Archive for August, 2007

\bullet Unapologetic John Armstrong discusses an economic model without once mentioning Category Theory. Also he appears to have started another blog. Oh, and here’s a thread on th’ Carny.
\bulletAnother thread on CoM, at Michi’s … and still another, at MathNotations. Michi himself is on vacation and hasn’t said much …
\bulletTom Lehrer Collides With the Periodic Table of the Elements, at Millard Fillmore’s Bathtub, via meeyauw. As for added verbosity, hmm … here’s a gratuitous self-reference.
\bulletTerry Tao on Danica McKellar.
\bulletGerbes in The Guardian.
\bullet Innumeracy Cannot Be Overestimated, in Language Log. See, I don’t just read math blogs …


Business as Usual

New Rules

Really a new rule: I have to publish a post every week that doesn’t consist entirely of links. Reading is a lot easier than writing, and there sure is a lot of great stuff out there … so my instinct is to just go on browsing and report on some of the highlights. And I mean to go right on doing quite a bit of that.

But there also oughta be something original from time to time. (Old joke. Consultant: Thanks for bringing me to your institution. You’ve got a great thing going here. All you need now is a Mission Statement, a Work Plan, and a Blogfor. Employer: What’s a blogfor? Consultant: A blog is for spouting your opinions and hoping for an audience. Dah-dum!)

So today, while I’m building up steam for yet another anti-graphing-calculator piece, I’ll share a few lines of code for the TI-*: the “integral checker” (INTCHECK).
1 \rightarrow I
While I<8
nDeriv( Y_1(X), X, I)  \rightarrow L_1(I)
Y_2(I) \rightarrow L_2(I)
I+1  \rightarrow I
Put your (tentative) solution in Y_1 and your integrand in Y_2; run program INTCHECK; look at the “lists” that the program has generated (these are kept in the STAT menu); if all has gone well, you’ll see seven matched pairs: the derivative of your solution equals the integrand (for X \in \{1, 2, 3, ... , 7\}). Of course this isn’t a proof (it’s merely very convincing). Worse than that, the set in question might very well not even be a subset of the domain of our functions (consider \int \sqrt{x-8} dx). But it’s been useful to me just the same: if there’s been a mistake, this’ll usually find it out.

Anyhow, my students have been required to buy these doggone things, so when I can find out any actual use for one of ’em, I’m pleased to share it. In particular, I like showing off the programming feature: these silly little boxes are hand-held computers, not mere calculators. In fact, a TI is more of a computer to me than the PC clone on my desk in at least one way, since I’m not up to speed on any of the programming features of the latter (I’m convinced that Microsloth clobbered BASIC so people wouldn’t learn how easy programming is …).

OK, that’s enough out of me. Back to reading everybody else’s stuff.

Weekend Reading


Jazz Math Ed

A lot of talented and effective lecturers prepare so carefully that every time they present a certain topic, it’s just like every other time they’ve presented that topic. So don’t get me wrong. I’ve seen it done, and done well, hundreds of times. Let’s call that “classical” style. You look at the printed score, and you pick up your instrument, and you practice like crazy until you could hardly ever miss a note except on purpose … and then, and only then, you go out in front of your audience, and, with any luck, you play your little bit and they love it.

On the other hand, there’s the “jazz” style (as I propose to call it here): you look at, not the score, but the chart: an outline of what’s supposed to go on in a given performance. Then you grab your axe and wail (and, with any luck, they love it).

Now, I have been known to prepare lectures carefully. Heck, once I even read a talk from a printed copy. Mostly, though, I won’t even think about doing things that way if I feel like I’ve thoroughly understood the material … which, since I’m mostly lecturing on subjects I’ve known pretty well for about thirty years, is most of the time. I get to have a whole lot more fun slinging the math this way … and I sincerely believe that the students (usually) get a better show. For example, I’m free to change everything around if there’s an interesting question. Moreover, since I’m working without a net, I make real mistakes in real time and have to recover somehow (just as every student will have to do before long if they’re keeping up with the homework); and more than this, I get to—have to!—model the quasi-obsessive “check it if you can think of a way to check it” behavior that typically separates the “A” students from the “B”s.

All well and good. The trouble is, it seems like whenever the subject (“how can we become better teachers”) comes up, all anybody ever wants to talk about is “classical” style. And that way madness lies. It’s possible to be too well prepared—I’ve sure as heck seen that done. There comes a point where you might as well just read the furshlugginer manual—except that that would be cheaper and more convenient. Also, quite often our bosses are enemies of the academy who would like nothing better than to find a way to eliminate their need for performing artists and replace us with canned “lectures” (or, worse yet, with computer programs) that can be paid for once and then controlled utterly. Now, I suppose I can be forced by economic pressure into training my own replacement … but I can’t be made to like it.

So: the charts don’t lie. For some reason, kids keep going to concerts. Even though the album is cheaper and you can play it again and again … heck, I don’t understand this fact myself. But record company executives seem to understand at some level that their livelihood depends on a bunch of wildly undisciplined misfits (i.e., performing artists). Wouldn’t it be nice if college administrators understood this too?

The Pencil Rant

Maybe you can do math in ink. I’ve seen it done. But most of us need to use the eraser quite a bit to get anything done at all. Why make things any harder on yourself than they need to be?

OK, now that’s out of the way. What I’m really here to talk about is the simple fact that until you’re moving your pencil (the eraser counts), you’re probably doing something other than seriously studying mathematics. This is presumably what everybody means by that wellworn saying “mathematics is not a spectator sport” (more here).

Students find this very hard to believe. Or think they’ll be the exception to the rule. Or something. It’s actually kind of amazing … and I say it even though I can remember some of my own efforts at avoiding exercises pretty well. In my first Abstract Algebra course, for example … I’d just keep reading the same passages over and over (from Herstein’s Topics in Algebra) and wondering why nothing would sink in …

Now, every academic subject is about learning to read certain documents and write certain others. But I want to suggest here that the “writing” end of things has even more importance in the context of learning math than it does in, anyway, most of the others (I’m willing to accept that, for instance, composition classes might give us a run for the money). Part of the message for beginners is that one must learn to trust the code: don’t wait for some mystic vision to descend and make everything clear all at once; get down in the dirt and calculate. You ought to be able to trust me on this …

Hmmm …

... but really you have to push this button. Or, let's say, this one, whereby (if all has gone well) you'll find that I was concerned with the kinds of difficulties I've just been complaining of as of my very first post. It now appears that there's a "code" button in the point-and-click "toolbar" that might meet my needs. Oh, and by the way: please don't bother pointing out that I get this stuff for free and shouldn't look gift horses in the mouth and all that. A poor workman blames his tools and I intend to exercise the option.