What I would continue to do the same in the course.
I feel strongly that one of the best things about this course is having students do semester long research projects. These projects are one of the few occasions for mathematics education students to do mathematics as it is done by mathematicians. Having run the course four times and the projects three times, however, there are certain scaffolds that need to remain in place, and probably a few new ones should be added. The weekly project reports need to be in place, with the heuristic questions. I found that answering these questions helped teach the students heuristic problem solving strategies, and they forced the students to meet regularly.
The main addition to the projects that I would like to make would be to have a time midway through the term for students to present their partially completed work to classmates. Such presentations should create several positive outcomes. Primarily, this would give the students a chance to see interactions between their projects, since the point of such a meeting would be to share thoughts rather than present results. Another effect might be to create a greater feeling of community around the projects. Finally, requiring a midterm presentation will provide another incentive for groups to push forward on their projects.
I also feel it is important to continue asking homework questions asking the students to try and think about how to apply ideas in the classroom. Neal reported that one reason he respected my opinion on what mathematics they needed to know and why it might apply to the high school classroom was the existence of these questions. In particular, he said:
...(the respect) had largely to do when on the first day, or the assignments when you said, OK, how can you apply this to a high school course or high school curriculum. I suppose it was reinforced throughout the semester in that way as far as how can you deal with this problem throughout the high school curriculum.
What I would change.
One of the main pieces of the class that I would change would be to spend even more time taking stock of where we are and why we are doing what we are doing. I probably stopped 3 times during the class that term to look back at what we had done and why. Right now, I would want to do so about 5 times if possible. These days often became starting points for some of the important conversations that took place in class. Moreover, I think that in many of our classes the students are used to getting into a mode of forgetting the big picture of what is going on. As the last math class many of these students will ever take, it is extremely important that they concentrate on the big picture. Concentrating on the big picture is something that is very difficult as a teacher in mathematics because it is so linear and because most interesting problems require lots of little steps. Without the big picture though, mathematics loses cohesion and it is difficult to motivate students.
On the small side, I would change several of the homework problems slightly. In particular, I would change around the homework set that has the students prove that the constructible numbers form a field. The one I handed out set the students up wrong on how to do multiplication. Instead, I would do division before multiplication since this is the natural idea out of the high school mathematics problem. Another homework problem that needs to be changed is the problem on proving that an angle inscribed in a semicircle is a right angle. This homework problem has a hint that suggests the most difficult way possible to solve the problem, whereas a simple solution that one should present to high school geometry students can be found relatively easily.
The last thing I would try to change is how I interact with malfunctioning student groups. I feel that there were two groups that could have gotten more out of their projects had I handled the group dynamics differently. One specific piece I will add in the original request for a prioritized list of projects students want to do, is a place for students to ask not to work with particular other students in the class. This might well have avoided some of the problems with Jill and Bill. I also may consider assigning roles at some point for students to provide in discussions with me, although I am not sure how this would work.