TEACHING INTERESTS AND EXPERIENCE

I have consistently received excellent student evaluations in my 16 years of teaching. I tend to be enthusiastic about the subject I'm teaching and to encourage and inspire students easily. Year after year, my desire to help students taste even a little of the "glories" of mathematics has kept me trying new approaches to present material and get students involved.

I have enjoyed teaching all levels of mathematics, from college algebra and trigonometry to advanced graduate courses in foundational areas of mathematics. I have tried a number of teaching methods and have discovered approaches that work and others that don't work. I have also had a number of non-academic experiences in the application of mathematics to business and computer science that have helped me appreciate the world which most math majors enter upon graduating.

**Some Successful Teaching Experiences**

Four highly effective strategies I have used are: (1) computer-aided teaching; (2) letting students work in groups for class projects; (3) the Moore method in undergraduate analysis; and (4) using a famous theorem as a focal point for the development of an entire course.

*Computer-aided Teaching.* Computer-aided instruction has consistently proven to be an invaluable tool for bypassing needless complex or tedious calculations in order to familiarize students with central concepts. My first experience with the use of software in the classroom was at the University of Wisconsin, when I was teaching an advanced business math course that required considerable use of matrices. Students were able to handle "real-life" problems using this software (*MatLab*), and as a result, found the material more interesting. Also at UW, I helped to develop some software for a textbook (which I co-authored) on mathematical logic (*Mathematical Logic and Computability*, McGraw-Hill, 1996); the software enabled students to work out the fundamental ideas of Godel's Incompleteness Theorems as if they were video games--needless to say, this approach was very successful.

At MIU, we began our involvement with computer-aided teaching by introducing graphing calculators (TI series) as a required part of the course (beginning in 1991). I developed a section of our syllabus that would use class time to teach the basics of how to use the calculator. The students were first exposed to calculators in this way in trigonometry, and they were required in all exams. Students gained a much clearer feeling for trig, log, and exponential functions by being able to work with them on their calculators. In 1993, the department became interested in the Calculus Consortium curriculum and began using it with our computer lab. Our use of this approach to the teaching of calculus was successful right from the beginning—students grasped the concepts more thoroughly, and fewer were left behind. I had the opportunity to teach the first quarter of calculus using this system, in conjunction with Mathematica.

* Working in groups. *In my three most successful undergraduate courses, I gave class projects that required students to work in groups. I always included one or more of the better students in each group. In every case, I communicated to the students that the projects were significant by assigning a significant grade weight to the project (when I tried to get group projects going but only let them count 10%, the results were less impressive). Also, in every case I allowed some class time for working on the projects, asking questions about them, and getting individual help from me. (I have also tried assigning projects but not providing class time or so much individual attention, and the students tended not to take them as seriously.) I would regularly whet their appetites concerning their projects: Lectures would often refer to the projects as providing the key to important parts of the subject as we were developing it in class. With these strategies, I found that I could motivate students well beyond the usual norms to produce superior mathematics and turn in excellent projects—and the resulting teacher evaluations would consistently indicate that the approach was rewarding for the students. I have been told many times by students in such courses, in some cases years after the course had ended, that their class with me was the best educational experience at their university.

My first effort along these lines was in a trigonometry course I taught as a graduate student. As part of their final project, I asked students to solve a few problems that had a calculus flavor, involving epsilon's and delta's; I gave extra time in class to discuss the significance of the problems and to help with the concepts involved. To my surprise, two of the groups successfully solved all the problems. I found out later that these groups had put an extraordinary amount of time into that project and had found the whole experience very rewarding.

I tried a similar strategy with a number of first-semester calculus courses. The group projects would consist of a combination of "word problems," graphing problems, and small theorems to prove. The problems were on the challenging side, and usually there would be one problem that only one group would solve correctly. These projects always served to create an ongoing focus for the students; the longer they would spend on project problems, the more comfortable they seemed to feel with the corresponding areas of the course. And they could count on getting "unstuck" (in contrast with the usual nightly homework assignments) because they could get help from their group. Many superb insights emerged out of these courses.

The other type of class that received this group approach was an Abstract Algebra course. In this case, the problems consisted solely of theorems (or questions requiring proof or disproof). The final project included a number of fairly challenging applications of the Sylow theorems, and the results were phenomenal. One group produced the best proof I had ever seen for a certain corollary to Sylow #1. In this course, the group projects were actually the focal point of the course. Students thought of the lectures as tools to help them solve the problems they found on their project sheets. This was perhaps the most attentive class I have ever had.

* The Moore Method.* As a graduate student, my first exposure to both real analysis and point-set topology was by way of the Moore method--an approach in which class time is devoted exclusively to the presentation by students of proofs of theorems. Students are given definitions and statements of theorems and, as they come up with proofs, they present their work in class. Often, errors turn up, and students attempt to refine their efforts until a correct proof emerges. The professor lectures only occasionally, usually only for the purpose of communicating some intuition about the definitions and theorems that are being written on the board. Books are not allowed. Students are "forced" to discover that they actually have mathematical talent, that they can create their own mathematics without relying on some perfectly presented treatment of a topic that they might find in a book. It also allows students to see what mathematical research is all about. Their experience of having just one evening to work a few problems in order to complete homework assignments often denies them the experience of spending extended periods of time with a problem and surviving the dry spells long enough for deeper insights to surface.

