I have begun teaching two, not one, but two sections of differential equations this quarter, and immediately, the classes are different from each other. In one, the students contribute, respond, emote; in the other, I feel like I’m facing a mute wall. This is natural, of course, and not anything to worry about, but what does occur to me is that what works with one won’t work for the other.
For example, I was thinking of what I might say to the first class about the complex numbers. Just a teaser—I won’t teach them in earnest for a few weeks, when we need them. But still, it’s such a natural question to ask: if we invent complex numbers to have a place for the square root of one (which we call the imaginary number i), what is the square root of i? Such a natural question. And then, I think of Slaughterhouse 5, and the aliens description of our 3-dimensional life as comparable to riding a roller coaster with only a tiny pinhole to see out of. That’s exactly what the real numbers feel like once you’ve gone to the complex. They’re just a tiny slice of the whole picture, and once you know what the whole picture is, and how beautiful it is, you can’t imagine living without that knowledge.
This is what I thought of telling my students. But maybe just the one class. There has to be a naturalness to it, improvising off the script, and I have to talk to each of these classes in the best way for them.