Celebrating Mathematics Author(S): Lynn Arthur Steen Reviewed Work(S): Source: the American Mathematical Monthly, Vol

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Celebrating Mathematics Author(s): Lynn Arthur Steen Reviewed work(s): Source: The American Mathematical Monthly, Vol. 95, No. 5 (May, 1988), pp. 414-427 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2322476 . Accessed: 14/03/2013 16:55

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This content downloaded on Thu, 14 Mar 2013 16:55:36 PM All use subject to JSTOR Terms and Conditions Celebrating Mathematics

LYNN ARTHUR STEEN, St. Olaf College, Northfield,MN 55057

LYNN STEEN is Past Presidentof the Mathematical Associa- tion of America. This paper is the text of his RetiringPresiden- tial Address presented January 8, 1988 at the annual meeting of the MAA in Atlanta. Steen received his B.A. fromLuther College in 1961 and his Ph.D. fromthe Massachusetts Institute of Technologyin 1965. He has studied at the Mittag-LefflerIn- stitute and with his St. Olaf colleague J. ArthurSeebach, Jr. authored Counterezamplesin Topologyand served as co-editor of MathematicsMagazine. Steen is editor of Mathematics To- day, Mathematics Tomorrow,and Calculus for a New Century: A Pump, Not a Filter. He is Chairman of the ConferenceBoard of the Mathematical Sciences and Chairman-electof the Council of ScientificSociety Presidents.

This is the year we celebrateAmerican mathematics. One hundredyears ago, Thomas Scott Fiske foundedthe New York Math- ematical Society,precursor to the AmericanMathematical Society. Then, mathematicsflourished in Europe but barely existed in the New World. Today, the AmericanMathematical Society sustains the world's strongest environmentfor research mathematics. It has givenscience and society- muchto celebrate. The MathematicalAssociation of Americais nearingits 75thanniversary as an organizationdevoted to the teachingof mathematics, especially at the collegiatelevel. Ever since 1915, the Societyand the Associationhave coop- erated on manyjointly sponsoredactivities. Our missions-mathematical researchand mathematicsteaching-are like braided strandsthat together forma strongfiber for the fabricof Americanscience. On behalfof the MathematicalAssociation of America,I salute our sister societyfor a centuryof accomplishment,for the creationof a rich tapestry of beautifuland usefulmathematics. The AMS centenarybeckons all of us in the mathematicalcommunity to look around to our colleagues,to look outwardto society,and to look ahead to the futureof mathematics. We join the celebration,to applaud the achievementsof Americanmathematics; to proclaimthat strongmathematics contributes to strongscience, to strong defense,and to a strongeconomy; and to challengeAmerican youth with the excitementof mathematicaldiscovery. Of course mathematicsdid not begin with the foundingof the Ameri- can MathematicalSociety. Two hundredyears before that-three hundred yearsago-Newton publishedPrincipia Mathematica,thereby establishing mathematicsas the methodologicalparadigm of theoreticalscience. From thisparadigm have emergedmany mathematical sciences, and manymath- ematicssocieties. The AmericanStatistical Association celebrates its 150th

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This content downloaded on Thu, 14 Mar 2013 16:55:36 PM All use subject to JSTOR Terms and Conditions CELEBRATING MATHEMATICS 415 birthdayin 1989; the Societyfor Industrial and Applied Mathematicscele- bratedits 35thAnniversary last year,at the same timethat the Association forComputing Machinery celebrated its 40th birthday. So "100 Years of AmericanMathematics" is a bit of hyperbole,a slogan that throwsimages beyondliteral meaning. It is in part a strategyto focus attentionon the need forsustained support for mathematics. We celebrate 100 Years of AmericanMathematics not because we equate Thomas Fiske withIsaac Newton,but because it is a timelydevice to set importantissues beforethe mathematicalworld, the scientificcommunity, and the attentive public. Despite a centuryof success, there are urgentmatters that need attention.

