winners of the early prizes havealready begun to make their mark. In-deed, three of the first four winners andone who was named honorable mentionhave received the Clay Mathematics In-stitute or American Institute of Math-ematics Long Term (five years) Prize Fel-lowships for researchers at the beginningof their careers. The first Morgan prizewinner in 1995, Kannan Soundararajan,received the 2003 Salem Prize given toyoung researchers for outstanding con-tributions to the field of analysis and the1996 winner, Manjul Bhargava, is a ten-ured full Professor at Princeton.

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February 2004

Be at work by 8:00 a.m., every day. Ob-serve the dress code. (Dress code!?) Havea cubicle as an office, but be out in theschools most of the time. Update myphone message every day. Carry a cellphone. This is definitely not the life ofan academic. But it was my life duringthe year I spent, away from my job as amathematics professor, as a Mathemat-ics Coordinator for the Denver PublicSchools (DPS).

The district’s decision to move towardsstandards-based curricula in grades 4–8increased my desire to step out of mycomfortable academic routine. EverydayMathematics and Connected Mathemat-ics Program are well designed, field-tested, sophisticated standards-basedcurricula. When they are used effectively,students will gain a broad, deep, and con-nected understanding of mathematics.To use them well, teachers must have asolid understanding of the importantmathematical concepts and be preparedto take on different roles in the class-room. I had changed my own teachingmethodology in most of my collegecourses and was excited to help othersmake similar transitions.

During the mid-1990’s at MetropolitanState, I was co-P.I. and MathematicsTeam Leader of our NSF-funded RockyMountain Teacher Education Collabora-tive. In this role I helped to hire middleand high school teachers to work on-campus side-by-side with college mathand science faculty on curricular revi-sions. I became a good friend of one ofthese Teachers-in-Residence, RosanneFulton, who had since become Directorof Curriculum and Instruction at DPS.One July evening last year she phoned.“Why don’t you take a year’s leave fromMetro and park your car over here atDPS where the real work is?” I decided itwould be a great opportunity, and my de-partment chair and dean concurred. DPScould benefit from my expertise and I’dlearn a lot about public schools… reallyfast!

My coworker, Debbie Hearty, had beendoing this job for several years and wasan expert. I studied her closely while Ihastily learned a whole new culture. AsMathematics Coordinators we directedthe Math Team, a group of nine teacherswho had been newly chosen to be dis-trict Mathematics Specialists. Togetherwith the Team, we designed and pre-sented curriculum implementationworkshops for over 300 fourth to eighthgrade mathematics teachers. Debbie andI regularly worked on math problemswith the specialists and supported themin the work with their teachers. We cre-ated curriculum pacing guides and vis-ited classrooms with school principals.

We had additional tasks. We conveyed thedistrict’s new vision of mathematics in-struction to various stakeholders: super-intendents, principals, parents, schoolboard members, the press, vendors, pro-fessors at various local colleges and uni-versities, and colleagues on nationalgrants. We explained the new curriculaat meetings with administrators andteachers from special education, giftedand talented programs, counselors, tech-nology, summer school, English languageacquisition, career education, alternativeschools, and student records. I had noidea how much effort was required justto inform all the non-teaching staff of theimplications that came with the changesin mathematics curricula.

What I Learned About the Schools

I learned that helping schools to improverequires very sophisticated thinking andexcellent management skills. The prob-lems are complex and resistance tochange is huge. Successful interventionstook time to plan.Carrying them out re-quired the right people in the right placesmaking good decisions.There are no easysolutions.

Everyday Mathematics and ConnectedMathematics Program places enormousdemands on teachers, both in terms oftheir content knowledge and their peda-

gogical flexibility. Providing adequatesupport for them in both areas will re-quire a significant effort for all of us foryears to come.

I learned that mobility of teachers is aproblem, especially in urban districts likeours. The teacher turnover within asingle building can be quite large fromyear to year. This underlying fluidity isvery different from my experience inhigher education. It cuts both ways in theschools. On one hand, school change ef-forts can lose momentum when the play-ers change. On the other, it can help, be-cause some of the troupe may be differ-ent next year.

I learned that the schools are not incharge of themselves. Many stakehold-ers know best about what should hap-pen. What happens inside schools is of-ten defined outside of the schools. Forexample, I talked with parents who didnot want mathematics instruction tochange despite admitting that their ownschool math experiences were not par-ticularly effective.

Each school exudes a unique culture andindividual personality. You can feel it themoment you walk into a building. Thisculture influences the nature andamount of learning that takes place.While this culture has considerable in-ertia, a strong principal can impact aschool and change the learning/teachingexperience for teachers and children.

The daily workload of K-12 teachers is“impossible.” Almost no one has enoughtime to do their job really thoroughly,although most do outstanding jobs.Their work schedules, the conflictingpush-pulls, and the slow pace of changeconspire to divert their attention fromthe business of teaching children. Ad-ministrators must be as wise andthoughtful as they can be, consideringtheir hectic schedule and breadth of re-sponsibilities. After working with themfor a year, I have profound respect for allof these people and the work they do.

By Jim Loats

Things I Learned in School Last Year

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FOCUS February 2004

Tracking

I was most disturbed by the conse-quences of tracking students. It may seemlike common-sense for a third gradeteacher to group children based onhow fast they read. Small, seeminglyobvious decisions like this one lead,over time, to a situation I observedacross this diverse urban district: APclasses were composed mostly ofwhite and affluent students, whileregular and low track classes hadmostly poor students of color. I was(and remain) very troubled to learnabout this reality in our schools, notjust in Denver, but all across ourcountry.

The issue is the balance betweenserving the needs of the whole class-room with what is best for any par-ticular child. There is good news.Schools that adopt these new so-phisticated curricula acquire a tool thathas been specifically designed to addressthis and other equity issues while teach-ing good mathematics, all without dis-criminating against whole groups of kids.These Standards-based curricula workespecially well in classrooms that containa broad spectrum of students. They relyon the rich variations of student experi-ences to provide multiple means to viewand understand the mathematics thatarises. They have multiple entry pointsto provide access for students with dif-ferent backgrounds and learning styles.

What I Learned About Me

I enjoyed spending time with teachers,both in their classrooms and at the work-shops. I loved helping principals andother administrators understand the de-sign of these new curricula. I felt like Iwas making a difference. I liked workingclosely with a team and pulling off nearlyimpossible missions. My work life as aprofessor is usually more isolated.

I valued being personally challenged ina series of workshops on equity. Usingprotocols from the National Center forEquity in Education, forty of us startedfacing our biases around race, gender,class, and sexual preference and becameclose friends.

I was proud to be working for leaderswho were smart and pushed the systemto change as fast as it could. I respect thedistrict’s administrative team, especiallythe Superintendent, Jerry Wartgow, the

Chief Academic Officer, Sally MentorHay, and my supervisor, Rosanne Fulton.I trusted that they were making wise,strategic decisions even if the details ofimplementation at my level occasionallyseemed hurried or impractical.

I truly fell in love with the people of theDenver Public School system. They lovethe children and give their best, often inless than optimal situations. Our missionwas to provide excellent schooling to “Ev-ery Child Every Day.” It was truly inspi-rational and compelling to be workingwith people who took that to heart dayin, day out.

I often felt ineffective because very fewthings changed directly or immediatelyas a result of my work. Back on campus,the feedback from students was alwaysmore immediate and direct.

Coming from the classroom, I wascaught off guard by the work life of anadministrator. I was not in charge of mytime. I was always on-call and account-able to my boss, my fellow curriculumteammates, and the public. My schedulefor the day was fluid and unpredictable.There was always something new to do.The tasks could not be done as fast asneeded. Any decision inevitably led to

unforeseen consequences. I missed thestructure, predictability, and freedom ofmy academic routine.

What This Means for Our Work asMathematics Professors

First, I urge all of us to continuerethinking our mathematicscourses so that our students fin-ish them with in-depth concep-tual understanding that makessense to them and sticks withthem long after the final exami-nation. Further, in each of ourclasses, we must regularly taketime to examine with our stu-dents how learning happens inthe class. For example, take briefmoments away from the con-tent to make public and explicitwhat we are doing to help themlearn.

I write this after observing over100 middle and high school classroomteachers who have been teaching formany years. Many “facts and skills” theylearned in their college mathematicsclasses have probably faded away. Nev-ertheless they model their own classes onrecollections of their experiences in col-lege mathematics classes. For most, thatmeant listening and taking notes, an-swering — not asking — questions,working alone and mistrusting opportu-nities for collaboration. It meant memo-rizing “how” to do a problem and notworrying “why” it works; and, impor-tantly, rarely being asked to write cogentexplanations of their own understand-ing of the important ideas of the class.

