I have been fortunate over the years to have taught at 3 schools where teaching was important and where I could learn from my peers and students while I developed my teaching skills and philosophy. At Ohio State University, where I was a graduate teaching assistant, there was a major emphasis on good teaching, and I had several helpful mentors. My last two years there I had complete responsibility for the course that I taught, so that when I took my first teaching assignment at Allegheny College I already had some significant experience. Allegheny was a wonderful place to teach, and certainly much of my teaching philosophy was developed there. CU-Denver represented a new challenge with a diverse student body, my first opportunity to teach graduate students, a chance to build and develop a complete program in applied math from undergrad through the Ph.D., and the chance to work with Ph.D. students and to teach them how to do research. There are many aspects to my teaching philosophy, and it continues to change and evolve through the years although there are some underlying principles that have remained fairly constant.
To start, one of my most important goals is to create an atmosphere in class that makes students feel comfortable and is conducive to doing mathematics. I wish that I had been clever enough to coin the phrase "math anxiety", for I was very much aware of this problem for students long before the term was used. And even though the issue of creating an anxiety free atmosphere is more important in the lower level classes, it persists right through the graduate program though to a lesser degree. So, for me, its always important to have the right type of classroom atmosphere. Much of what I do to create this atmosphere has either naturally evolved over the years or was learned from good teachers that I had and now pretty much happens without me having to think about it. Some of the specifics include presenting material clearly and slowly enough so that students understand what I am saying, involving the class in the lecture, poking fun at myself, laughing at mistakes that I make, making eye contact throughout the class, encouraging questions and comments from students, making analogies to other areas that humanize the subject, showing sympathy for the tension created by exams by discussing my experiences as a student both good and bad, etc.
I want students to realize that math is not a spectator sport but a participant sport. They can't just watch what I do and expect to understand the subject. They have to do it, and its ok to make mistakes, they can learn from their errors. I pose problems in class, have students ponder them for a few seconds or a few minutes either individually or in groups, and then solicit answers, sometimes even having a vote on the answer just to get everyone involved.
Perhaps one of the biggest changes in my teaching over the years is the extent to which I motivate new mathematical concepts by using practical applications. My graduate experience was pretty theoretical, and we learned to appreciate new results because of their intrinsic mathematical beauty or significance. While this is wonderful for mathematicians, it is one of the reasons that there has always been considerable justifiable criticism of most math teachers at all levels. Most students need to see the practical relavance of new mathematical concepts. As I pondered the difficulty of motivating students to want to understand theoretical concepts, I realized how much easier this is when you can demonstrate how the theoretical problem evolved from a need to solve a particular practical problem. Not only did this change my teaching, but with the help of an NSF Faculty Development Grant in 1979, I changed my research area from abstract algebra to applied graph theory. This led to my coming to CU-Denver and bringing applications not only to my class but to an entire program.
Another change in my teaching over the last few years is to use some cooperative learning in and out of the classroom. I encourage the class to form small cooperative groups to work on problems assigned to be done out of class but to be turned in for credit. I have even had a group component of some take-home tests. In class, from time to time I will have these same groups work on a problem that motivates a new concept. My experience is that this has dramatically incresed the level of problems that the students are able to solve, and students are spending considerable time outside of class talking about math. For many students this is a new experience as in the past they associated doing math with being alone in a quiet room.
For me one of my biggest and most exciting challenges is to bring to the classroom a sense that mathematics is alive and a subject where new things are being discovered all the time. Everyone has studied some math, but for most people they feel that nothing new has been discovered since Newton's time. Unlike the other sciences where new discoveries make it to the front pages of the newspaper, soon find their way into high school textbooks, and have a direct impact on peoples lives, this generally does not happen in math. Even students who get to calculus in high school are studying a subject that was developed by the eighteen hundreds. So what I do is talk about recent discoveries that are relevant to the particular course that I am teaching. But more than that I talk to them about my experiences in doing research, about some of the problems that I have solved, but also about the problems where I am stuck and don't know how to solve them, at least right now. What the students realize is that math is alive, that its ok not to be able to solve every problem, its ok to be stuck, but you want to keep working on it. They also realize that there are some exciting problems out there for them to solve if they stick with it. This is not only a message for my classes, but when I am invited to give a talk to high school students, I do the same type of thing and conclude with an unsolved problem which will perhaps be solved by someone in their generation.
One of my most important goals is to get more women and minorities interested in mathematics. This is an issue that I have pursued outside the classroom, but I will deal here just with my committment with my classes and students. This has been an important issue to me from the first moment that I taught at Allegheny College in 1971. My experience was that I would encounter women who were good students in math, but had come to college thinking that a more appropriate career for them was in the humanities or social sciences. I would not accept this, and over the years many women with my support and encouragement have pursued successful careers in math or science. In some instances, I know that my early encouragement of them was what made the difference. I have had the reputation both at Allegheny and here of being someone who supports women in math. This is true in the classroom where I consciously do the types of things that make women feel comfortable in an area that traditionally has been dominated by men. As a result of the positive environment for women in the classroom, many work with me on honors projects or their thesis. My first Ph.D. student was Kim Hefner, and I currently have 2 women doing their Ph.D. research with me. I have had some similar successes with the few minority students I have had in class, but unfortunately we don't have many minority students taking math classes. This is an issue I have worked on outside the classroom.
Another aspect of my teaching that has evolved over the last 20 years is working with Ph.D. students and teaching them how to do research in math. While I have done research myself since 1970, teaching someone else how to do research in such a way that not only can they get their Ph.D. but also they will have developed the ability to have a successful career is a real challenge. While I can assist the students by having good problems for them to investigate and by providing advice and support, research is a very personal thing and I have to encourage them to pursue their own strengths. It is a challenge to find the best way to work with each student, but so far its worked out well as since 1988 I have had 11 PhD students graduate, and I currently have 2 Ph.D. students starting their research.
In summary, I love teaching. When I was chairman, it was often the only thing that brought sanity and fun to my day. I have been fortunate to have had several memorable math and science teachers dating back to my days in high school, and much of what I do was learned from them. But I have also learned from my students, my peers and my own experience, so my teaching continues to evolve and I hope that I continue to improve.