Mathematics in 2001: Implications for today's undergraduate teaching

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MATHEMATICS lN 2001: IMPLICATIONS FOR TODAY'SUN DERGRADUATE TEACH ING*

Peter Hilton, Department of Mathematics, University of Binghamton, U.S.A.Dedication: "To my friend Ralph Nero, wishinghim a well-earned and happy retirement."l.lntroductionAlthough this conference is principally concerned withthe vital issue of the remedial teaching of mathematics,my own contribution will not center on this problem. Myconcernis that we designtoday's mathematics curriculumto take account ofthe changes which are takingplace, andwill continue to take place, in mathematics itself and theuses to which it is put in our society. I am further con-cerned that teaching styles and strategies be well adaptedto the cwriculum and to the fundamental purposes ofmathematics education.

Thus the plan of mytalk is as follows.I will first specu-late on what mathematics will look like in the year 2001.Such speculation could, of course, occupy the entireamount of time allocated to me for this lecture, but I willdiscipline myself and make this part fairly brief, confiningmyself to those aspects of mathematics and its use whichare relevant to my own principal theme. I will then drawinferences about how we should be thinking today aboutthe teaching of mathematics. In deriving these inferences,and in offering examples; I will not feel obliged to confinemyself to mathematical topics traditionally regarded asappropriate to the undergraduate level. My reasons forallowing myself this latitude are two-fold. In the firstplace, I am hoping to reach awide audience ofconcernedcitizens, and I would not wish to exclude those whose owneducational background does not include a concentrationon mathematics at the college level. Second if the talks atthis conference demonstrate anything (and they certainlydo), they show that it is no longer realistic to suppose thatcollege mathematics education can be predicated on theassumption that the students are adequately prepared inpre-calculus topics (algebra, geometry, trigonometry,elementary probability and statistics, elementary tech-niques of estimation and approximation, with applica-tions). I rather suspect that this particular situation mayget worse rather than better in the immediate future.Let me say a word about my choice of the year 2001

for my speculations. Of course, I am not the first to regard2001 as a significant date - Stanley Kubrick has pub-lished ahead of me! Nor am I unduly obsessed by the factthat 2001 marks the beginning of the third millennium (al-though the millenial rites will doubtless have been per-formed a year earlier). My main reason for choosing thisyear is that it seems to me reasonable to try to look 20years ahead. To look further ahead would be foolhardy ata time when mathematics, and society, are changing sorapidly; but to narrow one's sights would leave one opento a very real risk in trying to make predictions of any sub-stance. For it is reasonable, in looking ?n years ahead, toignore local fluctuations, (here we refer to a fluctuationwhich is'local'with respect to the time coordinate, ratherthan with respect to space coordinates), which may sim-ply be regarded as due to statistical variation. Thus whenwe try to assess movements in society and education it isperhaps easier to look ahead 20 years than, say, threeyears. Of course, it may be wrong to take the position thatwhat we are experiencing now in education is a local fluc-tuation rather than a trend; if, in fact, it is a trend then theprognosis is gloomy indeed. But I am discussing mathe-matics and matlematics education in this paper, ratherthan education in general, so I will not allow myself to getside-tracked onto that much bigger, and perhaps moreominous, theme.2. The future o{ mathematicsWhat then are the predictions about mathematics in 2001most relevant to our concerns today? First, with regard tomathematics itself, perhaps the most significant trend istowards unityin the science of mathematics. The wonder-ful flowering of mathematics in the last 30 years - a ve-ritable golden era - was characterized by the develop-ment and elaboration of several largely autonomousmathematical disciplines, and, in some cases, even theemergence of quite new ones. Characteristic of this peri-od has been the building of powerful mathematical struc-tures which have enabled mathematicians to conducldeep and penetrating analyses of existing problems andto formulate new ones. The subject to which I have my-

Editor's Note: Although this wo*ing document was prepared as long ago as 1981 by the author for a Conference onRemedial and Developmental Mathematics in Colleges hetd in the Roosetelt Hotel, New York in Apit of that year,many interesting and relevant points were raised for the tefiisry teoching of mathentatics in South Afica.

