Every great teacher has his or her own teaching style and philosophy but here are some thoughts on things that I think make for a pretty good teacher (some more math-specific and some less so).
I think good teachers respect their students. I think teachers can effectively engage students by treating teaching and learning as a collaborative process and by showing their students that their thoughts and opinions are valued. It can be really good motivation for students to feel that they're working with their teacher to develop their understanding.
Following on from that, I think good teachers can get their students talking. Having students ask and answer questions about the material being covered is a great way to get them really thinking about the ideas for themselves. Also, some teachers really encourage their students to talk to each other and I think that's a great way to show students how much they can learn from each other, independent of their teacher.
Along those lines, great teachers should challenge their students. They should encourage their students to get out of the mindset of "I only know as much as my teacher tells me". In math, I feel like many students end up expecting all the problems they encounter to be similar to some example they've been shown. An invaluable skill that great teachers pass on to their students is the ability to take their knowledge and skills and apply them in unfamiliar situations. It's important to show students that they can do things on their own!
Great teachers provide positive encouragement and credit where it's due. When you're in the middle of learning something, it can be hard to take stock of how far you've come or whether you actually know more than you did weeks or months ago. So it's important for teachers to keep track of their students' progress and to let them know about it.
Especially in math, I think good teachers provide motivation and explanation for the material. It's so easy to get caught up in formulas and theorems and simply ignore where they came from but it's so important to make sure that students realise that math doesn't just come out of nowhere. Even in cases where rigorous explanations are a little beyond the students, good teachers can appeal to their students' intuition and give basic ideas about why things are true. Being able to show students that math is all about logic and reasoning and that it should make sense I think is the mark of a really incredible teacher.
