The groups project was the "worm problem". This problem is from Graham, Knuth, and Patashnik's book Concrete Mathematics, and asks if a worm is on an elastic band that is 12 inches long, crawls 1 inch each day, and each night a boy stretches the band another foot, does the worm ever reach the end. In answering this question, the students must confront the harmonic series and then approximate at what term the series gets larger than 12. As is true with most of the projects, the problem can initially be approached using technology. A spreadsheet solution can be found to the problem, but then the students need to understand how to get that solution without the spreadsheet, and should ask (and answer) another question.
Early on, this group had answered the basic question using a spreadsheet (a common way to get an answer), but then they had decided to approach the question of whether it mattered how long the band was in the first place. For this latter question, it was important that they not rely on a spreadsheet solution since if the band is 100 inches, a spreadsheet solution will not work as the numbers become too large. At the start of the problem, all three students were working together, but once they had to prove the answer, Ellen and Teri started checking out of the process.
By mid-October (about 5 weeks into the project), it had become clear to me that this group had serious problems in working together. Unfortunately, I was not quite sure why the problems were occurring. I realized that Mary was doing most of the working on the problem and that she was frustrated that the other two students didn't seem to be worrying about completing the problem as much. In addition, it was clear to me that of the other two students, Ellen appeared more willing to work on the problem if she could find a role where she felt she would be helpful. Consequently, I tried to brainstorm with Mary what we might find for the other two students to do that they would find interesting. This broke into trying to find technological attacks for Ellen to work on and to find interesting questions for Teri to look at.
The other approach I took was to talk directly with Ellen and Teri whenever possible, asking Mary to listen and play the role of the summarizer. Unfortunately, at that time, I was not aware of the value of actually assigning such a role to students as Caroline Persell (Persell, 2001, personal communication) did in her Sociology capstone classes. Thus, the directions I gave Mary were far too vague to really force her out of her role as group leader.