### Compare and Contrast Activities

In a Compare and Contrast Activity, each small group solves a slightly different example of the same calculation. The focus of the activity, then, is often on the full-class discussion that happens after the activity.

Because each group is effectively doing the same calculation, it is possible, at any time during the activity, to pull the class back together briefly to give clarifying comments or to discuss problems that seem to be common to many groups. This strategy becomes particularly important as class size increases. It is not possible for an individual instructor to effectively and consistently talk to more than about four groups during a half hour activity. With a faculty member and a TA in the room, then, it is possible to manage 8 groups of 3, or about 24 students. The paradigms now typically have closer to 30 students with an instructor and a TA, so we pull the class together more often than we used to.

In the wrap-up, there are a number of things that need to happen:

1) Make sure that all the groups, even those that may not have reached the end, understand all the main parts of the calculation.

2) Make sure that students have a chance to reflect on what their calculation shows, what it means for physics, how it might be generalized, what the limitations are, etc.

3) Compare and contrast the results of different groups. What can students guess about possible patterns from what the different examples show? The instructor may want to confirm or deny the pattern or, immediately after the group experience, to give the general proof, if necessary. The activity is designed to give students enough experience that they are prepared to believe the result and to understand subtleties in the proof. This approach is backwards from the traditional lecture style that involves giving the general proof and then having the students work out examples for homework. Here is a sample narrative describing a day when the students are asked to come up with hypotheses for a rule about the geometric interpretation for determinants by looking at different cases, each calculated by a different group, as the groups present. The narrative and video show clearly how the students' ideas change as they have more and more examples to consider. (Add a link to videotape from Day 2 preface 06)

4) This is a great opportunity for students get some experience with mini-presentations. Every group has to present. There's not a lot of time to stress out about it beforehand or to worry if you do something silly because the pace is crisp. Since the class has many opportunities for such presentations, within just a few weeks most students are comfortable. Also, this is a great chance for the faculty member to make suggestions about students' presentation style, for example, we have been motivated to make the following comments in such wrap-up sessions: “We call that …”, “writing det without writing det of a particular matrix isn't right”, “speak loud enough for people to hear”, “break the chalk to keep it from squeaking”, “write big enough for people in that back to see”, etc.