I am apparently a good teacher. Well, I've won awards and I've got some good comments from my students. Now those indicators are external, but there's some good internal stuff as well. Sometimes I just know I've taught a superb class.

For all that, my view at the deepest level, is that I should be doing much much better. I have some general ideas of what the problem is but it's not all that clear what the course-by-course, class-by-class, topic-by-topic implications of that are. I know I want my students to have fun with the material, to let loose, take risks, and find themselves flying through an awesomely beautiful universe. At the same time I want them to understand that mathematics offers them enormous technical power, the possibility of some effective control over the small universe of ideas they are engaged with, a sense of real independence, of being in the driver's seat, and that mathematics offers them that in a way that no other subject can, and that this is a kind of ultimate freedom. But that this can only be theirs if they bear down on the technical ingredients of those ideas.

A big question has to do with the systematic nature of the subject. How crucial really is it that the students learn A before B, that they get everything they need (or most of what they need) at one level before moving on to the next? Do we really have to teach everything?