The goal of teaching is to let students learn about particular subjects as well as develop their critical and independent thinking. The simplest way to achieve this is through "good lecture" and "well-designed homework". To me, good lecture consists of concise exposition of a subject, idea or technique and working out relevant examples. The exposition is the way to introduce students to new things and I think it is good to include historical developments or real-life situations behind the subject. Lectures do not have to be too heavy and detail all at one time. After all, it takes time for students to digest. One could leave some gaps or questions in the lecture so as to stimulate students to think and find the answers on their own. Moreover, one can supplement it with homework problems. Working out examples has two purposes. First, it reinforces the subject matter in concrete terms. Second, it serves as an example on how to write down mathematical arguments which is quite important to students.

Talking about homework, it primarily serves to drill students on mastering the subject or technique as well as good mathematical writing. But as mentioned before, it helps students to discover and deepen understanding if carefully designed. More importantly, it provides opportunities for problem solving and critical thinking. Word problems and open-ended problems are excellent examples for this purpose. Also, group homework or project should be included because they allow students to communicate and share ideas as well as learning how to work together as a team.

Besides lecture and homework, in-class activities (usually in groups) can help achieving the teaching goals. Through a series of problems, the students could be led to discover new things on their own which would make a more lasting impression on them. Also, similar to group homework, it provides a time for them to think, discuss, communicate ideas and learn from each other. Furthermore, it gives teachers the opportunity to assess their students' understanding to a subject. Closely related to this is the use of technology like computers and graphing calculators. Often times they provide the means for activities ranging from discovering patterns to visualizing objects. I think that they should be used to facilitate learning. However, one should not rely too much on them. For instance, students should be able to do simple calculations or graph elementary functions by hand.

Last but not the least, a teacher should motivate, encourage and care about students and keep an eye on students' progress. If a student encounter some difficulties, I would invite him/her to my office and talk about it. On the other hand, if a student desires to learn more, I would let he/she read and work on something extra.