Imagine it is a year after one of yourclasses. Wouldn’t it be cool if the studentscould explain the central big ideas in theclass and understand how they fit to-gether? Even better if they had an explicitunderstanding of what you did as theirteacher to help them learn. For all of ourstudents, but especially the future teach-ers, it would send a strong message thatmathematics is about understanding andmaking sense; and that their task as stu-dents is to build, with our help, their owncoherent, connected structure of the fun-damental ideas.

Denver middle school teachers, Kelly Babcock and NathanPruss, solving a math problem together at a workshop.

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February 2004

Data from the recent Third InternationalMathematics and Science Study (TIMSS)video study show that teachers in theUnited States, unlike every other coun-try, always convert problem-solving op-portunities into procedural tasks andteach their students to do them that way.Somehow, in our college and universitycourses perhaps, these teachers havelearned that mathematics is about recall-ing, and performing procedures to theexclusion of thinking, conjecturing andproblem solving.

These new school curricula requireteachers to have the habit of asking, “Whydoes this work?” not just “How does itwork?” Most of the teachers I observedhad difficulty leading discussions aboutwhy things work. The prospective teach-ers in our classes need to see these typesof discussions modeled in their collegeand university classrooms by “real”mathematicians.

Second, it is important that we as mem-bers of higher education mathematicscommunity work to understand the im-portant, difficult work of the K–12 teach-ers and then be public with our supportand respect for them. Too often we aresilent while less informed voices get thesound bites and column inches.

Third, I urge each of you to find a way tohelp raise the math content knowledgeof our public school teachers. Manyteachers would really appreciate the at-tention of a mathematician who wouldsupport them in asking and answeringthe questions they have about the math-ematics they are teaching. Enjoy the op-portunity to share the mathematics youknow and value so deeply. Initiate a rela-tionship with them. You can find themjust down the street.

The author is once again professor in theDepartment of Mathematical and Com-puter Sciences at Metropolitan State Col-lege of Denver. He is an avid sailor, skier,musician, and co-author of Algebra Un-plugged and Calculus For Cats. He’d en-joy responding to inquiries atloatsj@mscd.edu.

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The 2004 IAS/Park City Mathematics Institute:

The Undergraduate Faculty Program

By William Barker

A unique mathematics event takesplace each July in Utah: the Park CityMathematics Institute, sponsored since1994 by the Institute for Advanced Studyin Princeton. For three intense weeksmathematics researchers, scholars, andstudents at the post-secondary level aswell as mathematics educators atthe secondary and post-secondarylevel gather for research and studyin Park City, Utah, a setting ofbreathtaking mountain sceneryand abundant hiking and outdooropportunities.

This year’s PCMI Summer Sessionwill be held from July 11 to July 31,2004, with the research theme ofGeometric Combinatorics and theeducation theme of From Policy toPractice: Partnerships with SchoolDistricts.

The central themes of PCMI aredeveloped in six separate but con-nected programs, each centered ona different component of themathematics community:

The research program components:mathematics researchers (Research Pro-gram), for graduate students (GraduateSummer School), and for undergradu-ate students (Undergraduate Program).The education program also has threeparts: for mathematics education re-searchers (Mathematics Education Re-search Program), for undergraduate col-lege faculty (Undergraduate Faculty Pro-gram), and for high school teachers(High School Teacher Program).

At the Summer Session these groupsmeet simultaneously, each following anindividual program of research andstudy. However, considerable interactiontakes place between the groups due to thefundamental structure of PCMI: lecturesand courses in each Summer Session pro-gram are open to all participants and, inaddition to the activities specific to eachgroup, there are daily events of general

interest designed for the full Institute.Further opportunities for informal andsocial interaction are also available, rang-ing from organized cross-program ac-tivities to informal conversations overmeals. Cross-program mentoring is alsoencouraged and nurtured to further en-

hance the sense of community amongparticipants.

The rich mathematical experience com-bined with interaction among groupswith different backgrounds and profes-sional concerns results in greatly in-creased understanding and awareness ofthe issues confronting contemporarymathematics and mathematics educa-tion.

Each of these programs will be attractiveand appropriate for many members ofthe MAA. However, the UndergraduateFaculty Program (UFP), with its focus oncollegiate mathematicians with a stronginterest in undergraduate education, isof relevance to a particularly large por-tion of the MAA. Co-sponsored by NSF’sChautauqua Program, the UFP offersopportunities for broad professionalgrowth and engagement with the excite-ment of mathematics by working with

peers on new approaches to teaching,tackling research questions, and interact-ing with the broader mathematical com-munity. Each year the theme of the UFPbridges the research and educationthemes of the Summer Session. Thisyear’s UFP theme is:

Combinatorics in Concert: forTeaching, Research, Outreach,and Recreation

The workshop leader will beFrancis Su, Associate Professor ofMathematics at Harvey MuddCollege.

The program will use geometriccombinatorics to weave togethermany aspects of a facultymember’s professional life —teaching, research, outreach, andrecreation — into a harmoniouswhole.

The many beautiful yet accessibleideas in geometric combinatoricsmake this topic perfect for: enrich-

ing a wide variety of undergraduatecourses with examples from this field;providing a source of research problems(for undergraduates or oneself); gener-ating topics for general lectures in thecommunity or local high schools; andsustaining recreation opportunities suchas puzzle solving.

Geometric combinatorics refers to agrowing body of mathematics concernedwith counting properties of geometricobjects described by a finite set of build-ing blocks. Primary examples includepolytopes (which are bounded polyhe-dra and the convex hulls of finite sets ofpoints) and complexes built up fromthem. Other examples include arrange-ments and intersections of points, lines,and planes. There are many connectionsto linear algebra, discrete mathematics,analysis, and topology, and there aremany exciting applications to econom-ics, game theory, robotics, and biology.

Andrew Bernhoff leads a session at the 2003Undergraduate Faculty Program at the Park CityMathematics Institute.

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February 2004

The UFP “concert” will feature two con-current parts, the Baseline and Melody.Each will generally last one hour per day.The Baseline hour will be a course onselected topics in geometric combinato-rics, open both to undergraduate facultyas well as other PCMI participants.

The Melody hour will build on theBaseline material to address issues ofparticular interest to faculty in highlyinteractive and participatory ways. Forexample, one topic will be how theBaseline material can enrich one’s ownundergraduate courses (such as discretemathematics, linear algebra, geometry,and topology) or how to teach a stand-alone course in the subject.

Not only does the UFP Program exploreboth the content and pedagogy of oneundergraduate topic per year, it alsohelps encourage and facilitate interactionamong PCMI’s constituent groups andprograms. For example, some UFP par-ticipants like to attend the GraduateSummer School courses. Some also work

actively with the High School Teachersprogram, particularly concerning peda-gogical, curricular, or articulation issues.A large number are attracted to the Un-dergraduate Program, both for the inter-esting mathematics in the courses and forthe kind of research experiences for un-dergraduates that are often available. TheUG courses for 2004 are From Polytopesto Enumeration, taught by Edward Swartzof Cornell University, and Groebner Basesand Polytopes, taught by Rekha Thomasof the University of Washington.

This summer’s UFP workshop leader,Francis Su, earned his Ph.D in Math-ematics from Harvard University in1995, and is now an Associate Professorof Mathematics at Harvey Mudd College.His research interests lie presently in geo-metric combinatorics and applications tomathematical economics. He has a dualpassion for mathematics education, andhas won several teaching awards,authored a popular Math Fun Factswebsite, and serves on the editorial boardof Math Horizons. In 2001 the Math-

ematical Association of America awardedhim the Merten M. Hasse Prize for ex-cellence in expository writing.

The Coordinator of PCMI’s Under-graduate Faculty Program, DanielGoroff, is Professor of the Practice ofMathematics at Harvard University andAssociate Director of the Derek Bok Cen-ter for Teaching and Learning. He is amember of the PCMI Steering Commit-tee, chaired by Herb Clemens of the OhioState University.

For more information and applicationinstructions for the Undergraduate Fac-ulty Program, consult the UFP web pageat http://www.admin.ias.edu/ma/program/ufp2004.html. Reviewing of applicationsbegins February 15, 2004, and financialsupport is available for participants in allthe programs. For more general informa-tion on all aspects of the 2004 SummerSession consult the PCMI web page athttp://www.admin.ias.edu/ma/.

Student Paper Contest in the

History of Mathematics

HOMSIGMAA, the History of Math-ematics Special Interest Group within theMAA, has recently announced a studentpaper contest. The contest is open to allundergraduate students. Its purpose is toincrease awareness of and interest in thehistory of mathematics among under-graduates. A grand prize and two run-ner-ups will be chosen. All three will re-ceive a one year student membership tothe MAA; the grand prize winner willalso receive a $25 dollar gift certificateto the MAA bookstore.

