I love reading and the idea of book clubs, and I have always wanted to be part of a really stimulating and fun book club, with bright, unpretentious people thinking hard about good stuff. I haven’t found the book club of my dreams yet, but I may have found the math-instead-of-book version: Philadelphia Area Math Teachers Circle (PAMTC). Math Teachers Circles are communities of teachers, professors, and education professionals who meet to work on interesting mathematics problems. The model is similar to that of Math Circles for students (see Adriana Salerno’s report about her math circle here). Someone presents a problem, and the participants work on it in groups, everybody has fun and learns new math. Since the participants are teachers, there is some discussion of how to connect the problem and ideas to the classroom, and it is possible to serve wine (all the more like a good book club!).
It seems that the best problems for Teachers Circles are “low-threshold, high-ceiling” problems (in the words of the National Association of Math Circles, and someone I talked to during the meeting). These are problems that everyone can get started on and figure out some examples, but which generalize to any degree of difficulty, and often have extensions that are unsolved. At this month’s meeting of the PAMTC, my awesome Villanova colleague Katie Haymaker presented the pancake sorting problem. Basically, you are a waiter with n pancakes of different sizes stacked on a plate. You want to arrange them from largest to smallest, but your only tool is a spatula. The one move you are allowed is picking up some pancakes from the top of the stack and flipping them over. How many flips does it take to sort the stack? The number of moves it takes a maximally efficient algorithm to sort the worst stack of n pancakes is called the pancake number of n. These are known for small n but open for n=20 and beyond. Katie and I heard about the problem in a talk last year by Ivars Peterson, and she thought right away that it would be great for a Math Circle. She was totally right.
Katie had developed a couple of very simple slides about the problem and talked to the participants for just about 5 minutes, explaining the idea, then challenged everyone to find the pancake numbers of n=3 and n=4. The groups got right to work and spent an hour or so figuring out the first few cases, working mostly on their own with some facilitation from Katie and the other organizers. There was no lecture, no explanation of the right techniques—just questions to help organize and spark ideas. It was a room full of people talking to each other and doing math essentially for fun.
After working, people shared their process, insights, and results. The teachers had developed some nice ideas about upper and lower bounds for the pancake numbers. Katie gave them a handout that she’d made explaining the problem and some more resources and directions. The meeting ended with one of the organizers, Josh Taton, leading a short activity on evaluating curricula.
Having spent a lot of time planning activities for my math classroom (and having them not always be as fun as I’d hoped), I was surprised by and excited about how well it all worked at PAMTC—kind of a dream of what math class could be like if everybody wanted to be there. Why did it work? First of all, I think this was just a good group, with some veterans scattered around the tables keeping things running smoothly. The organizers deserve a lot of credit for creating a good atmosphere, providing good food so that people could relax and do math at dinner time, and working with the presenter to make sure the problem was presented in an accessible yet exciting way. Katie chose a great problem and did the work of turning into an activity, by creating a handout and providing little disks to use as manipulatives (physical problem solving props). The teachers mentioned that the manipulatives had been really helpful.
One very helpful additional way to make Teachers Circles work is to find a way for teachers to get professional development credit for attending. The other side of this is the importance of professionalizing Math Teachers Circles for professors–finding a way for professors to also get credit for their work with the Circles when applying for tenure and promotion. After talking to friends who have been involved with Math Circles and Teachers Circles, it seems that not every department values faculty members’ contributions to Circles. I would love to (and plan to) start a Math Circle someday, and I hope that it will be perceived as worthwhile by my department. NAMC is actively working to raise the profile of Math Circles and to help make sure that contributions to Circles are professionally valued. They also have a ton of resources on their website (including Circle in a Box) making it much less time consuming and so theoretically possible for a pre-tenure professor to run a circle.
However, when I was at the Math Teachers Circle last night, I wasn’t thinking about professional credit, or that it was 5 pm, after a long day of teaching and everything else. I forgot about my big shopping bag of quizzes and homework to grade later, and that I still needed to calculate and report midterm grades for my students, as well as figure out what I was going to teach the next day. I was just digging the math. It would take something pretty cool to get me and 25 others, who all worked with people all day, to spend two hours sitting in small chairs in a middle school classroom, talking to each other about pancake numbers. I guess I can report that Math Teachers Circles are actually that cool.