Wednesday, August 15, 2007

Notes to self, part 2

We're just a few days away from beginning the new semester (Monday, August 20th: do you have your calendars marked?). In between periods of vegetative depressurization from the newly-ended REU and continuing research in probabilstic graph theory, I've been spending a bit of time during the past couple of weeks putting together various activities for MATH 280 and MATH 191. Much of my planning is outlined nearly illegibly in the margins of my copy of John C. Bean's Engaging ideas: the professor's guide to integrating writing, critical thinking, and active learning in the classroom (Jossey-Bass, San Francisco, 2001), but I really need to compile it all in one place. Ergo...

(An open exercise in academic free writing)

During the coming semester I will be putting greater emphasis on "decentering" activities that promote cognitive dissonance, unorthodox points of view, and healthy skepticism. In presenting students with counterintuitive mathematical ideas, I can imbue them with a sense of surprise, wonder, and curiosity. In asking them to develop the ability to see a problem from all perspectives (literally and metaphorically), I ask them to become stronger problem-solvers. In encouraging them to question unproven assumptions, I not only charge them to be more careful in their calculations; I also open up to them unexplored fields of inquiry. Where would geometry (and by extension, much of modern mathematics, not to mention the physical sciences) be had a number of brilliant minds not questioned the validity of Euclid's Parallel Postulate?

Now, how to do all of this?

I hope to introduce the students to the idea that mathematical discovery, along with its recording and transmission, is a dialectical process that involves the researcher in conversation with herself and with others. The act of discovery is almost never a burst of light illuminating the void, but rather is born from the steady nurture of an ever-growing spark. Discovery begins with the posing of a question, the wrestling with a problem. One's first thoughts on a problem are generally chaotic and messy, and harken back to solutions to analogous problems and inchoate modifications of earlier ideas. After long hours of talking with others and lying awake at night staring at the ceiling, and after countless pages of notes have been scribbled, studied, and redacted, more complete ideas take shape. Bean (p. 20) speaks on the nature of the written manifestation of this process: "the elegance and structure of thesis-governed writing -- as a finished product -- evolves from a lengthy and messy process of drafting and redrafting."

Below are a number of the exercises I plan on implementing in some fashion during the coming semester:
  • Response writing to Polya; possible guiding questions: "Is Polya's proposed process relevant in a modern problem-solving course such as MATH 280? Take a position on this question, and defend your point of view." "Have you ever applied Polya's process, knowingly or unknowingly, to solve a problem posed to you? Explain carefully." "Do you feel that intuitionism is a defensible mathematical philosophy?" "Use Goldbach's Conjecture to illustrate the difference between constructivist mathematics and nonconstructivist mathematics."
  • Decentering exercises focusing on puzzling phenomena such as various sizes of infinity, space-filling curves, fractal dimensions, et cetera.
  • For the 191 folks, to get them to take a position in a short thesis-governed paper: "Suppose you need to differentiate a function of the form f(x)/g(x). Do you prefer to apply the Quotient Rule, or would you rather rewrite the function as a f(x)(g(x))^(-1) before applying the Chain and Product Rules? Explain the reasoning for your preference."
  • For the 280 folks, to accustom them to "mathematizing" messy problems and developing intuition (skills I feel are overlooked in even the more discovery-learning oriented proofs courses): "Consider the game of Nelinurk (see Play the game for a half-hour or so to get used to the rules and the flow of the game. Once you feel comfortable playing, see if you can describe the game mathematically and develop a strategy for optimal play. Explain your strategy as clearly and as completely as you can. (It may help to develop your own terminology and notation as you write.)"
  • More "intuition-building" activities for 280 students: estimation exercises? Incomplete proofs? ("How big?...", "How many?...", "Give the outline for a proof of...") As I said above, I feel that the nurturing of mathematical intuition that's done in most proofs courses is woefully outweighed by the emphasis placed on learning how to do formal proofs. (And no, I don't think it needs to be put off until a "problems course" like our 381; good intuition makes for clearer, more succinct proof-writing, and clearly written proofs feed a healthy intuition like Wheaties feed Mary Lou Retton.)
  • Taking a page from my own playbook, five or six years ago at Vanderbilt: have 'em write a few "poems inspired by mathematics." It can't hurt, and it might be just what the more humanities-minded students need to get their creative juices flowing.
  • Have a "show and tell" day on which I ask everyone (myself included) to bring in all of the notes, scribbles, emendations, and so forth that went into the final draft of a given project. (I might simply ask them to save all of their homework drafts?)
  • Class-opening and class-ending one-minute essays: "where do we need to go today?", and "where did we end up in our travels?"
  • Mock trials: in 191, the obvious, Newton v. Leibniz. In 280, perhaps Brouwer v. Hilbert? Each side is taken up by roughly half of the class, certain individuals chosen to act as the given personages, with others as their supporting staff (i.e., legal counsel). For the 191 debates, I could even have the two sections square off in a "finals" round, one section taking the side of Newton, the other that of Leibniz.
  • Analogy games (cf. Bean, p. 111): ask the 280 folks to complete the following and elaborate upon it: "Writing proofs is like ________ ." For the 191 classes: "If differentiation were an Olympic event, it would be most like ________ ."
  • Precise proof-summarizing and theorem stating: require students to write a proof summary or a theorem statement using a precisely defined number of words, giving maximal credit only for using exactly that many words. Example: "explain the Axiom of Choice in precisely 25 words."
  • To give the students a taste of "original research," I can hand the 280 folks the data I obtained this past summer on consecutive inverses modulo p and ask them to describe the patterns they see. I can do the same for the Calc I kids, giving them the graphs of the sequences of expected degrees for various of the random tree construction algorithms, asking them to supply likely models (and concomitant analysis) for the shapes they see.

Hmmm...that's all for now. More to come, I'm sure.

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