Day One, revisited, or, Patrick very quickly gets off into pedagogical theory

Back to the grind. Today hardly felt like a school day at all, as my only MWF class is an 8:00-to-8:50 section of Linear Algebra I that was over nearly as soon as it started. (At least on Tuesdays and Thursdays I won't be done in the classroom until noon-thirty.) As first-days-of-class go, it was a good one, though. I've had better, but I've had far worse.

Plus ça change, plus c'est la même chose. Both other times I've taught this course (including the first time, the iteration of the course that occasioned the founding of this blog) I've started with some variation of the same game, a simulation of a Markov process in which the students shuttle some sort of token back and forth at each iteration of the game. The first run (Fall 2006), the students themselves were the tokens as the class participated in a great big single instance of the game; I switched to pennies (and smaller groups) the next time I taught the course (Fall 2010), and I stuck with that latter version today, though leaving a bit more room than I did before for students to discover and speculate upon the patterns their own damned selves. This time around I also asked the students to take bolder and more unassisted steps toward the next conceptual mile marker, solution of the linear systems that arise from the Markov process we investigate together: not only must the students experiment and then speculate on the outcome of their experiment, they must then find the appropriate mathematical model (a simple linear system in two unknowns) and then back-solve the "run the model in reverse." All in 45 minutes' time!

All in all, it went remarkably well. No one seemed lost ("one in a row!" as my colleague Tip would say), and everyone participated actively. I'm aided this semester by the fact that I've only got 23 student in the class (yay), though they're packt like sardines in a crushd tin box (boo), sitting at single-person-sized tables (yay) bolted together and arrayed in orderly rows (boo) in such a fashion as to discourage all but the most anachronistic teaching techniques (boo hiss).

Interesting facts (yes, there is a train of thought that took me from the previous paragraph to this one): recently, while reviewing the literature on the effect of class size on learning, I discovered that (1) said literature says almost nothing about college-level instruction, most research having been done at the K-12 level, and (2) a number of studies do not, strangely enough, suggest small class size improves student learning in mathematics. It was only after a bit of reflection that I realized why this might be: such studies, while controlling for class size, do not (and, methodologically, cannot) control for instructional method. Thus what I suspect is happening in these studies is large-section lectures are being pitted against small-section lectures, lecture being, until recently, about the only viable instructional paradigm for large-section classes. Of course, it is pedagogically retarded (in the literal...well...until recently literal...sense) to assume that one's instructional method remain the same when smaller class size permits more effective application of student-centered learning strategies: pit large-section lectures against small-section IBL and you're sure to see a difference.

Maybe more about that in a post soon to come (why on Earth was Patrick researching this topic? Edge-of-your-seat action!). For now, I've got reading to do for my first meeting of HON 479 tomorrow!