I have chosen this as one of the important "snapshots" in the course for several different reasons.
Sample of a Good Project: The project report turned in by this group is one of the best papers I have seen in teaching this course over the last three years, and as such, I think it provides an excellent example of what students are capable of doing as something beyond the basic content requirement of the course.
Student Learning: Moreover, the paper provides clear evidence of a deeper understanding of how problem solving in mathematics works and provides evidence that the students have learned to apply higher level mathematics to high school mathematics in a slightly deeper sense than they did at the beginning of the class as the following paragraph shows:
One other skill that we used in solving the worm problem was to estimate the area under the curve 1/x. Although students would not be introduced to integrals in most high school classes, they could certainly estimate areas geometrically. For example, we could estimate the area under the curve of 1/x between two numbers, say 1 and 5, geometrically by drawing rectangles, which are figures that all high school students should be able to calculate the area of.
Thus, the students have learned that there might be some way to bring ideas like the integral to the high school classroom, at least somewhat. I am, however, somewhat skeptical of whether they would actually mention the importance of finding the area under the curve, although students that have taken this class usually recognize that they could give a geometric definition for e, the base of the natural logarithm.
Groups That Don't Function Well: This discussion also brings to light some of the ways that I have tried to mitigate the problem of unequal student work on the projects, although this is something that I need to do even better. In particular, I suspect that early intervention in poorly functioning groups is extremely important. One way to work on this might be to modify the weekly project reports so that each student has a role like Caroline Persell does in her sociology class (Persell, 2001). Thus, I might modify the report so that one student writes it, and then another student provides a commentary. I am hesitant to do this in general since the students already view this class as a lot of work, and while I see the weekly report as a good way to help the students see the value of problem solving heuristics, I am not as sure that I see the same type of value in having the students write a commentary.
Projects Affecting the Content: Looking at this group also provides one illustration of how the project affects the rest of the class. Already I have mentioned how the weaker students in this group became a little less talkative throughout the class and how I attributed this at least in part as a result of feeling less confident in their group. This will contrast greatly with the next snapshot, that of a student's journey to mathematics. This was not the only place we saw an effect, however. On several occasions, I found myself changing how I presented material in the class (and what material I presented) because of this project group's needs. In particular, my October 4th journal entry states:
On the other hand, the worm group came in and they are getting close to a solution. I had to bite my tongue on several occasions to stop myself from explaining the last step to them. That said, after much prodding, they wrote down a different expression for the progress the worm makes, involving the ratio of the worm to the length of the band. I am hoping that they will soon see the harmonic series.
At the time, I thought the group was very close to solving the first part of their project. Weeks later, they still had not solved the problem because they were only writing down the ratio in one way. Thus, at a later point, I realized that I needed to alert this group (and others) to the importance and value of having multiple expressions for algebraic formulas. This realization led me to discuss the value of transforming algebraic expressions from one form to another (when covering the algebraic numbers in class). This led to a discussion of why it might be important in high school algebra to study symbol manipulation. Thus, the project led to a different way of presenting content, which in turn led to a discussion of the high school curriculum and what you might want to emphasize when teaching some of the symbolic manipulations like factoring.
Another place where the project influenced my teaching of the class was to cause me to bring to the fore some of the different ways of thinking about integration from a lower level perspective. In office hours discussion on the project, we worked through methods of understanding the harmonic series, and this led to discussions of how to prove the divergence of infinite series without calculus. In so doing, I came to a realization that the students did not know/remember the basic argument for the divergence of the harmonic series, namely breaking the sum into pieces each of which sums to at least 1/2. Since the harmonic series can be part of the high school curriculum, I then tried to work these ideas into class (with limited success).