Ivars Peterson's MathTrek

September 21, 1998

There's something about calculus that can evoke a mixture of both wonder and dread.

I was recently reminded of my first encounter with calculus by the announcement of the publication later this year of a new edition of the book Calculus Made Easy. More than 30 years ago, the same title had proved irresistibly inviting to me as a high school student facing the mysterious realm of limits, derivatives, and integrals. I ended up purchasing a copy of the 1965 reprinting of the third edition, originally published in 1946. I still have the book.

There were several things about Calculus Made Easy that made it immensely attractive. The author's name, Sylvanus P. Thompson (1851-1916), carried an air of authority and trustworthiness. At the same time, the book's bold, yet self-effacing subtitle was heartening: BEING A VERY--SIMPLEST INTRODUCTION TO THOSE BEAUTIFUL METHODS OF RECKONING WHICH ARE GENERALLY CALLED BY THE TERRIFYING NAMES OF THE DIFFERENTIAL CALCULUS AND THE INTEGRAL CALCULUS. The ancient Simian proverb quoted at the beginning provided just the right sort of encouragement: "What one fool can do, another can." I was hooked.

The text proceeded about its business of unveiling the secrets of calculus in a reassuringly straightforward, direct manner. The new edition has maintained that structure and style, preserving the many examples and exercises that proved so useful to me. The main change is the addition of three "preliminary" chapters, written by Martin Gardner, explaining the concepts of function, limit, and derivative. An appendix, also by Gardner, introduces some recreational math problems that involve calculus.

"It is true that [Thompson's] book is old-fashioned, intuitive, and traditionally oriented," Gardner writes in the preface to the new edition. "Yet no author has written about calculus with greater clarity and humor." Interestingly, Thompson himself was a physicist and electrical engineer, and his book usually doesn't get much respect from mathematicians.

Calculus Made Easyproved a valuable reference and source of examples when I taught high school calculus for a few years in the late 1970s. I also often consulted several 19th-century calculus textbooks that I had found. These books typically had few diagrams, instead featuring exercise sections with amazingly long lists of problems.

Part of the calculus education landscape has changed considerably since my teaching days. Students can use graphing calculators or computers to solve calculus problems and to explore many different kinds of mathematical behavior. It's easy to obtain pictures of what's going on.

Software products such as Calculus WIZ are advertised as homework solvers for students. "Are you worried that calculus will sabotage your GPA or, even worse, that you might fail it? Calculus WIZ actually does the calculus for you--and shows its work."

At the same time, leaders of a movement to reform calculus teaching have advocated a shift in emphasis from problem solving, which computers can do much faster and more accurately, to developing an understanding of what calculus can do and building an awareness of the richness and elegance of the subject.

Is there still a place for Calculus Made Easy? "Curiously, Thompson's first edition, with its great simplicity and clarity, is in a way closer to the kind of introductory book recommended today by reformers who wish to emphasize the basic ideas of calculus," Gardner notes. In addition to teaching his readers how to differentiate and integrate simple functions, Thompson explains the "philosophy of the subject."

Thompson's compact book also stands as a rebuke to the hefty, overstuffed volumes lugged around by today's college calculus students. One of the few current textbooks that actually gets right to the point is the slim volume Calculus Lite, written by Frank Morgan of Williams College.

Other antidotes to the contemporary norm are also available. Colin Adams of Williams College and Joel Hass and Abigail Thompson of the University of California, Davis have written How to Ace Calculus: A Streetwise Guide. A breezy, cheerful, conversational tone and humor, via outrageous puns and ridiculous jokes, sugarcoat the neatly packaged nuggets of calculus wisdom dispensed by the book. It's funny and irreverent, but it also tends to treat calculus as an unfortunate hurdle that must somehow be surmounted.

David Berlinski's A Tour of the Calculus is another sort of antidote. A cross between a textbook and a romance novel, it gleefully plunges into the sort of calculus taught in a beginning course. From a mathematical point of view, there's nothing new here. The book's appeal stems from its passionate voice and its sense of wonder at the amazing intellectual achievement that calculus represents.

In the end, calculus isn't easy. It doesn't automatically seep into the mind. It requires thought and hard work. The best guides--whether teachers or books--illuminate the way.

Copyright 1998 by Ivars Peterson

References:

Adams, C., J. Hass, and A. Thompson. 1998. How to Ace Calculus: The Streetwise Guide. New York: W.H. Freeman.

Berlinski, D. 1997. A Tour of the Calculus. New York: Vintage.

Morgan, F. 1995. Calculus Lite. Wellesley, Mass.: A.K. Peters.

Peterson, I. 1988. Calculus in the palm of your hand. Science News 133(Jan. 23):62.

______. 1987. Calculus reform: Catching the wave? Science News 132(Nov. 14):317.

______. 1986. The troubled state of calculus. Science News 129(April 5):220.

Thompson, S.P., and M. Gardner. 1999. Calculus Made Easy. New York: St. Martin's Press.

Thompson, S.P. 1965. Calculus Made Easy(Third Edition). New York: St. Martin's Press.

Information about Calculus WIZ is available at http://www.wizpower.com.

Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.