Here you can find my teaching statement and philosophy, which are quite simple and easy to read, following the same style I use in teaching.
General Attitude and Environment:
The very first thing I establish in the classroom on the first few lectures is a light, mutual, sense of humour. I guess this is one of the main reasons my audience never diminished by the end of semester.
If possible, I avoid using any tools, or any notes, just to emphasize how doable it is to construct topics, proofs, or to solve all possible problems, by sound reasoning only. This need not apply to all courses, but it is at least effective for first year courses.
Why-questions are welcome anytime, and no question is trivial. Trivializing one question psychologically leads students to restrain their questions, and hence to a postponed comprehension.
Learning names. Whenever a student is answering or asking for the first time, I ask them for their name(s). As the semester proceeds, I can usually call every student by their name(s), and they appear to really like that.
Style:
Teaching any math subject, my goal is always to make proofs seem self-contained and to explain the thinking sequence that leads to it. This kind of interaction with students adds some excitement to lectures. Besides, it creates an opportunity for their thoughts and brings space for fun when we decide to adopt one or another proposal for the next proof step.
I’m not convinced/ Are you?/ How many believe this is true? From the very beginning, I train students to seek justification. For example, at some lectures I may provide a proof, then stop shortly to ask: I’m not convinced! Why? Why would this be the best step forward? If they would not ask me: Why? I usually proceed by asking them. That way students can understand, reproduce, and develop by criticism.
Interdisciplinary Teaching. I try to keep students entertained by relating to topics that might be of their interest. For example, in teaching the first-year algebra course in the University of Waterloo, MATH135, to the Software Engineering students, we came to the point of studying prime numbers and factorization. Most students do not really know that there is still no polynomial-time algorithm for factoring, and since building algorithms is very vital for them as software engineers, I decided to present a good deal of Primality-Testing algorithms, starting from the classic probabilistic ones, and ending with the 2002-breakthrough deterministic AKS Primality test. Their joy for that lecture was phenomenal that I decided to include the topic every time I teach.