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FOCUS January 2004

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March April May/JuneEditorial Copy January 18 February 13 March 11Display Ads January 29 February 25 March 25Employment Ads January 15 February 11 March 12

Inside4 Pacific Northwest Section to Meet in Alaska

5 Scientists and Mathematicians Exchange Latest Resultsin SARS Research

6 The Vanishing Line Between High School and College MathematicsBy Michael Pearson

9 Statistics, Technology, and the Social Sciences:A Successful Interdisciplinary ProjectBy Catherine Bénéteau and June Rohrbach

11 Math Scores of Students on the RiseBy Harry Waldman

12 Fields Medalist Lectures on Planar AlgebrasContributed by Dale Rolfsen

13 Are Our Mathematics Natural?By Ivar Ekeland

14 What I Learned by Teaching Real AnalysisBy Fernando Q. Gouvêa

16 NSF BeatBy Sharon Cutler Ross

17 Faces of Mathematics Goes Online

18 Short Takes

20 Letters to the Editor

21 Employment Opportunities

Volume 24 Issue 1

On the cover: The Pacific Northwest Section will be holding its first meeting in Alaska thissummer. See the story on page 6. Photo courtesy of Explore Tours, Anchorage, Alaska,http://www.exploretours.com.

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January 2004

The American Mathematical Associa-tion of Two-Year Colleges (AMATYC)and the Mathematical Association ofAmerica proudly announce the creationof Project ACCCESS, a mentoring andprofessional development initiative fortwo-year college faculty, funded througha three-year grant of $475,000.00 fromthe ExxonMobil Foundation. ProjectACCCESS will be a program for new orrecently hired faculty interested in ad-vancing the teaching and learning ofmathematics in two-year colleges. Its goalis to develop faculty who are effectiveteachers and who engage in a full rangeof professional activities in the math-ematics community.

Two-year colleges provide post-second-ary education to students who might oth-erwise not have that opportunity. First-generation college students, people whowant to change careers and future work-ers in high-tech areas gravitate to TYCsto improve their skills in critical areas,especially mathematics. Many two-yearcollege students transfer to four-year in-stitutions to continue their academic ca-reers. The presence of a strong facultythat is well versed in mathematics andthat possesses a variety of effective teach-ing strategies is key to the preparation ofcompetent workers and informed citi-zens.

Project ACCCESS will support groups of30 fellows each year. Participants must betwo-year college mathematics faculty inthe first three years of a full-time, renew-able position. Fellows will be selected onthe basis of breadth of interests, motiva-tion for participation, plans for imple-menting project goals, and degree of in-stitutional support.

The goal of Project ACCCESS is to de-velop a cadre of new two-year collegemathematics faculty (Project ACCCESSFellows) who are effective members of

their profession. The objectives of thisprogram are for participants to:

• gain knowledge of the culture and mission of the two-year college and its students,

• acquire familiarity with the scholar- ship of teaching,• commit to continued growth in

mathematics, and• participate in professional

communities.

Fellows will attend two consecutiveAMATYC Conferences, where they willparticipate in pre-conference workshopsas well as regular conference activities. Inthe intervening year, Fellows will attendan MAA Section meeting near theirhome institution where they will partici-pate in both regular and specially de-signed activities. For the duration of the

project, an electronic network will linkProject ACCCESS Fellows with eachother and with a group of distinguishedmathematics educators. The develop-ment, implementation, and evaluation ofan individual project will play a key rolein each Fellow’s professional develop-ment experience.

The directors of Project ACCCESS areSadie Bragg (Borough of ManhattanCommunity College), Mary Robinson(University of New Mexico, ValenciaCampus) for AMATYC, Sharon Ross(Georgia Perimeter College, emerita),and Janet Ray (Seattle Central Commu-nity College) for MAA.

Application materials will be availablespring 2004. More information aboutProject ACCCESS can be found at http://www.maa.org/ProjectACCCESS.

Project ACCCESS: Advancing Community College Careers:

Education, Scholarship and Service

From left to right: Joseph A. Gallian, Sharon Ross, Co-director of Project ACCCESS;Philip Mahler, Truman Bell, Program Officer, ExxonMobil Foundation; Sadie Bragg,Co-director of Project ACCCESS; Mary Robinson, Co-director of Project ACCCESS.

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FOCUS January 2004

For the first time in its59-year history, the Pa-cific Northwest Sectionof the MAA will be meet-ing in Alaska — in thesummer of 2004, at theUniversity of Alaska An-chorage. Speakers for themeeting include RonGraham and Ken Ross(present and past MAAPresidents, respectively)and Pólya Lecturer I.Martin Isaacs (Universityof Wisconsin). Adding tothe traditional sessionswill be one on math-ematics in the arctic;other specialty sessions are beingplanned and will be described on themeeting’s web site. Workshops and the

Pacific Northwest Section Project NExTmeeting are scheduled for June 24th. Pa-

Pacific Northwest Section to Meet in Alaska

Anchorage, AK. The site of the next section meeting of thePacific Northwest Section. Copyright Frank Flavin/Anchor-age Convention and Visitor’s Bureau. Used with permission.

per sessions, invited addresses, banquet,etc. are planned for June 25 and 26.

This meeting will begin shortly after oneof Alaska’s most exciting days — thesummer solstice, with Alaska’s famousmidnight sun. To celebrate, members canhike Anchorage’s famous trails after a dayof mathematics. For those who decide tocombine the meeting with a vacation,travel web links are included in themeeting’s web site. (The “reasons to at-tend” explained at the meeting web siteare worth checking out even if you’re notplanning to go.)

To reduce travel costs, a block of univer-sity residence hall rooms will be avail-able at reasonable cost (families are wel-come). For additional information, con-tact Larry Foster (aflmf@uaa.alaska.edu)at the Department of Mathematical Sci-ences of the University of Alaska Anchor-age or visit the meeting’s web site at http://www.math.uaa.alaska.edu/pnwmaa.

Apostolos Doxiadis, author of thenovel Uncle Petros and Goldbach’s Con-jecture, has recently released a play calledIncompleteness: A Play and a Theorem.The play was first seen in a “workshopproduction” in Athens last summer.Doxiadis hopes that the play will be pro-duced in the United States in the nearfuture.

Doxiadis’ play centers on the last days ofKurt Gödel, when the great logician wasdying of malnutrition as a consequenceof a serious mental illness. Building onwhat is known about Gödel’s life duringthis period, Doxiadis constructs his story,which pits Gödel’s logical approach toeverything against the point of view ofhis fictional hospital dietician.

The workshop production was directedby Tony Stevens, designed by MariaPesmatzoglou and lit by Andreas Bellis.The actors were Judy Boyle, JonathanKemp, Alexandra Pavlidou, and IanRobertson. The play was well received bythe Greek and international press.Vivienne Nilan, writing in the HeraldTribune, described the play as “movingbut not pessimistic. Loss leads to illumi-nation and the beginning of hope for

another character.” Tefchros Michailidis,writing in Ta Nea, describes the play as a“splendid experience.”

Gregory Chaitin of the IBM Thomas J.Watson Research Center is one of the fewAmerican mathematicians to have seenthe play. “In Incompleteness: A Play andA Theorem,” he says, “Apostolos Doxiadishas achieved the impossible. This mov-ing and deeply human play manages tobring Gödel back to life and simulta-neously tell us why so many mathemati-cians, philosophers and post-modernartists are fascinated and obsessed byGödel and his infamous ‘incompletenesstheorem’. Even though it’s about a fa-mous mathematician, the play is an en-tertaining, life-affirming intellectualtreat.”

Uncle Petros and Goldbach’s Conjecturewas one of the first of the recent groupof novels dealing with mathematics andmathematicians. In his MAA Online re-view, Keith Devlin said that “not onlydoes [Doxiadis] get the math bits correct,he can write fiction as well.” (See http://www.maa.org/reviews/petros.html for thefull review.)

Apostolos Doxiadis was born inBrisbane, Australia but grew up and livesin Greece. He has always been interestedin fiction and the arts, but a “sudden loveaffair with mathematics,” he said, led himto do graduate work in mathematics. Inthe 1980s, he turned back to his first love,working in film, theatre, and literature.Uncle Petros, written and published inGreek then translated to English by theauthor, was his breakout work. For moreinformation about Doxiadis and his play,visit his home page at http://www.apostolosdoxiadis.com.

Incompleteness — the Theorem Becomes a Play

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January 2004

In early September, leading scientistsand mathematicians convened in Banff,Alberta, Canada, to present the latestfindings about SARS and to work to-gether to determine collaborative re-search topics. Their aim was to speed upthe process of finding effective tests, pre-vention, and controls in the preventionand treatment of the frightening disease.

Mathematical models were one aspect ofthe research effort. They are integratedwith statistical analysis of real data andare built on a solid epidemiological foun-dation. The results from this conferencewill be interpreted and translated intopublic health policy recommendations inthe various nations. The collaborativeapproach of this workshop, led by math-ematicians, hoped to offer insights intoresearch on the epidemiology of SARS.

Scientists and mathematicians from theUnited States, Canada, the UK, Taiwanand Australia participated in the gather-ing, which was being jointly supportedand coordinated by MITACS (Math-ematics of Information Technology andComplex Systems) and the Pacific Insti-tute for the Mathematical Sciences(PIMS).

