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FOCUS May/June 2004

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

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August/September October NovemberEditorial Copy July 8 September 16Display Ads July 9 August 20 September 24Employment Ads June 11 August 13 September 10

Inside

4 Convergence: A New Online Magazine from the MAABy Victor Katz

5 SIGMAA QL is FormedBy Rick Gillman

6 The Archives of Mathematics at the Center for American HistoryBy Kristy Sorensen

8 The Teacher as Artist: A Letter to My ColleaguesBy Peter Taylor

10 What I Learned by Teaching Proof and LogicBy Peter Ruane

12 Mathematics Magazine Editor Search

13 2004 Summer Short Course—Teaching and Doing Knot Theory

14 Remembering Bill ChinnBy Jean Bee Chan

15 Letters to the Editor

16 MAA Contributed Paper Sessions:Atlanta Joint Mathematics Meeting, January 5-8, 2005

21 MAA President Ronald L. Graham Testifies Before HouseAppropriations Subcommittee on Funding forMathematics EducationBy Harry Waldman

Volume 24 Issue 5

On the cover: The MAA has just launched Convergence, an online magazine about theHistory of Mathematics. See the article on page 4.

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The Norwegian Academy of Sciencesand Letters has awarded the 2004 AbelPrize jointly to Sir Michael FrancisAtiyah, of the University of Edinburgh,and Isadore M. Singer, of the Massachu-setts Institute of Technology. Atiyah andSinger will receive the prize, wrote theAcademy, “for their discovery and proofof the index theorem, bringing togethertopology, geometry and analysis, andtheir outstanding role in building newbridges between mathematics and theo-retical physics.” The Atiyah-Singer IndexTheorem is considered one of the greatlandmarks of twentieth-century math-ematics, influencing many of the mostimportant developments in topology,differential geometry, and quantum fieldtheory. An account of the theorem aimedat a general mathematical public can befound on the Abel Prize web site at http://www.abelprisen.no/nedlastning/2004/popular_english_2004.pdf.

A few decades ago, physicist FreemanDyson pointed out that “the marriagebetween mathematics and physics, whichwas so enormously fruitful in past cen-turies, has recently ended in divorce.” Thesituation has changed dramatically sincethen; Dyson’s “divorce” turned out to beonly a temporary separation. Atiyah andSinger, both jointly and individually, havebeen instrumental in repairing the rift,and the Norwegian Academy noted this

as one of their most important contri-butions.

Isadore M. Singer was born in 1924 inDetroit and was educated at the Univer-sity of Michigan and the University ofChicago. He joined the faculty at theMassachusetts Institute of Technology in1950, and has been there ever since. Hehas received several prizes for his work,including the Steele Prize for LifetimeAchievement. Accepting that prize, hesaid, “For me the classroom is an impor-tant counterpart to research. I enjoyteaching undergraduates at all levels, andI have a host of graduate students, manyof whom have ended up teaching memore than I have taught them.” Singerhas also written several important text-books.

Sir Michael Atiyah and Prof. Isadore Singer Win the 2004 Abel Prize

Sir Michael Francis Atiyah Isadore M. Singer

Michael Francis Atiyah was born in 1929in London and was educated at TrinityCollege, Cambridge. He spent most of hiscareer at Cambridge and Oxford. He oc-cupied the famous Savilian Chair of Ge-ometry at Oxford and later was Masterof Trinity College in Cambridge.

While in Cambridge, he helped create theIsaac Newton Institute for Mathemati-cal Sciences and became its first direc-tor. Atiyah received the Fields Medal in1966. He has been Sir Michael Atiyahsince 1983, and was made a member ofthe Order of Merit in 1992.

For more information on the Abel Prizeand on this year’s winners, visit http://www.abelprisen.no/en. See also the April2004 Devlin’s Angle on the MAA web site.

In 1998, Thomas C. Hales announcedthat he had proved Kepler’s Conjectureabout the best possible sphere packingin three-dimensional space. His claimgenerated controversy, mostly because ofthe crucial role of computers in the (verylong) proof. As Hales says in the mostrecent version, “nearly every aspect of theproof relies on computer verification.”The issue was put into sharp relief by theeditors of the Annals of Mathematics, whowere leery of publishing a paper whosecorrectness they could not verify.

The problem, for the editors, was to whatextent they should trust mathematicsdone on the computer without checking

it. An attempt to check everything byhand ended badly: “[The referees] havenot been able to certify the correctnessof the proof, and will not be able to cer-tify it in the future, because they have runout of energy to devote to the problem,”said Annals editor Robert McPherson.The Annals then considered publishingthe paper with a disclaimer, to the effectthat the editors could not certify thatevery detail was correct.

An article in the April 6 issue of The NewYork Times, entitled “In Math, Comput-ers Don’t Lie. Or Do They?,”reports thatthe Annals editors have decided to solvethe problem in a different way. The pa-

per will be split in two parts. The moretheoretical part will be published by theAnnals and the computer-intensive partwill be published in another journal, Dis-crete and Computational Geometry.

Hales himself has opted for a more radi-cal route. He has started the FlyspeckProject, an attempt to create a formalproof of Kepler’s Conjecture. The idea isto create (via computer) a version of theproof of the conjecture that can be au-tomatically verified line-by-line. You canread more about this at http://www.math.pitt.edu/~thales/flyspeck/index.html.

An Update on Hales’ Proof of Kepler’s Conjecture

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FOCUS May/June 2004

The MAA announced the launching ofa new online magazine on the history ofmathematics and its use in teaching, en-titled Convergence: Where Mathematics,History and Teaching Interact, with thefinancial support of the National ScienceFoundation. The target audience isteachers of grades 9-14 mathematics, bethey secondary teachers, two- or four-year college teachers, or college teacherspreparing secondary teachers. (“Grade9–14 mathematics” encompasses algebra,synthetic and analytic geometry, trigo-nometry, probability and statistics, el-ementary functions, calculus, linear al-gebra, and differential equations.) Theinitial editors of the magazine are VictorJ. Katz, from the University of the Dis-trict of Columbia, and Frank Swetz, fromPenn State University, Harrisburg.

Among the types of material that willappear in the magazine are the follow-ing:

Expository articles dealing with the his-tory of various topics in mathematicscurriculum. They will usually containinteractive components and color graph-ics, to take advantage of the capabilitiesof the Web. Frequently, articles will bedesigned, through use of hyperlinks, toappeal to multiple audiences. In addition,each article will have a discussion groupattached, where readers can share sug-gestions as to how the material can beused in the classroom and point outstrong points and possible pitfalls; au-thors would also have a chance to re-spond.

Translations of original sources. These willgenerally be accompanied by commen-tary from experts showing the context ofthe works. If possible, interactive com-ponents will be used to help with theunderstanding of these materials. But thegoal of the translation is always to showteachers how ideas were developed invarious cultures and how knowledge ofthis development is useful to teaching thesame ideas to today’s students.

Reviews of current and past books, ar-ticles, and teaching aids on the historyof mathematics of use to teachers, as wellas reviews of websites providing infor-mation on the history of mathematics.

Classroom suggestions. These may be self-contained articles showing how to usehistory in the teaching of a particulartopic or they may be materials closelyrelated to a main article, showing in somedetail how to use the article in a class-room setting.

Historical problems. These problems willappear in a section entitled “Problemsfrom another time,” with new problemsappearing frequently. After publication,the problems will be archived in sectionsbased on the main topic of the problem,such as algebra, geometry, trigonometry,or calculus. Answers will appear sepa-rately.

What Happened Today in History? Eachday, there will be a listing of 2-3 “math-ematical events” which happened on thatdate in history. Many of the items in thissection will have links to other websites,so teachers can find out more about theparticular person or event.

Quotation of the day. A new and interest-ing quotation about mathematics froma historical figure will appear in this sec-tion each day. The reader will also be ableto search our database of quotations tofind additional ones.

An up-to-date guide to what is happen-ing around the world in the history ofmathematics and its use in teaching. The

magazine will report on past meetingsand give notice of future meetings.Where abstracts are available for a par-ticular meeting, these will be included.We may also include copies of handoutsfor easy access, as well as links to theauthor’s webpage, if available.

Initially the magazine will be free to all.However, a subscription fee will be nec-essary after the initial period. We hopeto secure institutional as well as personalsubscriptions. Information as to cost andaccess will be provided on the Conver-gence site shortly.

Currently, we have a limited supply ofarticles in our pipeline. Because our goalis to bring out new material on a regularbasis, we need a continual flow of articlesand classroom suggestions. We thereforewelcome your ideas for articles as well asyour completed manuscripts. In particu-lar, we welcome short classroom sugges-tions that can immediately be imple-mented by teachers. Materials should besent both in hardcopy and electronically.Hardcopy should be sent to Victor Katz,Convergence, Mathematical Associationof America, 1529 18th St. N.W., Washing-ton, DC 20036. We can take articles inWord or TeX, but please include illustra-tions (in jpg format), applets, etc. as sepa-rate files, and give explicit instructionsfor both internal and external hyperlinks.If there are many illustrations or applets,it is best to send the electronic versionon a CD to the same address. Short ar-ticles, as well as ideas for articles, can besent electronically to Victor Katz atvkatz@udc.edu. If you have an idea foran article, but do not know how to pro-duce applets for it, we suggest that youcontact an expert on your own campusfor help. If necessary, however, we canprovide help in the editorial office, pro-vided you give us very explicit instruc-tions as to what you need. The editorslook forward to producing this magazinefor the mathematics community.

