Number Theory for Teachers

This summer I'm taking a course in Number Theory designed for secondary school math teachers, specifically those of us who went through the Traders-to-Teachers program at Montclair State University. As such, it will cover a subset of the topics usually found in introductory number theory courses, with, I suspect, less rigor. From the syllabus: "Our class discussions will relate number theory topics to middle and high school level curriculum."

At the beginning of the first class the instructor asked what we hoped to get out the course. I kept silent, as my situation - unemployed - and interests differ from the rest of the class. However, I continued to ponder the question and hope to answer it here. Aside from the obvious and trivial response - to satisfy a requirement for full teacher certification - here's what I want to get out of the course:

Errol Morris on Pythagoras, Incommensurability, and Whiggish History

In March 2011 Errol Morris, an award-winning documentary filmmaker and director of TV commercials, published a series of five opinion pieces in the New York Times with the collective title "The Ashtray". Those who know about Morris through his films and ads may not be aware of his philosophical writings about the nature of truth, knowledge, and history. If those topics put you off from reading Morris, perhaps it would help to know that the series title comes from the object thrown at the author by Thomas Kuhn, author of "The Structure of Scientific Revolutions", when Morris was a graduate student under Kuhn at Princeton. (If that's still not enough, then see his description of a Dan-Brownish library encounter, below.) The series as a whole is a critique of Kuhn's notion of "incommensurability" of scientific paradigms, but the part that appealed to me particularly is the third in the series, on the origin of the term "incommensurable" in mathematics and the legends surrounding the Pythagorean proof that the square root of two is irrational.