Friday, February 11, 2011

Abstract thought in applications

Sitting here in Para Coffee grading my students' first exam of the semester. It's not going so well, or at least not as well as I'd hoped. I feel like the last time I taught this course my students were a little more prepared for calculus. I guess what I have to remember is that this is spring term--a chance for those who really hate math to get this class out of the way. Oh boy.

As I grade, I'm realizing how silly it is when we break things down into "abstract" and "concrete." In math, solving concrete problems takes abstract thought--that's the whole point of even learning this stuff. You need to build something with a given amount of material and optimize the amount of space it contains. You need to maximize the speed of your vehicle. Whatever. The point is that if you just attacked the problem with no abstract thought whatsoever, you'd be stuck. If you can't imagine the problem as anything other than what it looks like on the surface, you can't solve it.

The real distinction, in my opinion, is between abstract and habitual. Students hate word problems. Why? Because they don't always look the same as every other problem they've ever done. They actually have to approach the problem on its own terms. That's frustrating, and students commonly express the opinion that their teacher "never gave us anything like this before." Students can do habitual assignments pretty well. It's much harder to get them to actually solve a problem. That's why they'd prefer to just take a bunch of meaningless limits and crank out a number or a "DNE" (does not exist) than solve real problems that someone in real life might actually ask. I worry about the implications for their future occupations. Will they be stuck just doing the same task over and over again, simply because that's easier? I suppose there's nothing wrong with that. It's just that you'd think a UVA student might imagine something more. I don't know.

The problem is that students have a hard time seeing the heart of a problem. They don't want to sit down and think about principles. It's so easy to treat everything mechanically. Yet it's so much easier to actually solve a problem by first figuring out what in the world they're looking for. And yes, that means abstractly. For instance, if you're looking for area, write down area = length times width. This is a thoroughly abstract statement, but it gives you a guide to solving the problem. You can't approach a concrete problem and solve it without identifying the abstract principles governing it. Indeed, one's knowledge of the abstract is tested best in application questions.

That's it for now--just a short rant on the difficulties of teaching math. I admit I can't help like feeling that my students' failure is my failure, even though that's absurd. I'm just some grad student the school throws up in front of these kids for less than three hours a week. Most of what these students need to know, there's no way I can teach them in three weeks before their first exam.

1 comment:

  1. I fairly recently discovered your blog -- I'm not 100% certain how, but at any rate, here I am -- and wanted to comment.

    First, Sorry about the students. Personally I think teaching could be ok some of the time, but maddening much of the time, so I admire teachers.

    Second, interesting observations about abstract vs. concrete. I think that actually, my own problems with mathematics were the opposite, and stemmed from the fact that my textbooks tried to show me concrete examples, but often that wasn't enough to show me the abstract principle, and it frustrated me. I couldn't see the big picture, so showing me lots of examples was futile. But maybe that's not a normal math experience?

    Anyway, I hope some math is getting through to the pesky students. :)



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