Mathematics and Technique

There are two sorts of truth, Bohr claimed. Ordinary truths, whose opposites are falsities, and great truths, whose opposites are also great truths. Now it should go without saying that one shouldn't take Bohr too literally, but these latter are the ones that interest me. Libertarians have some of these truths, I think, mixed together with a gross misinterpretation of the nature of the human condition. One of those confused libertarians links to this interesting great truth about the teaching of mathematics.

Briefly, Paul Lockhart's Lament, as he styles it, is that mathematics is not being taught as one of the arts, as it ought. An emphasis on technique and memorization, at the expense of ideas, makes students dislike mathematics and resist learning it. I'm not sure the comparison with music that he chooses really makes his point:

musician wakes from a terrible nightmare. In his dream he finds himself in a society where music education has been made mandatory. “We are helping our students become more competitive in an increasingly sound-filled world.” Educators, school systems, and the state are put in charge of this vital project. Studies are commissioned, committees are formed, and decisions are made— all without the advice or participation of a single working musician or composer. Since musicians are known to set down their ideas in the form of sheet music, these curious black dots and lines must constitute the “language of music.” It is imperative that students become fluent in this language if they are to attain any degree of musical competence; ...
Sadly, our present system of mathematics education is precisely this kind of nightmare. In fact, if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done— I simply wouldn’t have the imagination to come up with the kind of senseless, soulcrushing ideas that constitute contemporary mathematics education.

You get the idea, or, if not, read at the link. He isn't being original here. Every decade or two, somewhat similarly motivated people discover that math isn't fun for most students because students aren't taught to be mathematicians. Sometimes these ideas turn into a movement, usually with the result that kids wind up doing things that their parents and teachers don't understand, kids find they can't solve math problems, and a counter-revolution produces a back to basics movement and we begin again.

I am not entirely unsympathetic to Mr. Lockhart's argument, but I am pretty unsympathetic. I am unsumpathetic because I have seen the devastation wrought by previous similarly motivated "reforms" - new math, discovery math, etc. I also suspect that he has either forgotten or never knew how music is actually taught. I would also say that mathematicians should not and cannot be trusted to design a math curriculum for non-mathematicians. Not to be impolite, but a professional mathematician is a freak of nature - unnaturally inclined to live in a very narrow part of his/her head. He can't be trusted to guess what kids find interesting and can't be trusted to remember how he himself learned.

Put techniques for discovering the fun of mathematics in front of 100 school kids and 6 will say cool, and become interested, 34 will try to figure the thing to memorize to pass this new stupid class, and the remaining 60 will look blankly and uncomprehendingly on at yet another bit of tedium. For their teachers, the corresponding numbers will be 2, 10 and 88.

I also distrust this notion of mathematics as an art. Considered purely as an art, mathematics would rank in popular appeal somewhere between dog collar calligraphy and vegetable peeling collage - the audience would only include other mathematicians, and darn few of them. Mathematics is interesting to the wider world because it is useful, necessary and indispensable. We teach it in schools because so many people need to use it. That means that schools need to teach the kinds of skills and techniques that adults actually need to use. Nowhere in Mr. Lockhart's essay do I see any recognition of the way mathematics is used by 99.9% of the human race.

We should remember too, that much of art consists of learning technique. Musicians, or at any rate, most musicians, do have to learn how to read music, and all have to spend thousands of hours learning the technique of their instruments. If Mr. Lockhart wanted to argue that schools are still wasting far too much time on obsolete techniques of mathematics, he would find a ready listener in me.

Games, puzzles and the other fun activities he thinks should take the place of the current curriculum are fine, but is there any chance that any significant segment of the population would thereby learn enough mathematics to do their taxes, much less calculate statistics, solve differential equations or learn the much more difficult techniques needed to actually do mathematics professionally? I doubt it.

I have been a frequent critic of the math curriculum myself, but I regret to say that I see almost nothing valuable in Lockhart's suggestions. Anybody care to venture a contrary opinion?