Saturday, September 08, 2007


Boy howdy, was my brain ever in Friday mode yesterday. I was muffing this and that, one thing after another, and until late in the afternoon my to-do list was growing longer far faster than I could cut it back.

I've had a good weekend so far, though, so I guess things are evening out in the end.

So, yesterday, what of it?

The first misstep of the morning took the innocuous form of a forgotten stapler. I'd meant to make a clean start and begin bringing it with me to class Fridays so that my Calc I students could properly assemble their ever-unattached homework pages in some manner other than messily crimping the corners together in a sad and useless little lump. Of course, the stapler found itself left behind on the corner of my desk.

No biggie. 'S all good, 's all good.

Then I started handing out the Mathematica installation disks (finally). Six students in each section got a disk, and I calmly went through my little spiel about the installation process...completely omitting three crucial points: (1) they'll need passwords to register, (2) they won't get the passwords instantaneously, but rather those'll be sent to them by e-mail within a day or two, and (3) they must use their school e-mail addresses when registering, otherwise our license manager won't know any one of them from Moses on a pogo stick when it comes time to kick a password studentward. (I later remedied the oversight by sending the entire class an e-mail about the correct procedure.)

I soldiered on. Saturday, after all, was but a half-day away.

It was 280 that was truly frustrating for me, though I know it probably shouldn't have been.

The second committee report of the season (regarding the construction of a truth table purporting to demonstrate a certain tautology) went off without a hitch, and there was robust, respectful discussion on a number of salient points: "Do you have to include all relevant columns in the truth table, or can you omit a few if you can perform some of the operations (like simple negation) mentally?" (It was then determined that one ought to write not for one's own understanding, but for that of the reader, and error should fall on the side of liberality in column inclusion.) "Even though it's obvious if you're looking for it that the first column (containing statement P's truth values) and the last (containing the truth values of a logically equivalent compound statement R) are identical, so we're meant to conclude that R is true if and only if P is...but wouldn't it also be correct to answer the question 'what can you say about when R is true?' by comparing R's truth to some other column, if you didn't notice the equal columns?" (It was agreed that though technically another answer might be correct, indicating the tautology P <=> R would be a "more correct" response to the question.)

The third committee report (concerning a tricky proof by contraposition, asking for a verification that [not Q] implies [not P] in order to prove P implies Q) went a bit more roughly. Tamar and Cornelius led the charge, and all went well in negating the given statements P and Q. But things got a little dicey when it came time to prove the desired contrapositive. Perhaps due to the subtle nature of the negated statements (one was a conjunction requiring a DeMorgan Law), the team got [not Q] and [not P] turned around. I'm not sure they were completely comfortable with the proof they presented, though: Cornelius correctly indicated that he wasn't sure their proof would handle one of the three cases which might arise in [not P], but they weren't able to salvage the proof once it foundered.

At this point, I stepped in for a few minutes to try to patch together a proof of the corrected implication [not Q] => [not P], but my own hastily-assembled argument was a weak one. It was technically correct, but smacked of proof by contradiction, something we had yet to discuss (and in fact would begin discussing a few minutes later), and didn't explicitly use the DeMorgan Law I hinted at on the homework sheet.

By the end of the report, I think everyone (including me) was a little confused and wearied, and we had only twenty minutes to finish a direct proof from the previous class period and to begin tackling contradiction. We made our way through an outline of the proof technique, and now stand poised to construct our first contradictory proof, a feat we'll undertake on Monday.

After class had ended, Dewey came up to me and asked me to take a look at his solution to the contrapositive problem.

It was beautiful.

Aside from a small boo-boo in the negation of one of the statements, his proof was error-free, elegant, and made full use of the required DeMorgan Law. Best of all, it didn't have a whiff of contradiction about it.

Ah, c'est la verification.

So why was I "frustrated"?

Because I know what it's like to muff something in front of one's peers: it ain't pleasant, and I hoped that the folks on that third committee didn't feel overwhelmed by the problem they'd been given.

I also felt frustrated that at the time I'd felt obligated to step into the ring, when I probably should've just stayed the hell out. After all, the whole point behind the use of the committees is to (fittingly) commit its members to take authority over the task they've been delegated. I see my role as that of an ex-officio, advisorial member of each committee formed: though I might provide a little input behind the scenes if it's asked of me, it's not my place to usurp the committee's authority in the classroom. If I keep doing that, how can I expect them to grow more comfortable in wearing the crown? Until yesterday, I feel I've done a really good job in reining in my own reign, and my frustration is probably born from the fact that yesterday, improperly, I let slip my own authority.

In that regard, I definitely fucked up.

On the other hand, the experience provided an object lesson to everybody: we're all going to miss a few now and then, and as I've said many times before (and as many smart people have said before me), learning how to prove things and learning how to clearly record those proofs are iterative processes that generally make their progress (sometimes) painful fits and starts. The students all contributed their versions of the desired proof, the committee undertook the thankless (if you're reading this, thank you, all!) task of collating these versions and producing their own, incomplete version, I made a half-assed attempt at cleaning this up, and Dewey succeeded in showing me up with a nearly flawless feat of mathematical legerdemain.

And hell, isn't this sort of imperfection the essence of discovery learning?

Yeah, I'm learning, folks, I'm learning. Slowly, maybe, but I'm learning. I've certainly got a rough road to travel to my own mastery of that method. In the past several years I've come a long way down that road, but the had sun set on Friday before I could take another step.

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