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self mads a modest contribution, algebraic topolog5r,typifies this kind of mathematical progress. fhe subjeitbarely existed before the Second World War, but, in thelater years, it became an eiceedingly active independentbranch of mathematics, employing its or"n methods, itsownlanguage, its own terminology; and tremendous pro-gress was made. In the course of this progress offshootsemerged - homological algebr4 categorical algebra _which,-though related in obvious ways to existingspecial-ties of mathematics, duly became independent them-selves and were served bytheir own expeits.- Og" may say, speaking roughly but not too roughly,that the recent era has been one of vast progress in thetechnologyof mathematics - the develop-ent of power-firl new techniques. We are now seeing a distinct trend to.y*qr the synthesis of these individual disciplines.Mathematics is regaining its unity, but it is in many re-spects a nsv,/ rrni[y. Methods developed in one area ofmathematics are provingvaluable in another; algebra andtopologl are again being seen as playing a vital role astools in the study of the fundamental problems of mathe-mirtical analysis - and eve nin so,applied' a field as math-ematica! statistics. A young mathematician requires todaya very broad base in order to be properly prepared for aneffective career involving mathematical reseirch.

_ -There is one very important feature of this new unitywhich must be stressed before I leave this aspect of mytheme today. I refer to the new unity of ,p ure, aid ,appliei'mathematics. For certain historical .eisorrs, which manyhave analysed, there has emerged in this country a strongdistinction, amounting among many mathematicians to iveritable antagonism, between pure and applied mathe-matjcs (and-between pure and applied mathematicians!)At first, in the period u4der discussion, the pure mathe-maticians were in the ascendancy, but, in the latter yearsofthe period, the tables have been turned. There are nowmany_co_lleges and universities in this country where it isverydifficult for a pure mathematician to obtain a posi_tion in the mathematics department, and much easier foran applied mathematician, almost as if applied mathe-maticians were members of a minority group whose inter_ests need protection by affirmative action. Of coursethere are often absolutely valid professional reasons forpreferring the applied mathemaiician; but there are alsoexamples where the choice of an applied mathematicianhas about it something of the flavor of a public relationsexercise.Happily we can look, ih the near future, to the disap-pearance of this rigid distinction, so that we -ry uguinspeak simply of mathematicians, working sometimei onproblems coming from the real world, sometimes onproblems from within mathematics itself. So many partsof mathematics, hitherto regarded as the preserve of puremathematics, are proving of great value in the study ofreal-world problems, so that it is mathematics itself,rather than certain special disciplines within mathema-

tics, which is being recognized as applicable. Moreover,new concepts in mathematics are being evoked, and newmethods invented, to study the real world, so that we cansay, without, I believe, undue optimisrn, that we are on thethreshold of forging a new unity of the mathematicalsciences as a whole.What are the implications of this new synthesis with-in mathematics for the teaching of mathematics at theundergraduate level - or even earlier? I believe that theanswer must be that we should also be breaking down thebarriers between pure and applied mathematics. This hasbeen argued in detail elsewhere (Hilton:L981), so I willcontent myself with a brief description of the principalfeatures of such a change in teaching style and content.We should emphasize the interconnections between dif-ferent parts of mathematics, so that, for example, linearalgebra would appear primarily as a tool in the study ofdifferential equations and Euclidean geometry and much

of elementary complex variable theory would likewise bemotivated by the study of systems of linear differentialequations. We should also include in our treatment ofJusf as doing'pure' mathematicshas much in common with doing'ap-plied' mathematics, so should theteaching be closely related

mathematical topics, where appropriate, a discussion ofaccessible, and interesting, applications. These applica-tions should be genuine - and this is a severe conditionto impose! For an application to be genuine, two condi-tions must be satisfied. First, the problem must be a realproblem and, second, it must require the particular pieceof mathematics for its solution. However, we do not denythe validity of illustrations in helping to establish the sig-nificance of a piece of mathematics, provided that an il-lustration is honestly and openly presented as such. Anillustration can involve a highly idealized picture of theuniverse (for instance, a process which changes under theinfluence of one single variable rather than a number ofvariables); and it may not actually require the particularmathematics for its solution - it would suffice that themathematics was capable of providing a solution. Indeed;some of the best illustrations would involve problemswhich had already been solved by different methods, sothat they would constitute verification of the new piece ofmathematics under study.It is worthwhile pointing out that, just as doing,pure'mathematics has much in common with doing ,opplied'mathematics, so should the teaching be closely related.For we want to teach our students how to use mathema-tics; much of our teaching today is, however, concernedwith training students to regurgitate memorized formulasand routines and I assume it to be common ground thatthis sort of teaching is merely a response to administra-


