A mathematical idea did not start from a set of definitions. Mathematicians' thought processes usually begin with a problem, followed by the development of a theory for solving it. Along the way, terminologies are defined in order to present the idea. For the sake of logical rigor and formalism, they have to present their theories starting from definitions which is the format for the majority of literatures. Unfortunately, this format does not serve the educational purpose well. Therefore, as a mathematics teacher, it is our responsibility to unveil the secret behind them. Being an instructor in UGA for more than four years, I have been reminding myself not to start a topic with definitions. Instead, try to tell a little story beforehand. Being a formally trained mathematician in the mean time, it is not always easy nor natural. Whenever I start a new section with a rigorous mathematical statement, I usually get some strange looks on their faces. With a few years of classroom experience, I can surely tell that they just take what I wrote as something needed to be memorized.

        Quite a few students told me that mathematics is another language to them. I agree, it is indeed a language of science. Being grown up in Asia, I understand how hard it is to learn a secondary language (English) when it has nothing to do with our daily lives. However, this is not entirely true for mathematics. I must help them to bridge the classroom and the real world. For instance, when I talk about mean value theorem in a calculus class, I give my students a daily live example to illustrate.
 

        Imagine that you are driving on an interstate highway to a party that you are about to be late, so you go well over the speed limit 65mph. Suddenly you see a police car sitting next to the highway, then you step on the brake to reduce the speed back to 65mph. After you pass the police car, you speed up again. What a coincident, you see another police car next to the road again! Of course, you do the same trick. Slow down to pass it. The radars on both police cars did not detect the over speed. You may think you are safe this time. However, it is not totally true if the two police cars are communicating. Suppose you pass the first police car at noon and the second at 2pm. Moreover, they know the distance between them is 180 miles. They can set up a distance function S(t) for your car and set t=0 at noon. Thus the distance at 2pm is S(2) and the derivative of S(t) is V(t) which is the velocity function. Then by the mean value theorem, there exist a time t₀ between 0 and 2 (i.e. between noon and 2pm) such that (S(2)-S(0))/(2-0) = V(t₀). The polices know that the distance between them is 180 miles, i.e. S(2)-S(0)=180. Thus 90=V(t₀) for some t₀ which means at some point between noon and 2pm, your speed was 90 mph! By the end of the semester, when I ask them what the mean value theorem is, they would say if a car could go 180 miles in two hours, at some point the speed of the car is 90 mph.

        Classroom is not just a place for instruction as on-line courses are sufficient to do the job. It is a great place for discussion and knowing each other. Solving problems in class play a key role in learning. While the students are working on the problems I pose on the board, I would pay more attention to some weaker students and let others to discuss. In a class of 35-40 students, the background of students varies. Some students took calculus in high school while others did not. Some of them have not studied mathematics for years while some of them just took a quantitative course the semester before. During the time of in-class exercise, I am able to walk through the problems with the students who need help most. Since most of them just have trouble to initiate, a little direction can jump start their thought. For other students, I can observe how they approach the problems and identify their common mistakes so that I would address them right away on the board. Apart from that, this is an effective method to draw their attention in the classroom.

        In order to know more about my students, I always make effort to learn the name and some background of each student. The week after the add/drop period is ended, I would give my students a small piece of paper and ask them to write down something about them, such as the name they go by, their major, the year in college, etc. Then I would try to remember their face by calling out their names and have a little chat. Of course, I would not be able to remember all of them, but this can help to break the ice at the beginning of the semester and energize the classroom dynamic. Another good way to communicate with students is by giving weekly quizzes. I usually put two questions on each quiz that take them about 15 minutes to finish. The quizzes not only can keep them studying, but also allow me to keep track of their progress and provide feedbacks on their quiz papers. An extra benefit of weekly quizzes is that I get to give their quizzes back every week so that I have chances to recall their names. I believe that knowing students' names is the first step to establish the bond.

            My experience of the topology class in graduate school is not special. I have heard similar experience from others but on different subject or different period of time in school. In the worst case, some of them even think they are not able to study a certain subject after taking a certain class. Therefore, a teacher could make a crucial influence for some students. My bottom line as a mathematics teacher is to maintain or even enhance their interest in mathematics. Since if the passion is enervated, they will not continue the pursuit of knowledge even though they ace the course.