The reflections come in multiple parts and formats. While the students were doing their work, I wrote two entries in my journal.
After the assignments were graded, I wrote the following entry. This last entry also talks a little about the next assignment and how it ties together with this one.
Reflecting on the assignment now, I would say that there are several important points to make. The calculator program appears to be very important to the class on several levels. Interviewing the students after the course was over, one of the things they remembered most from the early part of the course was this assignment and how it was both difficult and showed that I had something to teach them. The students came away with a recognition of their lack of complete understanding of basic topics and how this might help them. Not surprisingly, they didn't remember much about the periods of fractions, but I am convinced that it allowed us to start discussing .9999... earlier than we might have, and to talk about whether period is well-defined.
The other problem that must be kept in the assignment is the problem not from the course text, that is the high school algebra problem that asks the students to do the problem and to then watch a non-expert solve the problem and reflect on the difficulties the person has based on the understanding of fractions that we are developing in the class. The results on this problem were quite surprising to me. Of the 14 students in the class, only 5 solved the problem correctly without help. Some of this was a result of not being careful (Tom's problem), but some of this was because the students were out of practice doing high school algebra. Grading the problem made me aware of the students' difficulty with some high school algebra problems. On the other hand, the students discovered that there could be a relationship between higher level mathematics and pedagogy. This tied into a goal that I had never stated for the class until after that term, namely to help the students learn how to transform advanced mathematical content knowledge into pedagogical content knowledge, what Shulman (1987) defined as the content (and process) knowledge used in teaching..
That said, one difficulty in the assignment was the question on showing that the period is the same as the smallest power of then that is congruent to 1 modulo n (if n is relatively prime to 10). This problem was extremely difficult for the students, and I think it bears thinking about. I believe that you need to keep difficult assignments in the early work, but I also want assignments that the students will understand the solution of, and I am not as sure whether this assignment fits that bill. Possibly, the assignment should be changed to provide some outline of the proof and ask the students to interpret and fill in the gap.