Friday, October 21, 2011

The Role Of Memory In Theory

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I have to say that I liked [a research problem] immediately. Dick and I continued talking about it in his office right after lunch, where we reduced a proof to a simple fact about Markov chains. We needed to know the expected return time of a certain random walk. Unfortunately Dick had to go teach class, so I told him I would search the library, which was downstairs from his office, while he was in class. I hoped that I could find a formula for the expected return time.

Recall this was in 1979 and there was no Google, no search engines, no Internet, so one actually looked things up by hand. I went into the math library and began a quick linear search of all journals that seemed like they had anything to do with random walks. After about an hour I found the formula we were looking for: the expected return time was just the reciprocal of the probability of being in that state.

As soon as Dick returned from his class I showed him the formula. We both were a bit embarrassed, since the formula was easy to prove. But worse, it was something that we had no doubt seen before, but forgotten. If we’d had better memory we would have proved the theorem instantly. In any event the conjecture became a theorem, and we all wrote up a paper together [..].

Should we stress more memorization in teaching and learning math [..] ?
I think education’s current stance, encouraging memorization as little as possible is the right one [1]. If a student can memorize however, good for him / her, they can go ahead and do that. But curricula, especially math curricula needs to be about deriving things, showing how pieces connect, how to move from one concept to another, so students understand how things are in relation to eachother, and that kind of comparative ability is what makes science tick.

In your story, you and your friend could not remember Theorem X or Y, so what? You had considerable skill in guessing where the answer would be, and you researched it, connected the pieces, and reached the conclusion. How is this not a success story?

Directing curricula can only be done in blunt moves, supporting memorization is one of those blunt moves, but once started, the details that follow can lead you in some dark places. Feynman saw this up close when he taught for a while in Brasil, and after he came back to US, he found the same thing in US curricula.

But I agree that having more things in memory (especially if they were understood first) can make a difference in research.

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[1] Obviously how much they succeed in implementing this, with average teachers, their concentrated modernist institutions are whole another matter. But at least the stance is correct.

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