Mathematics Today Those of us who are part of the mathematicalcommunity recognize the enormouscurrent vitality of the mathematical sciences. Applications, enthu- siasm, initiatives, opportunities, and unity are the "vowels" around which mathematicallanguage is formed,a language in whichmathematicians express solutionsto old problemsand explorenew areas of fruitfulgrowth. Explosivegrowth is a signof remarkablehealth, but it does leave the thou- sands of us who tryto keep up pantingbreathlessly as the researchleaders disappearover the horizon. Dozens of areas of activeresearch could be cited to documentthe vitality of contemporarymathematics (see [17], [18], [24]). For some of these,look at the programof this meeting.Better still,look at the wholeprogram for 100 Years ofAmerican Mathematics, especially at themarvelous symposium that is part of the 1988 AAAS meetingin Boston and at the special series of expositorylectures on contemporarymathematics that is featuredat the AMS CentennialMeeting in Providence. Today I want to highlightfour areas as examplesof the unityand appli- cabilityof mathematicalresearch: computationalstatistics, mathematical biology,geometrical mathematics, and nonlineardynamics. Each of these offersopportunities for initiatives in mathematicalresearch and forengen- deringenthusiasm among students. Think especially about thelatter, about opportunitiesfor initiatives in mathematicseducation, while I brieflyoutline each of thesefour areas. Computational Statistics. The statisticalsciences study problems as- sociated with uncertaintyin the collection,analysis, and interpretationof data. Not surprisingly,the increasinguse of computersto recordand trans- formdata has generateda host ofnew challengesfor the statisticalsciences. For example,analysis of data fromelectronic scanning devices (in tomog- raphy,in aircraftor satellitereconnaissance, in environmentalmonitoring) has produced an urgentneed for statisticalanalysis of data that has an

This content downloaded on Thu, 14 Mar 2013 16:55:36 PM All use subject to JSTOR Terms and Conditions 416 L. STEEN [May inherentspatial structure[26]. Research in this emergingfield of spatial statisticsemploys a wide varietyof mathematical,statistical, and compu- tationaltechniques: problems of separatingsignals from noise borrowideas fromengineering; ill-posed scattering problems employ methods of numerical linear algebra; and smoothingof data requiresstatistical techniques of regularization.Underlying all thisis the inherentgeometry of the problem, whichin many cases is dynamicand non-linear. Many applicationsof statistics(for example, to clinicaldata frominnova- tivemedical protocols) involve small data sets fromwhich one wouldlike to infermeaningful patterns. Bradley Efronand othershave pioneeredinno- vative,computationally-intensive statistical techniques that use the limited available data to generatemore data with the same statisticalcharacteris- tics ([4], [5], [14]). By resamplingthe givendata repeatedly,these so-called bootstrapmethods generate millions of similar possible data sets whichyield accurate approximationsto various complexstatistics. By comparingthe value of statisticsfor the given sample with the distributionobtained by these resamplingschemes, one can determinewhether the observedvalues are significant. Mathematical Biology. Nothingbetter illustratesthe potential for mathematicsin the biologicalsciences than the manytraces of mathematics behindthe Nobel prizes. For example,the 1979 Nobel Prize in medicinewas awardedto Allan Cormackfor his applicationof the Radon transformto the developmentof tomographyand CAT scanners. The 1984 Nobel Prize in chemistrywas awardedto biophysicistHerbert Hauptman, President of the Medical Foundation of Buffalo,for fundamentalwork in Fourieranalysis pertainingto X-raycrystallography [19]. Indeed, recentresearch in the mathematicalsciences suggests dramati- callyincreased potential for fundamental advances in the lifesciences using methodsthat dependheavily on mathematicaland computermodels. Struc- tural biologistshave become geneticengineers, capturing the geometryof complexmacromolecules in supercomputersand thensimulating interaction with other moleculesin their search forbiologically active agents. Using these computationalmethods, biologists can portrayon a computerscreen the geometryof a cold virus-an intricatepolyhedral shape of uncommnon beautyand fascinatinggeometric features-and searchits surfacefor molec- ular footholdson whichto securetheir biological assault. Geneticistsare beginningthe monumentaleffort to map the entirehu- man genome,an enterpriserequiring expertise in statistics,combinatorics, artificialintelligence, and data managementto organizebillions of bits of information.Ecologists-the firstmathematical biologists-continue to use theextensive theories of population dynamics to predictthe behavior and in- teractionof species ([9], [32]). Neurologistsnow use the theoryof graphsto modelnetworks of nerves in thebody and theneural tangle in thebrain [10].