Topics for the paper can be drawn fromany field of mathematics. Papers mayaddress a single person, topic, or period,or may survey the history of a broadertopic or a school of thought. They shouldbe approximately 5000 words (12double-spaced 12pt pages) long, shouldinclude a full citation list, should notdraw too heavily from internet sources

(except for web sites that give access toprinted material, such as JSTOR orGallica). Papers will be judged by a panelof specialists for content, presentation,and grammar.

The deadline for submissions is March15, 2004. Submitted papers should in-clude the student’s name, institution,their email and postal address, and thename of the supervising instructor ifapplicable. Submissions and questionsshould be directed to Amy Shell-Gellaschat amy.shellgellasch@us.army.mil or bypost at Dr. Amy Shell-Gellasch, CMR415, Box 3161, APO AE 09114. Resultswill be announced via email and on theHOMSIGMAA website in May 2004.

Information on the contest can also befound at the HOMSIGMAA web site athttp:/ /home.adelphi.edu/~bradley/HOMSIGMAA.

AAAS Elects Fellows

in Mathematics

In September, the Council of the AAAS(American Association for the Advance-ment of Science) elected 348 membersas Fellows of the AAAS. These individu-als will be recognized for their contribu-tions to science at the Fellows Forum tobe held on 14 February 2004 during theAAAS Annual Meeting in Seattle. Thenew Fellows will receive a certificate anda blue and gold rosette pin as a symbolof their distinguished accomplishments.The new Fellows in the Mathematics Sec-tion are Donald Ludwig of the Univer-sity of British Columbia, David W.McLaughlin of New York University,Maxine L. Rockoff of the New York Acad-emy of Medicine, Annie Selden of NewMexico State University, Las Cruces, andVictor L. Shapiro of the University ofCalifornia, Riverside. Both Shapiro andSelden are members of the MAA, andSelden is also a member of the FOCUSeditorial board.

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FOCUS February 2004

Students majoring in mathematicsmight wonder whether they will ever usethe mathematics they are learning, oncethey graduate and get a job. Is any of theanalysis, calculus, algebra, numericalmethods, combinatorics, math program-ming, etc. really going to be of value inthe real world?

An exciting area of applied mathematicscalled Operations Research combinesmathematics, statistics, computer sci-ence, physics, engineering, economics,and social sciences to solve real-worldbusiness problems. Numerous compa-nies in industry require Operations Re-search professionals to apply mathemati-cal techniques to a wide range of chal-lenging questions.

Operations Research can be defined asthe science of decision-making. It hasbeen successful in providing a systematicand scientific approach to all kinds ofgovernment, military, manufacturing,and service operations. Operations Re-search is a splendid area for graduates ofmathematics to use their knowledge andskills in creative ways to solve complexproblems and have an impact on criticaldecisions.

The term ‘Operations Research’ is knownas “Operational Research” in Britain andother parts of Europe. Other terms usedare “Management Science,” “IndustrialEngineering,” and “Decision Sciences.”The multiplicity of names comes prima-rily from the different academic depart-ments that have hosted courses in thisfield. The subject is frequently referredto simply as “OR”, and includes both theapplication of past research results andnew research to develop improved solu-tion methods.

Some key steps in OR that are needed foreffective decision-making are:

Problem Formulation (motivation,short- and long-term objectives,decision variables, control param-eters, constraints);

Mathematical Modeling (representationof complex systems by analytical ornumerical models, relationshipsbetween variables, performancemetrics);

Data Collection (model inputs, systemobservations, validation, trackingof performance metrics);

Solution Methods (optimization,stochastic processes, simulation,heuristics, and othermathematical techniques);

Validation and Analysis (model testing,calibration, sensitivity analysis,model robustness); and

Interpretation and Implementation (so-lution ranges, trade-offs, visual orgraphical representation of results,decision support systems).

These steps all require a solid back-ground in mathematics and familiaritywith other disciplines (such as physics,economics, and engineering), as well asclear thinking and intuition. The math-ematical sciences prepare students toapply tools and techniques and use a logi-cal process to analyze and solve prob-lems.

OR became an established disciplineduring World War II, when the Britishgovernment recruited scientists to solveproblems in critical military operations.Mathematical methods were developedto determine the most effective use ofradar and other new defense technolo-gies at the time. OR groups were laterformed in the U.S. to meet needs of war-time operations, such as the optimalmovement of troops, supplies, andequipment.

Following the end of World War II, in-terest in OR turned to peacetime appli-cations. There are now many OR depart-ments in industry, government, andacademia throughout the world. Ex-amples of where OR has been successfulin recent years are the following:

Airline Industry (routing and flightplans, crew scheduling, revenuemanagement);

Telecommunications (network routing,queue control);

Manufacturing Industry (systemthroughput and bottleneckanalysis, inventory control,production scheduling, capacityplanning);

Healthcare (hospital management,

facility design); and

Transportation (traffic control, logistics,network flow, airport terminallayout, location planning).

There are many mathematical techniquesthat were developed specifically for ORapplications. These techniques arosefrom basic mathematical ideas and be-came major areas of expertise for indus-trial operations.

One important area of such techniquesis optimization. Many problems in indus-try require finding the maximum orminimum of an objective function of aset of decision variables, subject to a setof constraints on those variables. Typi-cal objectives are maximum profit, mini-mum cost, or minimum delay. Fre-quently there are many decision variablesand the solution is not obvious. Tech-niques of mathematical programmingfor optimization include linear program-ming (optimization where both the ob-jective function and constraints dependlinearly on the decision variables), non-linear programming (non-linear objec-tive function or constraints), integer pro-gramming (decision variables restrictedto integer solutions), stochastic program-ming (uncertainty in model parametervalues) and dynamic programming(stage-wise, nested, and periodic deci-sion-making).

Another area is the analysis of stochasticprocesses (i.e., processes with randomvariability), which relies on results fromapplied probability and statistical mod-

Mathematics and Operations Research in Industry

By Dennis E. Blumenfeld, Debra A. Elkins, and Jeffrey M. Alden

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February 2004

eling. Many real-world problems involveuncertainty, and mathematics has beenextremely useful in identifying ways tomanage it. Modeling uncertainty is im-portant in risk analysis for complex sys-tems, such as space shuttle flights, largedam operations, or nuclear power gen-eration.

Related to the topic of stochas-tic processes is queueing theory(i.e., the analysis of waitinglines). A common example is thesingle-server queue in whichcustomer arrivals and servicetimes are random. Figure 1 illus-trates the queue, and the curveshows how sensitive the averagequeue length becomes underhigh traffic intensity conditions.Mathematical analysis has beenessential in understanding queuebehavior and quantifying im-pacts of decisions. Equationshave been derived for the queuelength, waiting times, and prob-ability of no delay, and othermeasures. The results have appli-cations in many types of queues, such ascustomers at a bank or supermarketcheckout, orders waiting for production,ships docking at a harbor, users of theinternet, and customersserved at a restaurant. Ex-amples of decisions inmanaging queues are howmuch space to allocate forwaiting customers, whatlead times to promise forproduction orders, andwhat server count to as-sign to ensure short wait-ing times.

An important mathemati-cal problem in manufac-turing is the performanceanalysis of a productionline. A typical productionline consists of a series ofworkstations that perform different op-erations. Jobs flow through the line to beprocessed at each station. Buffers be-tween stations hold the output of onestation and allow it to wait as input tothe next. A finite buffer can fill and blockoutput from an upstream station or canempty and starve a downstream stationfor input. Blocking and starving are key

mechanisms of the complex interactionsbetween queues that form in the line. Acritical measure of performance isthroughput, defined as number of jobsper unit time that can flow through theline. Throughput is reduced when sta-tions experience random machine fail-

ures, a common practical situation.Mathematical modeling is needed to cap-ture the impact on throughput of stationreliabilities, as well as processing rates

and buffer sizes. A model can supportoperating decisions, such as how to im-prove a line to meet a throughput target,how to identify bottlenecks, and howmuch buffer space to allocate in line de-sign.

Another real-world mathematical prob-lem, common to many industries, is the

distribution of material and productsfrom plants to customers. For a networkof origins and destinations, there aremany shipping alternatives, includingchoices of transportation mode (e.g.,road, rail, air) and geographical routes.Some key decisions are routing options

over the network, and shipping fre-quencies on network links. Asshown in Figure 2, routing optionsinvolve shipping direct, via a ter-minal or distribution center, andby a combination of routes. Theseoptions affect distances traveledand times in transit, which in turnaffect transportation and inven-tory costs. Shipping frequency de-cisions also affect these costs.Transportation costs favor large in-frequent shipments, while inven-tory costs favor small frequentshipments. Trade-offs betweenthese costs are complex for largenetworks, and finding the optimalsolution is a challenging math-ematical problem. In addition todecisions for operations of a givennetwork, there are major strategic

decisions, such as the selection and lo-cation of distribution centers.

