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.

*For students to be able to look at advanced mathematics topics and see how they are reflected in the high school curriculum and how these advanced topics might inform their teaching of high school mathematics.*- One of the requirements of the project is that they discuss how their project relates to the high school curriculum. This is shown in the grading rubric for the project under Conclusions Drawn. For the next time I teach the course, I will be changing the rubric slightly. In particular, I am changing the computations part to computations and statements of results.*For students to gain a deeper understanding of themes of mathematics, and how various subjects in mathematics can relate to a common theme.*- Most of the projects can tie nicely into a theme of mathematics and certainly require using ideas from different areas of mathematics. This is particularly true of some of the geometry projects and those projects that involve either pi or the exponential function in the answer.*For students to successfully experience mathematical research on their own.*- This is the main goal of the projects. For the students to experience solving a mathematical problem similar to how mathematicians might do so. That said, we should be careful in describing mathematical research. The main goal for me is that the students learn to attack problems**on their own**. Thus it is particularly important that they do not look answers up in outside sources. Of course, mathematicians do look at how other mathematicians have solved similar problems, but we also look for our own solutions. The students have a great deal of experience in trying to mimic "known" methods of solutions, and thus I feel that they need less experience doing this. On the other hand, when the students reach certain stages in their projects, I probably will make suggestions for them to look in certain books. For example, once the students doing the "worm project" (#6) have discovered the role of the harmonic series in the problem and have worked through one proof that the harmonic series diverges, I suggest that they might want to look at a calculus book. The idea is that they should arrive at their own work on this, but then look through other texts. Tied into this, I treat the students, much like I treat my graduate students. I help them through the projects, giving them some direction, but I also treat this as an apprenticeship to mathematics.-
*For students to learn the value of asking richer mathematical questions. -*Throughout the project, the students are supposed to ask questions of their own. Thus, I want them to see how the richer mathematical questions can provide them with greater understanding on working on the projects.

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.

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