Sunday, June 21, 2009

On Teaching Math

The Unknown Daughter is also known by most of my family as "The Math Princess." She seems to get math concepts more quickly than almost anyone else at her grade level. Well, there is one one boy in her class who is just about at her level, but she has a better Rear Naked Choke than him (knowing what boys can be like, I started teaching her various martial arts techniques at age 4 - I figure if she cripples the first boyfriend, my work is largely done).

She's really good at the rote learning/memorization/multiplication tables part of math, and that's important (hey - if you know your tables cold, many things seems to get easier). But what makes me happiest is that she really enjoys the math puzzles I give her. When she finally sees a pattern, she gets really excited. For example, when she realized that she could use the distributive law (i.e a(b+c) = ab + ac) to do mental math, she got really excited.

So, I enjoyed this piece titled "A Mathematician's Lament" by Paul Lockhart. Here's the best line(s) IMO:
Part of the problem is that nobody has the faintest idea what it is that mathematicians do. The common perception seems to be that mathematicians are somehow connected with science— perhaps they help the scientists with their formulas, or feed big numbers into computers for some reason or other. There is no question that if the world had to be divided into the “poetic dreamers” and the “rational thinkers” most people would place mathematicians in the
latter category.

Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or physics (mathematicians conceived of black holes long before astronomers actually found any), and allows more freedom of expression than poetry, art, or music (which depend heavily on properties of the physical universe). Mathematics is the purest of the arts, as well as the most misunderstood.

So let me try to explain what mathematics is, and what mathematicians do. I can hardly do better than to begin with G.H. Hardy’s excellent description:

"A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. "

So mathematicians sit around making patterns of ideas. What sort of patterns? What sort of ideas? Ideas about the rhinoceros? No, those we leave to the biologists. Ideas about language and culture? No, not usually. These things are all far too complicated for most mathematicians’ taste. If there is anything like a unifying aesthetic principle in mathematics, it is this: simple is beautiful. Mathematicians enjoy thinking about the simplest possible things, and the simplest possible things are imaginary.
He's an advocate of teaching by puzzles. like "if you inscribe a triangle in a rectangle, can you figure out how much of rectangle's area is captured by the triangle" (for an arbitrary triangle, that is). Read the whole thing - it's worth it.

HT: Kids Prefer Cheese