Other OR topics requiring mathemati-cal analysis are in-ventory control(when to reordermaterial to avoidshortages under de-mand uncertainty),manufacturing op-erations (what sizeof production runwill minimize sumof inventory andproduction setupcosts), locationplanning (where tolocate the hub toserve markets withminimal travel dis-tances), and facility

layout (how to design airport terminalsto minimize walking distances, maximizenumber of gates, allow for future expan-sion, and conform to government regu-lations).

OR analysts can model difficult practi-cal problems and offer valuable solutionsand policy guidance for decision-mak-

Figure 1: Single-Server Queueing System

Figure 2: Network Routing Options

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FOCUS February 2004

ers. Constraints involving budgets, capi-tal investments, and organizational con-siderations can make the successfulimplementation of results as challengingas the development of mathematicalmodels and solution methods.

In general, Operations Research requiresuse of mathematics to model complexsystems, analyze trade-offs between keysystem variables, identify robust solu-tions, and develop decision supporttools. Students of mathematics can besure there are plenty of uses for theknowledge and skills they are develop-ing. As the world becomes more complexand more dependent on new technology,mathematics applied to business prob-lems is likely to play an increasingly im-portant role in decision-making in in-dustry.

On a personal note…

All three of us developed an interest inthe mathematical sciences early on, andtook undergraduate degrees in math, ormath and physics. We each got into thefield of Operations Research as a resultof looking for practical ways to use ourmath training. Below, we each answer thequestion: “How did you decide on a ca-reer in math and decide to join GM?”

Dennis Blumenfeld: The math courses Iliked best were the ones on applied topics.I found Operations Research an especiallyappealing subject, since it uses basic math-ematical principles in clever ways to solveall kinds of complex problems in everydaylife — such as queueing, reliability, sched-uling, and optimization. I was intriguedby applications of OR models to traffic flowand congestion, and as a graduate studentat University College London I focused onmodeling of transportation systems. I con-tinued research on this topic in engineer-ing school faculty positions at PrincetonUniversity and University College London.I knew of the traffic studies and other re-search at GM R & D through meetings andtheir publications, and was interested togain experience of applied research in in-dustry. I joined GM R & D, where I havehad the opportunity to work in a varietyof research areas, including traffic safety,logistics, inventory control, and productionsystem design, and to see results used in

practice. It always impresses me how pow-erful even simple mathematical modelscan be in providing insight into system be-havior.

Debra Elkins: I took a lot of classes inmath, computer science, physics, andchemistry, and finally realized I liked sportcomputing and slick mathematics appliedto real world industrial problems. I endedup in Operations Research, which lets mecombine my interests in probability, supercomputing and high performance comput-ing, simulation, and so forth. As a gradu-ate student in the Industrial Engineering/Operations Research Program at TexasA&M University, I found out about work-ing at GM R&D when I was at a technicalconference. I decided to interview out ofcuriosity. I was really surprised and de-lighted with the people and the caliber ofresearch going on within GM. My firstmajor research project was to explore fi-nancial implications of agile machiningsystems for GM. While working on thatproject, I was poking around in risk analy-sis work, and connected with GM Corpo-rate Risk Management, a group thatwanted some help with probabilistic mod-eling of risks. Now I’m working on strate-gic supply chain risk analysis. I’m exam-ining how to model the GM manufactur-ing enterprise, exploring the frequency andseverity of business interruption events—anything that interrupts production opera-tions—and considering strategic mitiga-tion options that can reduce GM’s risk ex-posure. What excites me about my researchis combining ideas from different subjectareas, like math, computer science, statis-tics, and operations research, to developnovel modeling approaches and solutionsfor large-scale problems.

Jeff Alden: I basically pursued areas thatI liked, excelled in, and seemed good for afuture career. Since I really enjoy problemsolving, math modeling, and helpingpeople make better decisions, I naturallymigrated to Operations Research. So I wassure what I wanted to do, but not surewhere to work. Well, I attended a presen-tation about the research opportunities atGM R &D given by Larry Burns (now aVP at GM). It seemed like an ideal placeto work, so I gave him my resume and soonaccepted a research position at GM R&D.For the next decade, I researched produc-

tion systems looking at throughput, main-tenance, leveling, stability, agility, andcost-drivers. I’m now on a two-year rota-tion in GM Engineering managing devel-opment of decision-support tools andmethods for engineering in a variety ofareas that include test-scheduling, war-ranty cost drivers, and work load planning.

We wish to thank Dr. Devadatta Kulkarniat General Motors R&D Center forvaluable comments and suggestions.

Bibliography

Allen, A. O. (1990). Probability, Statistics,and Queueing Theory - With ComputerScience Applications (Second Edition).Academic Press, Orlando, Florida.

Cohen, S. S. (1985). Operational Research.Edward Arnold, London.

Chvátal, V. (1983). Linear Programming.W. H. Freeman, New York.

Gross, D. and Harris, C. M. (1985).Fundamentals of Queueing Theory. Wiley,New York.

Hillier, F. S. and Lieberman, G. J. (2001).Introduction to Operations Research(Seventh Edition). McGraw-Hill, NewYork.

Hopp, W. J. and Spearman, M. L. (2000).Factory Physics: Foundations ofManufacturing Management (SecondEdition). Irwin/McGraw Hill, New York.

Larson, R. C. and Odoni, A. R. (1981).Urban Operations Research.Prentice Hall, Englewood Cliffs, NewJersey.

Lewis, C. D. (1970). Scientific InventoryControl. Elsevier, New York.

Ross, S. M. (1983). Stochastic Processes.Wiley, New York.

Silver, E. A., Pyke, D, F. and Peterson, R.(1998). Inventory Management andProduction Planning and Scheduling.(Third Edition). Wiley, New York.

Vanderbei, R,. J. (1997). LinearProgramming: Foundations andExtensions. Kluwer, Boston.

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February 2004

Prizes and Awards at the Phoenix Joint Mathematics Meetings

Thomas Garrity was one of three win-ners of the Deborah and Franklin TepperHaimo Award for Distinguished Collegeor University Teaching of Mathematics.In his acceptance, Garrity described him-self as “the William Shatner of math-ematics: a smidgeon of talent and a hugenumber of lucky breaks.”

Thomas Garrity Andy Liu

Andrew Chiang-Fung Liu was one ofthree winners of the Deborah andFranklin Tepper Haimo Award for Dis-tinguished College or University Teach-ing of Mathematics. He began his re-sponse by saying “I think I’m entitled toa rebuttal.” For more on the HaimoAward winners, see the November issueof FOCUS.

Olympia Nicodemi

Olympia Nicodemi was one of three win-ners of the Deborah and Franklin TepperHaimo Award for Distinguished Collegeor University Teaching of Mathematics.In her response, Nicodemi thanked herstudents “who have put up with all thisnonsense.” She also observed that MAAPresident Ron Graham has (so far) failedto teach her how to juggle.

Underwood (Woody) Dudley received aCertificate of Meritorious Service fromthe Indiana Section of the MAA. The ci-tation noted his many years of service tothe Section and to the Association at anational level. In his response, Dudleysaid, “I turned in my first set of grades inDecember 1957 (Calculus I) and my lastset in May 2003 (Calculus II). Not muchprogress!”

Underwood Dudley

Stephen Ligh received a Certificate ofMeritorious Service from the Louisiana-Mississippi Section of the MAA. The ci-tation noted, in particular, Ligh’s work inestablishing the Student Team Competi-tion in 1988 and running it until his re-tirement in 1997. Ligh has also receivedhis section’s Distinguished TeachingAward.

Stephen Ligh

Deborah and Franklin Tepper Haimo Award for Distinguished College or University Teaching of Mathematics

Section Certificates of Meritorious Service

Richard Barlow received a Certificate ofMeritorious Service from the Nebraska-Southeast South Dakota Section of theMAA. The citation noted his many yearsof service as Section officer, chair, meet-ing host, and newsletter editor.

Richard Barlow

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Thomas A. Hern received a Certificateof Meritorious Service from the OhioSection of the MAA. The citation notedhis work as Section President, his workon many committees, his work on thesection web site, and finally as Governor.It also pointed out that Hern was a mem-ber of an MAA Delegation to the People’sRepublic of China in 1983.

Thomas A. Hern

John W. Kenelly received a Certificate ofMeritorious Service from the Southeast-ern Section of the MAA. Currently theTreasurer of the Association, Kenelly hasserved both his Section and the MAA inmany ways: as a leader in the reform ofteaching and the use of technology in theclassroom, as a fund raiser, and by hiswork as an officer and a member ofmany committees.

