I just spent a snowy afternoon by myself, thinking about my Cryptology/Number Theory class that is going to start at the end of January. There are a few things that I love at my school, such as my ability to choose whatever semester long, college level class I want to teach. (The downside- I do this on top of my regular teaching schedule and the class meets for two and half hours one night of the week.) But, they give me complete autonomy to do whatever I would like with the students.
This semester, I'm teaching Linear Algebra, which has been quite the experience for both the students and me. It's definitely a harder course that what I had in my undergrad, but I don't think that's too hard to do. We're spending quite a bit of time focusing on proofs which had been totally ignored in my undergrad class but was done in detail in my masters Lin Alg course so I'm trying to find the middle road.
Anyway, I like to do one "traditional" course a year and one fun course. I taught the Cryptology course two years ago and the students loved it and were begging for me to have it again, although it would have been the same kids in the same course. (I don't know what they were expecting I would do the second time around.) I'm trying to set these up so that they will rotate on a two year schedule. In the past, course offerings have been Multivariable Calculus (with college credit from one of the local state universities), Proof and Problem Solving, and math history.
I'm wondering where to go next. Multivariable is hard to offer because there are really only two or three students a year that have the background to take it and I hate to limit the students that can be in it. Many high schools offer differential equations but whenever I've talked to my students about it, they seem to cringe at the idea. They too are looking for something fun and engaging. At the same point, I would love to have some mathematics included that would help my students for math team, the AMCs and beyond. (One of my current students took the USA Math Olympiad last year and I think I have another student that may have a shot at it.)
Here are some ideas for other courses that I've come up with: Probability, Non-Euclidean Geometries, Combinatorics, Fractals, and/or Game Theory. The first two courses I would have no problems in offering because I already have taken the courses and so, I know the basics. The second three are pretty much completely new to me and I would have to teach myself the material before I could teach it to them. Does anyone else have any comments or ideas that could help me figure out my possible courses for next year?