My biggest problem as a math teacher?
I’m not sure anyone agrees on what math really is. Even among math teachers and coaches, I’m not sure I can get agreement. Forget it if we need to include parents and administrators.
Last week, I was talking to an administrator about math instruction (They’re planning on getting rid of the program I’m teaching right now.) His theory on math instruction? Make everything real world, applicable, and concrete. I’ve heard dozens of other math teachers say they teach how to translate every problem into money because the kids can understand that.
Here’s why I have problem:
I don’t think math is about being concrete. I think the whole point of math is abstraction. I am clearly out of step with a lot of people here.
Here’s a semi painful analogy that I hope will illuminate my belief:
Math (and by my opinion, abstraction) is like leaving the ground. It allows you to jump from one concrete reality to another, without having to contact the ground in between.
The early abstractions are easy. Neither the greeks nor romans managed to come up with the decimal system, yet our kids pick it up well. Multiplying gets a little bit more difficult, but it works because it extends the same model. Fractions, even more difficult, but you can still get them by constant concrete reinforcement.
We’re picking up speed, we’re running, but we haven’t jumped yet.You can come up with models that explain negative numbers, but you can’t do negative numbers on your fingers. It’s the first time you have to use an abstraction without a strong concrete reference to support it.
Negative numbers are jumping.
Algebra is when you strap on a hang glider. It’s the first feel of taking off in one very concrete place, flying with no concrete support, and landing in another very concrete place, but via a journey that was out of touch with the ground the whole time in between. It is the first serious brush with abstraction without safety nets.
After that come ultralights, piper cubs, beechcrafts, twin engines, small jets, all the way up to 747’s, bombers, high performance fighters. There is a whole world of abstraction, of leaving the concrete behind.
How many people do you know who say “I never got algebra”?
When I tell someone that I teach 8th grade, that’s usually the answer I get.
How do I explain to them that their desire to be on the ground, that their encouragement to help my kids stay on the ground, is the exact antithesis of math?
In a world where most people think that math means arithmetic, how do I lure my kids into making the leap?
1 Yes, you start understanding a basic concept in the concrete. You can’t learn without a real world tie in. But that’s just a running start. To do math, (to continue abusing my analogy) you need to leap in the air and flap your wings. It takes a lot of work, and it is uncomfortable if you’re not used to using those muscles. I’m not sure we are very effective at getting our kids to do that.