Teaching Statement -- Christopher Hardin …

                          Teaching Statement -- Christopher Hardin
     I have taught several different courses at Smith College, in addition to the teaching experience I had as
a graduate student. Being a Project NExT fellow has also improved my teaching, partly from the work-
shops, but mostly by connecting me to others who care about teaching. (Project NExT--New Experiences
in Teaching--is a professional development program of the MAA for new faculty, with an emphasis on
developing teaching skills. Being selected as a Project NExT fellow is already a recognition of commitment
to teaching.) Good teaching requires ongoing development, and what follows is a snapshot of where I am
now.
     Students learn best when they are interested in the math and when they do it themselves. One way that
I promote interest is by creating examples that have aesthetic appeal. To illustrate, I will share a favorite
here. After covering the chain rule in a calculus class, I go over how sounds are really functions of time,
and have students do an activity where they predict how f (t) would sound compared to f (t) (consider
f (t) = sin(t/10) + sin(t) + sin(10t), for example). They are able to predict that high frequencies will
be amplified when one takes the derivative, while low frequencies will be attenuated. Conversely, they are
able to reason that integration has the opposite effect. After they make this prediction, I produce a circuit
board with an integrator and a differentiator on it--just a few dollars worth of parts--wire it up messily to
a music player and a speaker, and we test the prediction, which is confirmed dramatically. (Also, wiring
the differentiator and integrator end to end, we get something close to the original signal back out.) This
reification of differentiation and integration delights and engrosses the students.
     It is important to get students doing problems on their own as soon as possible. To this end, I typically
have students work in groups of three for 10­15 minutes of class. I prepare worksheets that start with a
straightforward application of the material from class, and progress to more subtle questions that require
thought and discussion. This serves a number of purposes: immediate application of material from lecture
helps the students understand and remember it better; forcing students to verbalize mathematics sharpens
their understanding of it; and questions that do not arise during lecture often arise when students first try
to use the material. Group work brings out those questions while I am there--and when possible, other
students answer them.
     I had extremely positive experiences with the Moore method as an undergraduate, and variations of
the Moore method are the direction I would like to go with my teaching. I designed a course, Measure
Theory and the Banach-Tarski Paradox, that I am teaching next semester; last semester, I wrote the packet
of definitions and theorems (interspersed with exposition--I am not a purist) that the students are working
through. The students work in teams (which change over the course of the semester), and present the
theorems to each other in class. Some class time is also be spent having students share and refine drafts
of their proofs; this is concurrent with "coaching sessions", where I meet with teams and give them some
feedback on early drafts.
     Another course I designed was a half-semester course on parsing, which I taught as half of Advanced
Programming Techniques, a topics course in computer science. I also (voluntarily) designed and taught an
Interterm course on the mathematics of sound. We did activities such as recording the sound of a balloon
popping in a deep stairwell, and using convolution (made efficient via the fast Fourier transform) to process
another recording and make it sound as if it also came from the stairwell.
    This year, I have the pleasure of advising my first honors student, Katherine Peterson, who is working on
the Cable Guy Paradox and other infinitary questions in decision theory. Many of my research interests are
a good source of problems for research by undergraduates. Much of the work is in novel areas, providing an
abundance of questions that are unanswered but still accessible. (By accessible, I do not necessarily mean
easy; rather, they do not require much technical background.) In the infinite hat problems (see my research
statement for a summary), one can explore any number of variations: What if agents can hear other guesses?
What if each agent gets to ask a yes or no question (but there are more than two hat colors)? Or, related to
circular societies: What if on replaces the circle with a tree, or other graph?
    Many students go through high school feeling that their classes are just something to endure for the sake
of acquiring knowledge that will be useful later. Their environment reinforces this: "impractical" subjects
such as music and art are expendable, college is promoted only as a career enhancer, and classes are taken
for the sake of getting into college, not to satisfy curiosity. The message is that education is merely a
vehicle, not also a destination--mathematics especially receives this treatment--and many students bring
this attitude with them to college. While embracing the role of education, and specifically mathematics, as a
vehicle, mathematicians must share with students the joy of elegant concepts and beautiful ideas. Through
this, we help students find intellectual fulfillment in their work, making their stay in college more enjoyable
and productive, and we also make learning a habit that they preserve.




                                                      2