May 17, 2002

"A mathematician is a machine for turning coffee into theorems."--Paul Erdös

My Integrated 1 placement now consists solely of computer-generated multiple-choice testing. At this point I'm resigned to it. I don't know of any better way to handle students' disparate needs in this class that wouldn't also involve a much higher teacher-to-student ratio in the room. In that sense, the Accelerated Math program is a good thing.

However, I'm bothered by the fact that the questions seem solely aimed toward better test-taking rather than greater conceptual understanding. Given the necessity of test-taking skills for those who stay in the school system, it's not entirely bad, of course; but anything the students learn about mathematical inquiry is going to be stilted. They're not learning to "do mathematics" so much as "learning to do formulaic questions." If I had an alternative to suggest at this point, I'd suggest it. I don't. I'm resigned to the situation, but I wouldn't say I'm happy per se, either.

This day actually wasn't as hectic as it usually tends to be (though it varies depending on the attendance du jour). Most students will generally wait up to 5 minutes to have a question answered before they start goofing off in earnest out of sheer frustration. There are a couple who will not wait longer than a minute no matter how many other hands are raised (usually around 2 to 4). Since then, the addition of a printer in the classroom has helped enormously, as it means that neither my cooperating teacher nor I is always dashing into the "annex" area to pick up printouts every 5 minutes.

Because students are in different places on different units, and inevitably a few of them are testing at any given time, I'm not sure how one could logistically get them to pay attention to yet another decontextualized, and potentially more confusing, example. I find it difficult to get two students to focus on working through a problem them have quesions about in a one-on-two session; the thought of doing the same for an entire class just boggles the mind. Heck, I can't even grab more than 50% of students' attention in the Integrated 3 class for longer than 5 minutes (if even that).

Students don't learn by having teachers talk at them. Even in the Integrated 3 class, I've tried to cut back on any form of whole-class lecture as much as possible--limiting it mainly to introductory "road maps" and "here's a couple examples, now you try these other examples and learn this for real" sessions. I've very occasionally also used "exposure" lectures, where the point is to show students something "neat" (such as 3-dimensional Cartesian coordinates) without expecting them to "learn" it for anything to be assessed. The relevant point is that in that class, I've tried to force the students to ask each other through designated "experts" or just plain asking, instead of going to the board where half of them will fall asleep again, and need to ask me personally afterward; it's far more efficient than taking the burden of individualized explanation solely on myself or attempting mass-dissemination of understanding. This doesn't mean I don't answer questions; it means I answer once or twice, then direct other students to the first to ask.

Of course, in the other class I have had the luxury of structuring instruction as I pleased, at a pace that seemed reasonable. I was talking to Manny Medina about the trend in including matrices in high school math and we both suspected that, other than the exceptional case of "advanced" classes champing at the bit, it was due to more "upping the ante," as neither of us could think of things students could learn from a cursory study of matrices that couldn't be done by existing means. I imagine the murderous pace of Integrated 1 (and probably Integrated 2) is due to a related phenomenon.

Classroom norms and how to build them? It beats me. I think, even given a completely free hand--which would not have been reasonable in any case--I would have used a slightly different approach to this class than the other, a gentler and even more gradual one. They are different students with different needs, and unfortunately are subjected to a different curriculum (in the sense of the testing-every-6-weeks pacing, as though people learn on a schedule).

If I were in such a class next year, I would really, really rather start with a unit on problem-solving involving something a little "friendlier," something that didn't look like the algebra and geometry they knew they'd already "failed." Topology (for example) is not going to be anyone's panacea, but certain slices of it have the remarkable property of being approachable with very little prior background, yet also representing significant mathematics. The Jordan curve theorem, various map-coloring theorems, graph theory through the Bridges of Königsberg...now if only I could actually attempt such a course.

I've come to the conclusion that I don't even want to be teaching first-year algebra next year (though I doubt I'll have that choice), and the reason, paradoxically, is that I think it's far more difficult than second-year algebra. The foundation you're laying for subsequent study of the traditional, mainly sequential math sequence is enormously important, and you can assume far less about what students know.

By that reasoning elementary school teachers should have the most experience with math. I am comfortable with this line of logic, though I acknowledge that it's unlikely to see practical realization anytime soon. Perhaps I should shoot for college calculus at a community college?

Copyright © 2002 Yoon Ha Lee <requiescat@cityofveils.com>
Last updated on 8 June 2002