As I mentioned in a previous post, I’m terrible at remembering things—names, faces, dates, all become a blur for me very quickly. That is why, as a young child I was attracted to mathematics. There was very little memory work involved, and what little there was fit neatly into patterns, not as a jumble of unrelated factoids (which was my perception of all the history classes I ever had).
I recently read a critique of Lemov’s Teach like a Champion by The Number Warrior, that points out the similarity between Lemov’s Technique 13, Name the Steps, and Devlin’s satirical title In Math You Have to Remember, In Other Subjects You Can Think About It. I think that both Devlin and the Number Warrior are correct in pointing out a major flaw in much math teaching: emphasis on memorizing procedures rather than understanding them. Once you understand them most of the algorithms in elementary and secondary math are fairly trivial, but memorizing them without understanding is difficult and error-prone.
One of the great joys of math for me was that by learning only a few facts and some general thinking techniques, I could solve a huge variety of problems. I was the sort of kid who never bothered to memorize the quadratic formula, because I could rederive it when I needed it. (By college, I had used it enough that I did remember it, but I still derived most of the trig identities as I needed them, using only a few key facts, like that cos omega + i sin omega is the unit circle.)
In most of the courses I have taught, I am not interested in students learning many facts. What I want students to develop is skill at applying a few facts to a number of problems. The subjects that develop these skills best are math and computer programming. Computer programming is particularly good at developing skill at getting the details right (Lemov’s “Right is Right”), since the computer will do precisely what the students instruct it to, even if that is subtly or ludicrously wrong. The need to break big problems down into smaller pieces, work on the pieces separately, and still be able to put the pieces together again is particularly evident in programming.
I have little use for touchy-feely approaches to math education that result in neither memory nor understanding, or ones that insist that no child is capable of abstract thought so everything must be physically manipulable. I think that the best pedagogy starts with interesting questions (“the hook”), then helps the students learn the tools needed to answer those questions. What’s nice about math is that good craftsmen can get by with a fairly small toolbox, if they learn how to use the tools well.