Convergence can be accessed through theMAA home page, http://www.maa.org, ordirectly at http://convergence.mathdl.org.

Convergence: A New Online Magazine from the MAA

By Victor Katz

Frank Swetz and Victor Katz

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5

At its January, 2004, meeting, theBoard of Governors approved the forma-tion of SIGMAA QL — the special in-terest group for Quantitative Literacy.

Quantitative literacy (QL) can be de-scribed as the ability to adequately useelementary mathematical tools to inter-pret and manipulate quantitative dataand ideas that arise in an individual’sprivate, civic, and work life. Like readingand writing literacy, quantitative literacyis a habit of mind that is best formed byexposure in many contexts.

While developing a quantitatively liter-ate citizenry is the responsibility of amuch larger community, it is the obliga-tion of the collegiate level mathematicscommunity to take leadership in identi-fying the prerequisite mathematical skillsfor QL, finding innovative ways of de-veloping and implementing QL cur-ricula, assisting colleagues in other dis-ciplines to infuse appropriate QL expe-riences into their courses, and stimulat-ing the national dialogue concerning QL.

The purpose of this SIGMAA is to pro-vide a structure within the mathematicscommunity to achieve these goals. It willachieve this by hosting receptions atMAA meetings, organizing talks, panelsand paper presentations at MAA meet-ings, providing e-mail bulletins of timelyevents and issues via a listserv restrictedto members of the SIGMAA, and pro-moting submissions to publications thatfurther the QL movement. We will alsomaintain a web page of QL informationand distribute a semi-annual newsletterto the SIGMAA membership.

In addition, the leadership of SIGMAAQL will be coordinating its activities withthe many other groups, both internal andexternal to the MAA. Internally, thesegroups include the CUPM and its vari-ous subcommittees and other SIGMAAswhich have an interest in this area. Ex-ternally, the SIGMAA will be interactingwith the NCTM, AMATYC, and the Na-tional Numeracy Network.

The initial leadership for SIGMAA QLincludes Judy Moran (Trinity College) asChair, Caren Diefenderfer (Hollins Uni-versity) as Chair-elect, John Bukowski(Juniata College) as secretary-treasurer,

SIGMAA QL Is Formed

By Rick Gillman

and Matt DeLong (Taylor University ) aswebmaster. The SIGMAA will hold itsfirst electronic election for new officersin November, 2004, with the new offic-ers assuming their duties in January of2005.

SIGMAA QL will hold its first businessmeeting and sponsor an invited speakerat the Joint Mathematics Meetings beingheld in Atlanta in January 2005. In addi-tion, the SIGMAA will host an informalgathering of interested individuals atMathFest being held in Providence inAugust, 2004. Further information onthis gathering will be published in theprogram for that meeting.

Finally, members of the MAA are encour-aged to be part of a team from their in-stitution attending the PREP workshopon Quantitative Literacy being held thissummer from August 17-20, 2004, at theSleeping Lady Mountain Resort inLeavenworth, Washington.

Tucked into a major appropriationsbill this March was a request by Congressthat the Institute for Education Sciencesassess teacher education programs na-tionwide. According to the March 3 is-sue of Education Week, “Congress in-tends for existing data to be synthesizedon the consistency of requiredcoursework, how reading and math aretaught, and the degree to which pro-grams are aligned with scientific evi-dence on the subjects.” The study willlikely be conducted by the National Re-search Council.

The Institute for Education Sciences isa branch of the Department of Educa-tion created in 2002. Its purpose is “to

advance the field of education research,making it more rigorous in support ofevidence-based education.” Grover J.(Russ) Whitehurst is the current Direc-tor of the Institute. For more informa-tion about IES, see http://www.ed.gov/about/offices/list/ies/index.html.

It will probably be several years until thestudy is complete. Meanwhile, reactionsare mixed. Some education experts areconfident that the study will confirm theeffectiveness of teacher education pro-grams. Others fear that the study will beused against Schools of Education, anddoubt that the concept of “scientific evi-dence” is applicable to the assessment ofteacher education.

Congress Asks for Teacher Education Study

If you started counting your heartbeatsat midnight on January 1, 2000, whenwould you count the millionth beat? Howabout the billionth beat? This is one of the“math challenges for families” to befound at the Figure This! web site at http://figurethis.org. The rationale for the site,which was created by the National Coun-cil of Teachers of Mathematics, is ex-plained in the family corner: “Familymembers — as children’s first teachers— are crucial to student success. And themore adults become engaged in theirchildren’s education, the greater thechances that children will succeed.” Thesite provides mathematical challengesboth in a format appropriate for the weband in printable format.

Figure This!

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FOCUS May/June 2004

The Archives of American Math-ematics (AAM), a unit of The Uni-versity of Texas at Austin’s Centerfor American History, has been animportant resource for mathema-ticians, historians, and sociologistsfor nearly thirty years. Thanks tothe generous financial support re-cently provided by the Legacy of R.L. Moore Project, through theMathematical Association ofAmerica this valuable collectionwill be made more accessible forteaching and research.

In February, 2003, the Center forAmerican History utilized grantfunds to hire Kristy Sorensen, afull-time archivist, and TraciDrummond, a part-time assistant,to manage the growing collection.

The Archives of American Mathematics at the

Center for American History

Together they havetaken major stepsto enhance thevalue of AAM col-lections to re-searchers, includ-ing the placementof over 50 collec-tion inventories onthe Internet, wherethey can be ac-cessed by research-ers worldwide. InFebruary, 2004, theAAM was awardedan additional twoyears of fundingfrom the Legacy ofR.L. Moore Project.In March, AmeliaAbreu replaced

R.L. Moore, January, 1969. Photo taken by Homer G. Ellis,from the R.L. Bing Papers, Archives of American Mathemat-ics, Center for American History, The University of Texas atAustin.

By Kristy Sorensen

George Bruce Halsted, date unknown. From the R.L. Moore Pa-pers, Archives of American Mathematics, Center for AmericanHistory, The University of Texas at Austin.

R.H. Bing, ca. 1960s, from the R.L. Bing Papers, Archives of AmericanMathematics, Center for American History, The University of Texas atAustin.

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Traci Drummond as the archival assis-tant. She and Kristy Sorensen look fowardto continued work with the mathematicscollections in the coming years.

Colleagues of R. L. Moore, a prominentmathematician and long-time professorat The University of Texas (1920-1969),established the Archives of AmericanMathematics in 1975 with the donationof his papers. This core collection at-tracted other donors, and led to the ad-dition of the papers of many of Moore’sstudents and colleagues, including R. L.Wilder, R. H. Bing, and G. B. Halsted. In1978 the AAM became the official reposi-tory for the records of the MathematicalAssociation of America, expanding thecollection to include the administrativerecords of this important professional or-ganization. Adding to the breadth of col-lections in the AAM are the records ofthe School Mathematics Study Group,creator of the influential “New Math” pri-mary and secondary curriculum of the1960s. Other prominent collections in-clude the papers of Max Dehn, EmilGrosswald, and William T. Reid. Majorstrengths of the archives are in topology,mathematics education, analysis, num-ber theory, logic, and the mathematicalfoundations of physics. The AAM cur-rently consists of 70 collections measur-ing more than one-thousand linear feet.

The Center for American History is aspecial collections library, archive, andmuseum that facilitates research andsponsors programs on the history of theUnited States. The Center supports re-search and education by acquiring, pre-serving, and making available researchcollections and by sponsoring exhibi-tions, conferences, symposia, oral historyprojects, publications, and grant-fundedinitiatives.

Persons interested in conducting re-search or donating materials or who havegeneral questions about the Archives ofAmerican Mathematics should contactthe Archivist Kristy Sorensen,k.sorensen@mail.utexas.edu, (512) 495-4539. For more information on the Ar-chives, visit the archives’ web page ath t t p : / / w w w . c a h . u t e x a s . e d u /collectioncomponents/math.html.

Joint AMS/MAA Mathematics Meeting, Cleveland, Ohio, December, 1930. From left toright: Wilfrid Wilson, J.W. Alexander, W.L. Ayres, G.T. Whyburn, R.L. Wilder, P.M.Swingle, C.N. Reynolds, W.W. Flexner, R.L. Moore, T.C. Benton, K. Menger, S. Lefschetz.From the R.L. Moore Papers, Archives of American Mathematics, Center for AmericanHistory, The University of Texas at Austin.

C. Zarankiewicz, W.L. Ayres, and B. Knaster (with unidentified dog), undated. From theR.L. Moore Papers, Archives of American Mathematics, Center for American History, TheUniversity of Texas at Austin.

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In a time of diminishing resources, mustwe give up the attempt to offer a highquality mathematics education to ourstudents? Peter Taylor, recently chosen tohead the Department of Mathematics andStatistics at Queen’s University inKingston, Ontario, addresses this questionin a letter to his colleagues.

Department of Mathematicsand StatisticsQueen’s UniversityFebruary, 2004

Dear Colleagues:

Thanks to all those who have congratu-lated me and wished me well. We haveat Queen’s a versatile and dedicated De-partment and I look forward to guidingit through the next few years of itsgrowth. I say “growth” in the faint hopethat this means more than spiritual andintellectual growth, though the Deanseems unable to say more than that somebut not all of our upcoming retirementswill be replaced. These are challengingtimes.

