Friday, October 19, 2007

Long, long, long

As I'd suspected would be the case, the past week and a half or so has been absotively, posilutely insane, and I've hardly had a chance to keep on top of each day's work as it's come due. Between travel and teaching, City Council meetings, grading, grading, and a little bit more grading, Learning Circles, research, and reappointment shit, it's been a rough one. Ergo, no posts for a bit now. Many apologies, etc. Today's the first day I've not felt torn in several directions at once; once or twice this morning I actually had a chance to sit at my desk and ponder my next task before instinctively setting to it.

So what's up?

The first draft of the 280 students' Professional Proof Analyses, in which I asked them to apply our course rubric for superlative mathematical writing to three proofs of the same theorem, each written by three different authors, were fantastic. The students raised excellent points, made perspicacious observations, dug deeply into the "Four Cs" of the rubric (Correctness, Completeness, Clarity, and Composition) and applied these principles consistently and clearly. By one means or another most of them accounted for variable audiences, the evolution of exposition through time, the difficulties entailed in the analysis of a single proof extracted from within the context of the entire textbook, the subtle epistemological differences between putting a proof before a proposition's statement and the converse configuration, and so forth. And these were just the drafts! I was able to honestly say as I handed them back that those papers were among the strongest mathematical writing I've yet seen in any of the classes I've ever taught. How much of this is due to the students' inherent skill in constructing well-thought-out essays, how much to the clarity with which the assignment was designed and implemented, and how much to the fact that I feel I'm a much better teacher of mathematical writing than I've ever been before, is hard to say. I like to think it's a combination of all of the above.

I'll be polling the class more formally on Monday after the final drafts are handed in (this will be one of the assignments collected for the purposes of the Writing Assessment study, incidentally), but preliminary estimates show that the proof of the second part of the Fundamental Theorem of Calculus appearing in Hass, Weir, and Thomas's latest edition came out on top, beating Stewart's 2nd edition out in terms of completeness and composition, both of them beating out Abraham Schwartz's 1967 treatise that made use of outdated and relatively unfamiliar notation and awkward terminology. There was some dissent on this point, though: a few of the class's strongest students argued that in terms of completeness and composition, Schwartz's proof had the others beat. I believe the students' sense of completeness might come from Schwartz's explicit construction of Riemann sums; the other authors hide most of the messy details inside references to other theorems and corollaries contained elsewhere in the text, and so might feel a bit more scanty than Schwartz.

Class itself has had its ups and downs during the past couple of weeks. Last Friday I was in no mood to talk about permutations, so after a few minutes going over suggested corrections on the most recently-graded homework, I gave an impromptu lecture on Russell's paradox, proper classes, the infinity of infinities, and the cardinality of the reals. The topics are engaging, the students asked fantastic, insightful questions, and we all had a good time: that's how I wish every class could be. Quincy suggested that perhaps I should try to get my Chair to allow me to teach a "Random Seminar," in which topics are drawn from a fishbowl at the room's center, and teacher and students together spend a few weeks digging into the topic so chosen, convening as needed to fill each other in on the details. Sounds like fun, but it would be require an incredible amount of work on the parts of both the teacher and the students, and it would take some tweaking before it would fly.

The past week saw us slog through the remainder of our work on combinatorics, leaving us ready to tackle relations. On Wednesday equivalence relations proved a bit dodgy for the students, so I took some time yesterday to make up an additional handout that dealt more concretely with equivalence relations, asking the students to construct explicit examples of relations with certain properties, on small sets. I took several of the students aside after class, one at a time, and asked them if they felt the worksheet helped ground their understanding, and the consensus was that yes, it did. I'm glad.

Calculus, meanwhile, has been a hoot, but with the conference I attended all of last weekend, I've felt out-of-whack with regard to that class. I wasn't able to grade last Friday's homework over the weekend, as I nearly always do, so I didn't get it back to them until Wednesday this week, and that's made me feel as though I'm a bit behind. (Likely, the students couldn't give a rat's patoot.) I do feel more on top of things now that I've had a chance to catch up...just in time for this weekend's homework. Huzzah!

We've been talking about related rates, an ever-vexing topic that never fails to confound student understanding at first. The last couple of days we've been talking about exponential and logarithmic models, including the semilogarithmic model for network growth that my colleague and I came up with this summer. (I hold out hopes that by infusing my teaching with my research and vice versa I might catch a few students early in the game and entice them into considering a Math might be working: Tallulah seems open to the idea of undertaking a little undergraduate research soon.)

The most exciting events concerning Calc I have to do with the Newton v. Leibniz project. Role proposals were due on Wednesday, and every one of them made it across my desk before zero hour. The proposals were...entertaining. Some were quite formal, serious pieces of persuasive writing, offering solid arguments for why I should make one appointment over another. Others were simply silly. I had fun reading them. In assigning roles I attempted to balance the strength of the individual proposals with the cumulative "happiness" of the class as measured by the number of teams receiving their respective top choice. In Section 1 I was able to grant five of the eight teams their top choice for roles, while only three of the eight teams got their number one choice in Section 3. There were a few long faces in that section, I know a couple of the teams had high hopes for their first bid and had only half-heartedly lobbied for their second. On the other hand, there was genuine excitement on some people's parts in both classes. I think enough students are going to get something truly meaningful out of this project that it'll be worth the trouble it's taking to organize it.

What else? The conference in Charleston was a conference. I learned a bit, met some new people, managed to insinuate myself a bit more deeply into the graph theory and combinatorics community. (Today I was offered a chance to referee some of the papers for the proceedings, to appear next year in the Journal of Combinatorial Mathematics and Combinatorial Computing. Exciting stuff!) I came back with several interesting problems to think about, including one that I pitched to one of our brightest junior majors almost as soon as I got home. He picked up on the general idea almost immediately and is already working on the problem I gave him.

Ummm...what else? Hmmm...yesterday I finished my reappointment binder and got it in to my Chair a week before it's due. I'm happy with it. I tried to cut my "candidate's statement" down a bit, but I really wasn't sure how to. I'm certain I included more documentation than was necessary: at roughly 75 pages, it's probably about three times as long as it needs to be, but I don't do anything halfway (or a third of the way, for that matter).

What else? Hmmmm...

...I'm sorry, I'm really tired right now, and should probably get to bed.

I'll try to post again tomorrow, as I really do have many thoughts I'd like to commit to paper ( passes for paper this millennium) before they escape me for all time: Fabian's astute observation that a many-authored proof might more quickly than a solo effort reach a sound and stable equilibrium, my conversation with Tallulah about a student's authority to assess the rightness and wrongness of a mathematical computation, and so forth. But sleep would do me well tonight, if I'm to survive the onslaught of the Super Saturday kiddies tomorrow morning. I'll likely have several stalwart students by my side to help me out, but no such Saturday goes by without my renewed appreciation for the role played by our nation's middle school teachers.

Until tomorrow, then!

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