The second class day starts with a new topic, although we come back to working with equivalence classes (something that the students have already had, but is worth going through again). On the second day, I start with a simple question.
What is the 17th digit of 1/19? Please do this using your calculators.
The second half of this question is for the students to program their calculators to do long division. The idea of this project is threefold. For the content of the course, one of our early topics of discussion and early homework problems is on understanding the periods of decimals for rational numbers. This is something that the students see that can be brought to a high school classroom, particularly when you point out that they teach that any rational number has a repeating or terminating decimal expansion, and yet when they are used to relying on calculators, they have a difficult time convincing anyone of this for a simple fraction like 1/19th. (This is a proof that the students may have done once before depending on what they were taught in their first proof course, but even so, most students appear to have trouble explaining what is going on.) The second purpose is to force the students to confront the idea that I have something to teach them about high school mathematics, and that their learning may require understanding advanced mathematics. The first half of this is clear, the latter half only really comes clear after the students work on the questions in the first mathematical homework set. The third purpose of this problem is to force the students into a discussion of what understanding is. Ideally, this discussion can happen by e-mail, when one of the students having trouble with the program will complain that they understand long division, but cannot do the program.
All of this said, there are some potential pitfalls. In particular, many students have little experience with writing calculator programs, so I spend the second class day with the students working in groups, the hope being that we have enough students that have programmed their calculators so that no one will be completely lost. Even so, I usually have to work with some of the students to help them write a program. What I do with these students, usually, is to have them work through how to button push the algorithm on the calculator, and then I teach them the requisite knowledge. I also will hand out a short note to help them get started, and I allow them to use a spreadsheet instead of a calculator for the homework.
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