The event was held at the Banff Interna-tional Research Station (BIRS) in Banff,Alberta, on the site of the Banff Centre,from September 5-6, 2003. The interdis-ciplinary approach involved the collabo-

ration of mathematicians, statisticians,epidemiologists, and virologists.

The SARS workshop was organized byDr. Fred Brauer (University of BritishColumbia), Dr. Ping Yan (HealthCanada), and Dr. Jianhong Wu (YorkUniversity). “Working together withHealth Canada will enable this researchto have a real effect on health policyguideline recommendations,” said Dr.Arvind Gupta, Scientific Director ofMITACS.

Scientific experts participating in theworkshop included Dr. John Glasserfrom the Centers for Disease Control andPrevention, Atlanta, GA; Dr. Steven Rileyof Imperial College, London, UK; Dr.Ying-Hen Hsieh of the National ChungHsing University, Taiwan; Dr. NielsBecker from The Australian NationalUniversity; and Dr. Ping Yan of HealthCanada.

These researchers were joined by manyrespected scientists from North America,Europe, and Asia, all hoping to comparetheir experiences and to increase under-standing of SARS at a faster pace.

Established in 1999, the Mathematics ofInformation Technology and ComplexSystems networks (MITACS) is one of 19components of the Canadian Networksof Centres of Excellence. The 250 scien-tists in the network are working on 32

research projects in collaboration withmore than 80 organizations. Over 400students and other trainees work directlywith the scientists and private sectorfirms. MITACS works with organizationsto identify their problems, find scientistscapable of tackling those problems, andprovide significant funds towards theresearch that leads to innovative solu-tions.

The Pacific Institute for the Mathemati-cal Sciences (PIMS) was created in 1996by scientists in Alberta and British Co-lumbia. This collaborative internationalventure of seven universities in WesternCanada, as well as the University ofWashington, in Seattle, involves over 300scientists from these institutions, withadditional funding from the govern-ments of Alberta and British Columbia.

The Banff International Research Stationfor Mathematical Innovation and Dis-covery (BIRS) is a collaboration of PIMSand MSRI, the Mathematical SciencesResearch Institute in Berkeley, California.BIRS is operated in coordination withMITACS. BIRS provides an environmentfor enhanced opportunities for creativeinteraction and the exchange of ideas,knowledge, and methods within themathematical sciences, related scientificfields and industry. BIRS is located on thesite of the world-renowned Banff Cen-tre in Banff.

Scientists and Mathematicians Exchange

Latest Results in SARS Research

By Harry Waldman

New Math?

According to Air Force Print News, a 7thgrade student has just come up with anew way of subtracting mixed numbers.Here’s an example:

82

55

3

53

1

52

5

5

1

52

4

5− =

−= + =–

According to the report, the girl’s teacherhas never seen anyone use negativenumbers to do such computations. “Iwent home and tried to find fault withit, but I couldn’t,” he is quoted as saying.

You can see the story ath t t p : / / w w w . a f . m i l / s t o r i e s /story_print.asp?storyID=123006043.

Recent Reviews on MAA Online

The Curious Incident of the Dog in the Night-timeby Mark HaddonReviewed by Maria G. Fung

Four Colors Suffice: How the Map Problem Was Solvedby Robin WilsonReviewed by G. L. Alexanderson

Big Ideas for Small Mathematicians:by Ann KajanderReviewed by Laurie Johnson

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FOCUS January 2004

The Fall 2000 CBMS Statistical Abstractof Undergraduate Programs in the Math-ematical Sciences in the United States(CBMS 2000), released fall 2002 andavailable online at http://www.ams.org/cbms/, raised concerns about the increas-ing availability of “dual enrollment”courses that allow high school studentsto take college-level courses, either attheir high school or (principally) at lo-cal community colleges. According toCBMS 2000, approximately 14% of allsections of courses in college algebra,37% of precalculus sections and 15% ofcalculus I sections listed by two-year col-leges in fall 2000 were offered via dualenrollment, while 7% of elementary sta-tistics courses were available in this way.Concerns about such programs centeraround the quality of instruction and thelevel of control that the colleges, who areeventually asked to accept the credit, haveover the course content.

A closer look at mathematics courses of-fered at both U.S. high schools and col-leges reveals a much more complex pic-ture. In addition to dual enrollment asdefined above, increasing numbers ofhigh school students are gaining accessto Advanced Placement (AP) courses andInternational Baccalaureate (IB) pro-grams, both of which allow high schoolstudents the chance to earn college creditthrough exams based on courses that alsoprovide high school credit. Over 200,000students took the AP Calculus exams in2002. Some estimates suggest that nearlyhalf of high school juniors and seniorsare enrolled in some sort of programwhich offers an opportunity to gain bothhigh school and college credit. Almost allof these students will wind up in collegeclassrooms, so it is impossible to discountthe impact such programs have on math-ematics instruction.

Everyone agrees that providing challeng-ing, relevant instruction to our best highschool students is essential to maintain-ing U.S. competitiveness, and there issome evidence that offering advancedcourses even to “average” students in-creases their level of engagement and

achievement (perhaps primarily becauseof the quality of instruction and in-creased attention usually given to stu-dents in such courses). But while vari-ous forms of dual enrollment (exambased or not) continue to grow in U.S.high schools, enrollment in remedialcourses in postsecondary institutionsnow represents approximately 25% oftotal mathematics enrollment, while wellover half of current enrollment is at theprecalculus level. Clearly, there are toomany students being left behind andunder-prepared for college-level math-ematics. Thus there is the blurring of dis-tinction between high school and collegemathematics referred to in the title of thisarticle. It is no longer possible to clearlyidentify distinct roles in mathematicseducation for U.S. secondary and post-secondary schools.

Dual enrollment, AP and IB programsare all connected with the growing needfor thoughtful consideration of both thesecondary curriculum and the roles ofand relationships between high schooland college faculty. Our public schoolsare being called to task, through e.g. NoChild Left Behind, to prove, oftenthrough high-stakes testing, to show thatthey are answering the call to providequality education for all students. Par-ents and the public at large are question-ing the educational status quo, andrightly asking for a reasonable return ontheir tax dollars. We, as collegiate faculty,want students to come to us prepared tosucceed in our classes. For this last, it isimportant that we take a more inten-tional approach to articulating whatskills are necessary and desirable for en-tering students, and (at least as a disci-pline) a more active role in working withour colleagues in the high schools to pro-vide them the tools to educate their stu-dents.

Increasing calls for accountability at in-stitutions of higher education are alsobeginning to be heard, and I do not be-lieve that we can long rely on our claimsof expertise and academic freedom toavoid concrete, data-driven responses.

What is more important is that we edu-cate ourselves and direct the responses,rather than wait until standards are im-posed from outside.

Dual Enrollment as an Opportunityto Foster Improved MathematicsInstruction

As indicated above, the CBMS surveyraises concerns about the increasingnumber of students in dual enrollmentcourses. If high school students enrolledin such courses have regular high schoolfaculty as instructors, with little super-vision by the associated institution wherecollege credit is granted, the quality ofthe course may reasonably be questioned.On the other hand, a well-coordinatedprogram that involves mathematics fac-ulty at both the college and high schoolmay serve to enhance not only the stu-dents’ experience in a single course, butoffer long-term articulation benefits formathematics curriculum at both institu-tions.

One example of a program that uses dualenrollment programs to foster profes-sional development of high school teach-ers is the Nassau Community College(NCC) Precalculus and Calculus Part-nership Program. According to PhilChiefetz, a former president of AMATYCand director of the NCC program, thePartnership began in the 1997-1998school year with 41 students enrolled inthree precalculus classes in one schooldistrict. For the 2003-04 academic year,12 NCC faculty and 20 high school fac-ulty from 10 districts are involved, andover 500 student applications had beenprocessed by July 2003.

In order to participate in the NCC pro-gram, high school faculty must attend aweek-long workshop, as well as monthlymeetings through their first two years.NCC faculty partner with each highschool teacher, and the NCC facultymember assigned to a calculus or precal-culus class meets the class (at the highschool) 32 times during the course of theacademic year, administers three quar-

By Michael Pearson

The Vanishing Line Between High School and College Mathematics

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January 2004

terly and one final exam, and assigns thefinal NCC course grade for each studentusing a set weight on the students’ gradesthat includes 35% for quizzes/tests ad-ministered by the high school teacher.The students’ grades at the high schoolare assigned solely by the high schoolteacher. According to Phil, data suggeststhat when students go through the NCCprecalculus program and then take APCalculus (AB), 94% receive a 3 or betterand 67% receive a 5. The NCC programillustrates the opportunities for collegiatemath faculty to become more involvedwith high school mathematics that dualenrollment programs can offer, thoughclearly not without effort.

