I have the pleasure this year of working with a self-motivated, mathematically mature, and generally awesome high schooler for an independent study. He’s really comfortable learning on his own—both by reading and thinking through problems. It’s truly an independent study—I get to advise and suggest, rather than teach, guide, or cajole.
Having learned calculus and stuff about Taylor series last year, he’s interested in fractional derivatives (as in, what’s the 1/2 derivative of x^2+sin(x)?). He’s coming at the question from a vector space perspective, making an analogy with how multiplying by i is a quarter turn—it’s “half” of multiplying by -1. The analogy is a good one, but it breaks down in important ways. For instance, taking the derivative of a function isn’t distance-preserving, at least under the usual metric.
We’ve only met twice so far, but it’s been so much fun getting to hear him pitch his ideas, calculations, and sticking points to me. Today’s photo is of some work he showed me today, as well as a large drawing of a clock left over from someone else’s earlier fifth grade math class. A nice visual contrast, I’d say.