John W. Kenelly

Noam Elkies received the Levi L. ConantPrize from the AMS for his two-part ar-ticle “Lattices, Linear Codes, and Invari-ants,” which appeared in the October andNovember 2000 issues of the Notices ofthe AMS. In his response, Elkies ex-pressed the hope that the prize would en-courage more people to actually read thearticle.

Noam Elkies

Lawrence C. Evans (pictured) andNicolai V. Krylov (who was unable to at-tend) jointly received the Leroy P. SteelePrize for Seminal Contribution to Re-search for their work on the Evans-Krylov Theorem. The citation added that“while the Steele Prize for Seminal Re-search is explicitly awarded for thenamed works, it is noted that both re-cipients have made a variety of distin-guished contributions to the theory ofnonlinear partial differential equations.”

Lawrence C. Evans

John W. Milnor was the winner of theLeroy P. Steele Prize for MathematicalExposition, awarded by the AMS “in rec-ognition of a lifetime of expository con-tributions.” In his response, Milnor saidthat he has “always suspected that the keyto the most interesting exposition is thechoice of a subject that the authordoesn’t understand too well. I have theunfortunate difficulty that it is almostimpossible for me to understand a com-plicated argument unless I try to write itdown.”

John W. Milnor

Cathleen Synge Morawetz received theLeroy P. Steele Prize for Lifetime Achieve-ment from the AMS. She was honored forher research on partial differential equa-tions and applied mathematics, for offer-ing “guidance and inspiration” to manygraduate students, visitors, and post-docs,and for her leadership within the math-ematics community, which included aterm as president of the AMS. In her re-sponse, Morawetz thanked, among oth-ers, her high school mathematics teacher,“Buckshot” Reynolds.

Cathleen Synge Morawetz

Section Certificates of Meritorious Service Levi L. Conant Prize

Leroy P. Steele Prize for SeminalContribution to Research

Leroy P. Steele Prize forMathematical Exposition

Leroy P. Steele Prize for LifetimeAchievement

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February 2004

Bozenna Pasik-Duncan received theLouise Hay Award for Contributions toMathematics Education from the AWM.The award recognizes outstandingachievement in any area of mathemat-ics education. Pasik-Duncan was hon-ored for a wide range of education-re-lated activities, from teaching elemen-tary school students to working with un-dergraduates on research projects. Shededicated her award to those researchmathematicians who teach honors in-troductory courses in mathematics.

Bozenna Pasik-DuncanT. Christine Stevens received the Yueh-Gin Gung andDr. Charles Y. Hu Award for Distinguished Service forMathematics, the highest honor offered by the MAA,to a standing ovation from NExT fellows old and new.Stevens has had a long and productive career that in-cluded many forms of service to mathematics, but it isas co-creator and director of Project NExT that she hashad the most impact. In her acceptance, Stevens notedthat her advisor, Andrew Gleason, had also received thisaward (in 1996), making them the only advisor-adviseepair to have received the award (so far!). A fuller ac-count of Stevens’ work and influence will appear in theAmerican Mathematical Monthly.

T. Christine Stevens

Yueh-Gin Gung and Dr. Charles Y. Hu Award forDistinguished Service for Mathematics

Louise Hay Award forContributions to

Mathematics Education

Richard A. Tapia received the AMS Award for Distin-guished Public Service “for inspiring thousands of people(from elementary school students to senior citizens) tostudy and appreciate the mathematical sciences” and forhis dedication to “opening doors for underrepresentedminorities and women.” In his response, Tapia noted thathe often visits elementary school classrooms full of brightand interested students, and is saddened to think that agreat proportion of those who are African-American orLatino will never graduate from high school. He arguedthat this statistic should be a call to action.

James A. Sethian received the Norbert Wiener Prize inApplied Mathematics, sponsored jointly by SIAM andthe AMS, “for his seminal work on the computer repre-sentation of the motion of curves, surfaces, interfaces,and wave fronts, and for his brilliant applications ofmathematical and computational ideas to problems inscience and engineering.” In his response, Sethian men-tioned that his high school teacher once called him asideto point out that he was pretty good at mathematics,almost as good as the kid sitting next to him. This otherkid was Eric Schmidt, currently the CEO of Google. Healso pointed out that “there are closet mathematicianseverywhere” and urged his hearers to speak to them.Indeed, the citation noted that Sethian’s work is note-worthy in that it goes all the way from the formulationof a mathematical model to concrete applications inlaboratory and industrial settings.

James A.Sethian

Norbert Wiener Prize in Applied Mathematics

Richard A. Tapia

AMS Award for Distinguished Public Service

Mark Haiman received the E. H. MooreResearch Article Prize from the AMS forhis paper on “Hilbert Schemes, Poly-graphs, and the Macdonald PositivityConjecture,” which appeared in the Jour-nal of the AMS 14 (2001), 941–1006. Theprize is for an outstanding research ar-ticle appearing in one of the AMS jour-nals over a six-year period.

Mark Haiman

E. H. Moore Research Article Prize

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Kimberly Spears

Kimberly Spears received the Alice T.Shafer Prize for Excellence in Mathemat-ics by an Undergraduate Woman fromthe AWM. Spears is a senior at the Uni-versity of California, Santa Barbara. Inaddition to performing brilliantly in herclasses, she has done two significant re-search projects: a generalization of theQuadratic Reciprocity Law to non-abe-lian groups and a result showing that as-suming standard conjectures about theconnection of zeros of zeta and L-func-tions to the Grand Unitary Ensemble al-lows one to classify discriminants dwhose corresponding quadratic numberfields have one ideal class per genus.

David Gabai received the Oswald VeblenPrize in Geometry from the AMS for hiswork in geometric topology, in particu-lar the topology of three-dimensionalmanifolds. In his response, Gabai notedthat much of his work was based on andinspired by the work of former winnersof a Veblen Prize, and said that “It is hum-bling to be the recipient of the 2004Veblen Prize.”

David Gabai

Alice T. Shafer Prize for Excellencein Mathematics by an

Undergraduate Woman

Oswald Veblen Prize in Geometry

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February 2004

Eileen Poiani Receives

Several Awards

Eileen L. Poiani, Vice President for Stu-dent Affairs and Professor of Mathemat-ics at Saint Peter’s College and a long-time member of the MAA, received oneof New Jersey’s top honors when she washonored with the New Jersey Woman ofAchievement Award by the New JerseyFederation of Women’s Clubs. The pre-sentation was made on April 6, 2003, atDrumthwacket, the New JerseyGovernor’s residence in Princeton. OnMay 3, she received the Alumnae Recog-nition Award for outstanding service toDouglass College, her alma mater. Laterthat same month, on May 13, the Na-tional Conference for Community andJustice (NCCJ), New Jersey Region,Hudson County Chapter, bestowed uponDr. Poiani its Humanitarian Award. Theaward recognized her work with diversegroups to build harmony and respect.The year culminated with Dr. Poiani’sinduction into the inaugural Nutley Hallof Fame. Induction is based on anindividual’s extensive contributions out-side of the Township of Nutley, New Jer-sey. She is a graduate of Nutley HighSchool and a third generation Nutleyite.

Within the MAA, Poiani has most re-cently served on the Education Council.She was the founding director of theMAA’s Women and Mathematics Pro-gram and served as the Governor of theNew Jersey Section. She received theMAA New Jersey Section Award for Dis-tinguished College Teaching in 1993.

Note from the Editor

Because the Joint Mathematics Meet-ings were held early in January, we havebeen able to include some material onthe meetings (mainly prize winners) inthis issue. Expect much more Joint Meet-ings coverage, including a photo spread,in the March issue.

The transition points of a student’smathematical career have long been rec-ognized as critical junctures and havebeen a topic of concern and conversationin the mathematical community.Whether from high school to college,two-year college to four-year institution,undergraduate to graduate school, oreven graduate school to a post-doc fel-lowship or first job, each transition isknown to be critical to the developmentpractitioners in the mathematical sci-ences. Statistics on the loss of studentsat these points are well known. Othermore subtle transitions need scrutiny aswell; these include the move from com-putational courses to proof-focusedcourses in the undergraduate years andthe change from course-taker to inde-pendent researcher in graduate school.

A previous column (September 2003)described briefly a new NSF initiative,Mentoring through Critical TransitionPoints (MCTP). The first round ofMCTP proposals were reviewed in De-cember 2003, and these give some indi-cations about current thinking for deal-ing with the transitions listed above. Fora variety of reasons, the step from un-dergraduate to graduate school seems tobe the easiest on which to get a handleand generated the most proposals. De-veloping a program to help studentsthrough one of the other points may bemore challenging. Proposals may addressone or more transitions either within asingle institution or in a consortium.With some funding also available fromthe Directorate for Education and Hu-man Resources for MCTP projects, thearea of engaging talented high school stu-dents in mathematical research and thusattracting them to major in mathemat-ics as undergraduates is appropriate forproposals.