I have taught two real analysis courses using this approach. Both courses turned out well, but the second was especially good. We had just the right number of students (there were 6) and nearly all were of above average talent. Every student in the class described the course as the best mathematics experience he'd had so far. Most admitted that they had never worked so hard and felt very gratified with their achievements. I handed out weekly updates of the theorem list, and, for each theorem on the list, the person who successfully prsented its proof would have his name displayed by the theorem on the hand-out (this practice served to fire up their enthusiasm even more). The major challenge for all of these students (and this is typical in Moore method classes) was to get the logical details correct and to know when statements required further proof, but progress in these areas was very evident as the course progressed. It was the most passionate group of students I've ever had and one of the most rewarding teaching experiences of my life.

*Using a Famous Theorem as a Focal Point. *For a one-quarter undergraduate course in field theory, I decided to use the unsolvability of the quintic as a focal point. But rather than waiting until the last week of the course to unveil the proof of this theorem, I presented an outline of the proof on the first day of class. I made it clear that we would have to fill in the outline with a great deal of work and that even the definitions of some of the words used in the outline would require a fair bit of development. After a few weeks had passed, I handed out a more detailed version of the outline. As the course proceeded, each lecture would be presented in terms of its relevance to this general outline; this context made it possible for students to feel a sense of continuity and wholeness to the course. Eventually, we achieved our goal of proving the main theorem in every detail. One part of the final exam in this course asked students to produce their own outline of the proof and to fill in some of the details. This approach proved to be very successful; students felt much more in touch with the bigger picture throughout the course than is usually true in an Abstract Algebra course.

**Experience in Applied Fields** **and Advising Students**

My primary technical experience outside academia has been in applied computer science. One job involved development and maintenance of a software system for a health insurance agency; another involved research as a consultant for AT&T Bell Labs. And, as my CV indicates, I have been employed full-time as a Java software engineer for the past four years, working for telecommunications and agricultural insurance companies. In these positions, I have gained considerable experience in real-world implementations. This experience has proved to be a considerable advantage in my academic career when it comes to advising students--particularly students who have displayed mathematical talent. Certainly most mathematics majors will not, and should not, pursue a purely academic track; indeed, most will find work either in industry or in the education field at the community college level and below. My experience in industry has proven helpful in communicating to students what it's "really like" in such jobs; in many cases, I've been able to suggest what I think would be an optimal direction for a student based on my knowledge of his/her abilities and interests, and on my own experience in different fields.

Curriculum Development

While at MIU, I introduced three new courses into the undergraduate mathematics curriculum and provided a complete syllabus; in two cases, I also wrote up detailed documentation concerning the use of appropriate software.

*Statistics* Before this course was added to the curriculum, the math department had one probability course in which statistics was never mentioned and with which students typically had difficulty. Statistics courses for business, physics and psychology majors were taught within those departments. I designed a lower division statistics course that simultaneously provided an appropriate pre-requisite for the probability course and a service course for other majors. Using MathStat software proved to be extremely useful in helping students master the basic concepts without getting lost in time-consuming calculations.

*Introduction to Higher Mathematics* As in many colleges and universities, at MIU, prior to the introduction of this course, students were expected to pick up the skills of theorem proving as they went along, especially in linear algebra, abstract algebra, and real analysis (and in some cases, discrete math). The purpose of this new class was to provide a course that would focus on proof-construction--the goal would be to get across the concept of a proof; to develop confidence in the ability to see what is required in proving a theorem; to develop discernment concerning which lines of thought bear fruit and which violate logical rules. The course was very much welcomed by students; the faculty are in agreement that, generally, students come into abstract algebra and real analysis courses better prepared than before as a result of this new program.

*Mathematical Logic* While I was teaching at the University of Wisconsin, Madison, I wrote three chapters in a book entitled *Mathematical Logic and Computability* (McGraw-Hill, 1996). I have taught the logic course that flows from this work five times now. The course is generally a popular one because there is very attractive software that makes some of the most abstract material very accessible. Students who typically struggle in more abstract courses find the software a saving grace. The better students find it an appropriate finale to a math major as it serves to provide a mathematical analysis of the logical basis for all fields of mathematics.

Teaching Mathematics at MIU: Teaching Every Type of Student

My teaching experience at MIU from 1990-1996 has provided me with some rather powerful tools for undergraduate education. MIU places a rather high demand on its faculty for teaching excellence. One of the requirements is that big (and highly visible) charts be prepared each class day that list the main points of the day's lesson and make clear the connection between each day's lesson and the theme of the course as a whole. A course syllabus is always handed out the first day of class and must include the main points for each day of the course. Needless to say, the effect of this is that lectures tend to be well-organized and students tend not to get lost so easily.

In addition, each department is urged to create in-class exercises that involve students in writing. In the mathematics department, this recommendation has been implemented by having students write paragraphs describing in their own words various mathematical concepts, themes, ideas behind proofs, and relevance of mathematical ideas for other disciplines. Time and again, students express their appreciation for these writing exercises. I have found that, particularly in calculus, students seem to grasp the underlying concepts best--especially the limit concept--when they are asked to write about them regularly.

Because of MIU's open admission policy, I have taught classes in which student have widely varying backgrounds. This has given me the opportunity to try many more teaching approaches than I would have otherwise, and has enhanced my teaching skills tremendously.