A quick search on the internet will turnup a huge amount of information ondual enrollment programs across the U.S.One site offering a range of informationon dual enrollment, including state-by-state listing of dual enrollment policies,is the Center for Community CollegePolicy located online at (http://www.communitycollegepolicy.org)maintained by the Education Commis-sion for the States in cooperation withthe Department of Education. Moststates have passed laws that set rules for(and often mandating implementationof) dual enrollment programs. This isnatural given the funding ramificationsfor both public high schools and insti-tutions of higher education. Justificationfor such programs is found in studiesshowing that, statistically, students whoparticipate in either AP or dual enroll-ment programs tend to perform betteronce they become full-time college stu-dents. See, for example, the study “Com-munity College and AP Credit: An Analy-

sis of the Impact on Freshman Grades”published by the University of Arizona’sAssessment and Enrollment Researchoffice, available through http://aer.arizona.edu/AER/. A more recent ar-ticle, “Dual Enrollment Programs: Eas-ing Transitions from High School toCollege,” available through the Commu-nity College Research Center at theTeacher’s College of Columbia Univer-sity (http://www.tc.columbia.edu/ccrc/)points to similar results for other pro-grams, as well as cautionary notes aboutquality control and providing appropri-ate access to all students.

Your institution probably has articula-tion agreements with regional schoolsthat define what transfer credit is al-lowed. The mathematics departmentusually has some influence in determin-ing how mathematics credit is granted,both for transfer/dual credit and for test-based programs such as AP and IB. Per-haps it’s time for us to use the increasingpopularity of and demand for such pro-grams as an opportunity to strengthenour connections to the secondaryschools, working with local systems tohelp improve the quality of the programand help align goals so that more stu-dents can successfully make the transi-tion from high school to college.

AP, IB and the Role of Collegiate Math-ematics Faculty

The June 2, 2003, issue of Newsweek fea-tured an article on the “top high schoolsin America” and included a list titled“The 100 Best High Schools in America.”The list was actually compiled by JayMatthews, a writer at The Washington

Post, who calls his list “The ChallengeList.” In fact, the list was constructedsolely on the basis of the number of Ad-vanced Placement (AP) or InternationalBaccalaureate (IB) exams taken in 2002,divided by the number of graduating se-niors. While school administrators com-plain that the process of constructing thelist is flawed, parents often see the avail-ability of such courses as providing en-hanced opportunities for admission tocollege and access to scholarship fundsand are putting greater pressure on theirchildren’s schools to make such coursesavailable to them.

As mentioned earlier, participation inboth advanced placement (including IB)and dual enrollment programs is associ-ated with improved academic perfor-mance in later college courses. Moreover,both students and their parents are com-ing to view participation in such pro-grams as an important part of gainingadmission (and scholarships) at presti-gious universities. While never the solebasis for admissions decisions, enroll-ment counselors readily admit that stu-dents with a good AP background ontheir transcripts do indeed have an ad-vantage over students who do not. Withhigher education increasingly viewed asthe gateway to better employment op-portunities, we can expect the pressurefor expanded access to AP/IB and dualenrollment programs.

A recent report from the National Re-search Council, Learning and Under-standing: Improving Advanced Study ofMathematics and Science in U.S. HighSchools, identified the primary goal ofadvanced study (in any discipline) to be

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FOCUS January 2004

the development of “deep conceptualunderstanding of the discipline’s contentand unifying skills.” The report went onto recommend that AP courses not bedesigned to “replicate typical introduc-tory college courses.” Among the crucialissues identified in the report were ac-cess to advanced study courses, andalignment of curriculum, pedagogy andassessment with current knowledgeabout how people learn, and availabilityof substantial professional developmentopportunities for teachers.

The AP calculus course, redesigned dur-ing the 1990’s in response to the calculusreform movement, was singled out as agood example for other disciplines to usein the development of such a course. TheAP statistics course, though not as wide-spread, is actually providing a means forhigh schools to offer a course that has nottraditionally been a part of the highschool curriculum. Available data sug-gests that students who fare well on APexams also perform well in subsequentcourses (see e.g. the “21 Colleges Study,”available through the College Board’s APCentral, which is located at http://

apcentral.collegeboard.com/). From thepoint of view of course development,then, it appears that AP mathematics isin reasonable shape. However, some con-cern about professional development forteachers of AP courses remains. There islittle direct supervision of what goes onin the AP classroom. Quality control ismaintained largely through the examprocess. Another concern is that studentsmay use AP credit to fulfill core distri-bution requirements and avoid enrollingin subsequent courses altogether. Byreaching out to high school faculty whoteach AP courses, and their students, wemay be able to help support improve-ment in instruction while simultaneouslyencouraging further study of mathemat-ics.

I mentioned above the need for themathematical community to spell outwhat skills we want our students to de-velop. A variety of voices are now callingfor a concerted effort across secondaryand post-secondary institutions to moreclearly express expectations for all disci-plines. Some states are already forminggroups to promote improved articulation

between high schools and colleges, in-cluding not only enhanced expectationsfor students entering college but alsostandards for placement and exit skillsfor remedial courses. Our call as math-ematicians, then, is first to set goals as tojust what mathematics should be taught,and to which students, then to examinethe full range of mathematical experi-ences our students have, at both theprecollege and college level, assess theireffectiveness and participate in the pro-cess of refining or redesigning curricu-lum to meet those goals. Easier said thandone, of course, but worthy objectives,and necessary for us if we are to main-tain the level of control of our disciplinethat we now enjoy.

The complete CBMS 2000 report isavailable online: http://www.ams.org/cbms/

Michael Pearson is Director of Programsand Services at the MAA.

Tables on page 7 reprinted with permissionof the American Mathematical Society.

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January 2004

Recently we com-pleted an interdiscipli-nary project involvingMathematics and theSocial Sciences at Se-ton Hall University, asmall, suburban Lib-eral Arts college inSouth Orange, NewJersey. The studentbody consists of about10,000 students, whichincludes undergradu-ates, graduates, andlaw school students.The students are ethni-cally diverse, and manyof them commute toclass. Seton Hall hasalso been named oneof the top-25 mostwired universities inthe U.S. Each full-timestudent receives alaptop computer andthe faculty is encour-aged to integrate tech-nology into the class-room. The university tries to promotesuch integration by offering internal cur-riculum development grants.

All students at SHU are required to takeat least one mathematics course as partof their core curriculum, so the Math-ematics Department services the otherdepartments in the various colleges.There are about 2,500 students in theCollege of Arts and Sciences. We were ap-proached about applying for a three yearCurriculum Development Initiativegrant in conjunction with the social sci-ences in the fall semester of 2000. Nowthat we have completed our third yearon this joint project we have found it tobe professionally stimulating and haveformed good friendships in the process.

Students in the Social Sciences, whichcomprise sociology, social work, socialand behavioral sciences, criminology,anthropology and diplomacy majors, arerequired to take statistics as part of theircore requirement. This course, based onevaluating given data sets, serves as a pre-requisite for their Research Methodscourse, which teaches how to collect dataand design research studies. The facultyin the social sciences noted three prob-lems associated with our current statis-tics course, which was also required forNursing majors and certain Liberal Artsmajors. First, as with most undergradu-ate statistics courses, it was based prima-rily on quantitative data, which is ben-eficial for the Nursing majors, but some-what incomplete for the Social Sciencemajors. The students did not feel that the

material was rel-evant to their lives,their course ofstudy and to theirfuture professions.The material didnot always transferwell to the Re-search Methodsclass. Secondly,Statistics was takenin the students’freshman year andResearch Methodsin their junior year.Having three se-mesters betweenthese courses pre-sented a retentionproblem. Third, weneeded to stan-dardize our tech-nology. The Math-ematics Depart-ment was usingSPSS (StatisticalProduct and Ser-vice Solutions)

while Research Methods classes were us-ing Micro Case.

The solutions to our last two issues wererelatively easy. Research Methods wouldbe offered during the students’ sopho-more year, hopefully minimizing the re-tention problem. It was also decided thatboth Statistics and Research Methodswould use SPSS to standardize the tech-nology used in both courses. SPSS is win-dows based and user friendly. It is alsothe software of choice for many organi-zations and companies.

The real problem was developing a sta-tistics course that would be useful to So-cial Science Majors but still be rigorousenough to satisfy the Mathematics fac-

Statistics, Technology, and the Social Sciences:

A Successful Interdisciplinary Project

By Catherine Bénéteau and June Rohrbach

June Rohrbach teaching her class on Statistical Models for the Social Sciences. Theclass uses the computer about 20% of the time. The data used is from the 1996 GeneralSocial Survey.

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FOCUS January 2004

ulty. Finding a textbook that would sat-isfy these requirements was difficult. Aswe stated before, most statistics courses,and therefore most textbooks, are basedmostly on quantitative data. We neededa book that had social science applica-tions, that incorporated technology (spe-cifically SPSS), and that had an appro-priate mix of both quantitative andqualitative data. Social Statistics, by J. Ri-chard Kendrick, Jr. was the book wechose. As with most textbooks, it did notsatisfy all our needs, so we supplementedcertain topics such as the Empirical Rule,probability, and additional topics on hy-pothesis testing.

The resulting course, Statistical Modelsfor the Social Sciences, has been a hugesuccess. It stresses procedure, interpre-tation and computer application overrigor and emphasizes qualitative dataover quantitative data. SPSS is usedthroughout the course and is incorpo-rated into the classroom on a daily basis.Computer projects are assigned that usereal-life data from the 1996 General So-cial Survey. Students are required to ana-lyze data from this survey on social is-sues such as race, gender, and income,and write the outcome in article form.This incorporates another project impor-tant at Seton Hall: writing across the cur-riculum. The use of real data motivatesenthusiastic classroom discussions onstatistics and its social implications. Thestudents seem to like the more relaxedatmosphere this type of communicationencourages. They appear to have a deeperunderstanding of the interpretation ofoutcomes in statistics although theirmathematical skills are not as developed.And of course, their computer skills aregreatly enhanced.

