May 10, 2002

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

On 10 May 2002, I had students give presentations on their endless TREASURE HUNT group activity and give FEEDBACK on each others' presentations, which has been the standard modus operandi. The novel part was that in addition to verbal and whiteboard directions, I photocopied actual (handwritten) "feedback" sheets with prompts to guide the students' responses. This is, of course, entirely bass-ackwards, as the more sensible course would have been to provide the sheets as scaffolding, then wean students away from them. I can only say that the idea didn't occur to me until 20 minutes before class that day.

Afterward, students worked more-or-less individually on Cartesian-to-polar conversions, for the first time in quadrants other than the first quadrant. Frankly, when I introduced the conversions I thought that they would find the concept of arctangent (or inverse tangent) novel enough that I didn't want to push my luck any further. I will not claim to have instilled any real understanding of why arctangent works, and feel considerably guilty about this; but given that my attempt to develop understanding of inverses and composition of functions was a fiasco, I'm not sure what else I could have done but give them the magic formula. (I hate magic formulas and here I am using one.)

In any case, students had to plot the points, then estimate the conversion. I'm sure they were sick of the routine, but an earlier lesson in which I misdefined sine as x/r and cosine as y/r was a vivid demonstration of why the estimates are valuable--especially since most of them haven't had the opportunity to develop this semi-specialized type of "number sense." I can eyeball a conversion and decide whether it's ballpark reasonable, but most of them can't yet--and the only reason I can is long hours of drill.

The trouble with non-first (and fourth) quadrant Cartesian-to-polar conversions is, of course, the range of arctangent (from -pi/2 to pi/2 radians if I recall correctly--and that reminds me, I really need to introduce radians if they'll seem 'em next year), which gives them related but not-yet-correct angles. The way I learned to handle arctangent (and arcsine, and arccosine) was by rote memorization of domain, range, and rules for what goes where when. To this day I can't remember how it worked.

Instead of rote memorization, which is useless once you've forgotten it, I wanted them to have a built-in way of checking their work. As they wrestled with rusty geometry concepts (complementary, supplementary, and vertical angles--incidentally, now I see a use for the former two terms, even though it really all comes down to the fact that angle measures can be added linearly in Euclidean two-space, which is more generally useful), some of them may have developed internal "rules." Chances are good that three years down the line they'll have forgotten those, too; but I hope having them plot their points, draw auxiliary lines, and check against estimates will give them some idea of how to reconstitute their knowledge in three years, instead of scrabbling at the tatters of their memory.

Math should be a lot less about that frantic scrabbling, and a lot more about giving students "resilient" knowledge (in the sense that it comes with a way to rebuild itself from very little memory; if there's an "official" term for it I'm not familiar with it). My personal experience has been that I end up memorizing, almost painlessly, those formulae and equations that I have to use often. I have known a few people who functioned better by memorizing things straight-out and could retain what they gained that way, but most people don't seem to work that way; and even for straight-out memorizers, some understanding of the principels from which their formulae were derived is valuable.

It occurs to me now that "resilient" knowledge might itself have a fractal structure: far more efficient to store a generating pattern than to store each pixel, so to speak. As I'm neither a neurobiologist nor a psychologist, I shan't strain the analogy further.

Copyright © 2002 Yoon Ha Lee <requiescat@cityofveils.com>
Last updated on 28 May 2002