Teaching the teachers

Last week, I taught at the Exeter math conference for high school teachers. The teachers come for a week and take two courses, which each meet for two hours per day, on subjects such as using technology in the classroom, creative ways to teach math, and learning new mathematics.

My courses fell into the latter category. I taught one section of the Exeter math curriculum course, which someone else designed years ago, and then I created my own course, "Greatest Hits of Higher Mathematics." It was the first time I'd ever created my own course, and it was quite successful. In this post, I'll reflect on things that went well and things I'd like to change for next year.

The goal of the course was to introduce the teachers (hereafter, "students") to the college math major or graduate school math curriculum: specifically, analysis, algebra, topology and number theory. I chose these topics because the first three are standard core curriculum, and number theory seemed like the most compelling other topic (rather than algebraic geometry, partial differential equations, etc.).

When I chose the course materials and the exercises to assign, I had no idea what the participants' level would be. I was afraid that they would not be able to understand the textbooks that I used as an undergraduate, because they would only have one night to read it, and concepts like an "open ball" took me weeks to fully understand. Thus, I paired each textbook reading with an essay by Stephen Strogatz, which presented similar material in a way aimed at the general public.

I structured each class so that we spent the first half going over the previous night's homework, then had a short break, and then had an introduction to the next topic. This way, they were introduced to the ideas before doing the reading on their own.

As it turned out, the participants were really sharp. (This is probably because of the selection bias for people who sign up for a course on higher mathematics.) They understood the textbook readings and delved into the homework problems. Because I didn't want them to have no idea how to do the homework, I had assigned mostly easy problems, even some that were just repetition of examples from the text, such as proving that the fourth root of 2 is irrational after seeing the proof for the square root. I wish I had assigned some that were more difficult, because they could totally have handled it.

Here are my reflections on the individual topics.

Real analysis: This went well. On the first day of class, I asked them to prove (as a group) that the square root of 2 is irrational. Most of them had seen the proof before, but all had forgotten the details, so this was a good exercise in constructing a proof, and also in having a group discussion. Then we talked about open and closed sets (as intervals on the real line). Is the union of two open sets always open? How about the intersection? How about for closed sets? Can the infinite intersection of a collection of open sets be closed? Can the infinite union of a collection of closed sets be open? They loved thinking about these problems and were very engaged.

The homework also consisted of compelling problems, including some that were too hard for them to do for homework. We spent most of the class period going over solutions, and by the end we were able to solve even the hardest one (my favorite concept, "dense"). That was a good class.

Topology: I had trouble finding a good book to use. Everything I found started by talking about sets. I wanted something that talked about shapes and fundamental groups. I ended up using just the first chapter of a book on intuitive topology, which had a series of exercises that were pictures of knotted-up shapes that asked students to figure out how to unknot them.

My goal for the topology section was for the students to learn to imagine the shapes in their head, and to not use any numbers, equations or writing. This was successful, and we again spent most of the class period working on the hardest exercise, this time in small groups. However, it took the groups a very long time to untangle the object, and I was afraid that it wasn't the best use of time. But I decided that as long as they were thinking hard about math, it was a fine use of time.

Next time, I think I would actually do point-set topology, defining a topology and talking about the topology of various spaces with their open sets defined in various ways. I actually think the students might prefer that.

Algebra: My preview for algebra was to define a group, and then give a proof that the identity is unique. I didn't even have to give it, as someone told me what to write as I was writing it. The book I chose ended up being perfect for this course, as it was full of interesting exercises that were easy to state. I only photocopied small sections of the book, but I should have copied more, as some of the exercises I didn't assign referred to sections I hadn't included, so the students couldn't try them (because, for instance, they didn't know the definition of U(12).)

We went over the homework exercises quickly, and then spent the rest of the class period working on more problems in small groups. Algebra really is fun, especially at the beginning. This topic went well.

Number Theory: This was the weakest topic. I was using a book that explained the major theorems well, but lacked compelling exercises. I recently heard someone say that the exercises you choose make the course, and that was definitely true here. The homework exercises were not proofs, just verifications like showing that a whole bunch of even numbers could indeed be written as the sum of two primes, so we went over the solutions quickly in class and then had 90 minutes left without much to do. I will definitely choose next year's book based on the exercises.

Choice topic: Since the four topics I chose just scratched the surface of higher mathematics, on the last day I asked each student to pick a topic in higher mathematics and then give a 10-minute presentation telling the class what it is about, and maybe doing an example of the sort of problem the field looks at. I brought lots of books, and the students borrowed them throughout the week. They were also welcome to use other resources like the Internet, of course.

My hope was that we would see presentations on things like knot theory, graph theory, the real analysis behind calculus, and point-set topology. In fact, the presentations were about the game Nim, RSA cryptography, the math behind the win/loss predictor in baseball, the Euclidean algorithm, and subgroups. They were getting tired and overworked by the end of the week, and in any case, the presentations were good.

I'm looking forward to offering this course again next year, and I hope people sign up!