Mathematical Memories

A big battle in mathematics teaching concerns the importance of memorization. The latest incarnations of "new math" are adamantly anti-memorization. I have a small amount of sympathy for this point of view: rote memorization of algorithms not understood seems increasingly pointless in the age of the calculator. My hostility toward the modern approach is based on my experience with students who don't know their multiplication or addition tables. It's hopeless to try to teach children how to reduce fractions, determine primality, or almost anything else when they don't know the basic arithmetic facts, and need to add on their fingers.

When I was in seventh or eight grade, the new, 'hard' arithmetic we learned was taking square roots. I believe the algorithm was a variation on Newton's method, performed in a fashion to look like long division. I remember absolutely no discussion of how, or why it worked - it was just a succession of memorized procedures. It might have had some marginal utility in those pre-calculator days. It would be absolutely silly to teach kids to do this now.

Many of the algorithms we sweated over in school are a bit like that today. In the age of calculator, does it still make sense to teach kids to do column addition? How about multi-digit multiplication, or long division? I admit to some ambivalence on these. Let me try a wishy-washy middle of the road answer: yes, they should be taught, because the algorithms are useful objects for futher generalization, but only minimal competence is really required - speed and precision are overkill. The many pages of column addition I had to do as a kid would be absurd today.

I am more fanatical about the basic facts: every kid should drill to mastery on addition facts and the multiplication tables. Ditto on concepts like place value, the algorithms for simplifying fractions, adding fractions, and multiplying and dividing fractions. Without the multiplication facts, notions of primality and fractions are impossible to master.

Memorization is bad when it substitutes for understanding, but properly employed, it forms the skeletal structure upon which understanding can be built. Memory gives examples upon which understanding can be built. Facts are the signposts of our memory in every field, even an abstract one like mathematics.

Memorization of dates is a similar bugaboo in history, but it is attaching occurences to dates that permits us to make sense of historical connections. How could one understand the relationship of Islam, Christianity, and Judaism, for example, without knowing at least approximately when Mohammed, Jesus, and Moses lived?

I recently tried doing some elementary problems in a quantum field theory book. The first problem I encountered was that I had lost contact with certain neccessary technology: I had forgotten what little I had known about the traces of matrices. The key facts are pretty simple, and Wikipedia had them, but without them I would have been helpless.

It's the same in everything, I think. Everybody needs to remember a lot just in order to do his job and survive. Memorization needs to be understood not as an alternative to understanding, but as an essential ingredient of it. Medical students spend a lot of effort to learn names of bones, muscles, and nerves not because those names are important but because they need to be able to understand how all those parts work together, and because the names are needed to keep straight what is what and how it relates to the rest of the world.