Wednesday, January 16, 2008

Graph Theory: Day 2

As the snow storm descends on the Asheville area, I'll take a moment to briefly chronicle this afternoon's mathematical goings-on.

I felt a bit out-of-step in my first section of Calc II today. I never really got into my stride, somehow, and I felt awkward. The awkwardness carried over into the second section, with whom I felt more at ease, but still stretched thin. I'm looking forward to Friday in both of those sections, I'll be leaving much of the work up to them. Then Monday will bring the first of several food-based exercises, always favorites with the students.

These two classes were more than made up for by Graph Theory.

Right away the atmosphere was a positive one: before class, as people were still trickling into the classroom in dribs and drabs, everyone was chatty, jovial, open. The students joked, compared solutions. Everyone seemed relaxed, ready. I put some colored chalk on the front table and went to the side board, where I wrote "Correctness / Completeness / Clarity / Composition," urging the students to intone these words as a mantra as they prepared their presentations.

Then we began.

Things went well from the start: when called, each student took to the board to the sound of applause from her or his colleagues. Everyone was quiet and respectful during presentations, and each success was met by another round of applause and cheers.

The first few presentations went smoothly; it was Problem 4 that caused a bit of hullabaloo.

"Problem 4. Draw as many fundamentally different graphs as you can, each having order 4 and size 3, also writing each as a triple."

Its the fourth and fifth words here that brought down the house: there was (understandably! I'd somewhat hoped that this problem would provoke a discussion) a great deal of disagreement regarding what was meant by "fundamentally different"; it'll be another week, at least, before we define graph isomorphism. (Brigitte actually said a few words about "bijections" that were very close to the mark, but her quiet voice didn't carry so well amidst the hubbub.) The chimerical nature of this phrase, coupled with the immense number of graphs having the properties desired, led to uproar. Poor Joachim, attempting to answer the problem as fully as he could, was interrupted by a chorus of overly helpful classmates: everyone wanted a piece of the problem, and the next ten minutes were spent in taking unruly turns at trying to pin down the meaning of those elusive words, "fundamentally different."

Ultimately it became clear that we all had more or less the same idea as to what those words meant.

The discussion was lively, even heated, but ever respectful and supportive: no one attacked anyone else, corrections were friendly ones, and even when there was disagreement, the disagreement was civilly made.

The next three problems were relatively humdrum; Problem 8 caused a bit more furor, though without the controversy attending Problem 4. Quincy was called on the complete Problem 8 (asking for an enumeration of the maximal number of edges in an order-n graph without multiple edges), and he offered a nearly-complete proof of his (correct) formula, the sum 1 + 2 + 3 + ... + n.

"Did anyone have a different proof?" I asked. Sylvester offered that he did, and he went to the board to provide an inductive proof of his (equally valid) formula, Cn,2 + Cn,1. Throughout both presentations, everyone was quiet, attentive. Sylvester's proof brought us to the end of the period, midway through the first problem sheet.

Afterward Quincy characterized the mood of the class as "fun, but serious." "We all mean business, we're taking it very seriously," he said. "But we're having a good time with it." He had a blast, as did his friend Norbert, and as did Nadia, who spent some time after class trying vainly to convince Olivia to join our class.

I am positively delighted with the way class came off today: the students took control. They constructed their own mathematical meaning while engaging in lively, sincere debate about deep mathematical issues. If we can replicate today's success over and over again for the next several dozen class periods, I'm going to end this semester as the happiest man on Earth (not that I don't already hold claim to that title).

I'm already looking forward to Friday.

I'm also looking forward to tomorrow: barring too-hellish weather, I'll be trudging into campus to fulfill a number of bureaucratic commitments, and to meet with Sieglinde and Trixie, my budding freshperson graph theory research team. Trixie's progress on the problems I pitched her over break has been nothing short of astounding: I met with her yesterday and she showed me the pictorial essence of the results she's come up with, and they look solid. Sieglinde's indicated progress too, and I can't wait to see what she's got in store. They're both sharp are tacks and a kick to work with.

On that note, it is wearily but happily that I bid you a good night, I'm off to do some relaxing reading before calling it a day. Adieu!

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