Rick Simon gave some suggestions for applying the spirit of the Standards (in my opinion) in college level courses. He suggested getting students more involved in reading mathematics by giving "pre-quizzes" on material the students were supposed to read, but had not heard in a lecture.
I would like to hear more discussion about (a) reading mathematics as an aspect of mathematical communication; we hear a lot about having students write, but little about reading texts, which, after all, are a rich source of mathematical material. (Well, not all texts, but books are still good things.) (b) getting out of the lecturing rut at the college level.
I am teaching a course in topology for math majors. This is a hard topic for many students, and I despair when I see terrified blank stares from some of them. One thing I have tried that seems to help is, after a new concept, giving a simple example for the students to discuss in pairs for a few minutes. This is the kind of thing that one usually discusses by lecturing. Most students are in note-taking mode, which is often the opposite of thinking mode. By making them each say something, they have to wake up and start thinking about the new concept.
[Historical note: Point-set topology has been taught by discovery methods for many decades--the Moore method, or Texas topology. However, this isn't feasible for a 10-week course in which many new concepts must be introduced and understood.]
Susan Addington (email@example.com) Math Department, California State University San Bernardino, CA 92407 World Wide Web: http://www.math.csusb.edu/