Teaching Statement

Gareth E. Roberts


As a postdoc in the Applied Mathematics Department at the University of Colorado at Boulder, I have taught two semesters of Calculus III for Engineers and one semester of Matrix Methods with Applications. The calculus course is a large lecture class for sophomore-level students while the matrix methods course is a small-sized linear algebra class at the junior-senior level. I was responsible for maintaining the course webpage for both courses and for supervising a group of graduate TA's in Calculus III. For the matrix methods course, I wrote two computer labs to provide students with a better understanding of the material and to help them explore some of the applications of linear algebra. In addition to my duties as an instructor, I am supervising the research of an undergraduate student in the field of complex dynamics.

After obtaining my Bachelor's degree from Oberlin College in 1992, I taught for two years at a private high school in Princeton, New Jersey. I was responsible for all aspects of my courses and provided students with a daily extra help session after school. In graduate school, I worked as a teaching assistant in calculus and differential equations. My duties included lecturing in class, teaching discussion sections, holding office hours, tutoring, grading exams and writing/grading labs. In the summer of 1996, I was selected to teach a course in linear algebra at the sophomore level.

I also have a significant amount of experience in curriculum development having worked on the Boston University Differential Equations Project with Paul Blanchard, Robert Devaney and Glen Hall. I contributed to and supervised the production of their text Differential Equations and was invited to speak at two workshops focusing on teaching differential equations from a dynamical systems perspective.

I have received very favorable evaluations from students and have worked hard to polish my style and methods over the last several years. My hard work and persistence was recognized in graduate school when I received the Outstanding Teaching Fellow Award from Boston University in 1998.

A Flexible Teaching Style

Last spring I was giving a lecture in my matrix methods course on the Gauss-Jordan method of finding the inverse of a matrix. I was illustrating the algorithm on the board with an example, leading the students through the calculations and then demonstrating that we had indeed found the inverse. As I began a discussion on the uniqueness of the inverse, I noticed that several students were still copying down matrices off the board. Although I had a tidy lecture prepared for the remainder of the class, I decided to push off in a different direction. I gave the students another example and asked them to find the inverse on their own, using their neighbors as a resource if necessary.

Suddenly, the lecture became a discussion section. I headed for the back of the classroom, as often this is where the most confusion sits, and began working one on one with students. To my surprise, most of the students had no idea where to begin. Many of them were glancing back at their notes and trying to follow the steps of the previous example. I quickly went back to the board to help them start the problem. This was enough to get the brighter students going, which in turn helped their neighbors begin. I visited each of the 20 students in class that day until I was satisfied that the majority of them had learned the technique. Although, I was unable to finish the remainder of my planned lecture, it was time well spent.

The point of this story is to illustrate the flexibility in my teaching style. There are times when I prefer to lecture on concepts, proofs and theorems, and there are times when I prefer to do examples at the board or play wandering tutor while the students work problems at their seats. I think of this as shortening the communication gap between instructor and student. As I moved around the classroom that day, I obtained an excellent picture of the amount of learning and understanding that was taking place. Too often we discover that a certain concept or mathematical technique has been poorly understood only after the exams have been turned in. Although it can be difficult to combine the lecture style with the discussion section format, and it often appears that less material is being covered, I am increasingly taking the stance that less material learned well is better than more material learned poorly.

Mathematics is Alive

Another example I would like to share illustrates the importance of demonstrating interesting applications of mathematics. On my first attempt teaching Calculus III, I was criticized on my evaluations for not going beyond the material in the text. Students wanted to see more applications of the course material. Finding the center of mass of a region in the plane was not capturing their interest. Many of us have received complaints like this since we began teaching, but since most of my students were engineers (not math majors), it seemed to be a fair criticism. The second time I taught the course I decided to improve this aspect of my presentation.

To supplement the subject of Lagrange multipliers, I discussed one elegant example from my research on the N-body problem. To find relative equilibria, special periodic solutions where a configuration of bodies rotates rigidly about its center of mass, one needs to solve a Lagrange multiplier problem. In 1772, Lagrange discovered that placing any three masses at the vertices of an equilateral triangle yields such a solution. In class I derived the equations of motion for the N-body problem using Newton's laws and then illustrated examples of relative equilibria on a laptop computer.

The response from my students was wonderful. They were attentive and interested as they watched orbits of bodies circling around one another. As the symmetry of the solution broke down (after a few revolutions), it became natural to discuss my research on the stability of periodic solutions. This gave the students a glimpse into contemporary mathematics research and provided me with the ideal opportunity to encourage them to enroll in more mathematics courses and to consider becoming a major. Some students wanted to know more about celestial mechanics and asked if this material would be covered in their differential equations course. One student even sent me an email, asking if he could start attending the department colloquia. The presentation was very successful at generating interest in the course material and recruiting future mathematics students.

Women and Minorities

I believe that any eager student can learn the mathematics that we teach at the college level. This applies in particular to women and minority students, who are often discouraged from taking mathematics courses. I have reached out to members of these under-represented groups by taking the time to check in with them and monitor their progress as the semester unfolds. Last year I successfully recruited two of my top female students to become applied mathematics majors. One of these students was a high school senior who specifically decided to enroll at the University of Colorado because of the high-quality teaching she received from our department. This year, several of my female students in Calculus III are planning to enroll in my matrix methods course next semester and at least become applied math minors. I will have a semester to convert them to majors. Although many of the biases and stereotypes used to keep women and minorities out of mathematics are laid in at an early age, I believe it is possible to counter these false premises through persistent attention and encouragement.

I truly love to teach and I am committed to cultivating the minds of my students. I have been applauded by students for being an approachable and accessible instructor. My office hours are often overflowing, leading to an impromptu discussion section in the hallway. I bring a lot of energy and excitement to the classroom, striving to keep my students learning, interested and motivated. These attributes have been cited frequently on my course evaluations as a key ingredient to my success as a teacher.

I look forward to building on these ideas and experiences as I continue my career as an educator.