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 is irrational. For those who don't know it, here it is, in some detail:
If Sqrt is rational, we can find positive integers p and q having no common factor and such that Sqrt = 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 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 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