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Prepare
Reflect
Active Learning
Communicate
Talk to good Teachers
Individuals
Communicate
Experiment

Prepare
This is one thing EVERYONE can do, and is arguably the most important component of good teaching practice.

• Decide what is important and emphasize it
• Think about where students will get hung up (Reflect!)
• Know how to do homework problems
• Think about props, examples, activities that will bring the material to life (Communicate!)

Reflect
One of the sayings attributed to Mother Theresa is the following: "I have so much to do today, I'm going to spend two hours in prayer instead of one." This is not to say that prayer is the only thing that can save your teaching. But you will benefit by taking time to think about what you are trying to do and what the best way to do it is.

• Before class you can incorporate reflection into your preparation, as above.
• Right after class is an excellent time to ask yourself how you could have improved what went on.

Active Learning
As a successful math student, you probably enjoy listening to a well-done lecture on an interesting bit of mathematics. But you probably know that you don't "learn" the material in that way. To be able to use the mathematics and call it your friend, you have to engage it by working through the proofs yourself, and solving problems involving the results. The same is true of your students. No matter how brilliant your lecturing style, the students won't learn unless you coerce (force?) them to engage the material.

• You must assign and grade homework, or at least give quizzes on the homework.
• You should encourage student participation during classroom time in any way you can.
• Calling on a student by name is much more effective than a generic Are there any questions?   "Pat, what bothers you about this equation."

Communicate
As someone who has devoted considerable time and energy to the study of problem solving, I'm confident in asserting that communication is the world's biggest problem.

• Talk loudly
• Write large, write often, write neatly. Leave important material on a board at the side.
• STOP - Single Thought One Person. Look one individual in the eye as you make a point, then move on. Too much eye contact can make a listener nervous.
• Repeat yourself. Emphasize the important points by making them again, in as many different ways as possible.

Talk to Good Teachers
One can read countless (countable?) books and papers about teaching. One can get bored doing this, although some learning does occur. A more interactive way to engage the problem of good teaching is to talk to good teachers. If you do this, you will find that:

• Good teachers care about their students;
• Good teachers are happy to discuss teaching, happy to have you visit their classes;
• Good teachers are good teachers - you will learn good teaching from them.

Individuals
Ralph Waldo Emerson said "The secret to education is respecting the pupil."

• Learn your students' names: (1) make notes on your class list; (2) be in class 5 minutes early and talk to them
• Automathography. During the first week, have your students write a 1-2 page letter telling you about their math backgrounds, some personal details that distinguish them as individuals, and their goals and expectations for the course.

Communicate
Remember that I said communication is the world's biggest problem? Remember that I said to repeat yourself? There are so many aspects to good communication in the classroom that a second visit to this topic is mandatory.

• Goals, expectations, clear assignments
• Feedback : grade homework and tests promptly.
• Encourage all answers : we learn more from our mistakes than our successes.

Experiment
You are going to make mistakes, so you might as well learn something and have some fun doing it. Try new ideas once in a while. Tell the class when you do - they can provide helpful feedback. If you read or hear about a pedagogical device that interests you, such as using small groups in the classroom, give it a whirl. At worst, you will "waste" fifteen minutes. At best, you might turn your class from a moribund mass of note-takers into a galvanized group of active learners.

A Second Opinion
Another view of teaching mathematics can be found at:
http://www.math.uiuc.edu/~reznick/ciu.html

References

[1]  Case, B. A., (Ed.) You're the Professor, What Next? Ideas and Resources for Preparing College Teachers, The Committee on Preparation for College Teaching, MAA Notes No. 35, 1994.

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