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The Calculus Turmoil Challenges for College Mathematics Inside This Volume 10, Number 6 THE NEWSLETTER OF THE MATHEMATICAL ASSOCIATION OF AMERICA November-December 1990 The Calculus Turmoil Challenges for College Mathematics Paul R. Halmos Lynn Arthur Steen What's all this calculus turmoil about? Have all mathematicians This issue of FOCUS contains as a special insert the report of except me taken leave of their senses? Surely it isn't just a racket­ a joint Task Force of the MAA and the Association of American a way of getting money from the NSF? But if it is not, then why are Colleges (AAC) entitled "Challenges for College Mathematics: An otherwise honest and intelligent people screaming about it? Will Agenda for the Next Decade." This report was prepared as part somebody please explain to me what's going on? of a multidiscipline AAC project to explore effective approaches to undergraduate majors. Issues arising from the project will be the The two main symptoms (or are they the causes?) of this new subject of discussion at the January 1991 AAC meeting of college disease are that lots of students flunk calculus and that most text­ and university deans and presidents. books are bad. In August 1990, the MAA Board of Governors endorsed the Task Well, let's see. "More than a third of the 600,000 or so peo­ Force report "as a statement about the major on behalf of MAA" ple who study calculus in college every year fail to complete the and recommended it to MAA members "as a framework for cam­ course." (Peter D. Lax, "Calculus Reform: A Modest Proposal," pus discussion about the undergraduate major in mathematics." UME Trends, 2 (Number 2, May 1990). For a different view, see The MAA-AAC report follows in the footsteps of the "David II" re­ Leonard Gillman's "Two Proposals for Calculus," FOCUS, 7 (Num­ port "Renewing US Mathematics: A Plan for the 1990s," and pre­ ber 4, 1987).) I am prepared to believe that, and I am neither sur­ cedes by several months the publication of the MS 2000 report prised nor horrified. Teachers have been discussing failure rates "Moving Beyond Myths: Revitalizing Undergraduate Mathematics." with one another ever since the concept was invented, and 40% Together these three documents will provide an agenda for action is not unusual among the numbers that they bandy about. The to help stimulate departments of mathematics to improve the effec­ failure rate varies a lot, of course-it depends on the quality of the tiveness of undergraduate mathematics for all students regardless university in question as a whole, and, derivatively, on the quality of background, interests, or intended major. of the teachers and the students at the university. A part of the reason is the reason that students take calculus in the first place. Lynn Arthur Steen, professor of mathematics at St. Olaf College The word has a mystique; it's a Mount Everest to climb. It is an in Northfield, Minnesota, former MAA president (1985-1986), and essential part of the tool kit of many who wish to study science, chair of the Association's Committee on the Undergraduate Pro­ and science is a glamorous and well-paid(?) and highly respected gram in Mathematics (CUPM) chaired the joint Task Force of the (Turmoil continues on page two.) MAA and AAC that prepared the report described above. Inside This Issue Summer Institutes for Minorities and Women. ...... 4 Calculus, Computers, Concepts, and Cooperative Learning 4 Regional Technology Centers. .... 4 Board Actions in Columbus. 5 Nominations for Yueh-Gin Gung and Dr. Charles Y. Hu Award for Distinguished Service to Mathematics, .5 Goodbye. Columbus. 6 and 7 FOCUS on MAA Journals The American Mathematical Monthly. .8 Chaos in the College Mathematics Journal. .8 Mathematics Magazine . 9 1990 MAA Journal Awards. Paul R. Halmos . 9 2 FOCUS November-December 1990 (Turmoil continued from front page.) of the classical drill (that I for one went through in the 1930s) can profession. Students who take calculus "because it's there," or for be left to the button pushers-pocket calculators do very well what the glamor, the money, or the respect, are taking it for the wrong for a long time we haven't been wanting to do at all. But I am still in reason, and their failure is neither surprising nor worrisome. favor of teaching some (many?) rigged integration techniques and tedious calculations-they are as rigged and as tedious as finger What is the failure rate for people taking piano lessons? I don't exercises for the pianist, and just as indispensable. know, but I could make up some statistics as well as the next guy, and I wouldn't be surprised if the true statistics pointed to a situation Is computing to be "an integral part" of