Personal Reflections (1999)

William Thurston, a brilliant mathematician and Fields Medalist (the most prestigious award in math), wrote the following in a famous 1994 article in the Bulletin of the American Mathematical Society: 
 
How do mathematicians advance human understanding of mathematics? 

We are not trying to meet some abstract production quota of definitions, theorems and proofs. The measure of our success is whether what we do enables people to understand and think more clearly and effectively about mathematics. 

We mathematicians need to put far greater effort into communicating mathematical ideas. To accomplish this, we need to pay more attention to communicating not just our definitions, theorems, and proofs, but also our ways of thinking. We need to appreciate the value of different ways of thinking about the same mathematical structure. 

We need to focus far more energy on understanding and explaining the basic mental infrastructure of mathematics... This entails developing mathematical language that is effective for the radical purpose of conveying ideas to people who don't already know them.

Thurston charges all mathematicans with the task of doing whatever it takes to communicate mathematics to all learners, and that teaching should be approached, not as creating an elitist club, but rather as "developing mathematical language that is effective for the radical purpose of conveying ideas to people who don't already know them." This made a very great impression on me when I first read it, and I have used it ever since as a guiding principle to keep myself focused on finding good ways to communicate math that actually work with all the real people in my courses. So, I have developed an approach that uses math visualizations, kinetic modeling situations, and learning by doing with continous assessment of both the students and the course. Why? Not because I am hunting for a few students to tune in with my esoteric interests in math, but because my goal is to try to create a successful learning experience in terms of performanced-based abilitites for absolutely everyone that shows up and is willing to try. I can't imagine a more important goal to keep focused on for success in future.