I'm almost ready to start "integrating" the components of the design into a coherent teaching strategy built around a thematic structure for the course and meaningful learning activities.

Like I mentioned in an earlier post, I've had a hard time "categorizing" learning activities, but I have managed to come up with a few general schemata:

- Dialogues: students pair up and each pair constructs a script for a dialogue by means of which they explain to one another a tricky concept just defined, or work their way through an application.
- "What If..." exercises: students are asked to consider the effect of hypotheses on various mathematical statements. By adjusting these hypotheses (injecting a "what if" into the problem), students can explore generlizations of the problems already posed.
- "Calc Revisited" exercises: calculus topics related to linear algebra, such as Jacobians, change-of-coordinate transformations, cross-products, area formulas, and tangential approximations, are revisited in order to highlight the connections between the two subjects.
- Logic Games: various standard games involving mathematical logic (like the Prisoner's Dilemma) but no explicit mathematical formulas can be used to encourage participation by students who may be a little more unsure of themselves in performing math in front of others. By removing the formulaic, "formal" mathematical aspect of a problem, those who perceive themselves as weaker at math might step up to the plate with more confidence. (It's almost like tricking them into doing mathematics!) These'd be great for team-building exercises, too.
- Resource Scavenger Hunt: I've not worked up an exact format yet, but I hope this to be an exercise which encourages efficient and effective use of reference tools (on-line and off-line). The way I envisage it right now is something along the following lines: students are given a short list of topics on which they are asked to find information...maybe an article, website, or textbook...and are let loose for 24 hours to find the necessary material. The more relevant the sources they track down, the more appropriate, the better they'll do. Admittedly, there's some tweaking to do here, but I think I can make this a fun exercise.
- "Write a Bad Paper" exercise: perhaps the best way to learn how to do something right is to try to do it as poorly as possible. As I see it, you've gotta work pretty hard to write a truly awful paper, and my guess is that most people will notice palpable problems in composition, coherence, correctness, and citations as they try to make their papers bad.
- "Correct a Bad Paper" exercise: this exercise can be paired up with the previous one. Exchanging intentionally bad papers with a partner, each student can try her hand at righting wrongs in peer review.