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tive pressures and serves no really useful purpose for thestudents in their future lives. The way we use algebra tostudy geometric problems resembles very closely the waywe use, say, differential equations in gas dynamics or inthe studyof competitive populations; inboth cases we setup a mathematical model, reason within the model, andexamine the sigrrificance and validity of our conclusionsin the context of the original problem area. So we needmathematical courses with an applied flavor, whether the'applications' are within or outside mathematics.Let me readily cpncede that the proposals I am mak-ing here are not easy to implement. A curriculum of thekind eavisaged is much harder to devise than a standardcurriculum and.makes far greater demands on theteacher. Some attention is given to this aspect of the prob-lem in the NRC-report (1979:25), but I believe that thefaculty of each institution must decide how best to imple-ment these recommendations, assuming of course, thatit is decided that they have merit. The Tucker Panel of theCommittee on the Undergraduate Program of the Math-ematical Association of America will shortly be produc-ing its report and this document, gonlaining a number ofsample curricula in the mathematical sciences, should bea most valuable document for those wishing to adopt amore integrated curriculum. But the problem of ensuiingthat the teacher has the breadth of knowledge and theright positive attitude to teach such a curriculum effec-tively is highly non-trivial, and I have no glib solutiohs tooffer.A further aspect in which mathematics is undergoingradical change which will surely be reflected in substan-tial differences by 2001 is that of the impact of the com-puter. This must have an equally profound effect on ourteaching of mathematics"and deserves a greatdeal of ourattention. Thus I propose to devote a separate section toconsidering the impact and the effect.3. The computer and modern mathematicsThe computer is with us and will profoundly influence ourlives, whether.or not our educational system proves ca-pable of adapting to its presence and its potentialities. Itis not difficult to anticipate certain profound changes thecomputer (accompanied by its acolytes, the minicom-puter, the programmable calculator and even the hand-calbulator) will wreak in society within 20 years. Informa-tion accumulation, sorting, retrieval and display willbecome enormously more. efficient, so efficient that theindividual citizen and, less importantly, the corporationwill find themselves exposed to great risls of abuse.Check-writing and check-cashing will have disappeared.Video texts will be in common use. Each home will haveits supply of chips enabling domestic and family tasks tobe planned and executed and results made available inconvenient forms. Patterns of employment will undergochange. The working day and the working week will beconsiderably shortened and there will have been substan-

tial development ofleisure occupations. Indeed, whereastoday the average person's daily life is divided into twophases, work and rtcovery from wor\ we will, in the neKmillenium, be moving into a period where the two phaseswould naturally be described as leisure and interruptionof leisurr.The list could be prolonged; each of you could supple-ment it. But it is clear that the computer is going to playan enormous role in our lives, and I claim that it followsthat we should have some understanding of the basis onwhich it functions. This understanding should include amore than passing acquaintance with the computer as apiece of engineering; but even more important is an un-derstanding of its logical processes and its capacities. Forwe are certainly going to be using the computer; and if weuse it without this understanding, then we become slavesof the machine. It is bad enough to be a slave to the tele-vision set and the washing machine; it is incomparablyworse to be a slave to a machine which will be playing soubiquitous a role in our lives. For such slavishness will becharacterized by an abrogation of our control over ourlives and our decisions, a catastrophic drop in our level ofpersonal responsibility. The signs of these imminent dan-gers are already clearly visible; the awful scenario of thebook 1984 is nearer than we think.Thus the availability of the computer calls for a con-certed educational plan to teach our students to think,quantitatively and qualitatively, and this must mean bothto reason and to apply mathematical processes intelli-gently. Of course, many are aware of this; but there has, Ibelieve, been a tendency to concentrate attention onthose aspects of the learning of mathematics most close-ly related to the current activities of computers. Thusmuch has been written and done in the general area ofcomputers in education (I especially commend the pub-lication The Computing Teacher (1980-) and of com-puter-aided instruction, and I intend to say no more ofthis aspect here. My concern is principally with the im-pact which computer-availability should, in my view, haveon the teaching of mathematics - on what is taught andhow it is taught. Let me begin with a few remarks on therelation of computers to mathematics itself.The study of this relation has several components. Ofespecial interest, but of no particular relevance to thetheme of this talk, is the role of the computer in mathe-matical proof. This question has been highlighted by therecent computer-assisted proof of the celebrated FourColor Theorem, by Appel, Haken and Koch. An excellentdiscussion of the significance of this proof for mathema-tics, and mathematicians, is to be found in a special issueof the Tlvo Year College Mathematics Journal (198L,Vol.12).Closer to our concerns here is the influence of thecomputer on the direction mathematics is taking. Newareas of mathematics are arising, and new aspects of exist-ing areas are being thrown into prominencs. In the past,

