Teaching/research statement

John Kerl

Graduate student

Department of Mathematics

University of Arizona

Teaching/research statement

— draft and sketchy notes —

No separate teaching and research statements (not at this point anyway — as I am a second-year graduate student).

Tension in math between generality and specificity

y=mx+b good for all sorts of stuff. It's important that we do this.
Historically, specificity came first. Generalization came later once people saw similarities. (Cf. group theory).
For the learning, specificity must again come first. It's tempting to start a course with Consider the function y=f(x) ... (for freshmen), Let A be a ring ... (for advanced undergraduates), or Let M be a compact, orientable manifold ... (for graduate students). But when we do this we do our students no service whatsoever. We skip directly to the generality which we as instructors already possess, leapfrogging off the concrete examples which we have already digested. Our students have neither; they have nothing to work with.

Mention the exemplar model of cognitive psychology. Quote:

Suppose that you want to teach the "cat" concept to a very young child. Do you explain that a cat is a relatively small, primarily carnivorous mammal with retractible claws, a distinctive sonic output, etc.? I'll bet not. You probably show the kid a lot of different cats, saying "kitty" each time, until it gets the idea. To put it more generally, generalizations are best made by abstraction from experience. — R. P. Boas.

What do we retain after having read a 500-page book — all 500 pages, verbatim? Certainly not, nor even a single page verbatim. Rather, we remember a flavor, a diagram, a nicely worded turn of phrase — and the skills we have acquired by having performed themselves. All else is lost. As a result, in my writing and my reading, I care very little about wading through miscellany I won't retain. I care very much about organizing and refining those things for which I do expect myself to have 100% retention.

Mnemonic spiel

Something about prolrev?

Sob about: not much other than writing down and organizing.

Give the self-quote about improving the question. Cf. word problems.

Partial information: Smolin quote. Mine about what to do when we don't know what to do.

Give the "what do I know and how do I know that I know it" spiel.

Top-down spiel

Historical development: good and bad. Good because you see where things come from and why. Bad because sometimes the new way makes far more sense (cf. DG.), or perhaps has been morphed out of recognition. But, we do need applications.

Give many specifics about teaching and about research.

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