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I've been a math teacher for ten years, and I'm firmly in the "thinking-not-memorizing" camp. Of course, there are some things that have to be memorized: the multiplication tables through 9, the quadratic formula, and so on. But I see far more teachers erring on the side of too much memorization and too little deep understanding, than the other way around.

You may argue that standardized test scores prove me wrong, but that's a problem with the tests. It's very easy to test memorized facts. Testing deep understanding is not impossible, but it's a lot harder.

You may now want to dismiss me as politically correct, but I am the exact opposite. You'll believe me in a paragraph or two.

Step back and ask yourself, what is the purpose of all this math education? Your high school teacher lied to you when she told you that you'll need it in the "real world." You don't need to solve quadratic equations in order to bake a circular pie, calculate the tip at a restaurant, pay your taxes, or run a major corporation. You don't need anything beyond, say, fourth grade fractions. You do need math if you're going to get a degree in economics, or engineering, or a few other highly technical fields. But--here is my point, and it is the very politically incorrect point I warned you about--90% of my students are never going to go into those fields, and I already know which ones. (They know too.) I can get those students to memorize enough useless crap to pass a standardized test and then forget it all the next month, but it's not really doing anybody any good.

The students for whom math actually matters--the ones who might actually wind up using it after they graduate--those students need to actually understand what they're doing, and learn to "think in math," much more than they need to memorize trig identities. But "thinking in math" is not a squooshy, vague mathematical version of self-esteem. It means being able to set up and solve word problems. It means being able to give me a clear mathematical demonstration of why a negative exponent goes in the denominator, instead of just spitting it back. It means a lot of skills that are specific and testable, and are a lot harder than the stuff on the SAT. And (politically incorrect point #2 coming), the reason most teachers don't teach those things is because they don't understand them themselves. These teachers should be fired.

If I have your interest at all, I would love to get you to read my own essay on the topic:

and a much longer and cleverer one by Paul Lockhart: