I am teaching real analysis in the fall, and I am beginning to plan it out. Here is one more idea that I would like to record before I forget.
First, some background. I was never very good at real analysis. I like it a lot, but it was over my head as an undergrad (I got B’s in the course, largely because my relative difficulty with the material outpaced my study skills) and I only took one course as a graduate student (similar). Part of the reason why I am teaching it is that I want to learn more about this beautiful subject.
So I am not a great analyst. To use this to my advantage, I am planning on—perhaps weekly—asking my students to give me an analysis problem to solve “cold.” I won’t prepare for it at all; I will solve it on the spot.
I was inspired by a lost blog posting (please comment if it was you—I will happily edit this to credit you) about the difference between the way we discover mathematics and the way we communicate mathematics. Sadly, we normally teach by showing the “communication” rather than the “discovering.” By
- Doing problems “cold” in front of the students, and
- Not being very good at analysis, but
- Having general problem solving skills,
I can hopefully give the students a glimpse into the “discovery” world of mathematics. If I am really on the ball, I will (at least occasionally) take the time to re-write my “discovery” into a “communication” form.