I have answered this question with what I hope are some qualities a teacher can learn in order to improve their class. This is as opposed to listing qualities of a truly exceptional math teacher. I wrote this thinking of middle and high school level math, but most of it could be applied at other levels as well.

1) A good math teacher should motivate the math and engage the students. Take the example of solving linear equation. One can start by telling students the formal rules for how to manipulate an equation, but I think students will find this very dry, and won't understand why they are doing what they are doing. It becomes and exercise in memorization. Instead, one can start with problems that can be solved with such equation. One can first get students to solve them with other techniques (e.g. guessing and checking or using some sort of graph). After a while one realizes there should be an easier way, which turns out to be solving a linear equation. This way the students understand why the formal math was developed, understand how to apply it, and see how it is related with other ideas (like graphs). Right now you might ask where one can find good problems to use in this way. I think I will ask that as a separate question...

2) A good math teacher makes their students do math. I think it is crucial that every student, in every math class, every day, solve some math problems. Some of these should be easy (i.e. just practice solving equations, once they have been introduced), and some should require more creativity. It is of course a mistake to drill students with boring problems until they hate the subject, but it is also a mistake to let them do "interesting" or "discovery based" math all the time, and not make them practice the techniques they discover.

3) A good math teacher should convey the beauty of the subject. One of the other answers said ``infectious enthusiasm" was needed. That would be great, but in reality not all math teachers can be that charismatic. Even without a great deal of charisma, I believe it is possible to show students the wonder of extracting a simple answer from a seemingly difficult question, and the beauty of the tools that help one do this. Often it is enough that students see that their teacher believes this. So in particular, I do not think it is a good idea to say things like "I hated math when I was your age too, but we'll get through this".

`soft`

if only by dint of inheritance from the question that begat it. $\endgroup$ – Jon Awbrey Nov 14 '09 at 16:202more comments