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The problem with abstraction.

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.[1]

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.

{ 3 } Comments

  1. H. | January 26, 2008 at 11:30 pm | Permalink

    In a similar vein: Math is not about Numbers

  2. Zac | January 28, 2008 at 11:03 pm | Permalink

    Hi Mr. K

    Thanks for your post. You are right, mathematics is about abstraction but there are a lot of others things going on as well.

    Something that I often point out to math educators is the idea of readiness for learning. Most 14 year-olds are not ready to abstract and when they are forced to, they retreat into their “I just don’t get algebra” rationalization, where many stay for the rest of their lives. This is a great shame.

    A lot of people have flashes of insight when they study (say) calculus when they are older. “Ahh – that’s what was going on. Why didn’t they explain it like that at school?” Well, the teacher probably did explain it like that at school, but the adolescent students had many other things going on in their heads at the time – and it didn’t involve math abstraction, that’s for sure.

    You may be interested in a recent post on squareCircleZ: Math has to be meaningful, or why do it?.

    Don’t get me wrong – abstraction is important and math does involve stepping away from the numbers and the concrete stuff at times – but we need to be mindful of when that is appropriate… We also need to consider the vast range of needs in our students. Some really have to know how it is used before they will put effort into learning it. Actually, most of us are like that, aren’t we…?

  3. Mr K. | January 29, 2008 at 6:52 am | Permalink

    Some great points, which I intentionally skirted:

    - yes, the ability to abstract is in part developmental. I think it’s a shame that we push standards earlier and earlier. But I also think that there’s a problem in how we approach it – we don’t slowly ramp up the level of abstraction. By long efforts at keeping everything else concrete, we don’t develop the ability to abstract, and when algebra comes along it’s a lot to pick up too fast. For all of our talk of standards, it feels like we’re failing to address some of the longer term, deeper goals, and the lessons we teach have no long term focus.

    - To me, calculus was one of those landing spots, years later. I was lucky enough to go to a high school where the calculus course started in the middle of our junior year, and senior year involved a very tight coupling with the physics course. But the calculus would have been impossible to learn without the abstracting skills I’d developed in algebra and geometry (our geometry course was very heavy on proofs – everything we learned, we had to prove).

    As you say, we need to make stuff interesting. Some recent experiences suggest to me that they don’t really need to know how they’re going to use it in the future (any attempts to actually explain that have always fallen flat). Rather, you need to find a way to make that interesting right now. It doesn’t have to be real world – our kids eat up fake movies and video games with a vengeance. You just need to find the hook, though admittedly that’s usually easier said than done.