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Calculus Redux THE NEWSLETTER OF THE MATHEMATICAL ASSOCIATION OF AMERICA VOLUME 6 NUMBER 2 MARCH-APRIL 1986 Calculus Redux Paul Zorn hould calculus be taught differently? Can it? Common labus to match, little or no feedback on regular assignments, wisdom says "no"-which topics are taught, and when, and worst of all, a rich and powerful subject reduced to Sare dictated by the logic of the subject and by client mechanical drills. departments. The surprising answer from a four-day Sloan Client department's demands are sometimes blamed for Foundation-sponsored conference on calculus instruction, calculus's overcrowded and rigid syllabus. The conference's chaired by Ronald Douglas, SUNY at Stony Brook, is that first surprise was a general agreement that there is room for significant change is possible, desirable, and necessary. change. What is needed, for further mathematics as well as Meeting at Tulane University in New Orleans in January, a for client disciplines, is a deep and sure understanding of diverse and sometimes contentious group of twenty-five fac­ the central ideas and uses of calculus. Mac Van Valkenberg, ulty, university and foundation administrators, and scientists Dean of Engineering at the University of Illinois, James Ste­ from client departments, put aside their differences to call venson, a physicist from Georgia Tech, and Robert van der for a leaner, livelier, more contemporary course, more sharply Vaart, in biomathematics at North Carolina State, all stressed focused on calculus's central ideas and on its role as the that while their departments want to be consulted, they are language of science. less concerned that all the standard topics be covered than That calculus instruction was found to be ailing came as that students learn to use concepts to attack problems in a no surprise. As one conferee put it, "calculus is the course flexible and incisive way. mathematicians love to hate." Too many students seem to Will the discrete mathematics of computation render cal­ agree; at some schools fewer than half of those who enroll culus obsolete? Not according to the Tulane conferees. On pass the cou rse. Those who do pass, and thei r teachers, may the contrary, while the computer surely obliges us to rethink have to cope with poor pre-calculus preparation, inexperi­ what we want from calculus, it also presents new opportu­ enced or discouraged instructors, an over-fat text with syl- nities for applying and understanding calculus ideas, and for extending the reach of applications well beyond the tra- (continued on page 2) International Congress of Mathematicians Featured in Center Section The center section of this issue focuses on the Interna­ tional Congress of Mathematicians (ICM-86)to be held August 3-11, 1986 on the campus of the University of California, Berkeley. Members are urged to take advantage of this unusual opportunity to meet with many of the world's most prominent mathematicians. Employment Advertisements For the second time, FOCUS has included in the center Arnold E. Ross, Professor Emeritus at The Ohio State Uni­ section, employment advertisements. You will find the rates versity, was awarded the MAA's highest honor at its Annual and deadline information on page v. Meeting. See page 3 of this issue. 2 FOCUS March-April 1986 Calculus (continued from page 1) Wisconsin Section to Offer Short Course ditional closed-form methods applied to elementary func­ The Wisconsin Section is sponsoring A Short Course in tions. For example, calculus has much to do with the fasci­ Algorithms of Discrete Mathematics, June 16-20,1986 at Car­ nating interplay between the discrete and the continuous. dinal Stritch College, Milwaukee, Wisconsin. Computing in calculus embodies and illustrates that rich and Principal presenters will be Walt Meyer of Adelphi University deep relationship. and Doug Harris, Chairman of the Department of Mathematics, Already, inexpensive hand-held calculators approximate Statistics, and Computer Science at Marquette University. Other definite integrals and solve equations numerically. (Try to presenters will be John Dossey of Illinois State University who think of five really different arc-length problems that can be will kick offthe session and Karl Beres, Chairman ofthe Depart­ solved in closed form.) Still more powerful tools are just ment of Mathematics and Computer Science at Ripon College. appearing on the calculus scene. Computer algebra systems Among the topics discussed will be Data Compression and (MACSYMA, Maple, SMP, MuMath, and others) act as "super Huffman Codes, Graph Theory, Recursion, Machines and calculators", easily handling graphical, numerical, and sym­ Behavior, Codes and Communication, and Grammar and Lan­ bolic computations that would have baffled the best students