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.

Basic Information

• Course name: Number Theory (formerly, History of Number Theory)
• Course Code: THEO
• Subject Area: Mathematics
• Sites offered: Lancaster
• Prerequisites: Geometry, Algebra II

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. [FEEL FREE TO REWRITE THIS, but to set the record straight, in 96.2 Number Theory was taught by me as sole instructor with a TA, incorporating many of Pomm and Timmer's in-jokes and skits. I could not do two sessions a summer after that, so that was my only experience with Number Theory. Steinfeld] 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 nine 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 and 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 brightly 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..."

Happily, levity in the classroom is not restricted to the skits. Many class jokes vary from year to year, but others have persisted. Several props retain humorous significance outside their skits. The instructors play tricks on their students, sometimes with disastrous results. On one occasion in 2002, Pomm and Timmer, hoping to show their class that a familiar (true) fact was not as obvious as it might seem, unwittingly (and nearly irreparably) convinced them that it was utterly false, and they required the better part of an afternoon to disabuse them of this new-found notion. But perhaps the most frequent running joke in Number Theory is the naming of the theorems. Nearly every result proven in class is given a name that you won't find in any textbook. Some are named after students who suggested the particular result, (and so every student can go home and say that she got a theorem named after herself) and many, many other names have a gastronomical theme. This unusual nomenclature can be an obstacle when discussing number theory with someone who followed a more traditional course of study -- you don't know what the Chinese Remainder Theorem is, and he's never heard of Jennifer's Magic Lemma -- but it is always a source of amusement in the classroom.

The Students

Because Number Theory has both Geometry and Algebra II as prerequisites, the course mainly attracts older students. A large number of the students are nomores. In general, these students have spent a number of years at CTY, and feel quite at home in Lancaster. Also, because Number Theory is such an advanced subject, lying well outside the high school curriculum, students who sign up for Number Theory are truly interested in learning math; nobody gets stuck with THEO as their third choice. Perhaps their prior CTY acculturation, combined with a priori enthusiasm, leads them to be more receptive of the instructors' wackiness than a group of randomly chosen squirrels taking, say, Math Sequence.

The Instructors

Number Theory is one of the more difficult courses to teach: it requires extensive preparation, thorough mathematical knowledge, strong acting skills, occasional improvisation, and a sense of humor. Thus, every math instructor and his pet monkey seems to want to teach it. Unfortunately, teaching Number Theory is sort of like the Mob: in order for you to get in, someone else needs to die. So far, there have been no actual fatalities among THEO instructors, but graduate school, unaccommodating jobs, and other vagaries of life have provided temporary deaths that allowed nine instructors to join the Number Theory club:

• Pomm
• Timmer
• Teper
• Quimby
• Sideshow
• Stepper
• Zanmi
• Tortoise
• Birdie

These instructors have been paired up (97.2 was the only session with a teaching assistant (Teper) rather than two co-instructors) in many different ways over the years. [As mentioned before, 96.2 had an instructor with a TA, but I was the only instructor of the course to teach it without ever having worked with the original Pomm and Timmer. Steinfeld] You might also notice that every instructor is known by a single nickname. This is partially by design: Pomm and Timmer were so called before they started teaching Number Theory, but "Pomm and Timmer" sounded so good (if a bit derivative of a famous magic act) that "Teper" was a natural replacement for "Timmer," and when other instructors joined the fold and started pairing up in multiple arrangements, the "P&T" pattern was abandoned, but nicknames remained a requirement. Though many other staff members are known almost exclusively by nicknames at CTY, the THEO instructors have the possibly unique distinction of having their nicknames listed in documents as official as Lancaster site directories.

Most of the above instructors are long-standing veterans at Lancaster. Pomm and Timmer themselves are quite simply the most experienced staff members still working at Lancaster, (their CTY experiences both go back to being students in the mid-1980s) and adding up the combined number of sessions all eight have worked (including other math courses, such as Math Sequence, Math Modeling, and Cryptology) would be a depressing exercise for anyone else who fancies himself a seasoned veteran.

Concussion

Number Theory is a wonderful course. In over a decade, it has grown to be the crown jewel of Lancaster's mathematical offerings, and rightly so. Its unique blend of mathematical instruction and witty humor has left its impression on hundreds of students, and a number of staffers as well. It boasts some of CTY's most experienced instructors, and a brilliantly presented curriculum. If math interests you, then Number Theory is the course to take -- if you can get in. The author must confess a personal bias, however: it was Pomm and Timmer's magic from his student days that inspired him to return to teach at CTY.