calculus? (I can't help won­ as "lamentable" as the one in mathematics. dering whether that's an intentional pun.) Yes, of course, that's just what I've been saying. By computing, to be sure, I mean As for how the books are bad: they contain "inert knowledge," what calculus teachers have always meant-computing with ratio­ including "integration techniques that work only in specially rigged nal, trigonometric, exponential, and logarithmic functions, and get­ cases," "fake applications," and "tedious calculations." ting used to their behavior and their relations to each other. I do not Are those comments new? And, new or not, are they correct? Yes, mean computing in the sense that computing machines are good they are correct, and no, they are not new: they have been made for-horrors, no! for many decades (centuries?), and they point to something bad. The greatest "mathematical" invention of the last few centuries is To something that is all bad? I am not sure. not electronic but notational-I refer to decimals and their relation Very few people advocate epsilons and deltas in elementary cal­ to arithmetic. An electronic typewriter that can crunch numbers is culus, and, for sure, I am not one of them, I confess that when it useful-but I don't want to teach spelling in courses in literature, comes to arithmetic I am still in favor of the multiplication table being decimal calculations in courses on number theory, and computing drummed into innocent heads (at least through 9 x 9), together with techniques in courses on calculus. I like spelling and decimals and some of the elementary and almost universally needed techniques computers-but they should be kept in their places. of estimating orders of magnitude. (If each of 1,000 students writes 10 themes a week for each of the 30 weeks of the academic year, What then should a calculus course emphasize, if not integral and if, on average, the labor of grading each theme costs 20 cents, rigging and number crunching? Qualitative understanding of then we should be able to decide whether the money we need from functions?-asymptotics and approximation as implied by the dif­ the administration is $6,000, $60,000, or $600,000.) Similarly, in ferential equations they satisfy? Fine-sure-why not. Substantial calculus, I am still in favor of teaching the classical basic techniques applications-in engineering, in economics, in psychology?-yes, of the differential and integral calculus. I want a student who sees maybe-but our focus should always remain at the mathematical x 3 or sin x to snap back 3x 2 or cos x, and I want a student who center. We, teachers of calculus, are not usually practicing engi­ neers, economists, or psychologists, and just as we might worry sees f 1~:2 and f 1~:2 to know the difference between them about the competence of those people to teach our stuff, they have and to be able to come up with the indefinite integral in both cases, the full right to worry about our encroaching on their expertise. But even if a pocket calculator can do it more quickly. It is true that much I don't really think there is anything serious to worry about as far as the content is concerned. Most calculus courses (certainly the ones I took, the ones I taught, and the ones I hear colleagues tell about) are pretty much freewheeling-they emphasize the mechan­ ical techniques to the extent that drill is necessary (or a little too much, or not quite enough, but more or less to the right extent), and they contain the illustrations and applications that the teacher is competent to explain and the students are ready to receive. Ft'JCUS Do bad books contribute to making the courses bad? Yes and no­ mainly no. At the calculus level, in my experience, students simply FOCUS is published by The Mathematical Association of America, 1529 Eigh­ don't read the book-most of them cannot and none of them has teenth Street Northwest, Washington, DC 20036, six times a year: January­ February, March-April, June, September, October, and November-December. to. They cannot because they just don't know how to read-the formal language of the text is repulsive to them, and boring, and Editor: Andrew Sterrett, Interim Associate Director of Programs, MAA Associate Editors: Donald J. Albers, Menlo College and difficult-most of our class time has to be spent in repeating the David W. Ballew, Western Illinois University book in the vernacular. And they don't have to because most of Managing Editor: Harry Waldman our class time is spent in repeating the book in the vernacular. Advertising Manager: Siobhim B. Chamberlin Chairman of the MAA Newsletter Editorial Committee: A famous (infamous?), old (ancient?), bad (dreadful?) calculus Susan L.
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