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Cell biologistsstudy the replication of DNA usingthe newly discovered alge- braic classificationof knots ([12], [13]). Epidemiologistsmonitor the spread of AIDS withtechniques that blend innovativestatistics with classical analysis. And, finally,physiologists employ contemporary algorithms applied to nineteenth-centuryequations of fluiddynamics to determinesuch thingsas the effectsof turbulencein the blood caused by cholesterolor swollenheart valves [11]. Geometrical Mathematics. Ever sinceEuclid, geometryhas been one of the major pillars of core mathematics. Afterdecades of decline (especially in mathematicsteaching), the geometricalview in mathematicshas undergonea renaissance,assisted both by the developmentof new theoret- ical tools and by the power of computer-basedvisual representation.In a veryreal sense, geometryis once again playinga centralrole on the stage of mathematics,much as it did in the Greekperiod. Geometryclaimed two of the three 1986 Fields Medals, which were awarded to Michael Freedmanand Simon Donaldson for workin the geometryof fourdimensional manifolds ([1], [3], [21]). By exploitingproper- ties ofthe Yang-Millsfield equations that reflectthe wave-particleduality of matter,Donaldson showedthat the differentialgeometry of four dimensional manifoldswas vastlydifferent than that suggestedby theirtopological structure. Freedmanprovided the topologicalclassification. Together, their work yieldednot only deep understandingof four-dimensionalmanifolds, but the surprisinginsight that in fourdimensions there are differentiablemanifolds that are topologicallybut not differentiablyequivalent to the standardEu- cidean fourdimensional space. Already,insight from this workhas led to applicationsin stringtheory-the new super-symmetrictheory of elemen- taryparticles-thereby providing fresh evidence (see [34]) of what Eugene Wignercalled the "unreasonableeffectiveness" of mathematicsin the physical sciences. Computergraphics provides a powerfulnew tool that extends geometrical techniquesinto many parts of mathematics. Computers-especially supercomputers-cancalculate and displayvarious mathematical structures in visual form,thereby enabling mathematicians to "see" the significanceof abstractpatterns that beforecould be interpretedonly by formalmeans. For example,visual representationsof solutionsof differentialequations of- tenproduce conjectures that open up wholenew insightsinto the behavior of the systemwhich the equations represent.Geometrical studies themselves regularlyyield innovativechallenges in the design of new algorithmsand data structures,with spin-offbenefits to applicationsin computerscience (forexample, database systemsand wordprocessing) far removed from the originalgeometric problem. The newlylaunched Geometry Supercomputing Project at the Universityof Minnesotais one exampleof the growinginter- actionof researchers in geometrywith those in theoreticalcomputer science.

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Nonlinear Dynamics. Only in recentyears have we been able to provide mathematicalanalyses of problemsthat are essentiallynonlinear (for example,turbulence in fluids). These have been made possibleby novelana- lyticalmethods, clever numerical simulation, and visual displayon computer screens. Applicationsrange fromairfoil design to plasma physics,from oil recoveryto studiesof combustion([7], [15], [16], [22]). Nonlineardynamics has yieldedmany surprises (see [6], [8], [33]), includ- ing long-termlocalized structures(e.g., the Red Spot on Jupiter),determin- istic (ratherthan stochastically)generated chaotic motion (typical of some weatherphenomena), and fractalpatterns at the interfacebetween fluids (forexample, displacement of oil by water). The mathematicsof nonlinear dynamicsinvolves a greatdeal of traditionalanalysis (especially differential equations), reinforcedby iterativeprocesses, automata theory,and fractal geometry. Computerdisplay of nonlinearphenomena makes visible patternsthat would never have been noticed by analyticmeans alone. In researchon dynamicalsystems, on the transitionfrom order to chaos, and on the emer- gence of fractalshapes fromsmooth flows, computers are to mathematics whattelescopes and microscopesare to science: theyincrease by a thousand- foldthe portfolioof patternsthat mathematicianscan see and investigate. The Newtonianrevolution not only establishedmathematics as a para- digmfor scientific reasoning, but it also establisheddeterminism as a para- digmfor the behavior of physical systems. Nonlineardynamics-a direct descendantof Newtonianmathematics-shows how ambiguityand uncer- taintycan arise in even simple deterministicsystems, and how the onset of chaos itselfcan be predictable. In its power to change our Newtonian view of mathematics,nonlinear dynamics is as revolutionaryas quantum mechanics: each breaks the bond of determinismand reveals entirelynew structuresthat oftendefy what we have come to thinkof as commonsense.

Mathematics in the Classroom I choose these examples,drawn from diverse areas of the mathematical sciences,not just to illustratethe vitalityof the mathematicsthat we are here to celebrate,but to providea mathematicaland intellectualbackdrop forwhat is, unfortunately,a verydifferent portrait of mathematicsin the classroom. In researchwe see a lot of geometry,a lot of data, a lot of science,a lot of computation-togetherwith more traditionalmathematical tools. We see investigation,exploration, and a continualsearch for pattern. Contemporary mathematicscompels attention.It has the powerto excite the best minds of our youthand to stimulaterenewed creativity in teachingmathematics. But this mathematicsis not the mathematicstaught in typicalschool or collegeclassrooms. Far too often,mathematics in the classroomis a freeze-

