Down with “the” definition of the derivative

God said to Moses that ye shall define the derivative of a function as the limit of its difference quotient as the increment goes to zero. Or he might as well as far as modern calculus textbook are concerned, because they take it for granted as Holy Writ.

But Viktor, you say, modern books don’t take it for granted at all, they motivate it, you see, with various pictures of secant lines becoming tangents and something abut speed and whatnot. No they don’t. Such motivations do not lead unequivocally to “the definition” as the textbook authors pretend, because they are equally compatible with other perspectives such as the infinitesimal conception of the derivative that I use in my book. To pretend otherwise, as modern textbooks do, is an intellectually dishonest bait-and-switch.

The real purpose of such pseudo-motivation is not to stimulate critical thought but to preempt it. The use of the definite article---”the” definition, “the” proof---is a devious trick designed to preemptively stifle any traces of a critique of the status quo. It is hard to imagine that any printing presses are as active as those producing Stewart’s Calculus; excepting, perhaps, those printing the hundred-dollar bills for which the book is bought. But how could it be otherwise since this is “the” way to teach calculus?

Accordingly, people doing “research” in “education” ask themselves: How can we make students understand “the definition” of the derivative? How indeed? And how can we make them understand the benevolence of Chairman Mao?

To show you how fragile flowers are cleaved by the bayonets of this authoritarian regime, I need only point to the mutilated pieces of the sine. If you open, say, Stewart’s Calculus you will find in an early section a numerical “exploration” of the limit of sin(x)/x as x goes to 0. You may wonder what the point is of this dreadfully boring result, but then again you must realise that it is all very necessary for you to be able to “understand the definition.” Well over a hundred pages later, the time has come to differentiate the sine. “The proof” of course starts by plugging the sine into “the definition.” Unfortunately, though “the definition” is divine decree, this manoeuvre gets us absolutely nowhere. That is, until one recalls that limit calculation clouded in mystery from chapters ago. This result is now revived and for some reason in fact rederived using various complicated formulas that are soon forgotten. Altogether students are left with the impression that the derivative of the sine is not something one can see intuitively or reasons one’s way toward in natural steps, but rather that reaching this result requires numerical experimentation, obscure trigonometrical identities, and complicated algebraic manipulations of limits. This is an appalling state of affairs. Though there is a well-intended, and in a narrow sense sound, theoretical point at the basis of the standard approach, its actual net result is obfuscation and authoritarianism and the stifling of curiosity and thought.

This example also shows the extremely severe limitations of so-called “reform” teaching methods. Some decades ago a traditional book would have gone straight for the formal computation of the limit; nowadays “reform” winds have led to the preliminary “exploration” of the limit numerically. This is a very sad form of veneer reform. It takes for granted that the traditional approach is the “right” approach to the subject, theoretically speaking, adding only that it should be made more “accessible” through contrived “discovery” exercises that are fundamentally misguided since “discovering” one’s way to the traditional proof in this manner is intellectually nonsensical in the first place.