Indeed, we are called upon now to domuch more with many fewer resourcesthan were available when I was a studenthere. Our response to each new cut overthe years has been to squeeze harder andfit more in and I am now convinced thatthis response can no longer serve us orour students. It’s time to do old thingsin a new way. I’m not sure just what anew system might look like, but my un-derstanding of how significant advanceswere made in our evolutionary trajectoryas a species is that it was at times of tightconstraint that significant “saltations”were made.

I was glad that so many of you attendedthe colloquium last week — my thoughtson some possible changes in our largefirst-year service courses. It wasn’t re-ally meant as a directive, though someof you might have wondered about that.

Well maybe it was –– certainlyI am serious about this type ofchange: less breadth and moredepth, a few critical examples,the best we can find, powerful,beautiful, engaging, explor-atory, artistic, as many of thosefine words as possible, donethoroughly and well, hopefullyenough of these to capture theessentials of the subject, be thatcalculus or linear algebra. Suchexamples are not easy to find;but once you have them youhave pure gold. In the conver-sations I had afterwards, manyof you picked up on my phrasethat we should dare to be “lesscomprehensive and systematic.”You pointed out that whilemost of these students willnever encounter us again, someof them do stay to take MATH211 or MATH 232 and there aremany things they will need to know. It’strue that some will go on to other mathcourses and in fact I hope that thechanges I have in mind will encourageeven more to stay with us longer andlearn some more mathematics. In part,my purpose in writing this is to addressthis “prerequisite” worry that many ofyou seem to have.

On the other hand, perhaps I protest toomuch. Perhaps it’s not such a worry. Af-ter all, these students are majors in an-other discipline and this is an appliedcourse, and I am an applied mathemati-cian and apparently a good teacher so onthe whole you trust me with these stu-dents, and my schemes will probably turnout fine. And of course I did make it clearat the very beginning that I was not talk-ing about our honours courses –– those,we have a duty to teach “properly,” a dutynot only to our students, but to the sub-ject itself. No, the colloquium was verymuch about our service courses.

But now I owe you a confession. Thechanges I have in mind apply, in my view,equally to the honours courses. In thesecourses, I think we can do much less, havemore fun and engagement, and comeaway with a much higher retention rate.In saying this I am aware that ourhonours courses are generally very welltaught and our graduating students onthe whole look back on them with posi-tive memories. But less than half of thestudents (in fact much less) who enrollin those courses wind up with any kindof math concentration. I want more ofthem to stay. I believe that more math-ematics would serve them well, and thatthey in turn would serve our subject wellin whatever they chose to do in life.

I have already touched on this subjectwith one or two of you, and in fact, overthe years, with other mathematicians atprofessional meetings. It is a difficultdebate, partly because so much of it iswaged abstractly away from concrete ex-amples and explicit curriculum models.But the bottom line –– what my conver-

The Teacher as Artist: A Letter to My Colleagues

Peter Taylor

By Peter Taylor

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sations always seem to come down to ––is your worry that my “schemes” will failto properly prepare our students for up-per level mathematics courses. Think-ing about this, I wonder exactly what itis that you are worried about. Is it thestudents themselves and their future? Isit the reputation of the university? Or isit your own future courses, the readinesswith which the students will be able tograpple with the many neat things youwant to teach them?

Well, let’s think for a moment aboutthese students. They come in differentflavours. First there’s A and B who arenatural problem solvers and might wellwind up in research careers in mathemat-ics or science. These students will thrivewith your curriculum as well as withmine –– all they need is good problemsand someone to talk to about them. Socan we agree that they are not the prob-lem? Then there are C, D, E, F, G, H, Iand J. Some of them will go on to law orbusiness, some to government; some arein our teacher program. And then ofcourse there is (or will be) X, Y and Zwho started out in the service courses butbecause they were so interesting andlively (!) decided to switch to a mathmajor and find themselves now in thirdyear. You seem to suggest that these stu-dents, from C to Z, would have great dif-ficulty filling in the gaps on their own,that if we do not teach them a “proper”systematic course they will not manageto carry the subject effectively forward.

Well, in my experience, they do not nowcarry the subject effectively forward.Many of them take my MATH 382 and Ican attest that they are not very good atusing simple (basic!) ideas of calculusand linear algebra to solve new problems.[Actually I don’t mean to suggest thatthey should be able to do that in thirdyear; the point I’m making is that theydon’t come away (from the teaching andlearning) with what you might think,that while an alternate approach mightbe no better than the traditional one, itmight equally well be no worse. In fact Ihave reasons for thinking that a lighter,livelier approach might actually workbetter and gain us more students. As an

example I cite my way of treating eigen-values which (I feel) is much more funthan the standard approach but which Iam sure gives them a much clearer ideaof what an eigenvalue really is.]

For most of our math majors (from Con), it is enough that they learn a littlemath, enough math to see the power andbeauty, and learn it well and positivelyenough that they can feel it belonging tothem, feel it part of their domain. Afterthat, we simply have to trust the subjectitself to impel them forward as appro-priate, and trust them to respond.

I know that many of you have importantstories that you want to tell your seniorstudents, stories that they would not beable to appreciate or even understandwithout a “proper” grounding in the firstand second year, and so you are under-standably worried by my suggestion thatwe might lighten (that word in all itsmeanings!) our introductory courses. Ifind that the metaphor of “teacher as art-ist” is helpful here.

Briefly, the idea is that the curriculum beregarded as a set of integral works of art.Because such works have a wholeness inthemselves, it is not so crucial whetherthe student has mastered this technicalskill or that, rather what counts is thenature and level of sophistication of thetechnical skills he has mastered, and it isthis that should increase from year toyear. As each new work is brought for-ward, crucial technical skills can be, andindeed ought to be, reinvented in the newcontext, and the experience that isneeded for this to succeed is of a generalrather than a specific nature.

The artist and the poet do not to tell theentire tale as they know it. Rather theydisplay it on the canvas, or on the page,in a way that is true both to the story andto the constraints imposed by the me-dium. The response of the artist is there-fore highly conditioned, bowing to form,and to technique, but it is the more pow-erful for that; this is the wonderful para-dox of artistic expression. Good storiesbecome art when they are told in simpleways which capture their richness. It is a

challenge to find such ways, but it is thefundamental challenge of the teacher asartist.

In closing, these remarks of mine focuson what I believe are fundamental prob-lems of the area of study known as math-ematics education. For 30 years I haveattended professional meetings and readjournal articles in this field and followedthe big debates and you know what? ––except for an oscillation from right to leftand back again, little seems to change.One problem is that the system is com-pletely interconnected –– a powerful cur-rent of experience and ideas runs fromelementary school through high schooland university, to teacher’s college andright back to the beginning again. Per-haps the only hope is a decisive interven-tion at some critical point in the systemand my target these days is that crucialfirst year in which we play host to a sig-nificant fraction of the students at thisuniversity.

You’ve noticed a strong thread of ideal-ism running through this letter. That’show we’re going to start. We’ll temper itas we go.

All the best,Peter

Peter Taylor is Professor of Mathematics,Biology and Education at Queen’s Univer-sity. In July, 2004 he will assume duties asHead of the Department of Mathematicsand Statistics at Queen’s University. He isdetermined that this will not cut into hissquash.

Have You Moved?

The MAA makes it easy to change youraddress. Please inform the MAA ServiceCenter about your change of address byusing the electronic combined mem-bership list at MAA Online(www.maa.org) or call (800) 331-1622,fax (301) 206-9789, email:maaservice@maa.org, or mail to theMAA, PO Box 90973, Washington, DC20090.

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FOCUS May/June 2004

Most of my years in mathematics edu-cation have been spent in primaryteacher training in the UK. Naturally, thishas involved much time in primaryschools, working with teachers and do-ing maths with children between the agesof 5 and 11 years. By the phrase ‘doingmaths’ I mean that I always get the chil-dren actively exploring a problem or pro-vide them with heuristic exercises thatgenerate discussion and the exchange ofideas. It involves observation of pattern,forming hypotheses, explaining and jus-tifying (however informally).

On the question of ‘logic’ and ‘proof ’, Irarely embark upon explicit teaching ofsuch themes but they often arise as sideissues, and very valuably so. Lookingback, many instances spring to mind, butI recall some particular teaching episodesthat illustrate children’s capacity for logi-cal thought and proof. What I havelearned from this has consequently af-fected the way I have taught mathemat-ics to trainee teachers.

Below, I recount some past teaching epi-sodes as a means of illustrating this phi-losophy. Some of the conversations withchildren have been reconstructed frommemory, but the gist of the dialogue isauthentic.

Odds and Evens

This was a lesson with a class of childrenaged between 8 and 9 years, the aim be-ing to extend their understanding of ba-sic addition. The activity was centeredupon exercises that involved adding pairsof even numbers, or adding pairs of oddnumbers, or adding a mixed pair (oneodd, the other even). Initially, I didn’texplain what was going on, but I got thechildren doing the additions and, after awhile, asking them what they’d observed.

Nearly all the children had noticed thesimple patterns: E + E = E, O + O = E,E + O = O and O + E = O and this wasfollowed by quite a bit of discussion

around questions such as ‘How can wetell if a number is even?’ etc and then tothe matter of generalising results aboutthe addition of three or more odd num-bers etc.

