Math should be uncanny.
When I read E.D.’s piece on maybe teaching less math in elementary school, I figure that since I studied math in college and still do some recreational math, I should have something to say about math education in America. But whenever I dip into literature on math pedagogy, I come out baffled. There are a lot of people who have worked really hard to figure out how to make an effective math curriculum, but to students it all still seems arbitrary. I am not at all sure what the math curriculum ought to be like. So here are a few humbly presented thoughts on learning math, from someone who’s always enjoyed mathematics and had really great math teachers in high school. (If it matters, I went to a public high school.)
I’ve always found it very handy that I have my basic times tables and division tables burned into my brain, and it seems like elementary school is a fine time to do this kind of rote learning. I’ve tutored kids that will use calculators for single-digit multiplication, and it always makes me sad. A lot of the weirdest stuff I see in math textbooks for elementary schoolers seems to be in exercises designed to teach addition and multiplication by the kind of extremely basic reasoning that elementary schoolers are capable of, and this strikes me as a recipe for frustration and lifelong math fear. Maybe they don’t need to understand what’s going on until later. I don’t think my brain was developed enough to get the kinds of abstractions that math is really about until middle school.
Our high school math courses are a whole other mess. The problem with high school geometry right now, at least in the textbooks I have seen, is that it veers wildly between a Euclidean framework and the function-on-the-Cartesian-plane setup that leads into calculus. I don’t know what should be done about this, but the Euclidean stuff gets so watered down that it’s practically useless. Unless the math teacher is very good, the students aren’t going to learn mathematical reasoning in the section on proofs; they’re going to learn how to play a weird and somewhat arbitrary fill-in-the-blanks game that’s based mainly on guessing what the teacher wants to see. And then, SATs aside, they’ll probably never see Euclid again, even in college. When I learned how to write rigorous proofs in college, I had to prove number-theory statements rather than geometric statements.
But if I can end on a weakly Poulosian point, the political pressures to make kids good at math usually comes out of a view of math as a collection of techniques that will help them become awesome scientists or engineers. Or hedge fund quants, I guess. So boring! Very few teenagers, if any, will have the kind of drive to learn calculus just because they hope to be engineers or accountants in a decade’s time. You get excited about math when you manage to catch even a small glimpse of what’s so weird and special about mathematics: the abstract realm where numbers live, shimmering and clear, connected to each other in so many strange webs; where perfect flashes of inspiration disperse the thickest mists of confusion; where mysterious creatures — functions, complex numbers, rings and fields — dwell up in the high far-off hills. Not that most students would put it this way. In high school, it’s more like: “Isn’t it weird that the golden ration shows up in the Fibonacci sequence?” or “Isn’t it bizarre that I can find an exact value for the area between these two curvy lines?” Small moments of wonder can be enough to make the whole endeavor worthwhile — but if the teachers don’t know what’s wondrous about math, how will the kids ever see it?
Update: I wrote all of this last night, and I’ve got two things I want to add. First, the comment from T. Greer is worth reading, since it gives an actual educator’s perspective. Second, the book that first evoked number-wonder in me was Math for Smarty Pants by Marilyn Burns. I read it so many times. Buy it for your kids.