Eigenmumbles

When I was in grade school, I remember some of my classmates couldn’t wait to learn to write cursive. I didn’t sympathize at the time, though later in life I decided that there’s something very soothing about well-executed longhand. But though I might not have felt the same, I certainly understood the drive to learn that next big thing on the horizon, the one that looked so cool and yet so far away.

Calculus was my cursive. The public library near our home had a variety of popular math books, as well as introductory books on basic calculus. I gobbled them up, though it’s unclear to me now how much I really learned from them. Eventually, I actually took calculus, and that was that. Then I went to college, and spent a year obsessing over an introductory analysis sequence. Then it was tensors for a while — I knew what they were to a mathematician, but I wanted to understand how physicists thought about them, too. Compilers, too, were a minor obsession up until I took a compilers course. And so it went, and so it still goes: a topic just sounds cool to me, so I play around the outskirts of it for a while until I feel ready, then dive in. Usually, I come away from the experience still thinking that the topic was interesting, and excited about all the connections that I made while learning about it. Anyhow, I picked up a bag of cool tricks over the years, and I love ‘em all, even if some of them seemed cooler before I had them in my hand — a problem familiar to most of us, I suppose.

It has dawned on my recently that many of my students might not have gone through that preliminary bumbling-around phase where calculus was a seductive mystery. I got to start asking “what is a derivative, really?” around the end of middle school, and just kept on getting more nuanced answers as I learned more and more. I sometimes say, only half in jest, that one of my biggest lessons from grad school is that you can differentiate anything with respect to anything. So what would have happened if I’d started instead with someone telling me what a derivative is, then making me differentiate a bunch of boringly similar functions for a few hours before moving on to the next topic? I might have ended up like the dog in the Far Side cartoon, hearing “blah, blah, blah, differentiation, blah, blah.” I think that would have driven my nuts, and I was far better off starting with a slightly confused sense of wonder than I would have been with a more unprejudiced and unenthusiastic approach.

Now that I think about it, I kind of hope that someone in one of my lectures has at some point written “blah, blah, Taylor series, blah” in a set of notes. It would, at the very least, exhibit an emotional connection with the material, even if the emotion in question leaned more to irony than to awe. But if everyone just earnestly copies down what I’ve said without stopping to feel just a little bit of wonder or amusement at the material itself, I kind of think I’m not doing a very good job of conveying how I think about the material.