Most of the attempts to justify the claimthat O+O =E were based upon state-ments of the sort ‘It always works’. Prob-ing more deeply, I challenged the chil-dren to provide a counter-example,which, of course, they could not. Whatdid emerge were two particularly strik-ing contributions, one from Brendananother from Clare.

Brendan’s explanation was based upon alast digit argument. He pointed out thatsince the sum of any pair from 1, 3, 5, 7,9 would be even, then the sum of any twolarge odd numbers would also be even.For example, since 7 + 5 is even, then 127+ 105 is even, and so on. On the otherhand, Clare’s explanation was intuitivelybased upon that used in a formal proof.She held up both fists, each with just onefinger sticking out. She said ‘If this is odd,there’s one left over, and if that’s odd,there’s one left over. The left overs maketwo, so everything’s even’.

Formally, if x, y are odd then x = 2m+1(Clare’s left fist) and y = 2n+1 (Clare’sright fist), where m and n are non-nega-tive integers. So x + y = (2m+1) + (2n+1)= 2m+2n+2 and ‘everything’s even’.

Halving a Square

For this activity, the children were aboutten years of age and I provided each ofthem with a sheet with lots of squaresdrawn on it. After an opening discussion,I asked them to find different ways ofdividing a square into two equal piecesby use of a single line. I actually meant‘congruent’ pieces, but some of the chil-dren took this to mean ‘equal by area’ butnot necessarily congruent (so what?).

Many drew in the obvious dividing lines(vertical, horizontal and diagonal lines

of symmetry) and then ran out of ideas.Calling the class together for discussion,one of the boys (Glen), who had beenworking quietly in a corner of the class-room, told us that there were ‘lots’ ofways of doing this. I was amazed byGlen’s justification, which was basedupon rotational symmetry as follows:

• Find the centre, C, of thesquare by pencilling the diagonalsthen erase the pencil lines.

• Join C, to another point, P, on oneof the sides by drawing any line(straight, curve or squiggly).

• Using tracing paper, rotate thesegment PC through 180o (about C)to meet the opposite side at a pointQ.

• Line PCQ divides the square intotwo congruent parts.

• Check this by cutting out the twopieces and superimposing.

Of course, Glen didn’t word his responselike this, but he explained this methodin his own vernacular, showing us hisdiagrams. Equipped with scissors, trac-ing paper, coloured pencils etc, the restof the class then embarked upon Glen’sapproach and started to manufacturetheir own sets of examples.

Classifying by Symmetry

Symmetry is one of those topics that havelong been included in school syllabusesin the UK, but most teachers don’t knowwhy that is, so they end up treating thetopic by providing recreational colouringand cutting activities with no ostensiblegeometrical content. What I intendedwas to use symmetry to extend children’sawareness of properties of shape by get-ting them to investigate questions suchas:

• How many lines of symmetrycan a triangle have?

• Find, if you can, quadrilateralshaving 0, 1, 2, 3 or 4 lines ofsymmetry.

What I Learned by Teaching Proof and Logic

By Peter Ruane

FOCUS

11

• Which of these can have a right-angle? How many right anglesat most?

• Which of your quadrilateralshave rotational symmetry?

• Which quadrilaterals only havelines of symmetry that are di-agonals? etc.

The work was practical (templates, scis-sors, sticky paper, glue etc) and, for someof these questions, the children recordedthe results by cutting out and stickingtheir shapes into appropriate boxes of atwo-attribute Carroll diagram.

When it came to summarising the results,and reaching agreed conclusions, wewere really employing proof by exhaus-tion (deriving all possible cases). It wasagreed, for example, that a triangle couldhave 0 lines of symmetry, 1 line of sym-metry or three lines of symmetry (be-cause the children had found such ex-amples). However, as to providing a rea-son for not being able to find a trianglewith exactly two lines of symmetry mostchildren rather lost interest — except twoof the girls (both 11 yrs old).

Later, the girls came up with the argu-ment that, for a triangle with two linesof symmetry, all three sides must be equaland so it is equilateral and therefore hasa third line of symmetry. In other words,from an activity based upon exhaustingall possible cases, there emerged a nicepiece of deductive reasoning that wasaccepted as a convincing proof by mostof the class (and, of course, myself).

Sorting and Classifying

Working with children as young as fiveyears old, there is much scope for activi-ties based upon sorting and classifyingeveryday objects into sets. This facilitateslanguage development and particularlythe refinement of vocabulary connectedwith length, weight etc. By extendingsuch activities to work with 2d and 3dgeometric objects, shape names are in-troduced and various intuitive geomet-ric properties of shape begin to evolve.

Obviously, such work commences bysorting objects according to one attributeonly (e.g., using Logiblocs, sort accord-ing to thick/thin or square/non-squareetc) and leads to sorting by two or moreattributes. However, by using Venn orCarroll diagrams to record the results ofsorting, opportunity arises for the use oflogical connectives and the formation ofcompound statements from simple state-ments as a means of describing the re-gions of the diagrams (truth sets).

A typical lesson on this was based uponsorting a group of children according towhat sort of pets they may own. So, weput a large closed rope loop on the class-room floor, with the fifteen five-year oldsexcitedly gathered round. ‘WHO has aDOG?’ I ask in an exaggeratedly dramaticway. Eight eager responses are given, sointo the hoop those eight children go.

Qn: Why isn’t Alice in the hoop?’

Ans: ‘Because she hasn’t got a dog!’shouts Ruth (she isn’t in thehoop either)

Qn: ‘What pets do you have Ruth?

Ans: ‘A cat and a tortoise’ She replied.

Qn: ‘Who else owns a cat?’

There are about five positive replies tothis, including two from children in the‘dog hoop’. We put down another hoopfor the cat owners, non-overlapping withthe dog set.

Problem! Children with cats go into thecat hoop, but some of those are also inthe dog hoop. How do we get those chil-dren into both sets? After some discus-sion, it is decided to make the hoopsoverlap to produce a four-region Venndiagram (including the outside).

Each of the four regions contains at leastone child and we discuss why any par-ticular child is, or is not, in a particularregion. We have two logical connectives‘and’, ‘or’ and two simple statements andcompound statements are elicited fromthe diminutive participants. Replies tomy questions were typically like this:

‘Charlie’s in that bit ‘cos he’s got a dog,but no cat’

~ c d∧( )

‘Ruth’s got a cat and not a dog’

c d∧( )~

‘Hannah’s got nothing: she’s outside’

~ ~c d∧( )

This goes on for a while and we returnto this type of activity in various guises,always developing children’s ability todescribe and logically classify math-ematical or everyday objects. Really, thisis a prelude to symbolic logic, but, ofcourse, that stage is never reached andteachers confine the results to describ-ing subsets of truth sets by use of every-day language.

I have often applied the same idea tosymmetry activities for the classificationof triangles, quadrilaterals etc. For ex-ample, with two attributes ‘has rotationalsymmetry’ and ‘has mirror symmetry’,older children will be asked to insert atleast one type of quadrilateral into eachof the four regions of a Venn (or Carroll)diagram and give their reasons for plac-ing a particular shape in a particular re-gion. Descriptions will involve com-pound statements of the sort ~ m r∧(parallelogram), m r∧ ~ (trapezium),m r∧ (rhombus or rectangle) etc. Withsome children, it’s possible to establishequivalences like ~ ~ ~m r m r∨ ⇔ ∧( ) ,but only in the form of ordinary Englishand in a specific practical context.

Summary

In the past five years or more, primarymathematics in the UK has been basedupon the principle that teaching has tobe interactive. This means that children’sideas (right or wrong) must be incorpo-rated into lessons and teachers shouldencourage children to explain, questionand demonstrate. In this way, the capac-ity for ‘logical’ thought will graduallydevelop.

12

FOCUS May/June 2004

The Mathematical Association ofAmerica seeks to identify candidates tosucceed Frank Farris as Editor of Math-ematics Magazine when his term expiresin December, 2005. The Search Commit-tee plans to make a recommendationduring the summer of 2004 so that thenew editor can be approved by the Boardof Governors and begin handling all newmanuscript submissions in January,2005. The new editor would be EditorElect during 2005 and would serve asEditor for the five years 2006-2010.

Questions about the nature of the posi-tion and its workload can be addressedto Frank Farris (FFarris@scu.edu); ques-tions about MAA support for the editor’swork can be addressed to the MAA’s Di-rector of Publications, Don Albers(dalbers@maa.org).

Each applicant should submit a resume,names of references, and a statement of

Mathematics Magazine Editor Search

interest containing his or her ideas aboutthe journal. These can be emailed withattachments as Word or plain-text TeXdocuments to the chair of the SearchCommittee, Deanna Haunsperger(dhaunspe@carleton.edu), or mailed to:

Deanna HaunspergerDepartment of Mathematicsand Computer ScienceCarleton CollegeOne North CollegeNorthfield, MN 55057

Nominations are also welcome if youknow someone who would be an out-standing editor. Applications and nomi-nations will be accepted until the posi-tion is filled, although preference will begiven to applications received by earlyMay.

Moreover, the notion of ‘proof ’ has wid-ened from that used in formal math-ematics and we now acknowledge differ-ent levels of proof, beginning withchildren’s reasonable explanations as towhy they believe something to be trueor false.

In the examples that I have provided, Iam suggesting that many children arequite capable of achieving quite sophis-ticated proofs and that, even for youngchildren, deductive proof, proof by ex-haustion, and proof by counterexampleare not beyond the bounds of possibil-ity. Teaching activities such as thoseabove have also taught me that Booleanalgebra has its pedagogical roots in sort-ing activities with very young children.

