There are probably as many different opinions on how to be a good teacher as there are teachers; however, there are some things I believe most would agree on. It is important for the teacher to
In my experience as a student, I have found that acquiring new knowledge is a fragile process. This means that no learning really takes place until I can comfortably use what I've acquired in several different contexts. Recognizing that this may be a nearly universal phenomenon, I tend to assign and grade a large amount of homework problems, especially in the lower division courses. For the more advanced courses, I encourage students to construct their own problems, examples and counterexamples, so that they can feel they really "own" what they've "learned".
Much of my knowledge and enthusiasm for teaching stems from maintaining an active research program. One of my goals in teaching is to transform student's notion of mathematics as immutable, received wisdom to the notion that mathematics is an evolving, organic subject, and in whose growth they can partipicate. Thus, in addition to the necessary drill exercises, I like to include an occasional assignment problem with some depth. I encourage students to view such a problem as a mini "research project," requiring a creative synthesis of ideas from several areas. A significant component of research involves seeking patterns where none may be apparent, and formulating viable conjectures. Students enjoy this kind of activity, and quickly learn the art of making conjectures and testing them. What they come to realize is that this activity is a paradigm for how real mathematics is discovered.
Aside from exploiting the cross-fertilization of ideas that takes place between teaching and research, there are several other activities I'm involved in that have a positive effect on my teaching. For one, I subscribe to the MAA monthly, partly to keep on the lookout for new and insightful techniques that might be used in the classroom. Often when I go to the library I peruse back issues of the MAA monthly for possible classroom ideas. I also maintain an active membership in several scientific societies (Sigma Xi, New York Academy of Sciences, American Association for the Advancement of Science) in order to keep abreast of related developments in other fields. Students in mathematics courses are typically surprised and delighted to know that not all the results being taught were known to mankind since the 18th century. Thus, in my combinatorial computing course, student interest increased when I spent a few minutes discussing Adleman's 1994 experiment [1] in which DNA was used to find a Hamiltonian path in a directed graph.
Finally, everyone who teaches mathematics should be prepared to consider the question ``What can we use all this for?'' Like any good question, it calls not so much for the right answer, but for thoughtful consideration. For many years, I subscribed to a newsletter on educational issues [2] which forced me to brood on such questions in ways I might never have considered. There is a good deal of talk these days about communicating the value of mathematics to the general public. Thus, I find it remarkable that one of the most eloquent supporters of our discipline is not a mathematician, but a Professor of English. Regarding the passage from Plato's Gorgias quoted above, Mitchell [2] writes:
``For Socrates, the study of mathematics was nothing less than the soul's discovery of order and proportion, of permanent and essential relationships, of rightness. And the idea of Rightness revealed in mathematics provides understanding of the idea of Justice among men. That such an ancient understanding of the study of mathematics should seem to us at least unusual, if not downright quirky, is of course, the result of our schooling, and likewise a measure of the shallowness of our education.''
In support of studying algebra, Mitchell also writes
``Algebra is a world of principle, and a dramatic revelation of the power of principle. In fact, algebra, and even algebra alone, could provide a true and sufficient education out of which to understand the worth of living... I think you should learn algebra, not because I want you to solve algebra problems. You will find that algebra shows you some truths. The first great truth is that there can be something real, and complete, and harmonious, and even, in some strange way, absolutely perfect right in your own mind, and made by you alone. Algebra tells sad truths too. Where there is no balance, there is no truth. What is equal is equal, and between the equal and the unequal there is no conference table, no convenient compromise. In this terrible law there is a hinting question for all of life. Algebra will show you the inexorable, the endless and permanent chain of consequence, the dark thread of necessity that brought you to a wrong answer because of a tiny little mistake back in the second line. How unfair that seems, and how scary that what seems unfair is nevertheless justice. Is life like that too, as all of nature seems to be? How then shall we live? What are the laws of the algebra of our living, and where do they exist, where created? Who can show us how to learn them?''
Of course, it may not be prudent to speak of such things as Justice and the Laws for Life in a mathematics classroom. As Mitchell says, ``it takes some serious living to see the truth hidden in algebra.'' However, I am convinced that as teachers, we should consider these things, and brood on them while preparing our lessons. It can only reflect positively on our teaching. On the other hand, if we do not teach as if we know such things, and do not teach as though we have considered deeply the question ``What can we use all this for?'' then our students and the public will ask it for us, and they will not ask it in a favourable manner.