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Re: [math-learn] Pedagogy and Natural Ability (was Inquiry Method and
Motivat...
Posted:
Nov 30, 2003 1:52 AM
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Evidently, mail isn't going out from the main college server tonight,
because I sent this out 15 minutes ago and Yahoo hasn't yet returned
it. So here's another copy (with some typos corrected) by way of a
different server.
On Nov 29, 2003, at 1:09 PM, Ze'ev Wurman wrote:
> Can we also agree that:
>
> - Overwhelmingly, the ability to perform mathematics requires
> significant amount of memorization and practice.
> - Overwhelmingly, understanding mathematics requires significant amount
> of memorization and practice.
That would be like agreeing that getting very far in a car requires a
significant amount of air in the tires. Of course, memorization and
practice are required--though, I suspect, not in the quantities often
suggested, and, I also suspect, not of the things that the supporters of
Mathematically Correct want. The trouble with the curricula and tests
supported by the "correct" crowd is that supporting them is like
supporting a car maintenance program that involves no more than reading
the pressure in the tires periodically.
Tanner, for example, continually plugs his own favorite theorems; he
tells us repeatedly that knowing these theorems makes life so much
easier. I suggest that there is a great deal more to it than he thinks;
even it they were the best way for everyone to go, it would not be
enough simply to know those theorems and practice their
application--though that is certainly what he seems to mean to imply.
The theorems in question are at best summaries of important ways of
thinking about things, and--like the Laws of God--if those ways of
thinking are to be of any profit to the thinker, they must be written in
that thinker's heart.
Bishop continually plugs Saxon and standardized tests. This is
memorization and practice distilled and purified. And signifying
nothing. It's like using purified nitrogen in the tires and checking
the pressure hourly--at the expense of everything else that makes the
car go.
We rarely hear any of the "correct" crowd speak about the necessity of
giving good explanations of why things are the way they are. (In more
advanced mathematics, this becomes the necessity of giving a decent
proof.) We rarely hear any of the "correct" crowd speak about the
necessity of trying to solve difficult problems--problems that the
student has the tools to solve but has not been shown how to work.
You haven't gone far enough, Ze'ev. Overwhelmingly, the ability to
perform mathematics requires significant amounts of thought, which must
include committing certain things to memory, and understanding, which
can only be demonstrated through application of the principles involved
to appropriate problems upon which the learner has not yet had an
opportunity to practice.
Here is an example of what I mean. Consider the standard proof that
Sqrt[2] is irrational. For those who don't know it, here it is, in some
detail:
If Sqrt[2] is rational, we can find positive integers p and q
having no common factor and such that Sqrt[2] = p/q. Equivalently,
2 q^2 = p^2. This means that p^2 is even, because it is divisible
by 2. If p were odd, we could find an integer k so that p = 2 k +
1, and this would mean that p^2 = (2 k + 1)^2 = 4 k^2 + 4 k + 1 = 2
(2 k^2 + 2 k) + 1 = 2 M + 1, so that p^2 would be odd. Consequently
p must be even--say p = 2 N for a certain positive integer N. Then
p^2 = 4 N^2, so that 2 q^2 = 4 N^2. The latter is equivalent to
the equation q^2 = 2 N^2. But this last equation tells us that q^2
must be even. And as we have just seen, this means that q must be
even. Thus, both p and q are even, and the fraction p/q could not
have been in lowest terms. As this is a contradiction, it follows
that Sqrt[2] is not rational.
Give this proof to a collection of junior mathematics majors and tell
them that you expect them to study it until they understand it. Give
them ample time and answer any questions they raise. Then, a week or so
later, quiz them on the proof. Most will be able to recite it with
reasonable accuracy, though a small fraction will garble it beyond
meaning--thus demonstrating *their* lack of memorization and practice.
But that's just the preliminary. Now ask them to prove that Sqrt[3] is
irrational. Almost all of them will fail miserably. This means that
they didn't understand the original proof at all. They had simply
memorized it, perhaps even practiced reproducing it. But they had not
*thought* about it to any great extent. It suggests to me that they
haven't even learned what it means to understand an argument. The
memorization and practice that you suggest wasn't helpful.
--Lou Talman
Department of Mathematical and Computer Sciences
Metropolitan State College of Denver
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