One of the main features of the course is the requirement that student groups complete semester long research projects. The first day of the course, I hand out a list of 17 potential project questions (these come from a variety of sources - many of them from my colleagues) and ask the students to return a list of their top 5 choices from the list, together with students that they would like to work with. I then assign groups of three students to a question. Each group, however, does a different question. I try to pair up students that want to work together, although I make no promises about doing so. As part of the syllabus, they are handed out a rubric for the grading of the paper as well as instructions for the paper. One included assumption on the projects is that the question that is asked is only the starting point for the project. I want the students to understand that good mathematical questions/problems don't necessarily have final answers, but rather they lead to more interesting problems, and that part of the role of a mathematician is to ask those questions. Moreover, I want them to see that as part of their job as a teacher.
The projects play a major role in the class, as is shown by their 30% of the grade requirement. They work to satisfy four of the five process objectives of the class.
In addition, I hope that in doing the project and thinking about how it applies to the high school curriculum, they will also gain some experience in trying to transform content knowledge (the answer to their project) to pedagogical content knowledge (how it might apply to the curriculum).
For these projects to be successful, I have found (over three years of experimentation) that most students need a lot of support and scaffolding. The first piece of scaffolding is given by handing out a sample project that was written by David Meel and myself. The second scaffold was to require students to turn in a weekly project report in which they were to state several types of questions (mostly coming out of heuristics suggested by Polya). I believe that these updates are extremely important because they help the students ask me questions that I am willing to answer for them, they help the students ask the questions that will help them move forward, they provide an opportunity for me to help students investigate heuristics more deeply, and they help provide us with an important avenue of communication about the doing of mathematics rather than about the content of mathematics. I use the student responses to these questionnaires to help me when they come in and ask questions also. This time with me is the third scaffold provided. While they talk with me, I spend a great deal of time letting them talk to themselves. In some ways, I see the projects as reflecting Magdalene Lampert's analogy of teaching mathematics to teaching dancing (1990). The idea is that the students spend some time dancing with each other, some time watching me tell them how to dance, and some time dancing with me (the expert) so they can see what dancing well feels like.