Number Theory

You all know Number Theory, the branch of mathematics which deals with the properties of the natural numbers. What's not so well known about Number Theory is that it is a math course at the Lancaster site. In fact, it is perhaps the quintessential Lancaster course. It is quite popular with both students and staff, and has developed traditions nearly as rich as those of CTY itself. To reveal every detail of this course would be a three-week endeavor, and would ruin many surprises for the potential Number Theory student. Nevertheless, there is much than we can tell.

History

Number Theory or THEO, was started by Pomm and Timmer, two longtime math instructors at Lancaster, in 1991. Their reason for creating the new course was simple: they wanted an excuse to work together as co-instructors, and number theory (note: for the remainder of this article, "number theory" (uncapitalized) will be used to refer to the general branch of mathematics, while "Number Theory" (with initial capitals) will refer to the specific CTY course) seemed to be an ideal topic for a new math course. They realized early on that number theory, which is usually taught as an intermediate undergraduate course for math majors in college, would be a particularly challenging subject for younger students, even students of CTY's academic caliber. Thus, they sought ways to make the material accessible and entertaining, and it is safe to say that they succeeded. They peppered the curriculum with skits and jokes and other alternatives to the soporific collegiate lecture, and it is these additions that have made Number Theory such an enduring success.

Through the early 1990s, Pomm and Timmer taught Number Theory together for one session each summer. During this time, they honed their craft, refining skits and adding in-jokes. At the same time, Number Theory's reputation grew, and soon, demand for the course exceeded its limited supply. During the middle of the decade, CTY decided to meet the growing demand by offering Number Theory for both sessions of the summer. Unfortunately, Timmer could only manage to come to CTY for a single session, so extra help was needed. In 97.2, Pomm was aided by a young teaching assistant named Teper, marking the first time someone besides THEO's creators worked with the course. Teper soon became a fully-fledged co-instructor, and as the demands of Real Life drew Pomm and Timmer away more and more, other new co-instructors soon followed. To date, Number Theory has had six co-instructors, who have worked together in various pairings. Not content to merely carry the torch, these newer instructors have made their marks on the course as well, adding a new skit and compiling the scripts, which in the past were sometimes quite disorganized, possibly even improvised. But Number Theory has not been completely given over to the new instructors: Timmer has taught for one session every summer since the course's inception, and in 2002, Pomm returned from a three- or four-year hiatus to join his original partner in mathematical crime.

The Curriculum

The material covered in Number Theory is roughly equivalent to a semester of ungraduate-level number theory, as taught at a college or university. Students learn important results in topics such as integer factorization, continued fractions, modular arithmetic, and Gaussian integers. The presentation of these subjects is quite rigorous; very few stated results, if any, are left unproven. Additionally, students spend some of the last week researching independent projects on related topics, which they present during the final days of class. Although Number Theory lies well outside the traditional high school mathematics curriculum, its value is borne out by the fact that at least two past members of the United States team at the International Mathematical Olympiad have cited the course as a significant factor in their mathematical successes.

The Skits

Although the curriculum is important in Number Theory, it is the fun-filled presentation of the material that makes the course memorable. At the core of this presentation lie the Seven Skits of Number Theory. These skits are typically preludes to more serious lectures, and serve to introduce topics ranging from "What is a Formal Proof?" to...well, herein lies a problem. The element of surprise is an important part of many of the skits, and so the instructors are often quite secretive about them. (Interestingly, two of the Number Theory instructors have also worked for the NSA...coincidence?) They tend to speak in code when discussing skits in places where students might overhear, so I dare not say much about the names of the skits, or their topics.

You will have to take my word for it, then, that the skits are consistently funny, and surprisingly high in actual mathematical content. Pomm, Timmer, and the other instructors have shown considerable skill in crafting elegant unexpected metaphors to convey difficult mathematical topics, in such a way that the students might not even realize that they're learning something. The skits also draw on a wide range of props, from colored chalk to fake beards to a tuxedo. Many of the skits are centered on important mathematicians, such as Euclid or Gauss, and a familiar refrain from these skits is, "What's not so well known about [Name] is that he was also..."