To measure the impact of the course onthe students, a survey was administeredto each student, and the results weretabulated. The survey consisted of tenquestions. The questions were designedto measure course satisfaction, under-standing of the material, and student at-titudes towards their research as a resultof the course. Each of the first nine ques-tions had a scale of 1 to 5, with 1 indicat-ing strong disagreement and 5 indicat-ing strong agreement.

Course satisfaction was high: 77% of stu-dents were very satisfied with the course.The materials developed for the coursealso met with success: students agreedthat assignments and tests proved usefulin clarifying the concepts covered dur-ing the semester. We integrated the useof SPSS, a statistical software package forthe social sciences, into the every dayworking of the class. More than half theclass agreed that the use of SPSS made agreat difference in comprehending thematerial covered in the course. Studentsalso appeared to connect statistics totheir lives and their work in a positivemanner. For example, 75% of studentsagreed that as a result of the course, theyfelt much more comfortable understand-ing statistics reported in the popularmedia. In addition, they said that theywould be more comfortable using statis-tics in their research and writing thanthey did before taking the course, andthat as a result of the course, they wouldbe more likely to use statistics in their re-search and writing. Finally, we noticed

several negative comments about thetextbook used, although more than halfthe students felt that the textbook washelpful in understanding the materialcovered.

In conclusion, the results of the surveyshow that from the students’ point ofview, Statistical Models for the Social Sci-ences was a positive experience. The fac-ulty also benefited from developing thecourse. It has been an exciting and stimu-lating class to teach. We have learned evenmore about statistics and the SPSS soft-ware, which have lead to consulting po-sitions. We got to work closely with othermembers of the Mathematics and SocialSciences Departments, which improveddepartmental connections and commu-nication.

Catherine Bénéteau (beneteca@shu.edu)is an Assistant Professor and JuneRohrbach (rohrbaju@shu.edu) is a FacultyAssociate in the Mathematics andComputer Science Department at SetonHall University.

Course satisfaction on statistics course for social science majors.

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The nation’s fourth-graders andeighth-graders are getting better at math-ematics — that’s the surprising conclu-sion based on 2003 test scores. The re-sults come from the test considered tobe the nation’s report card: the NationalAssessment of Educational Progress,which shows that mathematics scores areheading in the right direction.

Compared to their peers in 2000, whenthe mathematics test was last given,fourth-graders and eighth-graders madesignificant gains at every level, from thelowest performers to the top achievers.Also encouraging is the fact that Blackand Hispanic students narrowed theirperformance gap with white students.The mathematics test covered probabil-ity, algebra, and mathematical reasoning,etc.

Nationally, more than 75% percent offourth-graders showed a basic under-standing mathematics, which translatesinto an understanding of the basics foracademic work. That’s up from 65%when the test was last given. Amongeighth-graders, close to 70% did well, upfrom 63%.

“The achievement levels represent verychallenging standards, so when you havethese kinds of increases, they’re to be cel-ebrated,” said Bella Rosenberg, an edu-cation policy specialist at the AmericanFederation of Teachers. “It’s important...to understand that ‘basic’ is a pretty highstandard.”

The results have become significant be-cause the national test is now used as abenchmark as to whether states are suf-ficiently challenging their students. Theyear 2003 marked the first time all 50states were required to make use of thetest as a condition for receiving federalmoney.

Every two years, national scores willcome out in grades four and eight formathematics and reading. Under federallaw, all students must be “proficient” inmathematics and reading by 2013-14,but there is one proviso: the states areallowed to define what that means.

Math Scores of Students on the Rise

By Harry Waldman

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We all learned algebra as a sort of lin-ear activity. Equations, like sentences inEnglish, are written from left to right ona line, with few deviations, perhaps forexponents and fractions. If you want tocompute, for example, x minus y, the or-der in which you write x and y makes abig difference. What if you could write xABOVE y, or to the northeast? What ifyou could turn the variables upside-down? A new version of algebra, PlanarAlgebras, developed by Fields medalistVaughan Jones, allows exactly that tohappen. A poet may think of this as aliberation of algebra from the tyranny oflinear thinking.

In the language of planar algebra, an al-gebraic expression might look like this:

On a more mathematical level, Jonesshowed how doing algebra in the planeis connected to a central idea in thetheory of operator algebras, the type II

1

subfactors which go back to the work ofvon Neumann. Jones argues that everyfinite index subfactor arises from a typeof planar algebra and vice-versa, a re-markable correspondence, which is notyet clearly understood.

Jones recently discussed these ideas atUBC in November 2003, in the CascadeTopology Seminar and as PIMS Distin-guished Lecturer in the UBC Mathemat-ics Colloquium. The work brings to-gether ideas from operator theory, quan-tum physics, statistical mechanics andthe more geometric theory of knots andtangles. The planar approach was in-spired by John Conway in his skeintheory developed in the late 1960’s,which converts geometric and topologi-cal pictures of bits of knots into algebraicobjects, which can then be studied moreprecisely and are more amenable to ac-tual calculation, using for example pow-erful methods of linear algebra. On theother hand, some arguments that mightbe quite complicated algebraically aremuch simplified by the geometric pointof view. This gives a fruitful interplaybetween ideas of algebra and methodsof geometry and topology.

Because of Jones’ earlier work—most fa-mously the Jones polynomial—thetheory of knots has undergone a revo-lution in the last two decades. His cur-rent work in planar algebras is but onedirection of the vast amount of researchcurrently going on in this fascinatingcircle of ideas.

Many of the new knot invariants can beunderstood in terms of planar algebras,and Jones’ approach has facilitated theuse of geometric and combinatorialmethods in solving algebraic problems.

Vaughan Jones was born on New Year’seve in 1952 in Gisborne, New Zealand.After undergraduate studies at the Uni-versity of Auckland, he went on to gradu-ate school at the University of Geneva,first in the Ecole de Physique and laterthe Ecole de Mathematiques, under thesupervision of Professor Andre Haefliger,where he received his doctorate in 1979.Following positions at UCLA and Uni-

Fields Medalist Lectures on Planar Algebras

Contributed by Dale Rolfsen, University of British Columbia

versity of Pennsylvania, Jones becameProfessor of Mathematics at the Univer-sity of California, Berkeley in 1985,shortly after his ground-breaking workin knot theory.

Jones was awarded the Fields medal in1990, and in the same year became a Fel-low of the Royal Society. Among otherawards, Jones was a Sloane Fellow, aGuggenheim Fellow, and has beenelected to the US National Academy ofSciences, the Norwegian Royal Society ofLetters and Sciences, the London Math-ematical Society and the American Acad-emy of Arts and Sciences. He has hon-orary doctorates from the University ofAuckland and the University of Walesand he is the winner of the Onsagermedal from Trondheim University. Jonesis also an Honorary vice-president for lifeof the International Guild of Knot Tyers.He is the Director of the New ZealandMathematics Research Institute and astrong friend of PIMS.

This article was first published in the Pa-cific Institute of Mathematics Magazine;reprinted with permission

Vaughan Jones

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Several years ago, David Ruelle (Bull.Amer. Math. Soc. 19 (1988), no. 1, 259-268) published a beautiful paper entitled:“Are our mathematics natural?” He con-tended that if somewhere there is aplanet, gravitating chaotically aroundthree suns, and where the physical con-ditions on the surface, although very dif-ferent from the ones we experience onEarth, do support some form of intelli-gent life, then the mathematics devel-oped on that planet are likely to be quitedifferent from the ones we know.

Not that there would be different ideasof what is correct and what is not. Therewould be a general agreement on what atheorem is, but not on what is an inter-esting result. Ruelle proves his point bystating a theorem which no one in hisright mind would want to prove, andthen disclosing that it plays an impor-tant role in equilibrium statistical me-chanics.

It strikes me that, although we are notlikely to travel out of our solar system inthe near future, we can still experimentwith different mathematics by movingout of our immediate intellectual neigh-borhood. Ruelle’s example comes fromphysics, but there are more distant solarsystems in economics, and in the othersocial sciences.

It certainly has been my experience thatbasic problems in economic theory give

rise to quite difficultmathematical problemswith the unmistakableflavour of the exotic.Who would ever havethought of restrictingvariational principles(such as Dirichlet’s prin-ciple) to convex func-tions? Except for some very early workby Isaac Newton, there is nothing like thisin the classical literature on the calculusof variations, probably because it was notconsidered an interesting or naturalproblem. And yet it has emerged in thepast years as a mathematical model of abasic problem in economics, namely in-formational asymmetry: people will lieto you if it is in their own interest, andcontracts should take that possibly intoaccount.

Similar things are happening in econo-metrics. Certain goods, houses for in-stance, are priced according to a complexbundle of qualities.

You buy only one house, but you takeinto account its location, proximity toschools, shops, public transportation, theview, the neighbors, the size and interiordistribution, and so forth. Is it possibleto observe the market price of houses,and infer the prices of all these differentqualities? For instance, could one findout in this way how much people arewilling to pay for an unpolluted environ-

Are Our Mathematics Natural?