What did I learn by teaching proof andlogic to undergraduate maths majors?That’s a different story!

Peter Ruane has taught mathematics inschools to children across the age-range (5yrs to 18 yrs) and he has been involved inthe mathematical education of teachers inEngland for over thirty years.

There is no basis for your independence

Sign next to a bridge in Beijing, China August, 2002. Photograph byColm Mulcahy.

FOCUS

13

The Ohio Section will sponsor a sum-mer short course on Teaching and Do-ing Knot Theory, which will be held June2–4 at Ohio Northern University in Ada,Ohio. Colin Adams of Williams Collegewill be the instructor.

Knot theory is a great topic for excitingstudents about mathematics. It is visualand hands on. Students can begin work-ing on problems the first day with theirshoelaces. Knot theory is also an incred-ibly active field. There is a tremendousamount of work going on currently, andone can easily state open problems. It alsohas important applications to chemistry,biochemistry, and physics.

This workshop is aimed at anyone whois interested in knowing more about knottheory. There is no assumption of previ-ous background in the field.

Participants completing the workshopwill learn how they can:

• Teach an undergraduate course in knot theory

• Do research in knot theory

• Direct student research in knot theory

Participants will have the opportunity toconjecture wildly, throw around ideas,and work on original research.

There will be morning and afternoonsessions on Wednesday and Thursdayand a morning session on Friday.

Colin Adams is the Francis ChristopherOakley Third Century Professor ofMathematics at Williams College. He re-ceived his Ph.D. from the University ofWisconsin-Madison in 1983. He is par-ticularly interested in the mathematicaltheory of knots, their applications andtheir connections with hyperbolic geom-etry. He is the author of The Knot Book.an elementary introduction to the math-ematical theory of knots and co-authorwith Joel Hass and Abigail Thompson ofHow to Ace Calculus: The StreetwiseGuide, and How to Ace the Rest of Calcu-lus: the Streetwise Guide, humorous

2004 Summer Short Course—————Teaching and Doing Knot Theory

supplements to calculus. Havingauthored a variety of research articles onknot theory and hyperbolic 3-manifolds,he is also known for giving mathemati-cal lectures in the guise of Mel Slugbate,a sleazy real estate agent. A recipient ofthe Deborah and Franklin Tepper HaimoDistinguished Teaching Award from theMathematical Association of America(MAA) in 1998, he was a Pólya Lecturerfor the MAA for 1998-2000 and was aSigma Xi Distinguished Lecturer for2000-2002. He is also the author of math-ematical humor column called “Math-ematically Bent” which appears in theMathematical Intelligencer.

Registration will begin in April and con-tinue until all seats are filled. The regis-tration fee is $150.00.

For more information contact Don Huntat d-hunt@onu.edu or call (419) 772-2351.

You may also visit: http://www.onu.edu/a+s/math/NewFiles/maa/shortcourse.htm.

Frank Thompson Leighton, professorof applied mathematics at the Massachu-setts Institute of Technology and anMAA member, has been elected to theNational Academy of Engineering “forcontributions to the design of networksand circuits and for technology for Webcontent delivery.”

The National Academy of Engineering(NAE) elected 76 new members and 11foreign associates this year, which bringsthe total U.S. membership to 2,174 andthe number of foreign associates to 172.Election to the National Academy of En-

gineering is among the highest profes-sional distinctions accorded to an engi-neer. Academy membership honors thosewho have made “important contribu-tions to engineering theory and practice,including significant contributions to theliterature of engineering theory andpractice,” and those who have demon-strated accomplishment in “the pioneer-ing of new fields of engineering, makingmajor advancements in traditional fieldsof engineering, or developing/imple-menting innovative approaches to engi-neering education.”

MAA Member Frank Leighton Elected to the

National Academy of Engineering

The first report of the 2003 AnnualSurvey of the Mathematical Sciences hasjust been released and is availableonline at http://www.maa.org/news/2003survey.html. It focuses on newdoctoral recipients and the faculty sal-ary survey. The second report will bepublished in August, 2004.

2003 Annual Survery

of the Mathematical

Sciences

14

FOCUS May/June 2004

Long time MAA member and sup-porter William (Bill) G. Chinn passedaway quietly on March 23, 2004, in SanFrancisco. Bill was an active member ofthe MAA. He served on national and sec-tional MAA committees, was the North-ern California Section Chair, and in1981–82 served as Second Vice Presidentof the MAA. He was a generous donorto both the MAA and the AmericanMathematical Society.

An alumnus of Lowell High School andUniversity of California at Berkeley, Billserved in Europe and Africa duringWorld War II, first as a pharmaceuticaltechnician and then as a meteorologist.After his discharge he resumed graduatestudy in mathematics. He was a Profes-sor of Mathematics at the City Collegeof San Francisco for many years.

Bill also taught in junior high and seniorhigh schools in San Francisco, where stu-dents enjoyed his humorous and wittyapproach to classroom teaching. Hemaintained a mathematics “laboratory”for gifted students, and served as assis-tant to the Curriculum Coordinator ofthe San Francisco Unified School Dis-trict. He authored and co-authored nu-merous books, including First Conceptsin Topology, with Norman Steenrod, and3.1416 And All That, with Philip Davis.

I first met Bill when he was NewsletterEditor for the Northern California Sec-tion of the MAA. Later he and I servedtogether on the MAA DevelopmentCommittee. Bill loved to take pictures ofpeople at meetings. Afterward, he wouldcirculate pictures with colleagues andstudents properly identified. On the oc-

Remembering Bill Chinn

By Jean Bee Chan

casions we met for dinner, the bill wasalways magically paid. It turned out Billalways paid for dinner while the rest ofus were enjoying ourselves.

MAA members Leonard Klosinski, JerryAlexanderson, David Sklar, and I were atthe memorial service for Bill Chinn inOakland, California, at the Chapel of theChimes on March 30, 2004. Bill was al-ways cheerful, always generous. He wasa great friend and a selfless contributorto the mathematical community. He wasa good role model for all of us, and hewill live forever in our hearts.

The Mathematical Association ofAmerica PRofessional Enhance-ment Program (PREP) seeks pro-posals for 2005 PREP workshops.PREP is a comprehensive, profes-sional career enhancement projectof the MAA funded by NSF DUEgrant # 0341481.

Programs for all aspects of careerenhancement are encouraged.PREP workshops that presentmodern mathematical ideas or currentinterest in research and interdisciplinaryapplications are especially welcome. Pro-grams that present new developments inteaching and learning, innovative ap-proaches to leadership and career en-hancement, or focus on innovative cur-

ricula and pedagogical strategies are alsowelcome. All programs must appeal to ageneral audience of mathematical pro-fessionals.

In addition to PREP-funded workshops,we invite directors of other funded

Call for 2005 PREP Proposals

projects who plan to of-fer a workshop to applyfor inclusion under thePREP umbrella.Through this process, aworkshop will gain ac-cess to the wide audi-ence, logistical supportand evaluation servicesdeveloped for the PREPprogram. Programsconsidered for inclusionin this way must meet

the same guidelines as core PREP work-shops.

More information regarding submissionof workshop proposals is availablethrough the program website, http://www.maa.org/PREP.

FOCUS

15

Letters to the Editor

Paul Monsky’s Real School

In FOCUS, March 2004, bottom of page28, it is mentioned that “The only per-fect score on the 1950 MathematicalContest was Paul Monsky, fromStuyvesant High School in New YorkCity”. This is incorrect. Paul was atBrooklyn Tech when I was there.

Peter KahnSUNY Stony Brook

What is an Acronym?

I was amused at the argument about the“acronym” SPSS in the Letters section ofthe March FOCUS. The point is, SPSScan never be an “acronym” for anything.A sequence of initial letters of a phraseor definition is an acronym, if it forms apronounceable word. The operative termis “pronounceable.” RADAR, for ex-ample, is pronounceable, hence is an ac-ronym. It would have been permissibleto include the “A” from “Package” (the RAfrom RADAR stands for “radio”), or the“A” from “and” from the 2nd examplegiving SPASS, an acronym.

Oren N. DaltonEl Paso, TX

Well, the OED seems to agree with you,since it defines an acronym as “A wordformed from the initial letters of otherwords.” The computer community, how-ever, seems to call any sequence of lettersan acronym. See, for example, the defini-tion of “TLA” in the Jargon File: http://www.catb.org/~esr / jargon/html/T/TLA.html.

The FOCUS Online section of theMAA web site includes a very active bookreview column called Read This!. You canfind the main reviews site at http://www.maa.org/reviews/reviews.html. Thearchive of past reviews is huge: more than400 books have been reviewed. Here is alist of some recently reviewed books:

The Art of the Infiniteby Robert and Ellen Kaplan

A Smoother Pebbleby Donald C. Benson

When Least is Bestby Paul Nahin

Recently Reviewed Online

Geometry: Euclid and Beyondby Robin Hartshorne

Dr. Math Gets You Ready for Algebraby The Math Forum

A Topological Aperitifby Stephen Huggett and David Jordan

16

FOCUS May/June 2004

The organizers listed below solicit contributed papers perti-nent to their sessions. Sessions generally limit presentations toten minutes, but selected participants may extend their contri-butions up to twenty minutes. Please note that the dates andtimes scheduled for these sessions remain tentative. See the endof this announcement for specific submission procedures andother details. Each session room contains an overhead projec-tor and screen; black boards will not be available. Persons need-ing additional equipment should contact, as soon as possible,but prior to September 14, 2004, the session organizer whosename is followed by an asterisk (*).