This content downloaded on Thu, 14 Mar 2013 16:55:36 PM All use subject to JSTOR Terms and Conditions 1988] CELEBRATING MATHEMATICS 419 dried mathematics-rigid,cold, and unappealing. Instead of exploration thereis drill; instead of investigation,imitation. From elementaryschool arithmeticto collegecalculus, mathematics in the classroomis dramatically differentfrom mathematics in practice. You've all heardthe litany of problems with school mathematics. It ranges frompoor test performanceon internationalassessments to declininginter- est amongAmericans in pursuingadvanced studyof mathematics.I won't repeat this evidence here since it has been widely publicized in reports, journals,and newsletters([20], [30], [31]). Not yet so well knownare the currentattempts by severalorganizations (for example, NCTM, MSEB, AAAS, and the Universityof Chicago) to reversethis decline ([23], [27], [28]). These projectshave engaged school teachers,mathematics educators, and mathematicsresearchers in collabo- rativework on the problemsof mathematicseducation in the schools. Al- thoughthese projects differgreatly in purpose and detail, theiremerging recommendationshave muchin commnonthat resonateswith the natureand practiceof contemporarymathematics: * Mathematicsshould be taughtin a natural context; * Studentsshould be encouragedto create,to invent,and to participate; * Calculatorsand computersshould be used throughoutthe mathematics curriculum; * New topics (forexample, algorithms, data analysis,estimation) should be introducedinto the mainstreamcurriculum; * Facilityin computationneed not be a prerequisiteto the studyof mathematics; * Mathematicsshould be studiedas an integratedwhole; * Mathematicsshould help build students'abilities to reason logically; * Communicationis an importantgoal of mathematicsinstruction. It doesn't take muchimagination for someone who is familiarwith examples of contemporarymathematical science to see how studentinvolvement in such mathematicscould contributeto achievingthese goals. Justthe ex- amplesI have cited-statistics,biology, geometry, dynamics-overflow with naturalcontext and withopportunities for students to use computersto dis- coverpatterns. These examplesreveal far betterthan the isolated morsels of the traditionalcurriculum that new mathematicalmethods are needed to solvenew problems;that communicationis importantfor one to just under- stand, let alone express,the subtletiesrevealed by mathematicalanalysis; and that mathematicsin action requiresnot onlycalculation and logic,but also intuition,imagination, and organization. Unfortunately,too fewof those who are mostknowledgeable about mathematicalresearch are workingwith teachers to translatetheir research into experiencessuitable forclassroom exploration. And far too fewteachers- even at the collegeand universitylevel--have the background,the interest,

This content downloaded on Thu, 14 Mar 2013 16:55:36 PM All use subject to JSTOR Terms and Conditions 420 L. STEEN [May or the time to learn enoughabout modernmathematical research to trans- late it successfullyinto classroom experiences. So longas a largegap remains betweenthose who create mathematicsand those who teach mathematics, we cannotexpect studentsto see in mathematicsthe challengeof an exciting and intellectuallyrewarding career. Amongthe anniversarieswe celebratethis winteris the centenaryof the birth of George Polya, who died just three years ago. Polya was one of the raremathematicians who made major contributionsboth to mathematics researchand mathematicseducation. AndreiKolmogorov, who died in October,was another. The December 1987 issue of MathematicsMagazine is devotedto Polya's life and work;I urge you to read it-I'm sure you'll findit as fascinating as I did. In that issue, Alan Schoenfeldwrote an interestinganalysis of Polya's theoryof heuristicsand its impact on teachingstudents to solve mathematicalproblems. Schoenfeld begins with Polya's dictumthat a good mathematicseducation is one that provides systematicopportunities for studentsto discoverthings. How oftendoes our teachingreally do that? Think about the contrastof the stylizedtwo-column proofs of high school geometry with the exploratory possibilitiesof three-dimensionalcomputer graphics, or of the linkingof knots,or of topologicaltransformations of commonsurfaces. Or think,as manydid at the NRC colloquiumon Calculus fora New Century,about the contrastbetween the fivethousand exercises in typicalcalculus books that mostlyask studentsto imitate calculatorbuttons, and the discoverypo- tentialin symboliccomputer systems or in visual presentationof nonlinear dynamics. Polya's discoverydictum was echoed (perhapsunconsciously) at the cal- culuscolloquium by OberlinCollege President Frederick Starr [29]. He cited research[2] on college studentcareer choices that shows "incontrovertibly" that the only institutionsthat are successfullyresisting the precipitousde- clinein the percentageof studentsentering careers in scienceare those that base theirpedagogy on a kind of apprenticeshipsystem. In these schools studentsare broughtinto the laboratoryto pursue real science under the directguidance of professorswho are th-emselvesactively engaged in the scientificquest. Traditionally,it has been the laboratorysciences-notably chemistry- that excel at attractingstudents by a styleof educationthat involvesstu- dentsin the discoveryof science. But now mathematicscan do the same. Withfrontiers as excitingas chaoticsystems and spatial statistics,there is no longerany reason for mathematics to fallbehind the more glamorous laboratorysciences in attractingthe interest and enthusiasmof our brightest youth.

