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Personal Observations about Teaching

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Overview of Experience

Over my career, I have taught a wide variety of mathematics and statistics courses for a wide range of audiences and have also been involved with other aspects of the university educational enterprise. I began teaching while a sophomore at Columbia when Lipman Bers made me the teaching assistant for his Calculus course. I worked as a teaching assistant for three years at Columbia. At Michigan, I worked as a teaching assistant all but one of my years there. During my undergraduate and graduate studies, I also tutored a diverse range of students in a wide range of courses. Starting as a new professor at Georgia Tech, I taught two classes a quarter. Many of these courses were aimed principally at engineering students. My time there included one of the highlights of my teaching career, which was the year I taught a group of freshman real analysis instead of calculus in an experiment that ultimately led to a textbook. When I moved to Colorado State, I was put in the position of teaching graduate courses almost exclusively, which provided me the opportunity to widen the subjects I could teach. My experience broadened considerably as I transitioned from the Mathematics Department to the Statistics Department over the last few years. At Simon Fraser University, I continue to teach in the Department of Statistics and Actuarial Science

Over my career, I have mentored a large number of Ph.D. students and postdocs using the support provided by a number of grants. Quite a few of my students and postdocs have had interdisciplinary interactions with either companies or national research laboratories. A number of my students and postdocs have moved into careers in academia.

I also became heavily involved in developing effective training programs in interdisciplinary research, through my involvement with a NSF IGERT sponsored graduate program in quantitative ecology (PRIMES) and a NSF UBM sponsored undergraduate program in quantitative biology (FESCUE) at Colorado State University and continuing to work with the computational mathematics community in education in computational science and engineering. I developed effective courses aimed at training students in the successful pursuit of interdisciplinary team-based research.

Teaching Philosophy

My ideas about teaching are all oriented around the belief that learning is essentially an individual process. In other words, the degree to which we can learn something depends primarily on our own innate ability and the amount of work we put into learning, e.g., through reading, thinking, and working problems. I believe that there is actually little possibility of transferring the understanding in one mind directly into another mind. This is perhaps more radical than seems at first, because in my experience, many university faculty believe that lecturing is the primary process for transferring the knowledge and understanding in their heads into the minds of the students and, consequently, view lecturing as the beginning and the end of educating students. This is not to say lecturing is unimportant. It is just that there is more to teaching effectively than just getting notes down into students’ notebooks.

In my view, the role of the teacher in the learning process parallels the role of a coach in the training of an athlete. First of all, of course, the teacher has the responsibility for choosing the topics to study and how the material should be approached. The best instructor can do no better than the material with which he or she works; an observation that is readily apparent in appraising the typical calculus courses taught around the country. Unfortunately, the choices are often limited in this regard because of the available instructional material and, at least in large state universities, by the inertia that builds up in course syllabi. Second, the instructor should provide the students with the “big picture” to help them get past the short-sighted blindness that often results from struggling to master details. An instructor should try to monitor the students’ understanding and lay down alternate paths to understanding at places that cause difficulty. Third, the instructor must provide support, incentive, and discipline for the students’ efforts to learn. Learning is difficult and requires discipline, and most students need supplements to their own self-discipline.  Grades are commonly used as both incentive and a form of discipline. But, an instructor should try to go beyond this by establishing a relationship with students that enforces their desire to do well. This is perhaps the closest analog between the role of the instructor and the role of a coach. It is important for an instructor to establish a relationship with the students in which they know that the instructor cares very much about how well they learn and perform and, in turn, in which the students care whether or not they disappoint the instructor’s expectations for their performance. Now, it is not possible in the typical single instructor-multiple student class to establish such a close personal relationship with each student on a one-to-one basis. But, it is possible to form one-sided relationships from the students to the instructor, e.g., by giving the students a glimpse of the instructor’s life as a student and as a mathematician and their own struggles with understanding mathematics.

On the topic of the need for discipline, I can narrow my belief about learning further in the case of mathematics in the sense that I believe the success in learning mathematics is determined primarily by the number of good problems one succeeds in doing. Thus the construction of good assignments and exams and a fair grading policy that gives students feedback so they can improve their performance is a critical part of good teaching. As a personal example, my general approach to classwork for graduate classes is to use take-home problem sets and to allow students to redo selected problems that they clearly fail to understand the first time.

My philosophy spills over into my approach to the textbooks I have written. In particular, texts are often written in a way that eliminates or masks the sense of discovery, which is counter-productive for learning how to create and use mathematics. In my texts, I attempt to blend proof and explanation and application into a seamless story that pulls the reader along the path towards learning how to discover new mathematics.

Development and Evaluation of Teaching

Aside from teaching, I have a deep interest in the development and evaluation of good teaching. Teaching is a craft for which there are myriad effective approaches. As with any craft, mastering good teaching is a never ending process. Open discussions about different ways that people teach, and what works and what does not, benefit all educators. Besides, it is obvious to me that placing a lot of emphasis on the evaluation of teaching without working on development is counterproductive. At Georgia Tech, I was heavily involved with the creation of development and evaluation procedures both for the graduate student teaching assistants and untenured faculty (I served as the first Director of Teaching Effectiveness). In Mathematics at Colorado State University, I was one of the main developers of training for new graduate students, which, among other things, is designed to help the students get a good start on their teaching careers. I am undertaking development of a similar program in Statistics with my appointment as Associate Chair of Graduate Studies. At Simon Fraser University, I chaired the department committee tasked on devising a comprehensive teaching evaluation system.

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