Monday, July 17, 2006

More meta, and then some

Well, I've got the first day's class clearly in mind now:

We begin with a very active "interpretive dance." The students will be divided into three groups of roughly equal size, and each group will be given a corner of the room to occupy. When the "music starts," a set percentage of the folks in group A gets herded off to group B, another set percentage shuffles off to join group C. The rest remain. Simultaneously, set percentages of groups B and C either stay put or head off to each of the other groups.

The percentages will be chosen so that one of the groups (Group A, say) ends up receiving far more than it grants.

Then, in computer science parlance, "wash, rinse, repeat." After a few more time steps have passed, the students should take note of how many people are standing in each corner: where'd everybody go? There's got to be something to the fact that most people wound up in Group A.

Try it again, this time starting out with everyone in Group C. What happens after several steps have been made?

At this point it'll be time to get the students to suggest means of making the exercise mathematically precise: what formulas accurately describe the game just played? What information can we extract from these formulas? What techniques might make manipulating these formulas easier? What realistic applications might involve computations like these?

This is a simple hands-on, and very concrete, introduction to Markov processes, a concept involving several of the key aspects of the foundational knowledge the students will be asked to store in their long-term memories: vectors and matrices (and their concomitant operations), linear transformations,, determinants, and eigenstructures. Of course, we won't be getting to all of these ideas immediately, but even on the first day we should be able to talk about the relationship between vectors, matrices, and the Markov game.

I've got some other learning activities in mind. This morning while walking in to campus I thought of an integrative exercise whose successful accomplishment demands that the students understand the interrelationships between the various key concepts of linear algebra: a knock-down, drag-out, no-holds-barred battle between vectors and matrices for the title of Most Important Linear Algebraic Concept. Somewhere near two-thirds of the way into the semester, one small group will take the side of vectors, a second that of matrices, and drawing upon all that they've learned about these concepts and how they relate to linear transformations, vector spaces, bases, and so forth, representatives of each side will do their best to persuade the remainder of the class that their respective point of view is superior. (The undecided hoi polloi will be free to ask probing questions of either party.) A solid defense of either position will require well-rounded knowledge of all related concepts.

I spent the last couple of hours typing up tentative versions of three of the semester-long projects. As planned, one deals with analysis of traffic flow, while a second, slightly modified version asks after applications of linear algebra to games such as Monopoly and backgammon. A third investigates the relevance of linear algebra to the structure of crystal lattices. While the first two draw heavily upon the theory of Markov processes as illustrated above, the linear transformations that arise most naturally in the third have to do with the symmetry of the crystal lattices considered.

Hot dog!

I've got to find four or five more suitable project topics...I'm thinking along the lines of some application to differential equations for one, and perhaps economics for another. Given the number of atmospheric science majors floating around, it would be nice if I can give some of them an introduction to linear models in atmospheric sciences...I'll see what kind of storm data I can find.

For now, it's time to head home.

To be continued...

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