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Causes for Celebration In celebratingmathematics, we point to the immensesuccess of mathematical researchin creatingan intellectualunderstanding of space and number,of orderarid chaos, of patternand disarray.Mathematics itself is beautiful,powerful, and deep; theprocess of doing mathematics is personally stimulatingand intellectuallyrewarding. Nevertheless,the professionof mathematics-as distinctfrom the discipline of mathematics-is not in good health. Decades of neglectin main- tainingclear communicationchannels-with education,with science,with the public-have leftmathematics isolated fromthe supportsystems that are vital to its health and well-being. Our celebrationmust become a commitmentto communicate. As we move into the second centuryof Americanmathematics-to continuethe hyperbole-we should build on the impressiveaccomplishments in mathe- maticsitself to bringthe excitementand powerof the mathematicalsciences to all Americans. Here are fivecauses to championas we celebratemathe- maticsin 1988.

. INVOLVE STUDENTS IN THE PRACTICE OF MATHEMATICS The evidenceis overwhelmingthat studentsreceive a better education, and are more likelyto be attractedto mathematics,when they are actively involvedin mathematicalexperiences [25]. Polya called it discoverylearn- ing; Starr describedit as apprenticeshipeducation. Althoughonly a few mathematicsstudents in the United States now receivethe benefitof this type of learning,there are many excellentmodels of such teaching: problems competitions,research experiences for students, exploratory computer graphics,innovative tutorials where students become teachers,and team- based internshipsin mathematicalmodelling. It is noteworthythat the Na- tional Science Foundation,under the researchdirectorates, has once again begunto supportprograms that provide research experiences to undergradu- ates. Especiallyfor undergraduate students, but also in appropriatedegrees for youngerstudents, we must tilt the balance of mathematicseducation towardsgreater student involvement in learning. Doing this will be expensive,and mightinvolve radical departuresin the way we financeundergraduate and graduateeducation. Since teacherstend to teach as they were taught,the most effectiveway to promotediscovery learningin the schoolsis to enhanceapprenticeship learning in the colleges, wheretomorrow's teachers are today learninghow to teach by the examples set by their college professors.The time is ripe fordepartment chairs to insistthat universitiesfund teaching at a sufficientlevel that all undergrad- uates can be taught by experiencedteachers who will involvestudents in the excitementof discoveringmathematics. Apprenticeshipeducation will require significantly better integration both

This content downloaded on Thu, 14 Mar 2013 16:55:36 PM All use subject to JSTOR Terms and Conditions 422 L. STEEN [May ofresearch mathematics and of contemporaryapplications in the experience of students. Currentresearch, new applications,and the emerginggoals of mathematicseducation could resonate in ways that would greatlyen- hance studentinvolvement in mathematicallearning. But such resonance cannot happen so long as researchersand teacherscontinue to operate in separate spheres-worlds apart in mathematicaloutlook, experiences, and expectations. Resonance requiressignificant connection between stimulus and resonator."Vertical integration" of mathematical knowledge that links researchand applicationswith education,and that bringsresearchers into active contactwith students and teachers,should become a major criterion in fundingdecisions that concernthe supportof research. We should celebratethe wealthof interestingnew mathematicsby bring- ing thismathematics into everyclassroom in the nation. To do that willre- quirechanges in the waywe judge teachingand in the waywe judge research: each shouldbe foundwanting if it does not includeappropriate linkage with the other. It is not enoughthat individualsbe competentas teachersand separatelyas professionals:separate but equal is as inadequate as a model forthe relationof teaching and professionalactivity as it is forracial compo- sitionof public schools. Our goal shouldbe professionalstandards that insist on apprenticeshiplearning and verticalintegration of mathematical research.

