Two junior boys came to me near the end of the school year with a request: could they take an independent study course with me in number theory? I was both surprised and pleased, in equal parts, by the request. Surprised because math teachers at my school rarely were asked to teach independent study classes. Pleased because number theory has always been a real interest of mine. I was pleased as well because Bob and Mitch were both excellent math students as I knew from my sophomore geometry honors class, and I was certain that they would keep me on my toes and make the class fun. Saying yes to them launched one of the most enjoyable episodes of my teaching career.
My first task was really no effort at all; I had to choose a good textbook. After finding a few titles online, I went to my favorite bookstore, which just happens to have a large math section. In just a few minutes I found a readable book on number theory with a good number of exercises (with answers), and the added benefit of a generous dose of math history. As luck would have it, the author proved helpful to me all year via e-mail advice on both structuring the course and answering questions about number theory.
My plan was to outline the course over the summer, read the chapters I intended to assign, and to work out as many exercises as I could. In the end I didn't do nearly as much as I should have. Despite a genuine interest in the subject, I am by no means an expert in number theory. An early fascination with Fibonacci numbers led to enough study for me to give a talk to colleagues and students several years ago. Then, when I wrote the geometry book we now use, I included one number theory lesson per chapter as enrichment. It was these lessons that interested Bob and Mitch in the subject.
Despite these factors, I wasn't uneasy about the class since I envisioned it not as two students and one teacher but three students. It is a cliché to say that teachers learn as much as their students but in this case that is exactly what I expected to happen. I would make the assignments, write the tests, choose quarterly projects for Bob and Mitch, and assign grades, but as much as possible I hoped we would act like colleagues, explaining new definitions and theorems to each other, solving problems together, and answering each others' questions. As it turned out, Mitch and Bob also had a hand in selecting homework problems and determining their projects.
We met twice a week for fifty minutes. Informality reigned. "So, what did you think of the reading?" or "How were the exercises?" I would say for starters, and from there Mitch or Bob would go to the board to explain a problem or theorem. We worked our way through five or six chapters in the course of the year. That may not sound like a lot of material but number theory entails some difficult concepts and I was determined to emphasize depth rather than breadth. Another reason for not rushing was my awareness that this class was an extra class for these students on top of their usual senior year course load. Both students were taking a "regular" math class in addition, and one was taking two "regular" math classes.
But the main reason we took our time was to be able to follow the many tangents and digressions that seemed to naturally occur because of Bob and Mitch's curiosity. Much of the work we did involved special sequences of natural numbers, such as digital root sequences, and numbers that satisfied specific requirements, such as Fermat numbers. Not only did Bob and Mitch eagerly verify properties and patterns but they defined new kinds of numbers and looked for new patterns. Their efforts were sometimes successful and sometimes not, but it really didn't matter because they were creating mathematics, which doesn't always happen in a math class, even an honors class. Quite a few classes were spent puzzling over just one problem or theorem, searching for a clarity that frequently seemed very hard to find. Both confessed to doing number theory in other classes. Nothing symbolized their passion for number theory as much as the shape of Bob's book after a couple of weeks. A thick paperback book, its covers were rolled back and its pages severely dog-eared and torn.
The topic they were most eager to study was cryptology. But when we got to that chapter in the book, our last, we soon realized that most of the codes were very difficult. We plugged away for a while and then Bob started bringing in a magazine that featured codes of varying difficulty that we enjoyed solving.
Have I mentioned that Bob and Mitch both have a wonderful sense of humor? Each class was punctuated with smiles, laughs, and puns. An example of their humor showed up at the end of the first quarter when they were doing a project that involved using a computer to generate a very large table of figurate numbers. When it was time to hand it in they proudly gave me the single 8.5" X 11" sheet of paper. All the numbers were there (I think) but the font was so small I couldn't read it. Then they handed in the real printout that ran to four pages. While they appreciated and valued what we were doing in our independent study class, their clear sense of proportion did not allow them to take our work, or themselves, too seriously. Nor did they treat me with undo deference; I was their teacher but the mood of the classroom was decidedly easygoing and egalitarian.
These days most teachers, even veterans, now believe that teaching has evolved from the idea of somehow bringing learning to students to the belief that students are largely responsible for their own learning and the teacher's role is to facilitate that learning. This class was a perfect example of that belief in action. Bob and Mitch were studying and learning difficult mathematics not because it was required and not to enhance their transcripts but for the fun and challenge that number theory presented. This is the situation we teachers always hope to achieve in our classes but rarely do.
Bob and Mitch are in college now and I am back to teaching four regular classes, with no independent study, but the experience of that class has been a shot in the arm for me. My enthusiasm for learning new mathematics has been revived; once again I enjoyed the challenge and mastery of difficult mathematics. I was also reminded that to be a good teacher one must be a dedicated student as well. I will try to be more of a student and less of a teacher (dispenser of knowledge) in my classes. And, like Mitch and Bob, I will try not to take myself too seriously, a particularly difficult thing for me to do. Our independent study class has also taught me everyone learns more when everyone in the classroom is both a teacher and a student.