MAA CP A1 Getting Students To Discussand To Write About MathematicsWednesday morning and Thursday afternoonSarah L. Mabrouk*, Framingham State College

This session invites papers about assignments and projects thatrequire students to communicate mathematics through in-classoral presentations, in-class discussions that they must lead andmotivate, and written assignments and/or papers. These as-signments/projects can include analysis and applications ofmathematics, presentations of and analysis of proofs, presen-tations about famous mathematicians and the mathematics thatthey studied, and assignments/projects that utilize creativewriting. Presenters are encouraged to discuss how the use ofthe assignment/project helps the student to gain greater un-derstanding of mathematics as well as to improve his/her un-derstanding of mathematics language and his/her ability tocommunicate mathematics. Of particular interest is the effectof such projects/assignments/presentations throughout thecourse on the student’s understanding of mathematics, his/hercommunication of mathematics, and his/her attitude towardmathematics.

MAA CP B1 My Favorite Demo—Innovative Strategies forMathematics InstructorsWednesday morning and Thursday afternoonDavid R. Hill*, Temple University, hill@math.temple.eduLila F. Roberts, Georgia College and State University

Mathematics instructors use a myriad of innovative techniquesfor teaching mathematical concepts. Technology readily avail-able in colleges and universities has provided a means to boostcreativity and flexibility in lesson design. Tools an instructorutilizes may include specialized computer applications, anima-tions and other multimedia tools, java applets, physical devices,games, etc. This contributed paper session will focus on noveldemos that mathematics instructors have successfully used intheir classrooms. Rather than focus on projects or student groupactivities, this contributed paper session will focus on theinstructor’s activities to facilitate learning. Mathematical con-tent areas will include pre-calculus, calculus, elementary prob-

ability, and selected post-calculus topics. This session invites1) demos that introduce a topic, 2) demos that illustrate howconcepts are applicable, 3) demos that tell a story or describethe development of a procedure, and 4) demos that lead to anactivity that involves the class. Presenters of demos are encour-aged to give the demonstration, if time and equipment allow,and to discuss how to use it in a classroom setting. Proposalsshould describe how the demo fits into a course, the use oftechnology or technology requirements, if any, and the effectof the demo on student attitudes toward mathematics.

MAA CP C1 Courses Below Calculus: A New FocusWednesday morning and Friday afternoonMary Robinson*, University of New Mexico-Valencia Campusmaryrobn@unm.eduFlorence S. Gordon, New York Institute of TechnologyLaurette Foster, Prairie View A&M UniversityArlene Kleinstein, Farmingdale State University of New YorkNorma Agras, Miami Dade Community CollegeLinda Martin, Albuquerque T-VI

An unprecedented collaborative effort has been developedamong members of MAA, AMATYC, and NCTM to launch anational initiative to refocus courses below calculus. The goalof the initiative is to encourage development and implementa-tion of courses that place greater emphasis on conceptual un-derstanding and realistic applications of the mathematics.Courses that better motivate students for and better preparethem to take subsequent mathematics courses, including cal-culus, statistics and quantitative methods, are needed to betterserve the needs of the quantitative disciplines and better pre-pare students to function effectively in today’s workplace, aswell as function effectively as citizens in today’s increasinglyquantitative society. Accordingly, for this session, we specifi-cally seek to address all of the college level courses below cal-culus, with particular emphasis on offerings in college algebraand precalculus. We seek proposals for presentations that offernew visions for such courses, discuss implementation issues(such as faculty training, placement tests, introduction of al-ternative tracks for different groups of students, transferabil-ity problems, etc) related to offering such courses, present re-sults of studies on student performance and tracking data inboth traditional and new versions of these courses and in fol-low-up courses, discuss the needs of other disciplines and theworkplace from courses at this level, discuss connections tothe changing school curricula and implications for teacher edu-cation. This session is co-sponsored by CRAFTY, the Com-mittee on Two Year Colleges, and the Committee on ServiceCourses.

MAA CP D1 Mathematics and SportsWednesday morning and Friday afternoonDoug Drinen*, University of the South, ddrinen@sewanee.edu

MAA Contributed Paper Sessions

Atlanta Joint Mathematics Meeting, January 5-8, 2005

PRELIMINARY ANNOUNCEMENT

FOCUS

17

Sean Forman, St. Joseph’s UniversityHoward Penn, US Naval Academy

When applied to the sporting arena, mathematics can provideboth compelling classroom examples and interesting researchproblems. Baseball has long been mined for interesting statis-tics examples ranging from regression and probability to thegame theoretic aspects of in-game strategy. Recent books onjai alai, football, and a few other sports have studied those sportsthrough a mathematical lens. The economics of sports is nowcovered by its own journal and the statistics publication Chanceroutinely discusses statistical examples from sporting events.This session invites papers describing interesting classroomexamples utilizing examples from sports and papers discuss-ing the application of mathematics to sporting events.

MAA CP E1 Mathematics in the Islamic WorldWednesday afternoonGlen Van Brummelen*Bennington College, gvanbrum@bennington.eduVictor Katz, University of the District of Columbia

This contributed paper session solicits presentations on all fac-ets of the history of the mathematical sciences in the Islamicworld, including the relationship of Islamic mathematics toWestern mathematics and to Indian or Chinese mathematics.We hope to elaborate both the unity and diversity of Muslimcontributions to both pure and applied mathematical disci-plines.

MAA CP F1 Mathlets for Teachingand Learning MathematicsWednesday afternoonDavid Strong*Pepperdine University, David.Strong@pepperdine.eduThomas Leathrum, Jacksonville State UniversityJoe Yanik, Emporia State University

This session seeks to provide a forum in which presenters maydemonstrate mathlets and related materials that they have cre-ated or further developed. Mathlets are small computer-based(but ideally platform-independent) interactive tools for teach-ing math, frequently developed as World Wide Web materialssuch as scripts or Java applets, but there may be many otherinnovative variations. Mathlets allow students to experimentwith and visualize a variety of mathematical concepts, and theycan be easily shared by mathematics instructors around theworld. The session is sponsored by the MAA Committee onComputers in Mathematics Education (CCIME).

MAA CP G1 Drawing on Our Students’ Thinking to Improvethe Mathematical Education of TeachersWednesday afternoonDale R. Oliver*Humboldt State University, dale.oliver@humboldt.eduMary Kay Abbey, Montgomery College

The MET document (The Mathematical Education of Teach-ers, CBMS, 2001) and the PMET project (Preparing Mathema-

ticians to Educate Teachers, MAA, 2003–2006) call for math-ematics faculty to re-examine what they teach and how theyteach in mathematics courses for prospective teachers. A keycomponent of this re-examination is careful consideration ofthe mathematical understanding and thinking of the prospec-tive teachers in our courses. Doing so informs faculty decisionsabout curriculum and pedagogy, and directs their instructionaleffort toward the individuals in the course. This session invitespapers on the mathematical preparation of teachers in whichwhat is taught and how it is taught is being informed by theunderstandings and thinking of the prospective teachers. Thissession is sponsored by COMET, the MAA Committee on theMathematical Education of Teachers.

MAA CP H1 History of Undergraduate Mathematics inAmerica, 1900–2000Thursday morningJack Winn*, SUNY Farmingdale, winnja@farmingdale.eduWalter Meyer, Adelphi UniversityJoseph Malkevitch, York College of CUNYAmy Shell-Gellasch, Grafenwoehr, Germany

This session will sketch how the last hundred years or so haveled us to today’s state of undergraduate mathematics. Ques-tions that are appropriate to discuss include: what curricularchanges occurred; how the changes depended on changes inmathematical knowledge; other reasons why changes in teach-ing occurred; what effects flowed from the changes; and howchanges affected student learning. Papers may focus on: im-portant individuals; important movements involving curricu-lum or styles of instruction; the evolution or disappearance ofparticular courses; case studies of particular institutions; thehistory and role of important organizations such as NSF andMAA; key events or circumstances external to the mathemati-cal community; etc. The speaker’s personal views about the bestway to teach certain topics are discouraged, unless those viewsare part of, or help explain historical issues.

MAA CP I1 Initializing and Sustaining UndergraduateResearch Projects and ProgramsThursday morningMargaret M. Robinson*, Mount Holyoke Collegerobinson@mtholyoke.eduSuzanne Lenhart, University of Tennessee

Papers are requested describing undergraduate researchprojects, courses, and programs. Of particular interest will bedescriptions of innovative ways to get administrative supportor other support that creates a sustainable program. Also ofinterest will be papers indicating where to find appropriateproblems and how to gauge the right level. Also descriptionsof courses with undergraduate research as the main goal wouldbe included. This session is sponsored by the CUPM Subcom-mittee on Undergraduate Research.

MAA CP J1 Projects and Demonstrations that Enhance aDifferential Equations CourseThursday morning

18

FOCUS May/June 2004

Rich Marchand*, Slippery Rock UniversityRichard.Marchand@SRU.eduShawnee McMurran, California State UniversitySan Bernardino

Differential equations is a diverse mathematical field that af-fords educators a great deal of flexibility in terms of content.The course can be highly theoretical, applied, or a combina-tion of each. This session invites novel projects or demonstra-tions that enhance a differential equations course either throughthe facilitation of mathematical theory or exposure to inter-disciplinary fields. New and interesting case studies are encour-aged, especially those that require computational or qualita-tive techniques. Demonstrations may be virtual, physical ormathematical. Examples include, but are not limited to, novelproofs, mathlets, or physical demonstrations.

