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reply to Lou Talman
Posted:
Jun 8, 1997 1:00 PM
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Howard Swann gave me permission to forward this to the AP-Calc group.
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Hello, Lou Talman;
David Klein forwarded a note you sent on calculus curriculum that mentioned
my article in the MAA Monthly.
You state:
``The whole point of the criticism so-called "reformers" level at the MVT is
that the standard presentation *isn't* a proof. ''
The points I attempted to make in the article are the following:
1. A glimpse of the power of mathematical proof is an exciting intellectual
endeavor, worth any number of group `discoveries' based on putting numbers
representing `real life problems' into a graphing calculator. By abandoning
the attempt to do this, we discard one of the strongest arguments in favor
of studying calculus.
However, in beginning multisemester calculus, self-contained rigorous
proofs are possible only when discussing limits at a point; `zooming'
graphic calculators are alleged to eliminate the conceptual difficulties.
Yet, when placed in suitable historical context, we can make a
discussion of the masterful pair <Dedekind's `real numbers', and
Weierstrass' proof that guesses for limits are *unique* under the
epsilon-delta definition> fascinating indeed. For this pair provided the
framework for humankind to finesse, (but not eliminate) the profound
mysteries of human attempts at understanding infinity. This is precisely
the point that Bertrand Russell made in the quotation in the Monthly
article; I urge you to read his entire essay.
Today's `reformers' presumably assume that students are incapable of
understanding the beauty and cleverness of these (admittedly difficult)
ideas. Certainly the conventional presentation often falls short, yet my
own experiments suggest that if the reformers got busy, they should be
able to come up with a way to do this. One has only to see, for example,
the current PBS programs on fractals and the Mandelbrodt set to realize
how fascinating the question of what `on earth' `infinitesimally small'
could mean. I invite you to `zoom' away on the example I cited in the
Monthly article, and welcome the chance to send you the current `handout'
I use when introducing the idea of limit if you are interested.
2. With regard to the MVT: I urged the following treatment in the Monthly
article:
(a) First a geometric argument;
(b) A discussion of why the geometric argument is not a `proof'
(c) A discussion of Rolle's theorem as a special case; a proof to be given
here by CITING the extreme value theorem and celebrating it as a triumph of
global results obtained from the pair <real numbers, Weierstrass' epsilons
and deltas.> The mystery of `number' can easily be re-visited here, for in
what way is there a maximum for -(x^2-2)^2, when no computer has ever seen
\sqrt{2}?
(e) A return to the usual proof of the MVT using Rolle's theorem; rigorous
this time by CITING Rolle's theorem.
Although lengthy, such a sequence of arguments is of value, for it gives a
paradigm for how we traditionally hope that most (non-mathematics-major-)
students should understand their calculus and post-calculus math courses.
A clear distinction should be made between what is and is not a proof;
important proofs should be given where accessible, which will usually
entail CITING (and celebrating) some global result.
If this experience is not encountered occasionally, subsequent math
courses (with their classical texts) will be extremely frustrating to
students and they never get a real glimpse of the power of mathematical
proof, although `reform' is at hand as new texts omitting any notion of
mathematical proof proliferate and it is asserted that somehow the
`meaning' of mathematics is made clear.
You continue with
``But, in point of fact (and Swann's mistaken "counterexamples" to the
contrary notwithstanding..N.B. DO TELL ME THE DIFFICULTIES HERE!), no proof
given in the traditional calculus sequence requires the full strength of
the MVT. ...the Increasing Function Theorem...or ...the Racetrack Principle
suffice in every case. ''
I return to the example of `arc length' cited in the Monthly article:
The traditional argument is the following:
The sum for approximating arc length of y = f(x) is
\sum_{i=1}^n \sqrt{(x_{i} - x_{i-1})^2 + (y(x_{i}) - y(x_{i-1}))^2}.
One then uses the MVT applied to assumed continuously differentiable f(x)
to write
y(x_{i}) - y(x_{i-1}) = f'(x_{i}^*)(x_{i} - x_{i-1})
for some x_{i}^* between x_{i} and x_{i-1}.
This is then converted to a RIEMANN sum of form
\sum_{i=1}^n \sqrt {1 + f'(x_{i}^*)^2} \delta x_{i}
We then CITE Riemann's central theorem, crucial to calculus, that if a
function is continuous, then Riemann sums converge to the integral no
matter where the x_{i}^* is located and the proof is concluded.
I can find no attempt to formally state Riemann's theorem in the Harvard
Calculus, for example; the argument for the formula for arc length is based
on the skilled use of the symbol `~', which is no proof whatsovever.
How can one contrive any `proof' of the formula for arc length using a
`Racetrack' argument?
Yours truly,
H. Swann
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