By Ivar Ekeland, University of British Columbia

Ivar Ekeland

ment? Models to do that have been de-veloped in the seventies, by the lateSherwin Rosen in Chicago. They arecalled hedonic models, and they havebecome very popular in recent years, be-cause of the needs of environmentalstudies. However, there are great math-ematical difficulties at all levels, and onlyrecently have they begun to be cleared,under the impulse of Nobel Laureate JimHeckman. The mathematical structure issimilar to the classical optimal transpor-tation problem of Monge andKantorovich, again with an added twistthat would have been judged uninterest-ing or unnatural if it had come frommathematics alone.

And there is, of course, the case of math-ematical finance: the first documentedappearance of Brownian motion in themathematical literature is in the 1900thesis of Louis Bachelier, who was tryingto model financial markets. Unfortu-nately, neither the financial nor themathematical community were ready forhim, and it was only in the seventies thatthe importance of his work became clear.It does not always pay to come in fromanother planet.

This article was first published in the Pa-cific Institute of Mathematics Magazine;reprinted with permission

Save the Date!

August 12 - 14, 2004

MathFest 2004

Providence, RI

The Annual Summer Meeting of

The Mathematical Association of America

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What I Learned by Teaching Real Analysis

By Fernando Q. Gouvêa

My main mathematical interests arein number theory and the history ofmathematics. So what was I doing teach-ing Real Analysis? We do that sometimes,my colleague Ben Mathes and me: I teachAnalysis, and he gets to teach Algebra. Wehave fun and vary our course assign-ments a little bit, and the students get thesubliminal message that mathematics isstill enough of a unified whole thatpeople can teach courses in areas otherthan their own.

I had taught Colby’s Real Analysis courseonce before. The first time I teach a newcourse, I tend to just dive in and see whathappens (it went all right). The secondtime, however, is when one starts to wantto think about the course. This article isone of the results of that process. It maybe that everything I have to say is wellknown to anyone who specializes inanalysis; if so, I’m sure they’ll write in totell me that. Still, maybe I can share a fewinsights.

Analysis courses can vary a lot, so let mefirst lay out the bare facts about our ver-sion. Real Analysis at Colby is takenmostly by juniors and seniors, with asprinkling of brave sophomores. It is arequired course for our Mathematicsmajor, and it has the reputation of beingdifficult. (This course and Abstract Al-gebra contend for the “most difficult”spot.) The content might best be sum-marized as “foundations of analysis”:epsilonics, the topology of point sets, thebasic theory of convergence, etc. As thetitle of a textbook has it, the goal of thecourse is to cover “the theory of calcu-lus.”

The first thing to do was to choose a text-book. The most common choice at Colbyhas been Walter Rudin’s classic, Principlesof Mathematical Analysis. It is a hardbook for students to read, but readingsuch books is a good skill for a math-ematics major to acquire, and Rudin’sbook repays the effort that students needto put into reading it. There is a lot ofbeautiful mathematics in it, and students

eventually come to respect, perhaps evenenjoy, the book.

But two serious problems have devel-oped, neither of which is really Prof.Rudin’s fault. The first is the price. Prin-ciples is now so expensive that I feel guiltymaking students buy a copy. The secondis the internet. Many instructors aroundthe world have used this book, and in thegoodness of their hearts have posted so-lutions to some of the problems for theirstudents. They have neglected to set uptheir sites so that only their students haveaccess to them, however. As a result, al-most any problem from Rudin’s book hasa solution online somewhere, and Googlewill find it for whoever wants it. Since Iwant my students to think about hardproblems rather than learn to find theiranswers online, I decided I should lookfor another text.

To my rescue came the MAA Onlinebook review column. Steve Kennedy hadwritten a glowing review of Understand-ing Analysis, by Stephen D. Abbott (youcan see it at http://www.maa.org/reviews/understand.html). The first paragraph ofthat review went like this:

This is a dangerous book. UnderstandingAnalysis is so well-written and the devel-opment of the theory so well-motivatedthat exposing students to it could well leadthem to expect such excellence in all theirtextbooks. It might not be a good idea tocreate such expectations. You might notwant to adopt this text unless you’re com-fortable teaching from a book in which theexposition will nearly always be clearerthan your lectures. Understanding Analy-sis is perfectly titled; if your students readit, that’s what’s going to happen.

And the price is very reasonable. So Idecided this would be my textbook. I alsodecided to encourage students to buy asupplementary book, and made two sug-gestions: A Course of Modern Analysis, byWhittaker and Watson, and A Primer ofReal Functions, by Boas. As I explainedto them, one can’t imagine two more dif-ferent books, and both of them are also

very different from Abbott. Plus, one canbuy all three for about the cost of buyingRudin’s Principles.

Now, I make no pretense of having care-fully examined the available textbooks,which is why this isn’t a “what’s the besttextbook” article. I went with what Iknew, and with Steve Kennedy’s recom-mendation. It worked out pretty well,though next time I’ll probably want tochoose different books for the supple-ment slot.

OK, so we launch into the course itself.Before we start, it is useful to ask whatwe want to achieve. The first answercomes easily: we want to introduce stu-dents to epsilon-delta proofs, and to usethis technique to prove all those theo-rems they took for granted in their cal-culus course. But the answer generatesquestions of its own. Why do this? Whatdo students gain from such knowledge?

That’s easy to answer if your students arelikely to be going to graduate school. Butmost of my students are not, at least notimmediately or not in mathematics. (Avast majority of Colby students do endup getting an advanced degree, but that’sdifferent from going on to get a Ph.D. inmathematics.) So the question becomesa little sharper. I decided to take it forgranted that my students wanted to learnmathematics, irrespective of their futurecareer plans. The sharper question is,then, are epsilon-delta proofs a crucialthing for them to learn, and, if so, why?

Now, it’s easy to mount an argument thatlearning this stuff is in fact not that im-portant. After all, lots of people learn anduse sophisticated mathematics withoutever having felt the need to delve into thefoundations of the calculus. Convergencequestions can usually be settled by “thisgets small, and that’s even smaller” ar-guments. Turning those into formal ep-silon-delta arguments is a nice partytrick, and one that professional math-ematicians have to know how to do, butno one should get too excited about it.

As a counter to that, let’s note that someof my students were planning to go tograduate school, and they would need toknow the trick. And the course is therein the curriculum, and listed as required.

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So it must be important after all. Thismakes it clear that in order to justify theformalism of real analysis, one needs tofind situations and problems in whichthe epsilon-delta approach is essential.

There is, of course, a very famous mo-ment (by no means the only such mo-ment, but probably the best known) inthe history of mathematics that serves asan example: Cauchy’s “proof” that thesum of a series of continuous functionsis continuous. As presented by Cauchy,this is a classic “this is small, and so isthis” argument. And, of course, it doesn’twork, because hidden in the argument isa uniformity assumption. I decided touse this example (or a simplified versionof it) fairly early on.

Abbott’s book turned out to be a goodfit, because the author was clearly wor-ried about similar issues. Each of hischapters begins with an example wherethings “go wrong” in ways that can onlybe understood by using the formal toolsof analysis. Some of these are more con-vincing than others, but I was delightedto be using a textbook that noticed thatsuch examples are crucial to the course.

One of the main threads in the course,then, was the idea of uniformity. The verydefinitions of uniform continuity anduniform convergence require the epsi-lon-delta formalism, and the notionssimply cannot be understood without it.Over and over, I showed students ex-amples of things that went wrong be-cause something failed to be uniform,and showed them how to fix it by add-ing the uniformity assumption.

To this, I added two other threads, bothstolen from articles written by wisermathematicians than me. The first wasthe problem of partitioning the realnumbers into two disjoint sets A and B,and then finding two functions f and gwhere f is continuous on A and discon-tinuous on B, and g does the reverse. Theeasiest case is when A is a singleton set(well, A empty is even easier, I guess), andstudents are usually able to push that alittle. The case A = Q is the clincher: inthis case the function g exists, but not thefunction f (i.e., no function can be con-tinuous at all rational points and discon-tinuous at all the irrationals). This fol-

lows from a beautiful (and fairly easy)theorem of Volterra that William Dun-ham wrote up in an article for Mathemat-ics Magazine some years ago. (The pre-cise reference is given below.)

This thread doesn’t really connect to theissue of uniformity, but it fits in well withthe point set topology. I also like it be-cause it shows that not every kind ofmonster exists. What I mean is this: whenstudents first start learning point set to-pology, they get a feeling that things canget arbitrarily pathological, that anythingcan be done. Cantor sets and similar ob-jects tend to reinforce that feeling.Volterra’s theorem, however, shows thatin fact not everything can be done, andit does it without having to introducesomething like Baire category theory.

The second added thread is the problemof constructing a “nice” function thatinterpolates the factorials. In otherwords, I posed to students the questionof how one should define x! when x isnot an integer. David Fowler wrote a se-ries of articles for the Mathematical Ga-zette on this topic, and I stole his ideaswithout remorse. For example, it’s easyto construct a continuous function thatdoes the trick: define x! to be 1 if x is be-tween 0 and 1, and then extend it by us-ing the functional equation x! = x(x-1)!.Unfortunately, the resulting functionfails to be differentiable. Finding the sim-plest way to define x! on [0,1] so that theextended function is differentiable is anice exercise. Eventually, of course, weended up with the Gamma function (onthe reals). To do that, we had to deal with

improper integrals depending on a pa-rameter. (Unfortunately, Abbott doesn’tcover improper integrals and integralsdepending on a parameter, so I had tofish this part out from other texts.) Thenice thing is that for most properties ofthe Gamma function (continuity, differ-entiability, etc.) we had to deal with is-sues of uniformity again, neatly closingthe circle. In fact, “as long as the conver-gence is uniform” came up so often inthe last few classes that it became clearto me that this was the core concept I wasteaching.