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we have regarded a mathematical problem as solved ifthere existed a theoretical procedur- for determining theanswer, and we would, broadly speaking, discriminate be_tween procedures for solving the same problem largely onaestheticgrounds. But today - and even mo.e i.,ihe fu-ture - we employ a different set of criteria (while, of

clearer to them when the computer, endeavouring to finda solution for an inconsistent system, actually produces arow consisting of a string of zeros followed by a non-zeroentry. Again, existence theorems for differential equa-tions often seem to the student abstract and academic.Yet, given the differential equation dyldy : f(afi wtthgiven initial values (a,b ),and with the function/saiis4ringsome suitable (e.g. Lipschitz) condition the Euier -"tt oIprovides a method of approximating to a solution, and thestudent can actually see the solution taking shape on thecomputer print-out (or, better stil[ on the computerscreen).- As an example of a subject which has lost virtually allof its importance, let us instance that of techniques oi in-tegration. Closed-form solutions are no longer central toproblems involving integration and cunning substitutions,to translate an unfamiliar integral into a familiar one, arenow little more than intellectual exercises. This aspect ofthe impact of mechanical calculators is, of couise, of

enormous significance in considering desirable reformsat the pre-college level - but this is so vast and import-ant a subject that it would be irresponsible to attempt totreat it here by a few asides.A word of caution is needed in considering the impactof computers and other calculators. The problem ofround-off error is a very serious one, and students musthave their attention drawn to it early in their studies.Computer arithmetic is not the same as human arith-metic; computer analysis and algebra are not the same as

Closed-form solutions arc no longercentral to problems involving integra-tion and cunning substitutions, fotranslate an unfamiliar integral into afamiliar one, are now tittle more thanint e I e ct u a/ exercisescourse, not abandoning considerations of taste and eleg_ance). For we now may use the computer to obtain nu-merical solutions and also to carry out logical task, andwe are interested in computational efficienry. It is of vitalconcern how rapidly the computer gets the answer, andnot only for financial reasons! Where a procedure de_pends on a number ofparametcrs, *e need to know howthe computer time required varies with the sizes of thoseparameters, and thus we are involved in a new area ofmathematics called the theory of computational complex-ity. There are already exciting resultsln this theory, fasci_nating and important open questions, and eue.,unexpected connections with some of the deepest ques_tions of mathematical logic. Automata theory and bodingtheoryare two other relativelynewbranches -of mathemaltics which owe much of their prominence today to theirconnection with computers, but which themselves makecontact wi_th important classical areas of mathematics, forexamgl_e, -lineq algebra, polynomial theory and the the_ory of finite fields, as well as with other more recent re_search areas, such as category theory. The implication ofthese considerations for the teaching of mat-hematics isclear - we must bear in mind the importance of the com_puter (and its baby brothers, down to the hand-calcula_tor) in designing and executing the mathematicalcurriculum, at all levels. But we should not simply regardthe machine as a means of swift, accurate and painless cal_cllation, important as that aspect is. We shouid recognizethat the computer can give the student significant inJghtsinto mathematics itself; and we should recognizethat thecomputer has affected the roles of varioui parts of thetraditionalcurriculum, causing us to change tire emphasiswe give to those parts in the curriculum.Let a few examples suffice. The study of systems of li-near equations is probably even more important todaythan.it ryas 20 years ago, because there are now so manyapplications of such a study and because the computerhas rendered procedures for solving such systems muchmore effective. Students often have ditrilulty under-standing the conditions under which a system of linearequations has a solution; the situation becomes much

Computer arithmetic is not the sameas human arithmetic, computer ana-lysis and algebra are not the same astraditionalsubl'ects

the traditional subjects. Ifyou try to sum an infinite seriesby machine then, in almost all cases, you will find that theseries converges if the nth term tends to zero _ a falsetheorem of real analysis. If you carry out an iterative pro_cedure on a machine, it will often become stationarv-in afinite number of steps, by contrast with .ordinary' -athe-matics - qnd the displayed stationary value may differfrom the value obtained by theoretical analysis. Here thenis another respect in which the availability of the com-puter changes mathematics, and gives rise to a new aspectof mathematics which we must teach our students.4. Presenting mathematics today for use tomor-rowLet us now consider how we can effectively teach mathe-matics today in such a way that our students will be readyto enter tomorro#s world. I take it that that world is part_ly characterizedby (a) an increased importance of math-ematical thinking, and (b) the general availability of thecomputer and similar machines. I will argue that not only