guages. of five years ago. Participants agreed that if due caution is The presentations will be aimed at high school teachers and exercised, using such devices to handle routine computation teachers of introductory college math courses. can help shift calculus's focus back to ideas, where it belongs. For more information please contact: Professor R.J.Mihalek, We can use computer technology badly or well, but we cannot Department of Mathematical Sciences, University of Wiscon­ ignore it; our choice is either to lead or to follow. sin-Milwaukee, Box 413, Milwaukee, Wisconsin 53201, (414) Conferees agreed that we must change not only what we 963-4516,or Professor Barbara Reynolds, Cardinal Stritch Col­ teach but also how we teach it and how we evaluate the lege, 6801 North Yates Road, Milwaukee, Wisconsin 53217, results. Calculus can no more be taught effectively on the (414) 352-5400. cheap than can, say, beginning French. Calculus students, like language students, must move along witl:! the course. To ensure this entails frequent and regular checking of prog­ ress and committing the necessary resources. (One partici­ pant, a university administrator, accused mathematicians of pressing their claims for resources ineptly, compared to their colleagues in other disciplines.) Not all our problems are 1••• financial. Again and again, conferees observed that too often, 1 we have gotten just what we asked for-shallow technical FOCUS (ISSN 0731-2040) is published by the Mathematical facility instead of deeper understanding-because the for­ Association of America, 1529 Eighteenth Street, NW., Wash­ mer is easier to teach and test for than the latter. ington, D.C. 20036, six times a year: January-February, March­ Their complaints notwithstanding, the Tulane conferees April, May-June, September, October, November-December. came not to bury calculus, but to praise it. Calculus, they Editor: Peter Renz, MAA Associate Director. agreed, is vital, as training for analytical thinking, as general Associate Editors: Donald J. Albers, Menlo College; David Ballew, South Dakota School of Mines and Technology; Wil­ education, as a foundation of higher mathematics, and for liam G. Chinn, City College of San Francisco; Stephen B. describing natural laws and deducing their consequences. Maurer, Swarthmore College. To give concrete, constructive shape to the group's findings, Chairman of the MAA Newsletter Editorial Committee: three subcommittees met, to draft detailed syllabi for the first Ronald M. Davis, Northern Virginia Community College. year, to recommend new ways of teaching, testing, and Readers are invited to submit articles, announcements, or administering calculus, and to devise strategies for bringing Letters to the Editor for possible publication in FOCUS. All all these changes about. materials should be sent to the Editor at the MAA Headquar­ ters in Washington, D.C. The content subcommittee, led by Thomas Tucker of Col­ gate University, drew up a two-semester syllabus that aims The annual subscription price for FOCUS to individual mem­ bers of the Association is $1, included as a part of the annual to build intuition and conceptual understanding, stressing dues. Annual dues for regular members (exclusive of annual numerical and geometrical ideas as well as algebraic tech­ subscription prices for MAA journals) are $22. Student, unem­ niques. To support this viewpoint, access to calculators with ployed, emeritus, and family members receive a 50% discount; definite integral and equation-solving keys will be assumed. new members receive a 30% discount for the first two years of membership. Within these guidelines, three alternative year-courses are envisioned, all sharing a common first semester: one cov­ Copyright © by the Mathematical Association of America (Incorporated), 1986. Educational institutions are welcome to ering single variable calculus through Taylor's series and the reproduce articles from this publication in quantities suffi­ beginnings of differential equations; another that cuts short cient for their own use, but not for sale, provided that the the single variable material to cover the basics of several following citation is retained: "Reprinted with permission from variable calculus in the second semester; and a third that FOCUS, the Newsletter of the Mathematical Association of America. Copyright © by the Mathematical Association of makes full use of the computer algebra systems soon to be America (Incorporated), 1986." widely available. Colleges and universities would offer one Second-class postage paid at Washington, D.C. and additional or more, as their needs dictate. Bringing new tools, ideas, mailing offices. and enthusiasm to an old and important effort, the new Postmaster: Send address changes to Membership/Subscrip­
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