Author: Clinton P. Curry <clintonc@clintoncurry.net>
Revised:November 10, 2010

August 30, 2010

Experience is what enables you to recognize a mistake when you make it again.

-- Earl Wilson

Terrence Tau said the following [MO2010].

The textbook presentation of a topology as a collection of open sets is primarily an artefact of the preference for minimalism in the standard foundations of the basic structures of mathematics. This minimalism is a good thing when it comes to analysing or creating such structures, but gets in the way of motivating the foundational definitions of such structures, and can also cause conceptual difficulties when trying to generalise these structures.

An analogy is with Riemannian geometry. The standard, minimalist definition of a Riemannian manifold is a manifold M together with a symmetric positive definite bilinear form g - the metric tensor. There are of course many other important foundational concepts in Riemannian geometry, such as length, angle, volume, distance, isometries, the Levi-Civita connection, and curvature - but it just so happens that they can all be described in terms of the metric tensor g, so we omit the other concepts from the standard minimalist definition, viewing them as derived concepts instead. But from a conceptual point of view, it may be better to think of a Riemannian manifold as being an entire package of a half-dozen closely inter-related geometric structures, with the metric tensor merely being a canonical generating element of the package.

Similarly, a topology is really a package of several different structures: the notion of openness, the notion of closedness, the notion of neighbourhoods, the notion of convergence, the notion of continuity, the notion of a homeomorphism, the notion of a homotopy, and so forth. They are all important, and it is somewhat artificial to try to designate one of them as being more "fundamental" than the other. But the notion of openness happens to generate all the other notions, and has a particularly elegant and simple axiomatisation, so we have elected to make it the basis for the standard minimalist definition of a topology. But it is important to realise that this is by no means the only way to define a topology, and adopting a more package-oriented point of view can be preferable in some cases (for instance, when generalising the notion of a topology to more abstract structures, such as topoi, in which open sets no longer are the most convenient foundation to begin with).

Ole Hald said the following [GT1990].

I view teaching at Berkeley as a wonderful challenge. Because there are so many courses in the Department of Mathematics, I can teach different subjects every year. I never teach the same course two years in a row, and I prepare every lecture from scratch, "reinventing" material, and constructing new examples to illustrate theory.

To teach mathematics well, you have to understand that students in lower-division, upper-division, and graduate courses are different. Lower-division students must learn how to calculate, and they may have great difficulty following even the simplest proofs. It's like learning to walk: you need to do it before you can make it effortless. Hence, I try to present the material so clearly that students can apply it without much difficulty.

I don't let my students take notes. My lectures are intended to be listened to, and notes will never be as good as required books. It makes some students very uncomfortable; they have had years of practice taking notes to avoid thinking. Instead, I am asking them to watch, think, question, and ask.

I am also old-fashioned in not using overhead projectors. There is a tendency to put too much information on transparencies, and typical students cannot get anything out of the lecture as it flies by. Instead, I use the blackboard, slowly creating ideas before their eyes. I keep them involved by insisting that they help me perform routine calculations. And students delight in catching my mistakes.

When students begin upper-division classes, the character of mathematics changes. The importance of calculations diminishes and the attention becomes directed toward theory. It is exhilarating but frightening: grinding examples take a back seat to the abstraction and generality of mathematics. Since it is important to know definitions by heart, I ask the students to repeat them in unison. Sometimes it sounds like religious service: "What is continuity?" (they answer), "What is a compact set?" (they answer).

In graduate classes, students should learn how to think. I often teach the graduate numerical analysis and scientific computing class. In this course, students learn how to solve complicated problems on the computer. Just as in real life, my problems may have several answers. This irritates everyone; students want precise, tidy problems. But my job is to teach them how to take messy, vague questions and transform them into a precise model that can then be attacked.

Sometimes, students get stuck. To unleash their creativity, I ask "What is the dumbest way we can solve this problem?" After I have asked the question two or three times, it becomes a game in which everybody is willing to offer suggestions because of the playful nature of the question. We can then sift through the suggestions, weed out the bad ones, and study the good ones.

[MO2010]Part of a Math Overflow discussion at http://mathoverflow.net/questions/19152/why-is-a-topology-made-up-of-open-sets
[GT1990]Quoted from http://teaching.berkeley.edu/goodteachers/hald.html