So far, my introduction to proofs class has met 5 times. We meet two times per week for 100 minutes, and I am continuing to use the flipped classroom model that I started last semester. The big difference is that this course is now a four-credit course and last semester it was a three-credit course. So last semester, we met two times per week for 75 minutes.
This is a great development but it has also created some challenges. Our goal in increasing this to a 4-credit course was not to add more content but to be able to spend more time working with the students as they attempt to do proofs and to spend more time on issues dealing with writing in mathematics. The basic model (times are approximate) for the flipped classroom last semester was:
- A short period (about 5 minutes) for questions.
- A short quiz (10 minutes).
- Introduction to the problems for the day (5 minutes).
- Work on the problems in small groups (45 minutes).
- Summarize the day’s work and look ahead to the next class (10 minutes).
My basic approach for the 100-minute class has been to slightly increase some of these times. In particular, I thought the students could work on the problems for up to 60 minutes. After a few times of doing this, I began to realize that this is just too long. As I walked around the room, I could tell the discussions were “deteriorating” in that they were not really about the problems or mathematics. So when I walked into class on Wednesday, I decided to try something a bit different.
I usually give the students about 3 problems for the day, have them work on them, and then collect one of the three problems to grade. This is done to insure that they will work seriously work on the problems and to make sure I do not have too much to grade. What I decided to do was to work on the problems one at a time and then discuss that problem as a class after students have worked on it in groups for some time. I will have to make a judgment about when to start the discussion. The idea is to have the same amount of total time students work on problems but to have it divided up into smaller chunks of time. This will hopefully keep the student discussions from deteriorating after a long time working on problems. I will still have to decide which problem to collect and I am not yet sure how I will do that. One option is to have one of the problems be collected before the discussion for that problem. Another option is to let the teams decide which problem they want to submit. We will see what happens.