What did my students learn? By the endof the semester they certainly had a goodenough understanding of uniform con-vergence and of the techniques used tounderstand and prove things related toit. I don’t know that that understandingwill last very long unless it is reinforcedin other courses. But they did under-stand, I think, that the difficult formal-ism of analysis is there for a reason, andthat it is needed if we are to do interest-ing things with functions. I don’t thinkthey’ll forget that. And that’s goodenough for me.

References:

“A Historical Gem from Vito Volterra,”by William Dunham. MathematicsMagazine, 63 (1990), pp. 234–237.

“A Simple Approach to the FactorialFunction,” by David Fowler. Mathemati-cal Gazette, 80 (1996), 378–380.

“A Simple Approach to theFactorialFunction — the next step,” by DavidFowler. Mathematical Gazette, 83 (1999),53–57.

A mock factorial function: define f(x) = 1 for x between zero and oneand extend using the functional equation f(x) = xf(x– 1).

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In October, the National Science Foun-dation and the National Science DigitalLibrary (NSDL) hosted a poster recep-tion in Washington, DC, in conjunctionwith the annual NSDL meeting. TheNSDL is an NSF-supported program tobuild an online, digital library for science,technology, engineering, and mathemat-ics education. Over 150 projects have re-ceived NSDL awards since 2000; this year35 new projects were funded. Five ofthese new projects are designed tostrengthen digital resources for math-ematics education.

CAUSEweb: A Digital Library of Under-graduate Statistics Education (D. Pearl,Ohio State University Research Founda-tion) will provide peer reviewed re-sources for undergraduate statisticsteachers. The project is an initiative ofthe Consortium for the Advancement of

Undergraduate Statistics Education(CAUSE), a group of thirty diverse in-stitutions. Collaboration Services for theMath Forum Digital Library (G. Stahl,Drexel University) will establish a modeland test case for direct collaborative useof a digital library. The project will al-low small, online groups to work bothasynchronously and synchronously onthe Problem of the Week.

Mathematics presents particular chal-lenges both textually and graphically forthe developers and users of digital libraryresources; the remaining projects addresssome of these projects. Techexplorer andMathDL: Robust Support for DynamicWeb-based Mathematics (D. DeLand,Integre) will construct a robust develop-ment and delivery environment for in-teractive mathematical content and webservices through a stable, easy-to-install

NSF Beat

By Sharon Cutler Ross

plug-in that works with a variety ofbrowsers and operating systems. En-abling Large-Scale Coherency AmongMathematical Texts in the NSDL (R. Con-stable, Cornell University) will extendcommon authoring tools so they can eas-ily produce semantically anchored docu-ments. A document is said to be seman-tically anchored when elements, such asdefinitions and theorems, are formallylinked to counterparts in formal logicaltheories that are implemented by com-puter systems. Finally, Enhancing theSearching of Mathematics (R. Miner, De-sign Science, Inc.) will develop searchalgorithms, markup guidelines, andmetadata practices with an emphasis onusing MathML. This project will alsosponsor a workshop and conference fortoolmakers, content providers, librarians,and other NSDL stakeholders.

Have you learned something whenyou taught a standard undergradu-ate mathematics course? Want to tellothers? Write your own “What ILearned by Teaching X” article! Seepage 14 for ye humble editor’s con-tribution.

What Have

You Learned?

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January 2004

An exhibit called Faces of Mathemat-ics can now be seen online at http://www.ma.hw.ac.uk/Endg/fom.html. Thephotographs, which feature British re-search mathematicians, are by MarcAtkins, a photographer and video pro-ducer. The goal of the project, which wascoordinated by Nick Gilbert of the He-riot-Watt University in Edinburgh, wasto display the creative side of mathemat-ics by focusing on the creators them-selves.

In his description of the project, Gilbertsays, “Who are the mathematicians?What sort of people are they? Faces ofMathematics offers a glimpse into thelives and personalities of some of themost successful and influential math-ematical researchers currently working inthe UK and shows mathematical researchas a creative, human endeavour. We seeengaging, passionate and enthusiasticpeople, with a shared commitment to the

development of mathematical ideas andwith a shared secret. They have lookedlong and hard at some of the most chal-lenging problems in modern mathemat-ics, statistics, and mathematical physics;and they share the secret of success, be-cause they have solved these problems

and made their own extraordi-nary contribution to the sub-ject.”

In addition to the photos, theexhibit included biographicalinformation and pages ofmathematics taken from thenotebooks of the researcher ineach photograph. These pagescan be seen online as the back-ground of the page containingthe main photograph.

Before going online, Faces ofMathematics was shown at theHighgate Literary and Scien-tific Institution, the OxfordUniversity Museum of NaturalHistory, and at the 2003 conference

of the Mathematical Association at TheUniversity of East Anglia. Project coor-dinator Nick Gilbert works in grouptheory and topology, and has a particu-lar interest in the public understandingof mathematics. Photographer MarcAtkins has exhibited in London, Paris,

Faces of Mathematics Goes Online

The Faces of Mathematics exhibit at Oxford.

Sir Michael Atiyah

Rome, and New York. His work has beenpublished in books and magazinesworldwide, and features in both privateand public collections, including the Na-tional Portrait Gallery, London

Timothy Gowers

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Are you coming to HRUMC XI?

The eleventh annual Hudson River Un-dergraduate Mathematics Conference willbe held at Mt. Holyoke College in SouthHadley, MA on April 3, 2004. The con-ference includes presentations on math-ematics by both faculty and students, andboth are encouraged to participate. Con-ference sessions are designed so thatsome presentations are accessible to un-dergraduates in their first years of study,and others are accessible to third- orfourth-year undergraduate mathematicsmajors.

The keynote speaker for this year will beDr. Nancy Kopell, co-director of the Cen-ter for BioDynamics at Boston Univer-sity. She will be speaking on DynamicalSystems of the Nervous System: DoRhythms help us think? You can find outmore about HRUMC by visiting the con-ference web site located at http://www.skidmore.edu/academics/mcs/hrumc.htm.

Those wishing to make a presentation atthe conference should submit an abstractvia the website by February 27, 2004. Wewould like to thank Mt. Holyoke Collegeand the MAA for their support ofHRUMC XI.

MAA Again Extends Free Membershipsto New Math Ph.D. Recipients

The MAA has just given complimentaryassociation memberships to 956 recentPh.D. and Ed.D. recipients. These com-plimentary memberships were given tothose who received their advanced de-grees between December 2002 and Au-gust 2003. This is another way in whichthe MAA encourages the participation ofnew Ph.D.s and Ed.D.s, which helps keepthe Mathematical Association vital andcurrent. After a year, these complimen-tary memberships can be renewed at alow, graduated promotional dues rate forthe subsequent two years.

Readers who received a mathematicsPh.D. or Ed.D. degree during this timeperiod and did not receive their compli-

mentary membership should contact theMAA Service Center by email atmember@maa.org or by phone at 800-331-1622.

DIMACS Reconnect ‘04 Conferences:Current Research Relevant to theClassroom

The Reconnect ‘04 Conferences, spon-sored by DIMACS (the Center for Dis-crete Mathematics and Theoretical Com-puter Science), are geared towards expos-ing faculty teaching undergraduates tocurrent research topics relevant to theclassroom, involving them in writing ma-terials useful in the classroom and recon-necting them to the mathematical sci-ences enterprise by exposing them to newresearch directions and questions. Thethree programs: “Experimental Al-gorithmics, with a Focus on Branch andBound for Discrete Optimization Prob-lems” at Lafayette College, June 20-26,2004; “Folding and Unfolding in Com-putational Geometry” at St. Mary’s Col-lege, July 11-17, 2004; “Topics in Com-putational Molecular Biology” atDIMACS/Rutgers University, August 8-14, 2004. Applicants accepted to partici-pate will receive lodging and mealsthrough NSF funding. For more infor-mation or an application form, visit theirweb site at http://dimacs.rutgers.edu/re-connect/ or contact the Reconnect Pro-gram Coordinator by email atreconnect@dimacs.rutgers.edu or byphone at (732) 445-4304.

HOMSIGMAA Events at the JointMeetings

The History of Mathematics SIGMAAwill hold its third annual meeting dur-ing the Joint Mathematics Meetings inPhoenix. Vicki Hill’s new documentaryon Constantin Carathéodory will be oneof the featured events. Check the JMMschedule for time and place.

In addition, HOMSIGMAA will sponsorits first annual guest lecture. PeggyKidwell and Amy Ackerberg-Hastingswill present a lecture on Making Sense ofYour Department’s Material Culture. In

many cases, math professors don’t needto leave their home institutions to “ex-plore the material culture of mathemat-ics.” Historic models, devices, and booksmay be tucked away in the drawers andclosets of their own departments. Kidwelland Ackerberg will discuss how to iden-tify, understand, and arrange such math-ematical objects and books.