. EDUCATE THE ATTENTIVE PUBLIC Far too manyeducated personsare ignorantof mathematics.More trou- blesome,most do not feeltheir mathematical ignorance to be a greathand- icap. Most successfullawyers, politicians, educators, business executives- and universityadministrators-have achieved positions of prominencewith only a minimal(and frequentlyarchaic) knowledgeof mathematics.More- over,many persons, whether well educated or not,harbor feelings of appre- hensionor even anxietyabout mathematicsdue to an unpleasantearly ed- ucationalexperience, often with something labelled the "newmath." When we tryto take the case formathematics to thepublic-or evenjust to universityadministrators-we face not onlythe healthyskepticism that naturally greetsany self-serving argument, but also ignorance,fear, and oftenhostility that is a legacy of our neglectof mathematicseducation. The state of mathematicsas a professioncompels us to findways to di- minishlpublicfear and ignoranceof mathematics,for without broad public support-for teaching,for research, for encouragement of students-there is no possible way forthe mathematicalcommunity on its own to sustain the momentumof the past half-century.Now, however, perhaps for the first time,the breadthof the mathematicallandscape makes it possible at least to imagineovercoming this pervasivepublic apprehensionof mathematics. Mathematicsnow touches people's lives in ways that matter and that can be describedand revealedin human terms.From symmetryand chaos

This content downloaded on Thu, 14 Mar 2013 16:55:36 PM All use subject to JSTOR Terms and Conditions 1988] CELEBRATING MATHEMATICS 423 to computersand cosmology,from AIDS epidemiologyand nuclearrisks to political polls and ozone depletion,mathematics lurks behind most mani- festationsof science and technology.The same mathematizationof society that makes the task of public understandingso essentialalso providesthe meansby whichthe task can be started-by buildingon mathematicalideas that are part of daily experience. Surelypart of our celebrationmust be to tell the storyof mathematicsto the attentivepublic. We are not publicists,but we are teachers. Suppose each departmentof mathematicsmade a commitment,just once each year, to arrangea publicevent that made mathematicsvisible in theircommunity: an outside speakerwho is workingon somethingin whichthe public might be interested;a studentproject that involveda practicalproblem of interest to the community;a forumon the changingnature of school mathematics; or an expositionof a slice of mathematicsrelated to some professor'sresearch. Since the public is always more interestedin people than in abstractions, thereare, in addition,many good opportunitiesfor news storiesin home- townpapers about the accomplishmentsof students.Someday some math- ematicsdepartment should try to put out as manypublicity releases on the accomplishmentsof theirstudents as the athleticdepartment does of theirs.

EXPLORE FUNDAMENTAL ISSUES IN MATHEMATICS EDUCATION Computersinfluence mathematicians not onlyby providingnew tools for researchand teaching,but also by posing deep questionsabout centralis- sues in our discipline. Now that calculatorscan manipulatesymbols and calculate answers, * What-if not arithmetic-shouldbe the core of elementaryschool mathematics? * What-if not manipulation-shouldbe the core of high school algebra? * What-if not calculation-should be the core of calculus? * What-if not calculus-should be the core of collegemathematics? At the same time that computersforce attention on issues that are deeply rootedin unexaminedtradition, mathematical research has transformedthe natureof mathematics,opening up new optionsfor what mightbe consid- ered centraland what derivativeamong the conceptsof mathematics. We need to findnew threadsof continuitywith which to weave a mathematics curriculumfor the twenty-firstcentury. Finding appropriate central themesposes an immensechallenge for the best mindsamong us, researchers and teachersalike. It gives commonpurpose to our diverseexpertise, and sets a commonagenda forthose in research,those in collegeteaching, and thosein schoolmathematics. This providesyet another occasion for celebration: the opportunity, joined withthe need, to transformschool mathematics in waysthat reflect the rich- ness and diversityof mathematical research and mathematicalapplications.

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Verticalintegration of research,applications, and teachingwill help bring about this transformation.But we need, in addition,structured opportunities forreflection in whichthe most synopticthinkers among us bringtheir experiencesin research,in applications,and in teachingto bear on the task of articulatingcentral themes for mathematics education as we move into the next century.