MAA CP K1 Countering “I Can’t Do Math”: Strategies forTeaching Under-Prepared, Math-Anxious StudentsThursday morningSuzanne Doree*, Augsburg College, doree@augsburg.eduBonnie Gold, Monmouth UniversityRichard Jardine, Keene State College

How can we create a comfortable learning environment forunder-prepared or math-anxious students and, in particular,how can we constructively assess student learning? What class-room practices are especially effective with such students andhow does research on student learning inform those practices?How might the recommendations of the 2004 CUPM Curricu-lum Guide influence our approach in teaching developmentalor introductory courses to better reach these students? Thissession invites papers on all aspects of “what works” in teach-ing under-prepared, math-anxious students.

MAA CP L1 Using Real-World Data to IllustrateStatistical ConceptsThursday afternoon and Friday morningThomas L. Moore*, Grinnell College, mooret@grinnell.eduJohn D. McKenzie, Jr., Babson College

Guidelines in statistical education emphasize the use of realdata instead of the small, contrived data sets that appear insome textbooks. Faculty who have used real-world data to il-lustrate statistical concepts are invited to submit proposals thatdescribe the data set, its location on the web, and their use ofthe data set to teach ideas related to an introductory course instatistics: (1) data collection (sampling, design of experiments,potential biases); (2) data description (numerical summaries,graphical displays); (3) sampling distributions; (4) elementaryinference (interval estimation and hypothesis testing); (5) otherapplications, such as ANOVA, regression, and chi-square tests.

MAA CP M1 Environmental Mathematicsand the InterdisciplinaryFriday morningDr. Karen Bolinger*Clarion University, kbolinge@mail.clarion.eduBen Fusaro, Florida State University

Bill Stone, New Mexico Institute of Mining & Technology

We seek presentations that deal with all aspects of the peda-gogy and the modeling of environmental problems suitable forgeneral education, calculus, and above. Readers are invited totake up the challenge of searching the natural sciences, as wellas economics, environmental science, and environmental edu-cation for problems that can be clarified, extended or solvedby undergraduate mathematics. We encourage contributionsthat emphasize computational, visual or qualitative approaches.

MAA CP N1 Teaching Visualization SkillsFriday morningMary L. Platt*, Salem State College, mplatt@salemstate.eduCatherine A. Gorini, Maharishi University of ManagementSarah J. Greenwald, Appalachian State University

The ability to understand, use, and create diagrams, graphs,and illustrations is essential for students in every area of math-ematics. Computers have made graphics of every form widelyavailable, so there is an increasing need to help students de-velop their ability to handle visual information. This sessioninvites papers on all aspects of visualization in the college class-room: which skills are needed for success in mathematics, howto train students to use visual information, examples of class-room activities that help develop visualization skills, and waysto assess a student’s visualization skills.

MAA CP 01 Teaching and Assessing Problem SolvingFriday morningAlex Heidenberg*, US Military Academyalex.heidenberg@usma.eduMichael Huber, US Military Academy

Developing problem-solving skills in the modeling sense is acentral component in refocusing courses to emphasize process,conceptual understanding, and student growth. Universitiesand colleges are now writing institutional goals that addressthe capabilities of their graduates. How do we measure successin teaching our students to be effective problem-solvers? Thissession invites presentations about courses that focus on theprocess of problem-solving as a vehicle to learning mathemat-ics at the pre-calculus / introductory calculus levels, with spe-cial emphasis on modeling. These presentations can includecourse composition, philosophy, teaching ideas, and/or pastprojects, examinations, or other successful methods of assess-ment where students have become competent and confidentproblem-solvers. Each presentation should address the spe-cific goals in developing problem-solvers as well as the assess-ment techniques used to measure attainment of those goals. Inaddition, presenters should address how technology (calcula-tors, computer algebra systems, etc.) is incorporated into theteaching plan.

MAA CP P1 Philosophy of MathematicsFriday afternoonCharles R. Hampton*The College of Wooster, Hampton@wooster.eduBonnie Gold, Monmouth University

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This session, sponsored by the SIGMAA on the Philosophy ofMathematics, invites papers on any topic in the philosophy ofmathematics except logic and set theory. Possible topics includethe nature of mathematics, the nature of mathematical objects,the nature of mathematical knowledge, the relation betweenmathematics and the physical world, the role of esthetics inthe development of mathematics.

MAA CP Q1 Using Handheld Technology To Facilitate Stu-dent-Centered Teaching/Learning Activities At The Develop-mental Algebra LevelFriday afternoonEd Laughbaum*, The Ohio State Universityelaughba@math.ohio-state.eduMaria DeLucia, Middlesex County College

Lecture is the predominate method of choice for teaching re-medial level algebra, but handheld graphing devices are oftenintegrated by faculty. However, in many cases, the teaching/learning is still instructor centered. Handheld devices offer theflexibility of enhancing teaching and learning through student-centered activities, which can be used outside of class, or dur-ing class through group work. Anecdotal evidence shows de-velopmental algebra teachers often supplement textbook ma-terials with “graphing calculator” activities because even “re-form” textbooks do not offer appropriate ancillary packages.Therefore, we invite developmental algebra faculty to submitproposals on creative teaching/learning activities that are stu-dent centered, provide a diverse learning environment, offeroptions for learning and teaching, and use handheld devices.

MAA CP R1 My Three Favorite Original Calculus ProblemsSaturday morningJ.D. Phillips*, Wabash College, phillipj@wabash.eduTim Pennings, Hope College

This session is for those who, while teaching single and multi-variable calculus over the years, have thought of a few clever ornovel problems with solid pedagogical value, which they wouldlike to share with others. In particular, we are looking for origi-nal problems suitable for homework assignments or challeng-ing test questions. (We are not looking for extended modelingprojects and open-ended problems since good collections ofthese already exist.) We hope to organize these into a bookletfor publication, which could be used as a resource for calculuscourses. Thus, we ask that each submission adhere to the fol-lowing template: i) Statement of the problem, ii) Brief expla-nation of why it is interesting and pedagogically valuable, iii)Complete solution leading to an answer in closed form. Sub-missions may include from two to four problems. Participantsshould bring copies of their problems to the session for distri-bution. Each problem should begin on a new page. To includeas many as possible, each participant will be given 10 minutesfor presentation of the problems.

MAA CP S1 MEETING the CHALLENGE: RelationshipBetween Mathematics and Biology in the 21

st Century

Saturday morningCatherine M. Murphy*, Purdue University Calumet

murphycm@calumet.purdue.eduG. Elton Graves, Rose Hulman Institute of TechnologyDavid A. Smith, Duke University

“Biology as Information,” the title of the 2004 Gibbs lecture byEric S. Lander, Professor of Biology at MIT, emphasizes thefundamental changes that the science of biology is undergo-ing, especially since the connections between biology and math-ematics are becoming broader and deeper. This contributedpaper session will provide a forum for mathematicians withexperience working with the interface of mathematics and bi-ology to present papers which discuss the mathematics neededby contemporary biologists, the opportunities for mathemati-cians and biologists to collaborate in teaching, curriculum de-velopment, student research projects, or professional research.Talks especially valued are those that make practical sugges-tions concerning how to establish fruitful communication be-tween mathematicians and biologists, and, how to stimulatemathematics and biology students to prepare themselves toparticipate in this swiftly changing field. This session is spon-sored by the Subcommittee on Mathematics Across the Disci-plines.

MAA CP T1 Mathematics Experiences in Business, Industryand GovernmentSaturday morningPhil Gustafson*, Mesa State College, pgustafs@mesastate.eduMichael Monticino, University of North Texas

This contributed paper session will provide a forum for math-ematicians with experience in Business, Industry and Govern-ment (BIG) to present papers or discuss projects involving theapplication of mathematics to BIG problems. BIG mathemati-cians as well as faculty and students in academia who are inter-ested in learning more about BIG practitioners, projects, andissues, will find this session of interest. This session is spon-sored by the MAA Business, Industry and Government SpecialInterest Group (BIG SIGMAA).

MAA CP U1 Mathematical Experiences for Students Outsidethe ClassroomSaturday afternoonKay Somers*, Moravian College, somersk@moravian.eduJody Sorensen, Grand Valley State University

Mathematics “happens” both inside and outside the classroomand, in fact, many mathematics majors are drawn to the sub-ject through a special event sponsored by a Student Chapter orMath Club. This session seeks presentations by academic, in-dustrial, business, and/or student mathematicians so that theaudience will be encouraged to organize and runspecial events for their students. Descriptions of non-class-room activities could include, but are not limited to, speciallectures, workshops for students, Math Days, Math Fairs, re-search projects for students, Math Career Days, student con-ferences, recreational mathematics activities, problem solvingactivities and contests, general community-building activities,and student consulting projects. Information on how such

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activities are organized and carried out, what activities espe-cially grab students’ interests, how students are contacted andencouraged to participate, and how the events are funded willbe especially helpful. This session is organized by the MAACommittee on Undergraduate Student Activities and Chapters.