It should be noted that of the three seg-ments of Enhancing the MathematicalSciences in the 21st Century, Vertical Inte-gration of Research and Education(VIGRE), Research Training Groups

(RTG), and MCTP, only MCTP acceptsproposals from four-year colleges. TheNSF hopes to see more proposals in thesecond round as the program becomesbetter known. Both the Foundation andthe U. S. Congress are committed to in-creasing the number and quality ofAmericans in the mathematical sciences.The deadline for new MCTP proposalsis mid-September.

MCTP is intended to generate commu-nity-wide best practices in promotingsuccessful careers in the mathematicalsciences for more students, in particularfor U.S. citizens, national, and permanentresidents. Because relatively modest ef-forts have been undertaken for dealingwith the critical transition points, muchremains to be done. Further discussionis called for. The MAA, with NSF-fund-ing, will host a conference in Washing-ton, D.C., April 30–May 1, Dialogue2004: The Division of Mathematical Sci-ences and the Mathematical SciencesCommunity. The conference will be aforum for the discussion of all three com-ponents of Enhancing the MathematicalSciences in the 21st Century.

NSF Beat

Mentoring Through Critical Transition Points

By Sharon Cutler Ross

Correction

The mathematics education team atUniversity of Michigan–Dearborn ispleased that MAA has noted our recentaward of an NSF CCLI Adaptation andImplementation grant. (NSF Beat, De-cember 2003 issue of FOCUS). There is,however, a misprint in the title of ourproject. It should be “A Problem and Rea-soning Based Curriculum for PreserviceElementary Educators.” (In the article“reasoning” has been replaced with “re-search.”) While our work certainly is “re-search-based” that is not part of our title.Also, please note that our affiliation isUniversity of Michigan–Dearborn.Thank you.

Rheta RubensteinDepartment of Mathematicsand StatisticsUniversity of Michigan–Dearborn

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FOCUS February 2004

My article “The Texas Octagon Massa-cre” in the December issue of FOCUSproduced (as expected) many reactions.Here is a selection of what we received.My comments are in italics.

Which is the Radius?

In the article The Texas Octagon Massa-cre, the author used the term “radius” todescribe the segment from the center toa vertex of a polygon. This is probablynot a good idea. The definition of a ra-dius should be a segment from the cen-ter of a regular n-gon perpendicular to aside of the regular n-gon. If we take acircle as the limit of a regular n-gon whenn tends to infinity, this definition of theradius of a regular n-gon would extendnaturally to a circle.

Fat LamGallaudet University

It’s a matter of convention, of course. Thesegment from the center to a vertex alsoconverges to the radius of the circle, afterall. I used “radius” because I was too lazyto say “radius of the circumscribed circle,”which was what I meant. I think the stan-dard name for the length of the radius ofthe inscribed circle is “apothem,” leaving“radius” to mean the radius of the circum-scribed circle. See, for example, http://mathworld.wolfram.com/Apothem.html.

Another Article

I also read the NASSMC Briefing andwrote an article about the Octagon prob-lem. I thought you might be interestedin reading it: http://learn.sdstate.edu/dwight_galster/Issues/TAKS8.htm

Dwight GalsterSouth Dakota State University

Thanks for letting me know. I thoughtreaders of FOCUS might want to take alook at it too.

Other Methods

I got a kick out of your “massacre” ar-ticle even if the story is very depressing.When I tried to work it out for myself, Icame to the same conclusion that youdid. Not having a calculator or tablesavailable I got the 26.5 answer by sim-pler methods. I pass the idea along foryour interest.

A circumscribed square will have fouredges of length 4 + 4 = 8 for a perimeterof 32 so clearly the octagon has less pe-rimeter, since it uses one side rather thantwo of the “ears” which are part of thesquare but not of the octagon. Thatmakes the 27 answer look about right;the 18 seems too low.

A more careful analysis shows that eachedge of the square consists of one octa-gon edge, x, and two edges, y, from theears at the corners. We get x y+ =2 8 andalso x y= 2 , leading tox = +( ) = …8 1 2 3 3137/ . With 8 edgesthe octagon has perimeter 26.5097.

I think this problem, which is clearly a“wrong problem,” could be an interest-ing exercise for students, teaching themto be willing to question the correctnessof claims that seem to lead to contradic-tion.

Carroll ZahnRetired, San Diego

I read with interest your write-up in theDecember 2003 issue of FOCUS. Youmight find it interesting I too instantlygot both answers, trying my best to takea second look at the problem as thoughit were a new problem!

Initially, I used the Pythagorean Theo-rem, did the math, and arrived at an an-swer of 36.3 cm. Fine: check “36 cm” andmove on.

Then I looked at the problem again, and,seeing the placement of the point in thecenter of the regular octagon. Not beingtold “drawing is not to scale”, it’s easy tosee the outer circle of radius 4.6 cm (pe-rimeter 28.9 cm) and the inner circle ofradius 4.0 cm (perimeter 25.1 cm), withthe average of these two perimeters ap-proximating the perimeter of the regu-lar octagon equal to 27.0 cm.

Remember, the question asked for ananswer “to the nearest centimeter,” so it’snot unreasonable to use the averagingabove to approximate the answer.

Michael RoundOverland Park, KS

Missing Assumptions

There is an octagon with the measure-ments shown in the illustration for yourarticle in FOCUS, but it’s not a regularoctagon. The horizontal and verticalsides have length 2 5 16. cm and the di-agonal sides have length 2 4 5 16– .( )cm. Its perimeter comes to about 27.95cm. The test taker is thus forced to rounddown, and select 27 cm as the answer.

George KangasUniversity of Kansas

Sure. The problem, however, did specifythat it was a regular octagon (as noted inthe January issue). Unfortunately, I failedto state that in my article.

You, along with those students who gotthe answer wrong (by failing to select thesingle correct answer of 36 centimeters),fell into a trap.

You are incorrect when you state that “nooctagon with those measurements ex-ists.” If the octagon is regular, there is onlyone. Where you (and those students whoused trigonometry to “solve” the prob-lem) went wrong was in assuming thatthe apex of the triangles (where the sidesof lengths 4.6 cm and 4 cm meet) was

Octagon Feedback

By Fernando Q. Gouvêa

FOCUS

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February 2004

the center of the octagon. Nothing in theinformation given (aside from a poorlydrawn figure) suggests such a conclusion.In any geometry course a student is re-peatedly advised to NOT draw conclu-sions from a diagram, but to use only thegiven information. (Of course, some-times a diagram might contain the in-formation through symbols, such as theright-angle symbol in this example.)

I take issue with your penultimate sen-tence also: “The truth is just that therewas a wrong problem.” A problem canbe ambiguous (this one wasn’t), or un-solvable (this one wasn’t), or misleading(this one was). But the only students whofell into the trap and used trigonometrywere those who made an unwarrantedassumption.

Was it a good question? Of course not,unless its goal was to identify those stu-dents who would assume a fact from adiagram when that fact could easily bedisproved by using the given informa-tion. But the identification of such stu-dents could be done with other much-better questions.

But I do agree with you that is was non-sense for the test makers to say that “therewere two right answers.”

Harry BaldwinAuthor of Essential Geometry

You are certainly correct. Looking at theproblem, I don’t see any explicit statementthat the apex of the triangle is the center.If we drop that assumption, then of coursethe only correct answer to the problem is36 (but the picture is very far from beingcorrectly drawn in that case, since theapothem of an octagon with perimeter 36is about 5.4).

The result is a problem that will confusethe better students, unless they are reallygood students. I kind of doubt that thiswas intended, but it’s hard to tell. Ofcourse, as you point out, this reading of the

problem seems to have been missed by thetest makers themselves!

How Often Does It Happen?

In regards to your question: “how oftendoes this happen?” in your article “TheTexas Octagon Masssacre,” I note thatthere were 5 errors in the NYS RegentsMath A Exam from June 2003. One ofthese errors forced the state to adjust thescores on the exam. I think [such errorsare] very common. I am hoping soon tocomplete an analysis of all the past MathA Exams.

David W. HendersonCornell University

David enclosed an article analyzing theJune 2003 Regents Exam. Most of the er-rors he finds are of the “missing assump-tion” kind. For example, students are giventhe total number of voters in an electionand the votes for candidate A. They arethen asked to compute the percentage ofpeople who voted for candidate B. Themissing assumption, of course, is that therewas no third candidate.

I’d like to address the question withwhich you ended your article in the re-cent issue of FOCUS: “How often doesthis happen?” You’re not going to like myanswer.

Speaking as a competent math teacher(I have an M.S. in mathematics) with 18years’ experience, it is my unhappy dutyto inform you that in every high-stakesmultiple-choice test I have had the un-pleasant job of administering there havebeen several — not one, several — obvi-ous howlers of the type you noted thismonth.

