Analysis

In analyzing what happened with the Dedekind cut section of the course, I find myself taking two separate tracks. First, there is what I wrote at the time that the homework assignments were turned in. Namely,

Tuesday December 5, 2000: I just finished grading homework number 11. I guess that I did not make clear enough what I wanted the students to do. I am somewhat concerned about the responses the students will have to the graded work. I am also concerned that my grading was somewhat arbitrary. I am convinced that if I had the time to go back through and see how I graded each problem, I would find many strange point assignments on each problem. I think part of the trouble is that I wanted the students to think critically about the problems, but between the time in the course (projects were also due soon), the lack of student understanding about the place of Dedekind cuts in the course ( I could have done a much better job on them), and the vagueness of the assignment, the students were left somewhat at a loss. I must say that I am particularly disappointed, however, in the responses on bounding the mistakes from the calculator. Students reverted to an empirical standard and discarded proof/algebra as being useless for the most part. The exception to this is Mary and the two students (Ellen and Teri) that work with her. Most other students missed the idea of bounding the error approximately. I think they would have done better had the question been more clear cut. Unfortunately, that would have changed the question and I would not have discovered the students' difficulties. I am also curious about the students responses to the last question on a different algorithm for addition. Many students seemed to misread the algorithm. Others didn't really look deeply into both sides of what the algorithm would lead to in both its strengths and weaknesses. On the other hand, Brad and Mary both gave wonderful responses to this, and I think that Jim's response was insightful.

The goal of the set was to have students use the higher mathematics of Dedekind cuts as a backdrop for the analysis of the types of problems that arise in doing the basic arithmetic operations with (irrational) real numbers. In class, we had discussed some of these issues surrounding repeating decimals, where a way out is to convert every number to fractions, and I had hoped that those discussions would help the students gain insights into the problems of using finite decimal approximations for calculation. The results from the students, however, seemed to show confusion on how to link these two ideas. Basically, I felt that here, where the students mostly felt out of their depth, with Brad, Tom, John, Mary, and Alan being exceptions to this for the most part, the students reverted to seat of pants work, rather than use proof as a basis for understanding the problems. That is, instead of framing the error question mathematically and deducing a result, the students framed the question experientially and simply tried things out on the calculator.

So what went wrong? I think one key issue was that the problems themselves needed a better framing. Not so as to get them to write what I wanted, but rather, I think I overestimated where the students were on understanding and proof on this concept. I suspect a better way to start problems 2 and 3 would be to ask them to use technology explicitly at first, and then force them to consider mathematically framing the arguments so as to establish generic bounds on errors.

All of that said, I think the Dedekind Cut section worked surprisingly well after seeing the final exams and having the exit interview with the students. In particular, three of the students chose to do the Dedekind cut question on the final, which is much higher than usual. Moreover, two of the students actually told me that they found the Dedekind cut section one of the best in the class, because it tied everything together. In particular, John said in a discussion of covering Dedekind cuts:

It (Dedekind cuts) was really cool, awesome,

and nobody challenged him. After many terms, most students would react quite negatively towards such a statement. Moreover, most students in the class recognized the need for the definition of the real numbers by the end of the class, and I think that this is an important step in understanding. Thus, while the original work on the homework set, was not as strong as I had hoped, I think that on the whole, I was successful in getting across some of the major ideas.

The changes I would then make in how I cover Dedekind cuts would then have more to do with refining the homework set, than refining how I covered them, except to make clearer the breadth of the reasons for why we cover them.