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do we need to modify the content of our mathematicscourses, as already suggested, but, even more urgently,we need to modify the style and approach - in short, theattitude towards mathemgtics which epitomizes today'steaching and today's learning.Consider todat's human being and today's computer;here there was no real need to emphasize today's humanbeing, because human evolution is a very slow process,but there was a need to emphasize today's computer, be-

Educators should now face squarclYthe fact that computers have been in-vented and are going to PlaY a verysubstantial role in our studenfs'llvescause computers pass through generations very rapidly.Thus some of the properties of cornputers I list in Tablel- are not necessarily permanent features of computersbut rather represent characteristics of their standardoperating function today. (I am not spe,aking here ofthecomputer as itself a subject for research by those workingin the field of artificial intelligence. For a popular pres-entation of this quite different, and fascinating, topic seeLondon Times, Feb 9, 1981-.) In Table 1, then I list sometypical and contrasting properties of human beings on theone hand and, on the other hand, of computers as usedtodayin everydaylife or as tools in the exact sciences. Thelist is, of course, incomplete.

One implication of the availabilityof the computer foreducation is obvious: leave to the computer what the com-puter does best. But what do we find in praclice? We finda determined effort in current mathematics education todo the opposite - to discourage human attributes and toreplace them by machine attributes. This would makesense if we were seeking to educate juvenile computers;it is perverse when our students are human. It had somesemblance of justification before computers had been in-vented but educators should now face squarely the factthat computers have been invented and are going to play

a very substantial role in our students' lives. We have nowthe opportunity and, moreover,.the pressing necessity torealize the dream of Norbert Wiener of. achieing'thehuman uses of hurnan beings'. But there are serious ob-stacles in the way. There are strong traditions and thereis an immense inertia in the system due to many factors(of which standardized tests constitute one particularlyconspicuous and vicious example). Arnong those factorsI again stress, as I did earlier, the important role of atti-tudes, and I will give examples of attitudes which must beswept away if we are to achieve an appropriate educationfor the citizens of a self-respecting, advanced society.Principally, the attitudes which must change are (a)the attitude to education, (b) the attitude to mathematics,and (c) the attitude to usingmathematics.I prefer to con-centrate on (b) and (c), since the problem of society's at-titude to education is so vast, so pervasive, and so difficultthat it cannot be treated adequately here; moreover, it isnot at all specific to mathematics, though its effect on pre-vailing views of what mathematics should be taught is pro-found. Suffice it here to say that there is a fundamentalconfusion in the minds of many people between educa'tion and training, and that this confusion tends to blindthese people to the true purposes ofeducation. Ifwe edu-cate properly, our students will also be trained. But if weonly train, then our students are deprived of the oppor-tunity of self-development and the capacity of self-ex-pression.Let us, then, speak of society's attitude to mathema-tics. Here we do not have in mind the attitude of the pro'fessional user of mathematics, but rather that of thelayman who does not regard mathematics as a significantpart of his or her life. We also do not wish to deal herewith the common confusion which identifies mathematicswith elementary arithmetic - this is not so much a falseattitude as a simple mistake. The attitude I have princi-pally in mind is that rvhich divides people into the healthygroup which consists of those who actually do somethingand the rest who merely think about what they might doand indulge in endless and often sterile discussion.

Human beings ComputersCompute slowly and inaccuratelyTire easilyGet distractedSometimes give upAre often intelligent and understandingAre interested in many things at the same timeHave ideas and imagination, make inspiredguesses, think.

Compute fast and accurateNever tireAre remorseless, relentless, dedicated, andcannot be divertedAre stubbornAre usually pedantic and rather stupidAlways concentrateCan carry oul'if ..., e/se ...' instructionsTable 1: Some contrasts between humans & computers

