As the course ends, the students' work on the projects heat up. Most students answer the initial question by the beginning of November and then have about a month to follow-up other ideas on their projects. Ideally, we see students getting a chance to ask and answer their own questions. As Polya (1973, p. 68) said:
The mathematical experience of the student is incomplete if he never had an opportunity to solve a problem invented by himself.
The project papers themselves show that some of this happened, although we should be careful here. In office hours, students would come in and hold discussions with me. Often they would ask me how to proceed. To the best of my ability, I would respond noncommittally. The one exception to this is if I thought the students were going to get off the path of good mathematics. The one group for which there is an exception is the Approximations group, where we ran into a great deal of difficulty.
The papers themselves are reasonably good, with two projects receiving a 4.0 grade, two 3.5s, and one a 2.0. This is higher than my average class, although that may have more to do with following a new rubric than with the papers themselves. On the other hand, my general feeling was also that these papers represented more mathematical thinking on the part of the weaker students than I have seen in other terms.
In the next five sections, I discuss each project group's work.