Welcome to the Independent Study in Problem Solving!

Greetings. I wanted to take a few minutes to introduce myself and the course. I've posted a brief welcome letter introducing myself, my family and my research interests here.

The following is from Nick Arnosti, captain of our math competition teams and co-organizer of this class, Carlos Dominguez, Wei Sun and Wen Han:

We're writing to advocate for the creation of an independent study class in mathematics next fall semester. The proposed class would focus on the techniques underlying mathematics competitions, with particular emphasis on the national William Lowell Putnam exam to be given in December.

As an avid participant in all math competitions I could find in high school, I am hardly unbiased with regards to this topic. I freely admit that I love competition math, and that the opportunity to spend a semester pursuing this love is enticing.  Objective success is also a motivator: Williams has come in the top twenty in the country each of the past two years, but I feel that we could have even more success with some formal preparation for the exam.

While these two factors motivate me, they are not the reason that I'm writing this letter. Every student here has a variety of interests and activities, and many of us devote countless hours to extracurricular competition. I would like to present to you the reasons that I believe the subject of competition math deserves to be an academic, and not merely extracurricular, activity.

At Williams, it is often said that the primary goal is to teach students how to think, rather than simply to recite facts and figures. I believe that this independent study course would fit perfectly with this objective. Too often in math classes, the techniques and approaches are somewhat scripted: "Well, we've been talking about the Theorem A in class this week, so that's probably going to apply to this problem," or "This problem is from the section of the book where they explain technique B, so I guess I'll try that."

In competition math, any topics can and do arise. There are no guidelines or hints about how to do the problem, so it truly is an opportunity to express and develop individual creativity and problem solving skills. In order not to disadvantage certain students relative to others, competition problems generally do not require knowledge beyond basic linear algebra and group theory. Instead, the emphasis is on the thought process. Students must find ways to combine well-known fomulas and techniques to do something they've never been taught. I believe that this sink-or-swim experience
often produces the most inspiring moments for students: at the end of the day, we can be truly proud of what we've done.

While the problems that we'd focus on don't assume an advanced background, they often can be generalized to shed light on deep mathematical topics. The Putnam problems, in particular, are selected not only to be interesting in their own right, but also to serve as illustrations of particularly elegant techniques in mathematical problem solving. By studying these problems, students will encounter ideas that will build their mathematical toolbox, and give them a broader context from which to analyze problems that they encounter in future classes.

Not only will working on competition problems help with future classes, I also believe that for those students who go on to do research, it will make them better researchers. It's often said that the best results in mathematics come when someone takes known techniques and applies them to a new field. Drawing connections between seemingly unrelated questions is a powerful technique that math competitions teach more effectively than anything else I've encountered.

Finally, because mathematics competitions tend to require that students prove their answers, this class will offer an opportunity to improve proof-writing techniques. Most students never receive any formal education in the area of mathematical writing. Some learn indirectly through their classes, but as a TA this year, I've had the opportunity to witness first-hand that many don't. Because competitions are graded without the student present, the problem solver must be able to express that answer convincingly in writing: if something is unclear, the grader will not go to the student to ask what they meant. As a result, the student must learn to write rigorous proofs. Conversely, because competitions are timed, there is no time to wander around stating irrelevant and/or redundant observations. This helps teach students to write concise and elegant proofs, avoiding excessive algebra when possible and writing only what is necessary to clarify the issue at hand.

Having outlined the many reasons that I believe this class would have great value to those who took it, it remains to show that there are a number of students who would, in fact, take advantage of the opportunity. The following students have expressly stated that they would be interested in taking this independent study were it to be offered next fall:

Nick Arnosti
Carlos Dominguez
Wei Sun
Wen Han

Looking forward to seeing you all.   //Steve

PS: My wife teaches at Boston College, 3 hours away. Thus for evening review sessions you'll often find my two children, Cam (he's almost 3 1/2) and Kayla (she's almost 1 1/2), the other TAs for the course (pictures will be updated in August).