For more information, go to theHOMSIGMAA web site at http://www.maa.org/homsigmaa or contactAmy Shell-Gellasch via email atamy.shellgellasch@us.army.mil.

Assessing Doctoral Programs

In 1995, the National Research Councilreleased Assessing the Quality of Research-Doctorate Programs: Continuity andChange, a report that evaluated andranked doctoral programs in the U.S.Now it has released a new report, Assess-ing Research-Doctorate Programs: AMethodology Study, which argues that theolder report should be updated as soonas possible using newer and better meth-ods. The full report, which was preparedby the National Academies’ Board onHigher Education and the Workforce,can be viewed or ordered online fromhttp://books.nap.edu/catalog/10859.html.

Two Mathematical Scientists NamedMacArthur Fellows

Among the 24 new MacArthur Fellowsnamed in October 2003, there are twowhose work in closely connected tomathematics. James J. Collins of BostonUniversity is a biomedical engineerwhose work “crosses the boundaries ofengineering, mathematics, and biologyto explore the complex mechanismsregulating biological systems.” Collins’work uses mathematics to understandhow organisms move and to suggestideas for treating disorders associated tomovement and posture. Erik Demaine ofthe Massachusetts Institute of Technol-ogy (and a member of MAA) is a com-putational geometer who “has already es-tablished a reputation for tackling andsolving difficult problems.” Demaine’s

Short Takes

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January 2004

work ranges all the way from parallelprocessing architectures and complexityof algorithms to geometry. He recentlycollaborated in the proof of a long-stand-ing conjecture that asserts that all closedpolygons with non-crossing connectionscan be made convex without breaking orchanging the length of the connections.He also proved that the computer gameTetris is NP-complete.

The MacArthur Fellowships, sometimesknown as the “genius grants,” are de-signed to recognize and support trulycreative individuals. Fellows “are selectedfor the originality and creativity of theirwork and the potential to do more in thefuture.” Each fellows receives $500,000over five years, with absolutely no stringsattached. To find out more about the pro-gram and to read about the other 22 fel-lows for 2003, visit http://www.macfound.org /programs/ fe l /announce.htm.

Articulation Continues to be a Concern

Various conferences and reports con-tinue to address the issue of the transi-tions facing students as they move fromhigh school to college. The Boston-basedJobs for the Future (http://www.jff.org) hasrecently released the results of a surveythat found that 57% of Americans be-lieve that students need to be better pre-pared for the transition. Large majori-ties favored more effective guidancecounseling and more challenging highschool courses. A second Jobs for the Fu-ture report, prepared jointly with the Na-tional Governors Association, makes rec-ommendations to state officials on howto improve articulation. The report isonline at http://www.nga.org/cda/files/031READY.pdf.

Two Mathematicians Named StateProfessors of the Year

Every year, theCarnegie Foundationfor the Advancementof Teaching namesProfessors of the Year,both at the state leveland at a nationallevel. Two math-ematicians are among

this year’s State Professors of the Year:Joe Gallian of the University ofMinnesota Duluth and Judy Kasabian ofEl Camino Community College inCalifornia. Gallian is currently SecondVice-President of the MAA.

The Professors of the Year program hasexisted since 1981. It recognizes out-standing teaching at the undergraduatelevel. Every year the program chooseswinners in each state and four winnersat the national level.

To read more about the program, visitthe Carnegie Foundation web site athttp://www.carnegiefoundation.org/POY/index.htm.

150,000 Students in AMC 8 Contest

MAA Director of Competitions StevenDunbar reports that AmericanMathematics Competitions hasregistered about 2800 schools world-wide for the AMC 8 contest. Theyestimate that about 150,000 studentsaround the world took the test. By thetime this issue comes out, we expect toknow the winners and high scorers forthis competition.

The AMC 8 is a 25 question, 40 minutemultiple choice open to junior high andmiddle school students. Its problems arebased on the concepts and ideas from thejunior high mathematics curriculum.That does not mean they are easy! Mostof the problems are designed to challengestudents and to offer problem solvingexperiences beyond those provided inmost junior high school mathematicsclasses. Students who do well in this examare invited to participate in the AMC 10competition.

To read more about AMC 8, visit theAMC web site at http://www.unl.edu/amc/.

Winter School on Number Theoryand Physics

The Southwestern Center for ArithmeticAlgebraic Geometry has been runningWinter Schools for several years now. The2004 session will take place at the Uni-versity of Texas at Austin on March 13 to

17. The topic will be Number Theory andPhysics. Invited speakers include VictorBatyrev, Philip Candelas, DavidMorrison, and Daqing Wan. Check outthe Southwestern Center web site athttp://swc.math.arizona.edu for more in-formation. Also available at this site arenotes and streaming video recordings oflectures from past Winter Schools andfrom the Southwestern Center Distin-guished Lecture Series.

NSB Sees Need for More Science andEngineering Professionals

The National Science Board has releaseda report entitled The Science andEngineering Workforce: RealizingAmerica’s Potential. The report arguesthat global competition for science andengineering professionals is intensifying,and that the United States may not beable to rely on an influx of suchprofessionals from abroad. It also arguesthat the number of American science andengineering graduates is likely todecrease unless something is done toencourage more students, and especiallymore students from currently under-represented demographic groups, topursue such studies. The report hasseveral specific policy recommendationsintended to help produce this outcome.The report is available online at http:/www.nsf.gov/nsb/documents/2003/nsb0369/nsb0369.pdf.

.Sources. HRUMC: the organizers. Freememberships: MAA headquarters.DIMACS: Reconnect coordinator.HOMSIGMAA: Amy Schell-Gellasch.Assessing Ph.D. programs: NationalAcademies news web site. MacArthur fel-lows: MacArthur Foundation web site.Articulation: NASSMC Briefing Service.Professors of the Year: AMS web site,Carnegie Foundation web site. AMC 8:

Steven Dunbar. Winter School: SWCemail, SWC web site. NSB Report:

NASSMC briefing, NSB web site..of the Year: AMS web site, C

Joe Gallian

20

FOCUS January 2004

Divisibility Tests Used

Professor Cohn’s article on divisibility tests (in the Novemberissue) particularly interested me since I have long found it use-less to attempt to overcome occasional insomnia by countingsheep. Not enough sheep left in Connecticut.

Instead, in order to put myself to, or back to, sleep, I amusemyself by factoring into its primes a bedside clock’s digital read-ing. One needs a clock that either is or can be illuminated.

Of course, if one delays bedtime until, or awakens after, 12:59AM, the job is easy. The times between, say, 11:01 PM and 11:59AM, inclusive, are more challenging.

Since the factoring of just one number does not take very long,I usually do several in succession, until I get bored, fall asleep,or loose track of where I am. To preserve my self-esteem, I havenot tabulated the relative frequency of each of these threeevents.

Ed RosenbergEmeritus Professor of MathematicsWestern Connecticut State University

Divisibility Tests Modified

I enjoyed Paul Cohn’s, “Divisibility Tests Remembered” in theNovember 2003 issue of FOCUS. I wonder why, however, hesubtracts a non-multiple of 1001 in two of the steps for check-ing the example number 479563. After stating “since 7x11x13= 1001, one can test for divisibility by 7, 11, or 13 by subtract-ing multiples of 1001,” Mr. Cohn goes on to give the example479563. He lists the steps as

479563→ 79163→ 76160→ 7616→1610→161 = 7 x 23.(400400) (3003) (68544) (6006) (1449)

(I have indicated the differences at each step.) In the third step,Mr. Cohn subtracts a multiple of 7 to obtain 7616. This par-ticular multiple of 7 (68544) is neither a multiple of 11, nor of13. Mr. Cohn does so without explaining why he has chosennot to continue to subtract a multiple of 1001. In the final step,he again subtracts a multiple of 7 (1449) which is neither amultiple of 11 nor of 13, to get 161. Had he subtracted num-bers that are multiples of 1001 in every step, readers wouldhave not only seen that 479563 ≡ 0 mod 7, but that 479563 ≡7mod 11 and 479563 ≡6 mod 13. I followed the procedure ofsubtracting multiples of 1001, in two different ways. The num-bers subtracted in each step are indicated in parentheses.

My steps (always use the leading digit to choose the multipleof 1001):479563 → 79163 → 9093 → 84.

(400400) (70070) (9009)

And note that 84 ≡0 mod 7, 84 ≡7 mod 11, and 84 ≡6 mod 13.Thus, 479563 shares those congruences.

Alternatively (following Mr. Cohn’s first two steps):479563→ 79163→ 76160→16100→ 6090→ 84.

(400400) (3003) (60060) (10010) (6006)

Margaret StempienMathematics DepartmentIndiana University of Pennsylvania

Why No Gender Data on Overall Membership?

On page 22 of the August/September 2003 issue of FOCUS thereis a table “Data on Gender For 2002.” Why doesn’t the tableinclude the percentage of female members of the MAA?

Albert VanCleaveVanCleave_Albert@colstate.eduColumbus State Univ., Columbus, Ga

Jim Gandorf, MAA Director of Membership, responds:

We’ve been gathering gender information on members for 2 years(renewal notices and new member surveys). Unfortunately, onlyabout 23% of our members have chosen to provide us with thatinformation.

Was it Regular?

I just received the December issue of FOCUS, a publication Ialways find very interesting. Your article about the Texasoctagon was well done. The possible explanation for thenumbers in the picture are sobering but certainly plausible. I’dlike to make one little comment, however.