MAA CP V1 Research on the Teaching and Learning ofUndergraduate MathematicsSaturday afternoonBill Martin*, North Dakota State Universitywilliam.martin@ndsu.nodak.eduBarbara Edwards, Oregon State UniversityDraga Vidakovic, Georgia State University

Research papers that address issues concerning the teachingand learning of undergraduate mathematics are invited. Ap-propriate for this session are theoretical or empirical investi-gations conducted within clearly defined theoretical frame-works, using either qualitative or quantitative methodologies.Of highest priority are proposals that report on completed stud-ies that further existing work in the field.

MAA CP W1 In-Service Training Programs ForK-12 Mathematics TeachersSaturday afternoonZsuzsanna Szaniszlo*, Valparaiso Universityzsuzsanna.szaniszlo@valpo.eduJudith Covington, Louisiana State University, ShreveportTamas Szabo, Weber State University

All over the country many small and large scale projects existaimed at providing in-service training for K-12 mathematicsteachers. The directors of these projects will share their experi-ences developing and implementing the projects, includingboth mathematical and organizational issues. Mathematicianscontemplating starting similar projects will be able to learnabout successful strategies and potential pitfalls for these out-reach activities. The session invites talks that showcase success-ful in-service training programs for K-12 mathematics teach-ers. The talks should reflect on every aspect of the programincluding funding sources, organizational details, informationon co-operations with the school districts, mathematical con-tent and methodology, follow-up, evaluation and dissemina-tion. Programs that are easily replicable will be given priority.

MAA CP X1 General Contributed Paper SessionWednesday, Thursday, Friday, Saturday mornings and afternoonsDaniel E. Otero, Xavier Universityotero@xavier.xu.eduThomas A. Hern, Bowling Green State UniversityJames K. Strayer, Lock Haven University of PennsylvaniaMichael A. Jones, Montclair State University

Papers may be presented on any mathematical topic. Papersthat fit into one of the other sessions should be sent to thatorganizer, not to this session.

Send your abstract directly to the AMS. At the same time,send a detailed one-page summary of your paper, preferablyby e-mail, directly to the organizer–indicated with an (*). Inorder to enable the organizer to evaluate the appropriatenessof your paper, include as much detailed information as pos-sible within the one-page limitation. The summary need notduplicate the information in the abstract. A proposal shouldnot be sent to more than one organizer. Participants mayspeak in at most two MAA contributed paper sessions. If yourpaper cannot be accommodated in the session it was sub-mitted, it will be automatically considered for the generalsession. Speakers in the general session will be limited to onetalk because of time constraints. The summary must reachthe organizer by Tuesday, September 14, 2004. Abstract mustreach the AMS by Tuesday, October 5, 2004.

The AMS will publish abstracts for the MAA talks. Abstractsmust be submitted electronically to the AMS. No knowledgeof LaTeX is necessary, however, LaTeX and AMSLaTeX arethe only typesetting systems that can be used if mathematicsis included. The abstracts’ submissions page is at http://www.ams.org/abstracts/instructions.html. Simply fill in eachfield as instructed. Submitters will be able to view their ab-stracts before final submission. All questions concerning thesubmission of abstracts should be addressed to:

abs-coord@ams.org.

Here are the codes you will need: MEETING NUMBER: 1003

The EVENT CODE is the seven characters appearing beforethe title of the sessions shown below, e.g., MAA CP A1

The SUBJECT CODE is the last two-character letter/num-ber combination from the event code list, i.e., A1.

Submission Procedures for MAA

Contributed Papers

Eastern PA/Delaware Doug EnsleyFlorida David KerrIllinois Richard WildersIntermountain David G. WrightIowa James FreemanLouisiana-Mississippi Roger A. WaggonerMaryland/DC/Virginia David CarothersMichigan Ruth G. FavroNorth Central Dan KempSouthern California-Nevada Arthur BenjaminTexas Elizabeth M. Bator

New 2004 MAA Section Governors

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On March 25, 2004,Ronald L. Graham, Presidentof the Mathematical Asso-ciation of America, testifiedbefore the House Subcom-mittee on Veterans Affairs,Housing and Urban Devel-opment, on NSF funding formathematics education.

Thanking the Subcommitteefor its steadfast support ofthe National Science Foun-dation in recent years—dur-ing which it provided in-creases well above theAdministration’s requestsfor scientific education and research—Graham urged continued support for theNSF’s investments that “have been andcontinue to be vitally important for thisNation’s long-term economic and na-tional security.”

Right from the start President Grahamfocused in on the one area of the NSF’sportfolio which, he said, “affects not onlyevery field of science and engineering butalso, the nation’s long-term economicdevelopment, national security, and gen-eral welfare. This area is mathematicseducation, particularly at the under-graduate level.”

He called undergraduate education “thecrucial link in the educational systempreparing mathematicians, scientists,health care professionals, engineers, tech-nological workforce, and the teachers ofthe next generation of students.” TheNational Security Agency, he pointed out,is the largest employer in the world ofmathematicians at all levels of educa-tional background.

The NSF funds undergraduate math-ematics education through two divisions:the Division of Undergraduate Educa-tion (DUE), located in the Education andHuman Resources Directorate, and theDivision of Mathematical Sciences(DMS), located in the Mathematics andPhysical Sciences Directorate. Both are

considered essential to ensure that thenumber of mathematically trained U.S.citizens does not drop below the levelneeded to maintain U.S. technologicalleadership.

DUE and DMS, for instance, collabo-rated in funding the Calculus Reformeffort of the first half of the 1990’s. Notonly did this program change and mod-ernize the teaching of calculus in highschools and colleges, the program waslargely responsible for spurring renewalacross science, technology, engineeringand mathematics undergraduate educa-tion throughout the country. InGraham’s view, he told the Subcommit-tee, no single educational program “hasever had such widespread impact.”

Further, a National Academy of Sciencesreport in 1998 concluded that, “based onpresent trends, it is unlikely that the U.S.will be able to maintain world leadershipin the mathematical sciences.”

In response to this report and others,NSF and Congress have increased fund-ing for mathematical sciences researchsignificantly in the past 5 years, doublingthe research budget from about $100million in FY 1998 to $200 million in FY2005.

“We at MAA,” Graham said, “stronglysupport this increased investment, and

believe that the nationwould be well servedby continuing to in-crease mathematicalsciences researchfunding. However, wealso believe that re-search funding alonewill not solve thenation’s difficulties inattracting and educat-ing students andteachers in the math-ematical sciences.”

The Division of Un-dergraduate Educa-

tion at NSF aims to improve mathemat-ics and science education through thereform of courses, curricula, and instruc-tional materials, and to increase the qual-ity and quantity of the mathematics andscience workforce. DUE programs fundinnovative efforts at colleges and univer-sities in every state and at every level, allin an effort to find the best ways to in-crease the scientific and technological lit-eracy of the nation’s youth.

DUE administers the Advanced Techno-logical Education program, which is aprogram that was created by a memberof the Subcommittee—CongressmanDavid Price. The ATE program enablestwo-year colleges to collaborate with in-dustry in the development of programsand courses for the education of thenation’s technological workforce. DUE isresponsible for programs in teachertraining.

“Not getting the education of teachersright the first time,” said Graham, “resultsin large expenditures for teacher train-ing later.”

While the NSF budget has grown over-all, Graham noted, “undergraduate edu-cation has actually declined.” In FY 2004alone, he said, DUE was cut by nearly $20million from the FY 2003 level. In FY2005, DUE would receive $159 million

MAA President Ronald L. Graham Testifies Before House

Appropriations Subcommittee on Funding for Mathematics EducationBy Harry Waldman

MAA President Ronald L. Graham testifying before the Appropriations Sub-committee.

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under the Administration’s request – anincrease of 2%.

“Report after report has stated that thekey to continued economic vitality is abetter educated workforce, particularlyin mathematics and science,” stressedGraham. Thus, the MAA “urges Congressto provide $200 million for NSF’s DUEprogram in FY 2005. This amount wouldreverse the nearly $20 million reductionthese programs suffered in FY 2004 andput it on a path that will enable thenation’s institutions of higher educa-tion—including our community col-leges—to more fully participate in NSF’sundergraduate programs.”

Another program that Graham urged theSubcommittee to support, which is of-fered by DMS, is VIGRE (Vertical Inte-gration Grants for Research and Educa-tion in the Mathematics Sciences), whichbrings together university professors,

“Not getting the educa-tion of teachers rightthe first time results inlarge expenditures forteacher training later.”

--Ron Graham

graduate students, and undergraduatestudents on research projects and edu-

cation. Universities with VIGRE grantshave reported a dramatic increase in thenumbers of highly qualified U.S. citizenspursuing graduate degrees in the math-ematical sciences. The program was re-cently broadened to support faculty andstudents from all types of institutions to

improve the pipeline from the under-graduate years through positions in theworkforce and academe.

Graham also briefly touched upon theMathematics and Science Partnershipprogram, which is slated to be trans-ferred to the Department of Educationunder the FY 2005 budget request. MSPfunds projects aimed at addressing thelackluster performance of U.S. childrenin K-12 science and mathematics by cre-ating partnerships among teachers andresearchers.

“The MAA believes,” Graham said, “thatif transferred to the Department, MSPfunds will likely be distributed via blockgrants, which could spread the moneytoo thinly to do any real good and whichwill, in all likelihood, result in much ofthe funding being redirected at the statelevel to programs outside the scope ofMSP’s original intent. We at MAA urgeyou to keep MSP at the National ScienceFoundation.”