What gets you hot?

What is it that makes one love math?

And can one teach another to find whatever that is inside of oneself?

My ex-student and good friend Mariposa, now a middle-school teacher in Virginia, today told me a story about one of her sixth graders. During class she'd given the students a brief history of science, in which she'd lingered on Newton, indicating that he'd found a way to compute the area of objects that were squirrely and squiggly, and not at all so well-behaved as prim and proper objects like squares and triangles, and that the way he took led him to calculus.

The student in question clearly thought long and hard about how one might go about finding such areas, and in class the next day he submitted to Mariposa an unsolicited brief that told how to find the area of a curvilinear figure: one needed to trace a graph around it and then divide it up into little boxes whose side lengths you knew ("they can either be centimeters or inches"), counting the boxes so enclosed to compute the area.

Not far off. Not far, at all.

Remember, we're talking about an eleven-year-old here.

Right now I've got a handful of students in whom I see that sort of passion for math, that willingness to think. They're spread pretty evenly between my 280 class (a class more-than-usually-heavily-weighted in the direction of the Atmospheric Science department) and my Calc I sections...feeding that passion proves hard, though.

Most of the time the Calc students simply don't seem to have the time to spend on "extra" math problems, so even if they seem naturally predisposed to think about math and I pitch them an odd problem from graph theory that's well within their reach, they don't have a chance to follow up on it.

And by the time they get to 280, many students (busier now, but generally with a more well-tuned work ethic and time management skills to match) look on math as a job, and not something to be taken lightly, for its own sake and for nothing more.

I should be happy that the Problem Group meetings are still drawing a solid core of five or six students, even though we're scrubbed from the Putnam list this year and the Virginia Tech Exam has come and gone. And that we're still getting a small several students coming to the research seminars on Tuesday afternoons.

Still and all...

...How does one best make math intriguing, interesting, exciting, sexy?

Through relevance? Is it enough to point out how it's useful, where it naturally arises? This seems to satisfy some.

Or aesthetics? Expander graphs, fractal drumheads, Julia sets, tournament diagrams...is mathematical beauty in the eye of the beholder?

What about mystery? I suppose that as many as are turned off by the challenge of the unplumbed depths of mathematics are spurred onward by it.

To this end, the counterintuitive is a big draw: that chaos can be quantified, that infinity can be counted. There's something of the numinous in the statement that there is no such thing as the set of all sets, or that every set can be well-ordered, or that with a period of three, every period is possible.

Even in calculus, there is profundity: I fear many mathematicians today fail to appreciate the beauty of something as simple as the Fundamental Theorem, so familiar is it, and so little changed over the past three hundred years. Something so old, so well known, and so well accepted can't be interesting, can it? Does it take its rederivation by an eleven-year-old to make us remember how fucking incredible it really is?

Discovery is something we should demand from every one of our students. Maybe that's what I'm getting at here: without the chance to discover anything at all, it may be damn near impossible to discover passion.

I'm tired of walls between research and teaching.

The next person to invoke the triumvirate of Teaching, Research (or its less formidable little sister, Scholarship), and Service gets a smack upside the head with a wet herring.

There's that famous shot of the Berlin Wall, hundreds of elated German citizens crawling over its face like ants, pounding away at it with sledges, hammers, fists, watching it crumble below them in front of them, no more to split their city in two.

No more.

So tell me, readers (I know I've got a few of you, judging from conversations with colleagues, students, and assorted others in my life): what gets you hot? Why math, or why not?

I'd really like to know.