Has anyone written a good essay on the "two cultures" of high school mathematics and university mathematics? The former is not just a subset of the latter. For one thing, I think high school teachers tend to be punctilious about some matters that university mathematicians are sloppy about. A case in point is the meaning of "imaginary part". Is the imaginary part of a+bi equal to b, or to bi? I don't think that I myself am consistent about this. I got the impression that Barry Mazur, in his book "Imagining Numbers", used a convention at odds with what's being taught in high schools these days, because one irate amazon reviewer of the book opined that Mazur was not qualified to write a book on advanced math because his misuse of the term "imaginary part" showed that he didn't even know basic high school math! I also think that high school math culture enshrines some 19th century usages that have fallen into disfavor. For instance, the word "cone" in three-dimensional analytic geometry refers to a complete locus, not just the half that lies on one particular side of the vertex (and some high school teachers are rather pedantic about that); but from calculus onward, "cone" in university math means what high school teachers would call a half-cone. Part of what's going on is a time-lag between developments in research and the training received by high school math teachers, and the nature of the courses that future high school teachers take. To give just one example: I've seen more than one course in "modern geometry for future high school teachers" in which the discussion of non-Euclidean geometry is extremely axiomatic and historical, and hyperbolic geometry is illustrated by way of a single picture (the pseudosphere), maybe supplemented by pictures of the Klein or Poincare models, but with no upper half-plane (and no Escher pictures!). Who would write such a book? Someone who had read such a book at a formative age, and who had a very static view of mathematics. It would probably be easy to write a one-sided essay that simply contended "high school mathematics is bad mathematics" (cf. the essay on applied mathematics by Halmos). But I think there's more to it than that. Comments? Jim Propp