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Topic: On Reform, Part 3
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Posts: 1,838
Registered: 12/6/04
On Reform, Part 3
Posted: Jun 23, 1997 11:44 AM
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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 about--if 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 counter-intuitive 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 behavior--among other things. It is, of course, incomplete. They
can complete it at much the same time that they complete the reals--as juniors
or seniors in a strong introductory analysis course--which 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 Left-hand Rule, the Right-hand Rule, the Mid-point Rule, the
Trapezoid Rule, and Simpson's Rule, as a very general way of approximating
definite integrals. (But I do Riemann-Stieltjes 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 definite-integral-approximation 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 later--if 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 pre-calculus 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 situation--say, 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 "Yeah--that'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)

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