Wednesday, November 28, 2007

Same proof, different theorem

It's been a heartening day since I last checked in.

I'm happy to say that today's installment of 280 proved a useful one, as far as I can tell.

A few weeks ago I spent a bit of time designing a suitable peer review component for the third and final exam for the course. Since time permitted neither exam revisions (as I'd allowed for the first exams) nor a by-now-typical committee-based peer review of one of the exam problems, I decided to allow those who completed a draft of a particular problem (namely, the first on the exam) to take part in an in-class peer-review activity in which participants were divided into groups of three at random, allowed to discuss their approach to the problem within these small groups, and finally given the chance to share their groups' discussion with the reconvened class at large.

Without fail, everyone completed a draft of the indicated problem, and group discussion was lively and apropos. After ten minutes, we met again as a class, and several of the most eager folks in the class took turns presenting their solutions to bits and pieces of the problem.

At one point Quincy scrawled on the board both a certain proposition and a "proof" of this assertion. Although the proof was a flawless justification of the proposition we really sought, the proposition as stated was incorrect. One of his peers pointed out the error, and with a slight modification, the theorem read correctly.

"Same proof, different theorem," I said. "I need a t-shirt that says that."

I'm impressed with how willing these folks have become to get up to the board and perform math in front of their peers: even Dewey, a relatively reticent soul, spoke up once or twice today when he believed his friends to be in error. And Fiorello didn't skip a beat before taking the board to slam down a nearly perfectly composed proof of one equivalence relation's transitivity.

I'm going to miss this class, it really has been one of my favorite so far at UNCA. I've learned as much from them as they've learned from me.

Today I've had several other things to be happy about pedagogically, professionally, philosophically.

This afternoon I had a brief tête-a-tête with my 280 student Keiko, who over the past couple of weeks has made tremendous strides in coping with equivalence relations. It's clear to me that she's truly understanding them, not just going through the motions. Though she's still making little errors here and there, the mistakes are typographical and not logical. I'll take an armload of typos over a single conceptual slip-up any day.

In an e-mail from Barrymore, one of my first-section Calcsters, I got some of the most useful teaching ideas I've ever received from a student. A veteran of several mock trials in high school, he offered me some advice on how to make the trial experience a more useful one, a more intense one, a more authentic one. His advice centers on introducing the instructor as an actor in the drama, perhaps as the defendant, or perhaps the plaintiff. As students are called on to challenge not their peers (who may be more or less knowledgeable about the subject at hand, depending on their level of preparedness) but rather their assumed-proficient professor, the care with which they must construct their arguments is concomitantly heightened, and the stakes are upped.

Barrymore therefore suggested having faculty play the roles of Newton and Leibniz, while students are asked to play the lawyers, witnesses, and colleagues.

I'm not sure how I feel about this advice. It's solid, to be sure...I'm only wondering if the benefits described above would be outweighed by the loss of the students' opportunity to play the leading roles.

Barrymore also suggested that the various experts should be asked to meet with the respective legal teams in order to perform "depositions" of sorts, to make sure both sides agree on a consistent set of evidence. I like this plan.

The specificity with which Barrymore was able to offer advice showed that he has really thought about this project. I respect his judgment and will certainly consider his input when I put this project together again.

Just half an hour ago I got an e-mail from Bethesda, eight pages into her final paper on the issue of computer proofs she's writing for our independent study on the history of math technology. She's frazzled. She feels uncomfortable making claims about the proof of the Four-Color Theorem, the proof of which she can barely understand (especially its migraine-inducing implicitness). She's questioning the validity of computer proofs, questioning what it truly means to be able to prove something in the first place. In one long-running paragraph she spat out a dozen or so insightful observations whose perspicacity made me want to weep with joy.

I'm going to have to think for a bit before offering a robust reply.

It's days like this that make me glad that I do what I do.

I'm off to bed now. It's nearly eleven, I've been up since four this morning, and I spent nearly thirteen hours on campus getting "caught up" after a "break" during which I worked for nearly a day altogether.

What's wrong with me that I work so hard and yet love that work so much?

One parting note: I've received the go-ahead from several Calc I students to quote from their reflections. Excerpts to come soon!

1 comment:

Anonymous said...

Yes, the t-shirt same proof different theorem would be perfect for you! Maybe even a Math Department shirt for sale! Certainly the rest of the campus would want to take many classes in math if they really knew how cool and versatile the classes were. I agree the 280 class is fabulous. Makes the brain hurt but there are days the "math fumes" are purely bliss! Keep up the great work! One day you will look back and marvel at the wondrous days of your beginnings at UNCA.