Calculus Made Easy
Book Preface
Introductory courses in calculus are now routinely taught to high school students and college freshmen. For students who hope to become mathematicians or to enter professions that require a knowledge of calculus, such courses are the highest hurdle they have to jump. Studies show that almost half of college freshmen who take a course in calculus fail to pass. Those who fail almost always abandon plans to major in mathematics, physics, or engineering-three fields where advanced calculus is essential. They may even decide against entering such professions as architecture, the behavioral sciences, or the social sciences (especially economics) where calculus can be useful. They exit what they fear will be too difficult a road to consider careers where entrance roads are easier.
One reason for such a high dropout rate is that introductory calculus is so poorly taught. Classes tend to be so boring that students sometimes fall asleep. Calculus textbooks get fatter and fatter every year, with more multicolor overlays, computer graphics, and photographs of eminent mathematicians (starting with Newton and Leibniz), yet they never seem easier to comprehend. You look through them in vain for simple, clear exposition and for problems that will hook a student’s interest. Their exercises have, as one mathematician recently put it, “the dignity of solving crossword puzzles.” Modern calculus textbooks often contain more than a thousand pages-heavy enough to make excellent doorstops-and more than a thousand frightening exercises! Their prices are rapidly approaching $100.
“Why do calculus books weigh so much?” Lynn Arthur Steen asked in a paper on “Twenty Questions for Calculus Reformers”
that is reprinted in Toward a Lean and Lively Calculus (Mathematical Association of America, 1986), edited by Ronald Doug-las. Because, he answers, “the economics of publishing compels authors. . . to add every topic that anyone might want so that no one can reject the book just because some particular item is omitted. The result is an encyclopaedic compendium of techniques, examples, exercises and problems that more resemble an overgrown workbook than an intellectually stimulating introduction to a magnificent subject.”
“The teaching of calculus is a national disgrace,” Steen, a mathematician at St. Olaf College, later declared. “Too often calculus is taught by inexperienced instructors to ill-prepared students in an environment with insufficient feedback.”
Leonard Gillman, writing on “The College Teaching Scandal” (Focus, Vol. 8, 1988, page 5), said: “The calculus scene has been execrable for many years, and given the inertia of our profession is quite capable of continuing that way for many more.”
Calculus has been called the topic mathematicians most love to hate. One hopes this is true only of teachers who do not appreciate its enormous power and beauty. Howard Eves is a retired mathematician who actually enjoyed teaching calculus. In his book Great Moments in Mathematics I found this paragraph:
Surely no subject in early college mathematics is more exciting or more fun to teach than the calculus. It is like being the ringmaster of a great three-ring circus. It has been said that one can recognize the students on a college campus who have studied the calculus-they are the students with no eyebrows. In utter astonishment at the incredible applicability of the subject, the eyebrows of the calculus students have receded higher and higher and finally vanished over the backs of their heads.
Recent years have seen a great hue and cry in mathematical circles over ways to improve calculus teaching. Endless conferences have been held, many funded by the federal government. Dozens of experimental programs are underway here and there.
Some leaders of reform argue that while traditional textbooks get weightier, the need for advanced calculus is actually diminishing. In his popular Introduction to the History of Mathematics, Eves sadly writes: “Today the larger part of mathematics has no, or very little connection with calculus or its extensions.”
Why is this? One reason is obvious. Computers! Today’s digital computers have become incredibly fast and powerful. Continuous functions which once could be handled only by slow analog machines can now be turned into discrete functions which digital computers handle efficiently with step-by-step algorithms. Hand-held calculators called “graphers” will instantly graph a function much too complex to draw with a pencil on graph paper. The trend now is away from continuous math to what used to be called finite math, but now is more often called discrete math.
Calculus is steadily being downgraded to make room for com-binatorics, graph theory, topology, knot theory, group theory, matrix theory, number theory, logic, statistics, computer science, and a raft of other fields in which continuity plays a relatively minor role.
Discrete mathematics is all over the scene, not only in mathematics but also in science and technology. Quantum theory is riddled with it. Even space and time may turn out to be quan-tized. Evolution operates by discrete mutation leaps. Television is on the verge of replacing continuous analog transmission by discrete digital transmission which greatly improves picture quality. The most accurate way to preserve a painting or a symphony is by converting it to discrete numbers which last forever without deteriorating.
When I was in high school I had to master a pencil-and-paper way to calculate square roots. Happily, I was not forced to learn how to find cube and higher roots! Today it would be difficult to locate mathematicians who can even recall how to calculate a square root. And why should they? They can find the nth root of any number by pushing keys in less time than it would take to consult a book with tables of roots. Logarithms, once used for multiplying huge numbers, have become as obsolete as slide rules.
Something similar is happening with calculus. Students see no reason today why they should master tedious ways of differentiating and integrating by hand when a computer will do the job as rapidly as it will calculate roots or multiply and divide large numbers. Mathematica, a widely used software system developed by Stephen Wolfram, for example, will instantly differentiate and integrate, and draw relevant graphs, for any calculus problem likely to arise in mathematics or science. Calculators with keys for finding derivatives and integrals now cost less than most calculus textbooks. It has been estimated that more than ninety percent of the exercises in the big textbooks can be solved by using such calculators.
Leaders of calculus reform are not suggesting that calculus no longer be taught, what they recommend is a shift of emphasis from problem solving, which computers can do so much faster and more accurately, to an emphasis on understanding what computers are doing when they answer calculus questions. A knowledge of calculus is even essential just to know what to ask a computer to do. Above all, calculus courses should instill in students an awareness of the great richness and elegance of calculus.
Although suggestions are plentiful for ways to improve calculus teaching, a general consensus is yet to emerge. Several mathematicians have proposed introducing integral calculus before differential calculus. A notable example is the classic two-volume Differential and Integral Calculus (1936-37) by Richard Courant. However, differentiating is so much easier to master than inte
grating that this switch has not caught on.
Several calculus reformers, notably Thomas W. Tucker (See his “Rethinking Rigor in Calculus,” in American Mathematical Monthly, (Vol. 104, March 1997, pp. 231-240) have recommended that calculus texts replace the important mean value
theorem (MVT) with an increasing function theorem (IFT). (On the mean value theorem see my Postscript to Thompson’s Chapter 10.) The IFT states that if the derivative on a function’s interval is equal to or greater than zero, then the function is increasing on that interval. For example, if a car’s speedometer always shows a number equal to or greater than zero, during a specified interval of time, then during that interval the car is either standing still or moving forward. Stated geometrically, it says that if the curve of a continuous function, during a given interval, has a tangent that is either horizontal or sloping upward, the function on that interval is either unchanging or increasing. This change also has not caught on.
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