Teaching math 2008/05/04

A Mathematician's Lament, by Paul Lockhart (via DHall)

I'm writing this response as I read through this fascinating paper on math in schools.  I have for many years held the belief that (at the high school level at least) anyone can make straight A's, even if they're doing everything in their power to challenge themselves.  This paper, if anything, strengthens my belief.

When I entered college, I thought I was pretty good at math.  Then I met quite a few people that were good at math, so I stopped saying as much.  I still very much believe it, though.  A more surprising lesson was that I actually hate math classes.  What I enjoyed were classes that relied on constructing and evaluating hypotheses and algorithms based on their mathematical merits.

Lockhart and the many others who've called BS on the way math is taught to students today have a point: math is inherently creative.  Beyond that, though, they get a little overzealous.  Painting and poetry (my go-to counter-example for the alliteration and audible distinction from "math") are commentary about the world and its people, inspired by the painters and poets coming before.  Math is an observation about the world and (less frequently) its people, based on the work of mathematicians coming before.  Inspiration does not equal groundwork.

"[Math] has relevance in the same way that any art does: that of being a meaningful human experience." This discounts the most beautiful thing about math: it elegantly describes the world in which humans experience.  Economics is a fantastic framework for teaching this very thing.  Economics is about building up simplified mathematical models of repeated observations about our world.  Teaching math through economics will work because the types of observations economics sets out to describe are far more tangible than the concepts hand-waved away by formulas memorized in math classes.

"'What do they want me to do?  Oh, just plug it in?  OK.  [said the fictional average student.]" This is an all too common conversation, so much that teachers frequently reached for the canned reply, "plug and chug!" I had the distinct torture of taking a physics class in high school where the teacher instructed us to "just take the cosine" to find the length of one side of a triangle that didn't contain a right angle.  My classmates were happy to follow along.  I felt the need to disagree.  Teachers are not infallible and that must be accepted before the intellectual dialogue Lockhart describes can begin.

The lack of infallibility opens the door for other liberties, too.  Paul Graham wrote that schools are primarily holding pens: "In fact their primary purpose is to keep kids locked up in one place for a big chunk of the day so adults can get things done." Without teachers' infallibility, the signal to noise ratio in the classroom would approach zero.  Operating within the constraints of order first, learning second, it isn't at all surprising that school is about following protocol.

Joy of joys, there's a whole section on High School Geometry (capitalism his).  Like the author, I found the material presented in High School Geometry fascinating and the scaffolding erected around it suffocating.  I do lots of math in my head and made absolutely no accommodations for my teacher that insisted I "show my work." I showed my thoughts, not my arithmetic.  But due to the infallibility discussed above, I could not tell her where to shove it.

Math is not an art form the same as painting or poetry.  But it is creative.  It is an experience.  And to teach it, it must be a dialogue.  We need better teachers.  To get there we need to make teaching noble again.  More realistically, we need to make teaching a financially sane decision.  Capitalism is cruel, I know, but it'll get a whole lot worse if students learn only process and not how to progress.