The paragraph, which begins “Well, I figured it out”, continueswith “The truth is that no octagon with those measurementsexists.” In fact no regular octagon with those measurementsexists, because there are three given facts about the right triangle,an angle and two sides and those figures conflict. However,there are any number of irregular octagons that one could makewith the triangle as pictured, if the assumption of it being“regular” is discarded. Was “regular” in the original problem?If not, any of the answers could be considered possible I shouldthink.

Delene Perleymathperley@earthlink.netRetired

Good catch! You had me worried there for a moment. But theoriginal problem did say “What is the perimeter to the nearestcentimeter of the regular octagon drawn below?” So yes, a regularoctagon was intended. Otherwise, as you say, almost any answerwould have been acceptable.

Letters to the Editor

FOCUS

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January 2004

EMPLOYMENT OPPORTUNITIES

KANSAS

Tabor CollegeFaculty Position: Mathematical SciencesDepartmentFull-time teaching and advising in math-ematics and computer science, startingFall 2004. Minimum master’s degree incomputer science with strong mathemat-ics. Prefer doctorate. Teaching experi-ence required. Affirm Tabor’s AnabaptistEvangelical Christian college perspectiveand articulate personal Christian com-mitment.

Applications accepted until March 1,2004. Letter, vita, and three referenceswho can be contacted to Dr. HowardKeim, Vice President of Academic Affairs,Tabor College, 400 S. Jefferson, Hillsboro,KS 67063. See Tabor College at http://www.tabor.edu

NEW JERSEY

Ramapo CollegeAssistant Professor of MathematicsTenure-Track, Fall 2004JOB DESCRIPTION: Teaching respon-sibilities encompass a wide range of un-dergraduate mathematics courses, in-cluding General Education mathematicscourses.

REQUIREMENTS: Ph.D. in Pure or Ap-plied Mathematics by September 1, 2004is required. Teaching experience at thecollege level is preferred. The applicantmust have a proven research record orresearch potential.

Faculty members are expected to main-tain active participation in research,scholarship, college governance, service,academic advisement and professionaldevelopment activities.

Ramapo College of New Jersey is a four-year undergraduate college located in thebeautiful foothills of the Ramapo ValleyMountains approximately 25 milesnorthwest of New York City. Establishedin 1969 as a state-supported, coeduca-tional college of liberal arts, sciences and

professional studies, this institution of-fers an array of undergraduate, gradu-ate, and post baccalaureate programs fo-cused on the four “ pillars ” of theRamapo College mission – international,intercultural, interdisciplinary, and ex-periential education. The College iscommitted to global education. It is aFulbright Center and houses the NewJersey Governor’s School for Interna-tional Studies.

All applications must be completedonline at: http://www.ramapojobs.com.Please attach resume, cover letter, a state-ment of teaching and research interests,and a list of three references to the ap-plication. Since its beginning, RamapoCollege has had an intercultural/inter-national mission. Please tell us how yourbackground, interest and experience cancontribute to this mission, as well as tothe specific position for which you are

applying.

Supporting documentation in non-elec-tronic format can be sent to ProfessorMarion Berger, Search Committee Chair,School of Theoretical and Applied Sci-ence. Review of applications will beginimmediately and continue until the po-sition is filled. Position offers excellentstate benefits. To request accommoda-tion, call (201) 684-7734.

RAMAPO COLLEGE OF NEW JERSEYDept. 42505 Ramapo Valley RoadMahwah, NJ 07430

“New Jersey’s Public Liberal Arts Col-lege”

Ramapo College is a member of theCouncil of Public Liberal Arts Colleges(COPLAC), a national alliance of lead-ing liberal arts colleges in the public sec-tor.

EEO/AFFIRMATIVE ACTION

SOUTH CAROLINA

Wofford CollegeTenure Track Appointment beginningAugust 2004. Qualifications: Ph.D. inMathematics, excellence in teaching un-dergraduates, continuing professionalgrowth, ability to use technology to im-

prove understanding, and reflective con-cern for the teaching and learning ofmathematics. Salary commensurate withqualifications and experience. Send a let-ter of application, vita, graduate tran-scripts, and three letters of recommen-dation, at least one of which assessesteaching ability and potential, to Lee O.Hagglund, Chairman, Department ofMathematics, Wofford College,Spartanburg, SC 29303-3663. Go tohttp://dept.wofford.edu/mathematics/facultyposition.html

TEXAS

Legacy of R. L. Moore ProjectExecutive Directorpersonnel@edu-adv-foundation.orgAustin, TXThe Educational Advancement Founda-tion, a nonprofit organization support-ing mathematical education through in-quiry-based learning, seeks a full-timeExecutive Director for the Legacy of R.L. Moore Project.

Applicants should have a strong math-ematics background (preferably a currentor former faculty member), includingrecent administrative or leadership expe-rience in academic societies or profes-sional associations. In addition, some“entrepreneurial” experience (i.e. soleproprietorship or small business) is de-sirable.

Duties: Strong “hands-on” managerialleadership is essential, with strong per-sonal involvement in carrying out non-routine activities. The Executive Direc-tor will coordinate workflow and assign“priority” while working actively with asmall staff and a large, diverse group ofoutside consultants/volunteers/constitu-ency members. The candidate shall haveexceptional communication, organiza-tional, and executive skills as well as anability to articulate a mission-focusedvision to diverse constituencies. It is es-sential that the Executive Director be ableto manage actively and conclude multiplenon-routine projects on a timely basis.

Requirements: Advanced degree inmathematics, science, or MBA/Businesswith over 10 years educational/adminis-

22

FOCUS January 2004

trative experience. The prospective Ex-ecutive Director must demonstrate ac-complishment in combining leadershipwith administrative controls and sup-port, while sustaining entrepreneurialinitiatives. Applicants must be adaptableand flexible to rapidly changing priori-ties; a self-starter with a successful his-tory working with both small groups andlarger organizations/associations.

Excellent benefits (health, life, and long-term disability insurance, and retirementplan) and a stimulating cross-disciplineworking environment with demandingworkload. Compensation includes anearly merit bonus for exceptional perfor-mance.

The Educational AdvancementFoundation is an equal opportunityemployer. U. S. Citizenship required.

Email cover letter and resume to:personnel@edu-adv-foundation.org

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Contact the MAA Advertising Depart-ment toll free at 1-866-821-1221, fax:(703) 528-0019. Ads may be sent viaemail to ads@catalystcom.com.

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January 2004

ALLEGHENY MOUNTAIN

March 26-27, 2004–West VirginiaWesleyan College, Buckhannon, WV

FLORIDA

February 20-21, 2004–University ofCentral Florida, Orlando, Florida

ILLINOIS

April 1-3, 2004–Roosevelt University,Schaumburg, IL

INDIANA

April 2–3, 2004–Goshen College,Goshen, IN

INTERMOUNTAIN

March 26-27, 2004–Weber StateUniversity, Odgen, UT

IOWA

April 16-17, 2004–Central CollegePella, IA

KANSAS

March 19-20, 2004–BenedictineCollege, Atchison, KS

KENTUCKY

April 2-3, 2004–Murray StateUniversity, Murray, KY

LOUISIANA-MISSISSIPPI

March 4–6, 2004-SoutheasternLouisiana University, Hammond, LA

MD-DC-VA

April 23-24, 2004, Salisbury StateUniversity, Salisbury, MD

METRO. NEW YORK

May 2, 2004–Nassau CommunityCollege (SUNY), Garden City, NY

MICHIGAN

May 7-8, 2004-Oakland University,Rochester, MI

MISSOURI

April 2-3, 2004–Southeast Missouri StateUniversity, Cape Girardeau, MO

NEBRASKA-SOUTHEAST SOUTHDAKOTA

April 2–3, 2004–University of Nebraskaat Kearney, Kearney, NE

NEW JERSEY

March 27, 2004–Rutgers University,Busch Campus, New Brunswick, NJ

NORTH CENTRAL

April 23-24, 2004–Winona StateUniversity, Winona, MN

NORTHEASTERN

June 3-4, 2004–Roger WilliamsUniversity, Bristol, RI

NORTHERN CALIFORNIA, NEVADA,HAWAII

February 24, 2004–California StateUniversity, Hayward, CA

OHIO

March 26–27, 2004–University ofCincinnati, Cincinnati, OH

OKLAHOMA-ARKANSAS

March 26-27, 2004-University of CentralArkansas, Conway, AR

PACIFIC NORTHWEST

June 24-25, 2004–University of Alaska,Anchorage, AK

ROCKY MOUNTAIN

April 16–17, 2004-Colorado College,Colorado Springs, CO

SOUTHEASTERN

March 26-27, 2004–Austin Peay StateUniversity, Clarksville, TN

SOUTHERN CALIFORNIA

March 6, 2004–University of San Diego,San Diego, CA

SOUTHWESTERN

April 2-3, 2004–Northern ArizonaUniversity, Flagstaff, AZ

SEAWAY

April 23-24, 2004–SUNY College atCortland, Cortland, NY

TEXAS

April 1-3, 2004–Texas A&M University,Corpus Christi, TX

WISCONSIN

April 16-17, 2004–University ofWisconsin-Platteville, Platteville, WI

MAA Section Meeting Schedule 2004