In the first week of my very brief career as a regular full-time high-school math teacher, my students and I somehow started having a conversation in the lower-right corner of the greenboard at the rate of one statement by each party per day. Their first two comments were very complimentary, and I responded as shown above. Of course the math in the first sentence is Fermat’s Last Theorem. Not that I expected them to understand it (despite the fact that I had all honors classes), or even to be interested in understanding it! But they definitely were interested. So I had the delightful experience of talking to my Algebra II students about number theory for a few minutes.
This is one of several experiences that convince me that a significant fraction of high-school math students — at least of honor students, but probably of ordinary students as well — haven’t had all their natural curiosity about things mathematical squeezed out of them, despite the worst efforts of the American math-education system. It’s well worth a few minutes of classtime now and then for a teacher to try to take advantage of it.