. ENSURE FOR ALL STUDENTS EQUAL OPPORTUNITIES FOR MATHEMATICAL SUCCESS Despite the culturallyneutral status of mathematics (as compared,say, to biology),the last decade has produceddistressing evidence of class and eth- nic distinctionsarising as a resultof the waymathematics education is prac- ticed in the United States. One-thirdof U.S. students-Blacks, Hispanics, and Native Americans-providesfewer than 10% ofmathematics graduates, despitethe evidencefrom isolated model programsthat excellentretention and successrates can be achievedwithin a suitableeducational context. An- otherthird of U.S. students-whitefemales-drops out of advanced degree programsin the mathematicalsciences at twicethe rate of male students. Compoundingthese problemsof class distinctionsare the political, ed- ucational, and social side effectsof large numbersof foreign-bornteaching assistantsin our major universities.It is easy to make a strongcase for havingmany foreigngraduate studentsin our universities;I join the many scientificleaders who defendthis practice which has led to the UnitedStates being,in the wordsof Robert White,the "schoolhouseof the world." It is harderto make a case forplacing inexperiencedforeign graduate students in the classroomas instructorsfor American students who are not prepared to cope simultaneouslywith the challengeof a foreignculture, a foreign language,and a foreigndiscipline-namely, mathematics. The consequenceof these two unrelatedtrends is that both majorityand minoritystudents in collegeclasses oftenreceive mathematics instruction in a contextthat is culturallyalien to them. Studentswho have too littlein commonwith theirteachers are unable to see themselvesas futuremathe- maticiansor mathematicsteachers. In thiscontext, it is not surprisingthat even whiteU.S. males are no longerchoosing careers in mathematicalsciences. As I am sure you are well aware,the numberof U.S. males receiving Ph.D. degreeseach year in mathematicsis less than 40% of what it was fifteenyears ago. The problemof opportunityin mathematicsis so seriousand so difficult that it is hard to even imagine a solutionthat is feasible,much less opti- mal. However,the mathematicalcommunity has an enormousresource that can be broughtto bear on this problem-namely,the strongand cultur- ally diversecommunity of researchmathematicians that provesby its very existencethe universalityof mathematics. We need to findeffective ways

This content downloaded on Thu, 14 Mar 2013 16:55:36 PM All use subject to JSTOR Terms and Conditions 1988] CELEBRATING MATHEMATICS 425 of conveyingthe rich and worldwidenature of mathematicsto youthfrom the many subculturesthat contributeto the Americanmosaic, and then to providea contextappropriate to theirbackgrounds in whichto nurture mathematicaltalent. The ProfessionalDevelopment Program at at Berkeley,led by Uri Treis- man and Leon Henkin,has transformedthe success rates of minorityfresh- men and enabledmany to finishBerkeley and proceedto advancedor profes- sional degrees. The Universityof Michiganhas had considerablesuccess in interestingminority students in careersin sciencethrough a special program that providesexperiences in undergraduateresearch. These examples-and I'm sure there are others-show that it is possible to successfullyattract talentedminority students to careersin mathematicsand science. Making progressin this endeavorwould provide just cause fora true celebration.

INVEST IN TODAY'S EDUCATION TO STRENGTHEN TOMORROW'S RESEARCH Currentdebate about supportfor mathematics too oftenpits research against education,when in realitytoday's educationis the pipelinefor tomorrow'sresearch. We read in the Noticesof mathematicians who are under pressureto get researchgrants on pain of "beingfired, having their teaching loads raised, or not gettingraises." We hear continuingconcern in the re- searchcommunity that, in timesof limited budgets, new fundsfor education mightbe subtractedfrom the alreadylimited amounts available for support ofbasic research-despitethe factthat the percentageof federal support for scienceand mathematicsthat goes to educationhas slippedduring the past fourdecades fromnearly 50% to around 10%. These argumentsfollow the same patternthat has led to the enormous growthin the federaldeficit: by givingpriority to the immediateneeds of those in positionsof power,we in effectsupport adults at the expense of children.Mathematicians, of all people, should be able to plan strategies that will optimizethe strengthof mathematicaland scientificresearch over the long term. Part of that strategyis the recognitionthat educationis not an alternativeto research,but the foundationfor future research. Our celebrationof the bounty of mathematicalresearch must entail a commitmentto educationas the wellspringof research. We need to use mul- tidimensionalcriteria in decidingon prioritiesfor our community,seeking strategiesthat lead simultaneouslyto improvementin schoolmathematics, in collegiatemathematics, in graduateeducation, and in research.

All One System Despite appearances to the contrary,mathematical research is inextri- cably entwinedwith mathematicseducation at all levels, with science and engineering,and withpolitical, econornic, and sociologicalaspects of society

