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On Reform, Part 3
Posted:
Jun 23, 1997 11:44 AM


Definitions are central to mathematics. Even at the upper division undergraduate and begining graduate levels, the majority of the proofs that the student is asked to invent amount to little more than verification of definitions. It is essential that advanced students see the role that definition plays in mathematics.
But as a general rule, I don't believe that we should introduce definitions unless we plan to use them in arguments and, moreover, plan to require that students be able to recite both the definition and the arguments. (As every playwright knows: You don't drag a cannon onto the stage unless you're going to fire it.) I think that one of the reasons juniors and seniors have difficulty coming to see the importance of definition lies in our failure to observe this principle in our calculus sequences. When we introduce definitions that we do not ask them to recite or use, we teach them that definitions in mathematics are like those in Webster; determined by the context of a language and to be looked up when we don't know what somebody's talking aboutif then.
Moreover, in choosing the definitions we deal with, I believe we should make use of students' serviceable intuitions. Owing, at least in part to the effective demise of proof in high school geometry courses, today's beginning calculus students have very little of proof in their personal histories. I think we need to accomodate them to proof and its necessity before we start discussing counterintuitive things, trample on the correct intuitions that do have, or undertake to prove what they consider obvious.
Students do have serviceable intuition for continuity, for area, and for exponential behavioramong other things. It is, of course, incomplete. They can complete it at much the same time that they complete the realsas juniors or seniors in a strong introductory analysis coursewhich we should require of all majors. But we should not think of beginning calculus as an introduction to analysis.
As I have suggested earlier, I ask them to deal with questions of approximation: How close must x be to 2 in order that we can be sure that x^2  6 x + 11 is within 1/250 of 3? To how many digits must we know \sqrt{2} if we want to be sure of the first 7 digits to the right of the decimal in \sqrt{\sqrt{2}}? I ask for correct and complete reasoning from them in their dealings with these questions. The student who has dealt carefully with questions like these will be ready for epsilons and deltas later on.
I prefer to take area as a primitive concept, and introduce Riemann sums after discussion of the Lefthand Rule, the Righthand Rule, the Midpoint Rule, the Trapezoid Rule, and Simpson's Rule, as a very general way of approximating definite integrals. (But I do RiemannStieltjes integrals in my Advanced Calculus course. See, e.g., Widder's _Advanced Calculus_, republished a few years ago by Dover.) I pay careful attention to the error estimates for the standard definiteintegralapproximation routines just mentioned. I do so not because those routines are particularly useful in practice, but because they are tied up with the notion of using a parameter to control closeness of approximation: That's the whole issue that underlies limits and continuity. Moreover, this discussion gives me good opportunities to ask them to show me that they know how to establishs bounds for functions, and this is a fundamental idea I think we've paid too little attention to in our moe traditional calculus sequences.
I think that giving the definition of the natural logarithm by means of a definite integral is one of the biggest mistakes we ever made in the traditional calculus sequence. That definition belongs in the strong introductory analysis course they will take laterif they major in mathematics. To begin with, that definition flies in the face of what we told them logarithms are just one course ago in their precalculus courses. To make matters worse, they have a very hazy idea of what a definite integral is (especially if one has taken the definite integral to be the limit of the Riemann sums as the mesh goes to zero) at the time that integral definition appears, and so it is, in effect, meaningless to them. (Ask them, in an informal situationsay, at a departmental picnic, what a definite integral is. Most of them will give you an operational definition: It's what you get by finding an antiderivative and then evaluating it at the limits and doing a subtraction. But their "definition" won't be nearly as succinct or to the point as the language I've just used. If you summarize their remarks in the language I just used, they'll say "Yeahthat's it.")
I'm quite happy with an intuitive extension of the exponential function from the rationals to the reals by continuity followed by definition of the logarithm as its inverse. The number e appears quite naturally as the value of a for which the function x > a^x has derivative x > a^x (or as the base for which the derivative of the function x > log_a x becomes x > 1/x).
Lou Talman
(To be continued)



