October 31, 2001

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

The class on Wednesday, October 31, 2001, was not as well-organized as it could have been--I refer here to my own preparation rather than that of the students. It's pretty much my fault for arranging to shadow my Adolescent Development case study student on the same day that my supervisor observed me--I should have stuck with Friday, instead of asking him whether Wednesday or Friday was better. I don't know what I was thinking, but by the time my brain woke up and smelled the coffee, I'd already sent out the letters to his teachers with the date on it. Lousy scheduling, which led to me getting flustered.

I didn't get the class' attention as effectively as I could have. I'm still learning how to be aware of the whole class at once, instead of only segments at a time. It requires a sort of gestalt of attention that I still haven't gotten down, which isn't too surprising (to me, anyway). My short-term attention span is pretty lousy (I fidget, read, doodle, etc.) though my long-term attention span is awesome (I decided in 3rd grade that I was going to be a writer and 10 years later, after a lot of crappy writing, I got my first story published). My sister used to be able to call my name to get my attention (usually when I was reading or at the computer) and then count the seconds until I realized someone had spoken to me. This is not in the least a desirable trait for a teacher, though it worked out well when I was tutoring (or reading in the lunch lines of noisy cafeterias, for that matter) since I could tune out just about everything else. I don't know how you improve something like this other than being aware of it and making an effort to notice everything, though.

What I really lust after in terms of class-attention is a rainstick, though I've considered bringing one of the many portable musical instruments in my collection. A harmonica, perhaps? (You don't get much more portable than that.) Too bad I left everything in New York except the soprano recorder (which was a gift from my sister, and was never in New York to begin with). Clapping number patterns (and getting the class to guess what's next)? Maybe that's too cheesy for high schoolers. Hmm. In any case, the spreading radius of student-attention wasn't intentional.

I wish I'd been a lot clearer in that explanation of the distance formula--and it's true that it's essentially a Cartesian application of the Pythagorean theorem. I did want to review it because the textbook specifically required its use, though that's not necessarily the best reason to mention something. I think not-teaching the distance formula explicitly is a good idea, except that by this class they'd already seen it. (And let's not even mention non-Euclidean geodesics.) It was clear to me where I was going, but judging from the class' reactions, a lot of them were confused. I'm also not sure that the steps of the derivation were appropriately paced (going from a nicely-labeled right triangle to a triangle placed on a Cartesian coordinate system), though I'm at a loss as to how it could have been better paced.

(I maintain, as I have before, that if you think about all the things actually involved in teaching--math or any other subject--it starts to seem like an impossible task.)

In C&I we've been spending some time with Wiggins & McTighe's Understanding by Design, and I feel as though I need to be applying those principles a lot more than I am. If I'd thought about what I wanted the class to walk (or limp, or skip, or run) away with, it would have been obvious to me that I needed to clarify the lesson's goals. (Whether the attempt actually succeeded would be another story, but there would have been an attempt.)

Randomly, I also now understand why teachers fall prey to the temptation to question "bright" students less, because I just fell prey to it with the whole "AAA" triangle business. The student who came up with that as a justification has one of the more mature mathematical minds and so I just took his word for it (though in all fairness, I might have taken another student's word for it, considering how flustered I was feeling).

The group-poster-proof, which culminated on Friday with the entire period spent on presentations, went pretty well despite a rough start, though. Yes, I biased the proofs: there were four pairs of propositions to be proved; or rather, four propositions, each set up two different ways to "bias" the resulting proofs toward either an analytic/algebraic or geometrical method. The posters weren't intended so much as final products as visual aids for the students' presentations (and also to give some sense of solidity to results that might not have been "complete" in the sense of a total proof). A lot of great work and questions, not to mention frustration.

The frustration, I think, was in some sense a good thing because a) it didn't seem to reach the point where students felt like giving up on the problem as too difficult or irritating and b) it resulted from the sensation of getting stretched in their mathematical reasoning. (God only knows how frustrated I got in a lot of my later college classes, mathematical or otherwise.) Given these premises, it also has the potential for leading into that "aha!" feeling you get when you finally "get it." (I have a vivid memory of my best "aha!": I stared for 15 minutes at Penrose's explanation of Cantor's diagonal slash proof on the cardinality of the real numbers vs. the integers, and then suddenly it all made sense. That was 11th grade, when I was reading Penrose's The Emperor's New Mind.)

The presentations themselves--well, I don't know why I thought they'd only take about 20 minutes. (A fellow steppie's master teacher said to me afterward that presentations will take a long, long time. Why didn't I remember that from high school? Possibly because we almost never did presentations in high school, other than the time I "guest-lectured" on medieval warfare for a teacher's Western Civilization class, since I was the resident expert, which doesn't count as far as my own learning went.) G.H. was the catalyst for a lot of great questions, but others were vocal in expressing confusion, too.

One thing Bonnie did before students began presenting was to have everyone draw a chart on a piece of paper to rate the presentations from 1 to 5 according to organization, presentation, content and clarity (I think I may have the wrong words). It was a great idea and obvious in retrospect, but when I conceived of this activity I hadn't thought of it and she didn't mention it as a possibility during our planning session (though I was happy to have it happen). I wish I knew more about concrete, specific things like this that go into supporting student presentations (since, again, they never happened in my mathematical apprenticeship except in that one honors seminar at college).

The crossfire of questions and ideas was great; the majority of the class seemed focused the majority of the time (a couple were almost asleep toward the end, but it was the end of a long Friday, so I can't quite blame them). I stepped in occasionally to clarify ideas or ask for clarifications--students were great at speaking up when they got lost, but sometimes there were mathematical gaps that were obvious to me and not to them. I hope I wasn't too intrusive, but I wanted to make some of the connections explicit, especially concerning how you can have different ways of doing a proof.

Also, a student who never does anything in class, was great in his group's presentation. He was expressive, clear, and downright eloquent. He was even paying attention through most of the class. My God. We need to find more different ways of doing things. I'm not particularly happy with the lecture/seatwork routine myself, though the next activity they'll be doing in pairs. But if we can tap his mathematical knowledge through his verbal ability and get him more interested--hey, every little bit helps.

The notion that ways-of-thinking rather than a finished product (here, a proof) was the goal of the lesson-sequence seemed a bit beyond them. That's okay. It's one of those things that startle me at first, but become obvious and reasonable in retrospect. They were especially upset that I was willing to not show them how to do the median-intersection proof, since the algebraic-proof group hadn't "finished" it, and move on.

What did really surprise me was that the whole class wanted to know how to do it! I put it off until all the groups had presented, and then set it up; a lot of messy and somewhat sleep-inducing algebra, but I'm not too concerned about the ones that started tuning out, since they got the point, which was that a) sometimes you need to figure out how to do something in the course of a proof (a subproof, if you will), b) generalization (variables instead of numbers) is a pain in the butt to work with but can sometimes save you time, c) there's more than one way to prove a given proposition, and d) sometimes one of those ways is easier or harder than another (for a given person)!

Well, I shouldn't exaggerate. What they probably have now is some hazy idea of some of those points, but dammit, it took me until my junior year of college to have anything resembling a solid notion of proof, and even now it's not absolutely solid. (It remains a fairly major problem in the philosophy of math, so why should it be solid?) If I've opened some doors, that's the important thing.

Copyright © 2002 Yoon Ha Lee <requiescat@cityofveils.com>
Last updated on 21 May 2002