Every year you hope that the powers thatbe have a reasonably howler-free test;every year you are disappointed. Forpeace of mind in the last couple of yearsI’ve stopped reading the tests. No mat-ter: within a day my colleagues are talk-ing about the latest boo-boo in theteacher’s lounge.

Look, the people who write the tests arenot math teachers. The people who or-der the tests are bureaucrats in the stateDepartment of Education: they’re notmath teachers. Neither the people whowrite them nor the people who orderthem value the services of math teach-ers; the result is predictable.

Michael MeoBenson Polytechnic High SchoolPortland Oregon

Well, if you’re right then this is an unfor-tunate state of affairs. The question weshould pose ourselves is how we can helpmake the tests more effective. It seems veryreasonable to me that those who pay forpublic education should want to seewhether they are getting the education theyare paying for, so I don’t object to testingper se.

But if the tests are going to give useful in-formation they should be designed well,and good mathematics teachers can andshould do what they can to help that hap-pen. I don’t know enough about how thesetests are designed to make concrete sugges-tions about how that might be achieved. Iam pretty sure, however, that if we just stayon the sidelines and criticize we won’tachieve all that much.

What is the perimeter to the nearest centi-meter of the regular octagon drawn below?

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FOCUS February 2004

The Mathematical Association ofAmerica will hold its annual MathFest,Thursday, August 12 through Saturday,August 14, 2004 in Providence, RhodeIsland.

The complete meetings program willappear in the April 2004 issue of FOCUS.This announcement is designed to alertparticipants about the contributed papersessions and their deadlines. Please notethat the days scheduled for these sessionsremain tentative.

The organizers listed below, indicatedwith an (*), solicit contributed paperspertinent to their sessions. Sessions gen-erally limit presentations to ten or fifteenminutes. Each session room contains anoverhead projector and screen. Personsneeding additional equipment shouldcontact the organizer of their session assoon as possible, but prior to Tuesday,May 4, 2004.

Submission Procedures for Contrib-uted Paper Proposals

Send the name(s) and address(es) of theauthor(s), and a one-page summary ofyour paper directly to the organizer in-dicated with an (*). In order to enablethe organizer(s) to evaluate the appro-priateness of your paper, include as muchdetailed information as possible withinthe one-page limitation.

A proposal should not be sent to morethan one organizer. If your paper can-not be accommodated in the session itwas submitted, it will be automaticallyconsidered for the general contributedpaper session. E-mail submissions arepreferred.

Your summary must reach the desig-nated organizer by Tuesday, May 4, 2004.Early submissions are encouraged. Theorganizer will acknowledge receipt of allsummaries. If your paper is accepted, theorganizer will provide you with an e-mailtemplate and directions on how to sub-mit an abstract for your presentation.

Contributed Paper Sessions

MAA CP A1 Uses of the WWW that En-rich and Promote LearningThursday and Friday afternoons

This session seeks to highlight uses of theWeb and its tools that engage studentsin the learning process. Tools such ascourse management systems, digital re-sources, tutorial systems, and hybridsthat combine these functions on the Webcan make a difference in student engage-ment, understanding, and performance.Talks should demonstrate how thesetechnologies are being integrated into thelearning process. The session is spon-sored by WebSIGMAA and the MAACommittee on Computers in Mathemat-ics Education (CCIME).

Roger Nelson*Department of Mathematical SciencesBall State UniversityMuncie, IN 47306-0490Phone: 765-285-8653 (Office)765-747-7921 (Home)FAX: 765-285-1721Email: rbnelson@bsu.edu

Marcelle Bessman, Jacksonville Univer-sity Kirby Baker, UCLA

MAA CP B1 ExtracurricularMathematicsThursday afternoon

Most MAA sessions are focused on dif-ferent ideas and strategies to incorporateinto various classes. On the other hand,some of the most creative and diverseaspects of our work with students are theways in which we incorporate mathemat-ics outside the classroom. This sessioninvites presentations of mathematicalactivities outside of the classroom. Thisincludes, but is not limited to, MAA stu-dent chapters’ activities, field trips, ac-tivities with schools and in the generalcommunity, presentations and work-shops. Submissions are encouraged notonly from faculty, but also from studentswho have organized extracurricularmathematical activities.

Jeff Johannes*SUNY Geneseo1 College CircleGeneseo, NY 14454-1401Phone: 585-245-5403FAX: 585-245-5128email: johannes@geneseo.edu

Melissa Sutherland, SUNY Geneseo

MAA CP C1 Putting Some AnalysisInto Introductory Real AnalysisThursday afternoon

Introductory Real Analysis is a requiredcourse for most mathematics majors. Assuch, it offers unique opportunities togenerate real student interest in advancedmathematics by challenging their intu-ition and pushing them to think at ahigher level. All too often this goal is notreached and the course ends up as onethat presents the theory behind the re-sults in first year calculus—essentially acourse in epsilons and deltas. Papers sub-mitted for this session should presentapproaches that generate student inter-est and encourage them to explore math-ematics further while at the same timefocusing on the key theoretical compo-nents underlying first year calculus

Richard J. MaherDepartment of Mathematicsand StatisticsLoyola University Chicago6525 N. Sheridan RoadChicago, Illinois 60626Phone: 773-508-3565FAX: 773-508-2123rjm@math.luc.edu

MAA CP D1 Cooperative Projects Be-tween the Mathematical Sciences andthe Life SciencesSaturday afternoon

Emerging areas in the life science areopening up new careers for biology andmathematical sciences majors. Bothgroups of students need preparatory un-dergraduate experiences. This sessionwill consist of talks by faculty who,within the existing structures of theirdepartments and schools, have devel-oped such experiences with colleagues inthe life sciences. The presentations willdescribe the type of experience, how itwas initiated, how it has evolved and,

Call for Papers

Contributed Paper Sessions at MathFest 2004

FOCUS

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February 2004

most importantly, evidence of its impacton students. This session is sponsored bythe Subcommittee on MathematicsAcross the Curriculum.

Catherine M. MurphyPurdue University CalumetMathematics, Comp Sci, & StatHammond, IN 46323-2094Office: 219-989-2273FAX: 219-989-2165Email: murphycm@calumet.purdue.edu,

Bill Marion, Valparaiso University

MAA CP E1 Innovative Approaches inMathematics EducationThursday afternoon

This session invites papers dealing withinnovations in mathematics educationcourses for both pre-service and in-ser-vice teachers K–12. Topics of interestmight include: new courses; use of tech-nology or web-based modules; onlinetools and activities; courses aligned tonational or state standards; courses formaster’s of arts in teaching programs;relevant assessment techniques; interac-tions with local school districts; useful-ness of such topics as history of math-ematics or collaborative learning strate-gies; difficulties encountered in teachingmath ed courses. We look forward to in-teresting responses and hope that discus-sion will enhance our future teaching.

*Elias DeebaDepartment of Computer and Math-ematical SciencesUniversity of Houston-DowntownOne Main StreetHouston, Texas, 77002Phone: 713-221-8550FAX: 713-221-8086Email: deebae@uhd.edu

Carol Vobach, University of Houston-Downtown

MAA CP F1 Advances in RecreationalMathematicsFriday afternoon

There have been many recent advancesin recreational mathematics, some ofwhich have involved the use of comput-ers. This session is designed to give youan opportunity to explain your recent

work in the field. While the organizerencourages submissions that involvecomputers, that is not essential for con-sideration. For the purposes of this ses-sion, the definition of recreational math-ematics will be a broad one. The primaryguideline used to determine suitability ofsubject will be the understandability ofthe mathematics. For example, if themathematics in the paper is commonlyfound in graduate programs, then itwould generally be considered unaccept-able. Supplemental computer programscan be written in any language, howeverthey must be clean and WELL docu-mented. Any source code used to createthe paper must also be submitted forverification. Papers where existing pro-grams such as Mathematica® were usedwill also be considered.

Charles Ashbacher*Charles Ashbacher TechnologiesBox 294Hiawatha, IA 52233Phone: 319 378-4646FAX: 928-438-7929E-mail: cashbacher@yahoo.com

MAA CP G1 Model Lessons from First-Year CalculusFriday afternoon

This session will explore best practicesin teaching first-year calculus. In theirbook, The Teaching Gap: Best Ideas fromthe World’s Teachers for Improving Edu-cation in the Classroom, education re-searchers James W. Stigler and JamesHiebert point to “lesson-study” as thedriving force behind educational reformin Japan. Lesson-study is a “research-and-development system” in whichteachers collaborate to design, imple-ment, and refine mathematics lessons.The premise is that continual teachingimprovement results from understand-ing how the daily lesson promotes stu-dent understanding and learning. Thissession invites presenters to contributeto the scholarship of teaching by sharinga lesson given in the last three years on asingle topic from first-year calculus. Pre-senters should describe their studentprofile and address these questions: Whatis the topic of your lesson? What is (are)the main idea(s) behind the topic? Howdo you present the topic? How do stu-dents learn the topic? What are the stu-

dent mistakes and confusions in learn-ing this topic? Where do the teaching dif-ficulties lie? How did your lesson evolveover time? How did you measure the suc-cess of this lesson in promoting studentlearning?