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According to this view the pioneering spirit requires menand women of action and the tendeniylo reflect, to pon-der, to see two sides to a question, symptomizes an un-healthy unAmerican trait of hesitaiio; and inaction.Mathematics is not seen - as it should be - as a meansto.doing something, but as an alternative to doing some-thing.People holding this view often have a remarkablywarped view of mathematicians. Mathematicians areviewedas sterile, passionless persons, lacking any inter-est in what is best in life, like personal relationi and sport.Those holding this view maywell believe, incorrectly, thatmathematicians are exclusively male; but they are hardlylikely to believe them sturdily male. Following, or evenpreceding, Spiro Agnew, they will believe them effete andelitist.These absurd views spring, ofcourse, from a false viewof mathematics, of both its nature and its purposes. We

have to dispel that view vigorously and rp"baity. The na-ture and purposes of mathematics cannot be summarizedneatly in a few phrases corresponding to our particularprejudices - the question is far too subtle for tliat. Thosewho hold that mathematics is mere drudgery are onlyWe should show our studenfs nofonly how to answer interesting gues-tions but also how to ask them

marginally more wrongheaded than those who hold thatit is mere fun. It is, in some of its aspects, simple and natu-ral, in others subtle and sophisticated. But, above all,mathematics is not something to be avoided. We shouldnot seek to avoid it from distaste, for we can change ourattitudes if we understand better its nature and we shouldnot wish to avoid it since we need it the better to conductour daily lives. Since mathematics is so important, to eachintelligent individual, we must aspire to the attitude thatit is simplynot respectable to be ignorant of mathematics.How should we change popular attitudes to the usesof mathematics? Much attention has already been givenin this article to the affinity between pure and upin"amathematics; and our attitude towaids mathemiticsshould include the view that it is useful to learn to thinkmathematically whether the particular problem underconsideration is pure or applied. Thus pait of this new at_titude must be a twin respect for mathimatics and for itsapplicability. The increasing emphasis on applications isa healthy trend, but it must not be allowed to obscure theimportance of theoretical mathematics, nor must it beused to justify sloppiness of thought or statement. It isnecessary to say yet again (see Hilton:1981) for a previousversion of this remark) that problem-solving is not an al-ternative to setting up mathematical structures. One canreadily approve of the discussion of problem-solvingstrategies in the hands of such experts as George polyi

and Alan Schoenfeld; but it is naiVe to suppose that prob-lems suitable for attack by mathematical methods can besolved instead by a display ofgeneral intelligence and na-tive wit. These attributes are valuable, but they are inade-quate to the task. We must learn mathematics to be ableto apply mathematics and be able to recognize the mostpromising mathematical approach to a given problem.Moreover, we must fully understand the mathematics inorder to appreciate when a problem is ripe for attack. Ourmathematical knowledge, in other words, does more thanenable us to solve already formulated problems; it tells ushow to formulate questions, what questions to ask, andwhen to ask them.The prescription of the National Council of Teachersof Mathematics in the Agenda for Action is well-inten-tioned but could be dangerous precisely because it givesthe false impression that we always proceed in the se-quence - find the problem, formulate a problem, solvea problem. Thus it is argued: 'The mathematics curricu-lum should be organized round problem-solving ... Thecaniculum should mointain a balance between attention tothe applications of mathematics and to fundamental con-cepts'. The first sentence gives the false impressionreferred to, the second is banal, since it would not bepossible to argue the contrary. The point at issue is wherethe balance is to be found. The result we want to achieveis that our students have the ability and the will to domathematics and to use it.If we achieve it, we will have overcome the sterile an-tagonism between pure and applied mathematics, andshown their interrelations and the importance of both.We will have destroyed certain traditional sacred cows ofmathematics education and curriculum - and we willhave shown our students not only how to answer interes-ting questions but also how to ask them (thus we will cer-tainly have emphasized the importance of the geometricalviewpoint, today so often woefully neglected).I repeat - we must give our students the ability andthe will to do mathematics. The ability without the willleads to sterility; the will without the ability leads to frus-tration. We avoid both these pitfalls by teaching all ofmathematics as a unity, emphasizing both its uniquegenerality and its immense power.ReferencesAgenila for Action. (1980). ilational Council of Teachers of Mathe-natics, Reston.ii$lil"g::"tii*Tr3t*t&tol* and re have selr lot'hiDs Ft'fitto.n,.f (1981). Th enphasis o,n applied Dathenatics today ardlts inplications for the nathenatics -curriculun. In leceot DErrel-oprenta ia tbe Application of lrhtheratica, Springer.Ihe Corputing. feacher, (1980-).- I-nteraational Council for Conpu-ters in Education, lJniversity of 0regon.Ihe role of rpplicatims ia the uadeorgraduate nthecticr currisu-fur, (1979). Aa Hoc Comittee on appfi-ea Uaitrenailcs ftiinins-,-[ii-rport, pp 25.l{hat is_a P^roo^f Today?. (1981). Iro lan Coltege l.hthenticr Journal, Vol 12, 2.

Mathematics in 2001: Implications for today's undergraduate teaching

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