This content downloaded on Thu, 14 Mar 2013 16:55:36 PM All use subject to JSTOR Terms and Conditions 426 L. STEEN [May at large. Educators and researchers,teachers and professors,mathematicians and scientists-we are all part of a single systemof knowledgeon whichcontemporary society depends. It is in this spiritthat we join in celebratingthe centenaryof the Ameri- can MathematicalSociety. "100 Years of AmericanMathematics" provides the occasion forbuilding new mathematicalscience on the firmfoundation ofthe past century'sresearch. Our centenarycauses shouldbe as broad and sweepingas our discipline: . To involvestudents in the practiceof mathematics. . To educate the attentivepublic. . To explorefundamental issues in mathematicseducation. . To ensurefor all studentsequal opportunityfor mathematical success. . To investin today's educationto strengthentomorrow's research. What transformsthese causes fromempty rhetoric to concreteoptions is the opportunityfor education and communicationimplicit in the advances oftoday's mathematical sciences-in such areas as computationalstatistics, mathematicalbiology, geometrical mathematics, and nonlineardynamics. It is in the frontiersof mathematicalscience-not in currenttextbooks or today's classrooms-that one can findthe innovativeand intellectuallyre- wardingoptions needed to transformeducation, to excite our youth, to educate the public, and to reach all Americans. This should be our centennialcause, not just for1988 but forthe rest of thiscentury. It is a cause that can uniteresearchers and educatorsin a com- mon challenge:To let the powerand beautyof mathematics speak foritself.

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This content downloaded on Thu, 14 Mar 2013 16:55:36 PM All use subject to JSTOR Terms and Conditions 1988] CELEBRATING MATHEMATICS 427

11. , Editor,Mathematical Aspects of Phyjsiology, Lectures in AppliedMathematics, V. 19, AmericanMathematical Society, 1981. 12. VaughanF.R. Jones,A NewKnot Polynomial and vonNeumann Algebra, Noticea of theAmer. Math. Soc., 33 (March1986) 219-225. 13. Gina Kolata, SolvingKnotty Problems in Mathand Biology,Science, 231 (28 March 1986) 1506-1508. 14. , The Artof Learning from Experience, Science, 225 (13 July1984) 156-158. 15. MortLa Brecque,Fractal Symmetry, Mosaic, 16:1 (Spring1985). 16. , FractalApplications, Mosaic, 17:4 (Winter1986/7); 18:1 (Spring1987). 17. MathematicalSciencea: A Unifyingand DynamicResource, National Academy of Sci- ences, 1986. 18. MathematicalSciences: Some ResearchTrenda, National Academy of Sciences, 1988. 19. Mathematics:The UnifyingThread in Science,Notices of the Amer. Math. Soc., 33 (1986) 716-733. 20. CurtisC. McKnight,et al., The UnderachievingCurriculum: Assessing U.S. School Mathematicafrom an InternationalPerapective, Stipes Publishing Company, 1987. 21. JohnMilnor, The Work of Michael Freedman,Proceedinga of the Interantional CongreJJof Mathematicians,1986, American Mathematical Society, 1988, pp. 13-15. 22. IvarsPeterson, Packing It In: FractalsPlay an ImportantRole in ImageCompression, ScienceNewa, 131 (2 May 1987) 283-285. 23. AnthonyRalston, et al., A Frameworkfor Reviasion of the K-12 MathematicaCurricu- lum,Task ForceReport submitted to the MathematicalSciences Education Board, NationalResearch Council, January 1988. 24. RenewingU.S. Mathematica:Critical Resource for theFuture, National Academy of Sciences,1984. 25. LaurenB. Resnick,Education and Learningto Think,National Academy Press, 1987. 26. WernerC. Rheinboldt,Future Directiona in ComputationalMathematica, Algorithma, and ScientificSoftware, Society for Industrial and AppliedMathematics, 1985. 27. ThomasRomberg, et al., Curriculumand EvaluationStandards for School Mathemat- ica, WorkingDraft, National Council of Teachers of Mathematics, October 1987. 28. JamesRutherford, et al., WhatScience ia Most WorthKnowing? Draft Report of Phase I, Project2061; American Association for the Advancement of Science, Decem- ber 1987. 29. LynnArthur Steen, Editor, Calculuafor a New Century:A Pump, Not a Filter, MathematicalAssociation of America, 1988. 30. , MathematicsEducation: A Predictorof ScientificCompetitiveness, Science, 237 (17 July 1987) 251-252, 302. 31. HaroldW. Stevenson,et al., MathematicsAchievement of Chinese,Japanese, and AmericanChildren, Science, 231 (14 February1986) 693-699. 32. E. Teramotoand M. Yamaguti,Mathematical Topics in PopulationBiology, Mor- phogenesia,and Neuroaciences,Lecture Notes in Biomathematics,Vol. 71, Springer- Verlag,1987. 33. M. MitchellWaldrop, Computers Amplify Black Monday,Science, 238 (30 October 1987) 602-604. 34. EdwardWitten, Physics and Geometry,Proceedinga of theInternational Congreaa of Mathematiciana,1986, American Mathematical Society, 1988, pp. 267-303.

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