Upon analyzing the evidence from this class, I discovered several surprising features. One surprise was that the project work the students did, influenced how I taught the content portion of the class far more than I expected it to. This reflection concentrates on that aspect of the content-research project interaction.
It is well accepted (NCTM Standards, 2000) that changing how one assesses a class will change how one teaches a class. There are two main reasons for this. First of all, changing assessment techniques leads to gathering different information from the class. In the case of the research projects, I received weekly updates from the students detailing their mathematical difficulties with the projects. Moreover, the students would spend time during office hours discussing their projects with me and with each other. These additional inputs gave me a richer knowledge of student misunderstandings about mathematical ideas. Additionally, the need for the class to be successful on the assessment also changes how instruction is given. In this class, for example, it became far more important for me to discuss how discoveries in mathematics happen. The change in assessment also affects students. Students' desires to be successful on assessment may change what they believe to be important in the class. Thus a change in assessment can cause students to view the class through a different lens, which in turn will change what they pay attention to in the class. All of this said, at the beginning of the course, I saw the projects and the course as a“layer cake”. That is, the course had multiple layers formed by the in-class student work, the projects, and the lectures. Each of these related to each of the others, but also occupies its own space. Moreover, I saw the relations between them as fairly clear-cut. In particular, I saw the lectures influencing the in-class student work, the in-class student work influencing the lecture and the project work, and the project work possibly influencing the in-class work. After the course was over and I started analyzing what happened, however, I prefer a “strudel” vision of the interactions. That is, the three pieces wind together, each influencing the other. Moreover, one of the roles I took on constantly was tying the pieces together in an attempt to make all three successful.
The major data I am analyzing for this reflection is my Instructor's journal, the pre- and post-surveys, the grouped student interview, and a comparison of the text for the preceding term of the course and the text that was mostly written during the course in concert with the weekly student updates. This work is exploratory in nature, and for this note, it is based on a preliminary analysis of the data.
Before continuing with the main analysis, it is necessary to discuss the ways that I set the project group work up. Some of this can be found by reading the second artifact on the Projects. Other pieces are strewn throughout the portfolio, although here I hope to tie them together. The mainstay of setting the projects up to be successful is the time I spend in my office with the students. Before this, however, I need to get the students to buy into the idea of doing research projects. This student buy-in is important as in the past I have noted that students not seeing a benefit of doing a research project tend to treat them as a long homework set, and consequently they appear to avoid exploring the mathematics.
Consequently, the first day of class, I spend 15-30 minutes discussing mathematical research and my experiences with doing such research. Typically, I begin discussing with the students the idea that mathematics problems are not to be done in 5 minutes or never and then comparing this to their own experience. The purpose of this discussion is to help the students confront the idea that the types of questions they have been asked are somehow not authentic to the doing of mathematics. After this, I mention my own experience with research problems. I make clear that I do have a research record as a mathematician, and then I hit them with the key statement: “I was given my first research problem 15 years ago as a Ph.D. research problem. I have never solved it, but I have gotten three papers out of it.” I then mention, that I am hoping to answer the original question in another 10-15 years. I tell the students all of this so that they might begin to see that not having an answer to a mathematics problem is okay, as long as you are doing a good mathematical investigation. After this I spend some time talking about how mathematicians usually solve problems in a social context. That is, even on papers that are by one author, it is usually the case that the author has been talking with other mathematicians. At this point, I might bring up the case of Fermat's last theorem, and how even though Wiles is said to have worked alone in his attic, he did talk once a week with John Conway about the work (Fermat's Last Theorem, 1999).
The first true homework assignment is also meant to help foster them in doing their research project. Most of the research projects can be started by gathering data using technology. I make use of this idea in the first homework set (see Artifact 1, Assignment 1, Pedagogical Reasoning). Thus, my earliest homework set helps get the students prepared for doing the project. The last place that I anticipated making basic classroom changes is that I usually make time for the students to present their projects at the end of the class. This usually happens during the final exam period, and I like to think of it as a mini-conference.
Analyzing the above paragraphs, we see why I saw the projects as being layered onto the class at the beginning of the course. This, however, is somewhat different from what actually happened in the course. In particular, I didn't think about the implications of discovering student misunderstandings. The projects force the students to confront mathematics in a different way than they typically have before. This in turn creates a new way for student misunderstandings to become apparent.
The major effects of the student projects on the class in two main ways: they changed student attitudes about the relationship between the students and mathematics and they provided me with richer information about student misunderstandings of mathematics, which I could then address. These changes had their own effects. Among these were that they caused students to concentrate more directly on how one does mathematics than they naturally do and they caused me to change how (and sometimes when) content material was covered in the class. I also believe that these effects led to a change in the nature of classroom interactions and discussions, but the link here is more tenuous. A discussion of this can be found in the next reflection section.
These effects also changed how we dealt with new material in the class. As students started wanted to take ownership of mathematics and saw themselves as discoverers, it appeared to me that they were more likely to ask questions, and better able to function on the in-class group assignments. Unfortunately, I have no data to support this one way or the other, but I believe it bears further looking into. These changes in turn caused me to find more creative ways to use in-class group work, and to allow it to be more free form, since I was confident of good mathematics being done by most students. This last is something I have had trouble with in the past in my classes as large numbers of weaker students would check out of discussions.
Thus, it appeared to me that the projects caused the course to change dramatically from where it had been. All of this, however, may be dependent on the students and other complicated factors. While most students disagreed with him on this, John stated this position clearly when he suggested:
I think one of the things that altered the way, sort of the pedagogy of the class is that almost all of us were people who were very interested in and capable about talking about mathematics and talking through mathematics. I know in a lot of other math courses, where, I know in the math courses I teach, and also in the other math courses I have taken is the different group of kids, and I think it has a lot to do with people who are interested in math education as opposed to just mathematics I think those people are less comfortable with talking and with trying to work through those sort of ideas.