Jana Comstock                                    Teaching Statement

I am fascinated by the flexibility of mathematics; it can be entirely practical or sublimely useless, concrete or mind-bendingly abstract, and, of course, routine or as complicated and challenging as you could ever wish it to be.  This variety makes the study of mathematics enthralling, but even more so it provides a set of intriguing pedagogical challenges: most notably, whether to emphasize its utility or its capacity to broaden minds and suggest new ways of thinking.

When teaching linear algebra to a group of future engineers and scientists, for the most part, I'm forced to ignore its lovely potential as a pathway to proof writing in favor of the mechanics of matrices.  Similarly, the Math Sequence course I've taught at Johns Hopkins' Center for Talented Youth is largely populated by students who simply want to complete the required material in order to spend their time on other studies, so we spend five or six hours out of seven focusing on procedure and only occasionally engage in activities which might broaden their definition of what “math” is.  On the other hand, the Math Reasoning course I've taught at CTY consists entirely of material outside the usual curriculum.

The class I'm currently instructing at Berkeley is precalculus, the most basic math course offered at the university, and most students are attempting to transition from an unsatisfying high school experience with math to a more productive one at university.  My goals are to solidify their understanding of basic concepts and procedures they'll need as tools while simultaneously encouraging them to achieve a deeper understanding of the material, as will be expected routinely in later courses.  To this end, I tend to demonstrate relatively simple ideas and problems in lecture which students may assume they understand completely and then invite them to question that assumption and enhance their comprehension.  For example, on the first day of class I talked about multiplication and addition, pointed out common mistakes students make by confusing these two, how subtraction and division are implicit, and demonstrated how these operations suggested the progression from positive natural numbers to non-negative natural numbers to integers to fractions.

Apart from the actual content, another important aspect of my teaching philosophy is to attempt to make math less intimidating.  Recently much consideration had been given to the participation of women in mathematics, and I am fortunate to be able to encourage such participation through the simple fact of my existence.  However, the fear and hatred of math is widespread among men, women, rich, poor, whites, blacks, and hispanics alike.  Accepting that math is hard and that's it's okay to be wrong or confused -- and even okay to admit to being wrong and confused -- is an important skill to acquire.  Many people I know, including myself, only acquired it in math grad school, but the sooner one learns it the easier collaborating and tackling difficult problems becomes. 

During my most recent stint at CTY, one of the administrators passed out a copy of the article “How Not to Talk to Your Kids” by Po Bronson, published Feb 11th 2007 in New York Magazine.  This article contains findings suggesting that an emphasis on effort rather than innate ability encourages students to challenge themselves and improves their academic results.  This is a mindset I attempt to foster in all my classrooms by entertaining all questions, listening carefully to wrong answers and finding ways to praise students for offering them, and by pointing out that I and other mathematicians get confused as well.  I occasionally point out that a trial-and-error approach is perfectly acceptable, and that being wrong can eventually steer you towards being right if you are willing to think about your mistakes.

Graduate school has been educational and hugely enjoyable.  I've met great people and had a chance to indulge in the luxury of focusing on my own intellectual development.  However, as the novelty has worn off, the fact that this is a luxury with no palpable benefit to others has become more obvious, and the teaching, which is incidental to my degree, has actually become the most rewarding aspect of it.  I am looking forward eagerly to finding a position where I am encouraged to make teaching my first priority.