22 March 2008

The state of mathematics education

Keith Devlin recently published in his MAA column an essay by Paul Lockhart titled A Mathematician's Lament, a critique of American primary school mathematics education. It's a fascinating read, and the quickest 25 pages I've read lately.

Marvin Minsky has written an essay, What makes Mathematics hard to learn? In it he suggests some ideas for teaching math effectively, and relates a couple of poignant anecdotes about math education, my favorite of which is this one:

...that child [who had been struggling with learning to multiply] had a larger-scale complaint: "Last year I had to learn the addition table and it was really boring. This year I have to learn another, harder one, and I figure if I learn it then next year there will be another one and there'll never be any end to this stupid nonsense." This child imagined 'math' to be a continuous string of mechanical tasks— an unending prospect of practice and drill. It was hard to convince him that there would not be any more tables in subsequent years.

These got me thinking about my mathematics education. Perhaps the single most lasting skill I've picked up from doing math is learning how to prove things: not those two-column geometry proofs that everyone is tired of, but really knowing how to put together a line of argument that is both easy to follow and watertight. And there is not just one way to do that. Like writing, there is both art and science in it: choosing terminology and metaphors carefully; designing the structure and scaffolding of an argument; putting in enough details, but not too many; being economical with words.

Moreover, from mathematics I picked up an inclination to ask why things are true. In math— unlike in any natural science— you can always get down to the bottom of things by asking "Why?" until you get back to the axioms. Frankly, that's an amazing idea.

Yet, I saw very little of the "art of proof" and the "mathematician's skepticism" until I started going to a math circle in high school. Most of my public school math classes focused on learning recipes rather than talking about the why. When we learned, say, the quadratic formula, we did the derivation (based on completing the square) in class. But that derivation was soon squirreled away so we could do "applications", which is a fancy name for plugging numbers in ad nauseum. Personally, I find this tragic: the trouble is that in mathematics, the process is far more important and interesting than the result. More pragmatically, a student who remembers the idea of completing the square can derive the quadratic formula (and more) anytime, but students are given little incentive to think about, or remember, that beautiful idea. Most students will forget the quadratic formula once they have stopped using it for a few years, and if their experience with mathematics is that it is just a bunch of facts, they are more than likely never going to get that quadratic formula back.

The result of these methods, as Lockhart notes, is that pretty much no one today knows what mathematics is (not even math teachers). And students are being led to think that they dislike math when they actually just dislike whatever it is that they're being taught in math classes. Doing math is a deeply creative and enlightening process, but those aspects of mathematics are very rarely seen by the general public.

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