Donna Beers*Department of MathematicsSimmons College300 The FenwayBoston, MA 02115Phone: 617-521-2389FAX: 617-521-3199Email: donna.beers@simmons.edu

MAA GP H1 Mathematical ModelingModules and MaterialsSaturday afternoon

Courses in mathematical modeling havegrown in popularity in recent years, en-compassing a diverse range of offerings.The interdisciplinary nature of model-ing courses allows for the creative use ofa wide range of materials ranging fromwriting and reading in the curriculum toinnovative uses of technology. Modules,self-contained instructional lessons, havebeen popularized by the Consortium forMathematics and its Applications(COMAP) and have been disseminatedin several publications. This session willfocus on modules and other materialsthat would be useful for a modeling class.

Kyle L. Riley*Department of Mathematicsand Computer ScienceSouth Dakota School of Minesand Technology501 East Saint Joseph StreetRapid City, SD 57701-3995Phone: 605-394-2471FAX: 605-394-6078Email: kyle.riley@sdsmt.edu

Laurie J. Heyer, Davidson College

MAA CP I1 Theory and Applications ofGraph TheorySaturday afternoon

This session solicits papers on the teach-ing of undergraduate courses in graphtheory as well as papers on research inthe field of graph theory. Research pa-pers should be accessible to a reasonablywide audience. Both papers in theoreti-

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FOCUS February 2004

cal and applied graph theory are wel-come. Papers on teaching should presentnew and innovative methods of teach-ing the course.

Howard L. Penn*Mathematics DepartmentUnited States Naval Academy572C Holloway RoadAnnapolis, MD 21402-5002Phone: 410-293-6768FAX: 410-293-4883Email: hlp@usna.edu

Carol G. Crawford, United StatesNaval AcademyT. S. Michael, United StatesNaval Academy

MAA CP J1 Getting Students To Ex-plore Concepts Through Writing InMathematicsFriday afternoon

This session invites papers about assign-ments/projects that require students towrite about mathematical concepts, toexpress concepts and to interpret sym-bolic mathematics in their own words,and to write about mathematics, in gen-eral. These assignments can include con-ceptual papers such as having the stu-dents explain a concept in their ownwords as an answer to a question, in theform of a letter to a friend, a poem, oreven a short story, project reports thatrequire students to explain fully all con-cepts used as if to someone who knowslittle or nothing about the mathematicsused in solving the project problem, as-signments that require students to ex-press theorems in plain English so thatone of their friends could understand, oreven simple assignments that require stu-dents to explain the meaning and the useof the variables and notations that theyuse.

Each presenter is encouraged to discusshow the use of the assignment/projecthelped students to improve their under-standing of course concepts and how theuse of writing in the course helped stu-dents to understand and to learn math-ematics. Of particular interest is the ef-fect of such projects/assignmentsthroughout the semester on the students’understanding of course concepts andnotations, the ability of students to com-

municate mathematics using words andsymbols, and the attitude of students to-ward mathematics.

Sarah L. Mabrouk*Mathematics DepartmentFramingham State College100 State Street, PO Box 9101Framingham, MA 01701-9101Phone: 508-626-4785FAX: 508-626-4003Email: smabrouk@frc.mass.edu

MAA CP K1 Strategies for TeachingMultiple Audiences in One ClassSaturday afternoon

Mathematics classes across the curricu-lum are often populated by students whohave different levels of preparation,widely diverse mathematical goals, orboth. Providing a meaningful and appro-priate learning experience to multipleaudiences simultaneously presents aunique challenge to the instructor. Oneexample is an Abstract Algebra class withtraditional graduate school bound stu-dents as well as prospective high schoolteachers. Another example is College Al-gebra whose students typically have avery wide range of preparations as wellas diverse goals varying from generaleducation credit to preparation for a longsequence of mathematics classes.

This session solicits presentations thataddress how multiple student audiencescan be served in one class. Presentationsmight include: details of particularcourses that have dealt with diverse stu-dent audiences, descriptions of programstargeted to meet the needs of differentgroups of students in the same class, orexamples of tactics or approaches thatcan be used in a variety of mathematicsclasses. Presentations that include assess-ment of desired outcomes (whether suc-cessful or not) are strongly encouraged.

Carl Lienert*Fort Lewis College1000 Rim DriveDurango CO 81301Phone: 970-247-1067FAX: 970-247-7127Email: lienert_c@fortlewis.edu

Christopher Goff, University ofthe Pacific

MAA CP L1 General Contributed PaperSessionThursday, Friday, and Saturday after-noons

This session is designed for papers thatdo not fit into one of the other sessions.Papers may be presented on any math-ematically related topic. Papers that fitinto one of the other sessions should besent to that organizer, not to this session.

Lucy KimballBentley College175 Forest StreetWaltham, MA 02452Phone: 781-891-2467FAX: 781-891-2457Email: lkimball@bentley.edu

CALL FOR STUDENT PAPERSStudents who wish to present a paper atMathFest 2004 in Providence, Rhode Is-land, must be nominated by a faculty ad-visor familiar with the work to be pre-sented. To propose a paper for presenta-tion, the student must complete a formand obtain the signature of a facultysponsor.

Nomination forms for the MAA StudentPaper Sessions are located on MAAOnline at www.maa.org under STU-DENTS, or can be obtained from Dr.Thomas Kelley (tkelley@hfcc.net) atHenry Ford Community College (313)845-6492.

Students who make presentations at theMathFest, and who are also members ofMAA Student Chapters, are eligible forpartial travel reimbursement. Travelfunds are limited this year so early ap-plication is encouraged. The deadline forreceipt of applications is July 2, 2004.

PME student speakers must be nomi-nated by their chapter advisors. Appli-cation forms for PME student speakerscan be found on the PME web sitewww.pme.math.org or can be obtainedfrom PME Secretary/Treasurer, Dr. LeoSchneider (leo@jcu.edu). Students mak-ing presentations at the Annual Meetingof PME are eligible for partial travel re-imbursement. The deadline for receipt ofabstracts is July 2, 2004.

FOCUS

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February 2004

EMPLOYMENT OPPORTUNITIES

TRINITY COLLEGEClare Boothe Luce Assistant Profes-

sor in Mathematics

Trinity College, Washington, DC is oneof thirteen institutions named in ClareBoothe Luce’s bequest to receive fundsin perpetuity to support women in Math,Science and Engineering through schol-arships and professorships. Trinity Col-lege is a comprehensive University em-phasizing the education of women in itsundergraduate programs. The collegeinvites applications from outstandingwomen committed to teaching math-ematics at the undergraduate level for atenure-track position at the assistant pro-fessor level, beginning Fall 2004. Thesearch for the position is restricted by theLuce Foundation to women who are U.S.citizens or U.S. permanent residents.

In addition to a competitive salary, thesuccessful candidate will have access todiscretionary funds for professional ac-tivities. Doctorate in Mathematics re-quired. Review of applications will be-gin February 1, 2004 and continue untilthe position is filled. Send cover letter,resume, transcripts, a statement of teach-ing philosophy in a liberal arts setting,and three letters of recommendation to:Carole King, Trinity College, 125 Michi-gan Ave, NE, Washington, DC 20017;

email: humanresources@trinitydc.edu; orfax to 202-884-9123. Trinity College is an

EEO employer.

WASHINGTON, DCMONTANA

Montana State UniversityMathematics – Assistant Professor. TheDepartment of Mathematics of MontanaState University-Billings invites applica-tions for a tenure track position begin-ning Fall Semester 2004 (August 2004).A doctorate in the mathematical sciencesis required, but ABD will be consideredif the completion of a doctorate is im-minent. The successful candidate will beexpected to teach a wide variety of un-dergraduate mathematical sciencescourses, pursue a research agenda, andcontribute to the department and the

University through service. Excellence inteaching is expected, both in the class-room and in mentoring students outsidethe classroom. The qualifications andability to teach statistics or computer sci-ence as well as mathematics is desirable.Preference review given to applicationsreceived by March 1, 2004; however, theposition is open until filled. Submit aLetter of Application describing teach-ing and research interest, CurriculumVita, Copies of Graduate Transcripts, andThree Letters of Reference to: Search#2004-10, Human Resources, MSU-Bill-ings, 1500 University Drive, Billings, MT59101. Phone: (406) 657-2278; Fax:(406) 657-2120;http://www.msubillings.edu/humres.EEO/AA/ADA.