The following three sections outline some of my main ideas about being a good teacher. They are divided into categories labeled "Essential", "Important" and "Helpful"---which is how I consider them.

**Teaching should be interactive.**I always learned best in those classes where the students felt free to answer and ask questions. When the teacher engages the students in the learning and teaching process, then the material they see becomes*their own.*For these reasons, I make it clear that class participation is not only helpful for them, but necessary in order to keep the lesson moving. But for this to happen, the students must feel comfortable addressing the teacher. Thus I begin each semester's classes with an activity that requires them all to talk to me and to each other. I find this atmosphere very productive for my students. They get to know one another, and this has good effects. For example, they explain things to one another during class (sometimes a student who just learned the material can explain it better than the teacher...go figure), they form spontaneous study groups, they make new friends, etc...

Sometimes I forget what was hard about the material I am presenting. Then this interactive style acts like a pedagogical barometer, helping me to adjust appropriately. It can be difficult not to get bogged down when teaching this way, and I'm still learning how to avoid that. Many of the ideas below are designed to foster a good interactive teaching atmosphere.

**Learn students' names quickly.**It is my experience that students appreciate this immensely, as they often indicate on teacher evaluations.

**Welcome questions.**At the beginning of the semester it may be necessary to allow awkwardly long periods of silence while waiting for someone to ask the question that you feel they should be asking. But it is worth it once they become more willing to speak up when they don't understand.

**Material should be motivated by a good problem.**I believe very strongly in this. One of my observations in Hungary was that lessons tended to be geared toward solving some sort of problem. For example, a teacher didn't put the law of sines before the students and then give them some problems to try. Rather, a problem was given which could be solved with the law of sines, and then the teacher directed the students to its discovery. This way, by the time the students found the formula, it was*their*formula! Realistically, this takes longer, and isn't always feasible---especially when the syllabus is too crammed. The next time I teach integration, I am going to tell my students a story about a wife who is trying to figure out where her husband has been taking the car during his lunch hours. She knows it's either the race track, McDonald's or the Pub, each of which he's not supposed to visit. Unfortunately, the micro-camera she installed in the car slipped down, and instead of recording through the windshield, recorded only the speedometer and the clock (show 4 minute video.) But given this information, and the distances to the three locations, can she determine where he goes? This would be a good introduction to Riemann sums.

**When possible, let students discover the theorems.**I enjoy acting skeptical when a student proposes a theorem or method---and then acting surprised when they "convince" me that they are correct.

**Students learn math by doing it.**I stress that it will be impossible for them to learn the material unless they think about it and work with it on their own at home. Thus I like to assign homework which will be graded. I ask them if they think someone could become a great juggler simply by watching hours of videotapes, but never picking up the balls.

**Turn everything into a learning experience.**Students learn amazingly well immediately following an exam, so I give them solutions to the exam as they leave the room. If the class does poorly on an exam, giving a similar make-up exam helps focus the students' studying efforts, and gets them into office hours. I like to give*group quizzes.*I break the class into groups of sizes 4 and 3, and give a different version of essentially the same quiz to each student in a group. They are free to talk within their own groups. Since the questions are the same type, they can ask someone in the group for a hint on how to solve that type of problem, but since the numbers are different, they can't just ask for the answer. I walk around and stamp out abuses when they occur.

**Be accessible.**I like to keep my office door open, be present during office hours and encourage students to come. I mention my office location and hours at least every other week for the whole semester. It still surprises me that, near the final, a student will ask where my office is!## Important

**Use technology.**When I taught EXCEL (intensive calculus) to minority engineers at Rutgers, the use of graphing calculators so facilitated conceptualization that I enthusiastically advocate their use.

**Call on students.**This helps them keep from falling behind, and acts as a form of assessment for me*and*for them. Of course, the threat of being called on is also good for keeping students alert.

**Have students come to the board.**This can revitalize a class that has gotten drowsy. When I was teaching a "proofs" class for math majors, I required each student to present 2 proofs to the class, and I encouraged the class to be critical. This helps them realize that you truly understand something only when you are able to explain it.

**Group projects and activities should be encouraged, when appropriate.**Mostly for non-math majors, this helps prepare them for the real world.

**Students should learn how to write math.**At most levels, some sort of written work, where accuracy and*exposition*is criticized, can be appropriate. This worked well in the EXCEL course mentioned above.

**Drill can be appropriate.**"Drill" has gotten a bad name in some education circles of late. I think it has its place, and can be apropos for mastery of certain skills. Theory gets a juggler only so far.

**Give quizzes and tests.**While it is true that students are responsible for their own educations, it is helpful for the teacher to give the students many chances to discover if they are falling behind. Frequent assessment opportunites are valuable for the student as well as the teacher.

**Be visual.**I remember being unable to prove a theorem until I went and got myself some colored chalk. Then, when showing it to my advisor, he couldn't see what I was talking about until I used my colored chalk! I like to use models whenever possible to make a point---dominos to teach induction, a beachball for the Riemann sphere, graphing calculators (like the TI-82) to teach limits, paper folding, etc...

**Teach general problem-solving skills.**There are some techniques for problem-solving that apply very generally. Notions such as "look for a pattern", "try a simpler problem" and "think laterally" apply as well to introductory algebra, abstract algebra, chemistry and nursing. If possible, we should tell this to our students. I am thinking of Marvin Levine's book*Effective Problem Solving.*

**Have a clear grading policy early on.**Many students need this, and it is important not to change it during the semester, except, perhaps, to the students' benefit.## Helpful

**Tell one joke per semester.**This one has served me well for a long time now: Two hydrogen atoms are walking down the street. One says "I think I lost my electron." The other says "Are you sure?" He replies "Yes, I'm positive!"

**Show an occasional gem.**Occasionally the students will be particularly attentive, and I'll have the opportunity to tell them some really beautiful theorem. Something like Fermat's little theorem, or his last one, the closed formula for the Fibonacci numbers, or my favorite---the fact that every*4k+1*prime is the sum of 2 squares.