Ellen and Bob Kaplan, Jordan Hall, University of Notre Dame, 7/7/10

I just got back from Notre Dame, and boy, is my brain tired! Well, actually not. Despite a lot of walking for this out-of-shape math educator and a lot of strenuous mathematical thinking, I'm exhilarated after my first Math Circle Summer Teacher Training Institute with the wonderful Bob and Ellen Kaplan.

This was the third such institute held at the University of Notre Dame, a campus that seems as isolated from time as a medieval European monastery (though the food and accommodations are vastly better). I had wanted to go two years ago but couldn't get released from a relatively new teaching position to go. Last summer, I simply didn't have adequate funds to do it. But the third time was indeed the charm, and I can say without hesitation that this was the best-spent $800 I've invested in a very long time.

The Institute ran from July 4th through July 9th, with five days of sessions in the morning and afternoon from the 5th until the 9th, plus evening informal sessions at the dorm in which Leo Goldmakher, a number theorist from University of Toronto, and Amanda Serenevy, a mathematician from South Bend who organized the Institute and runs local Math Circles groups at the non-profit Riverbend Community Math Center, led interested participants in the pursuit of various math problems (some held over from our daily sessions).

I can't begin to express my pleasure at spending this much concentrated time with people who are really passionate about quality mathematics content, teaching, and learning. Some were K-12 teachers across the spectrum of grade bands. Some taught community college mathematics. Some were interested home-schooling parents. We had non-teachers who have a love for math and who have started or plan to start local math circles in their communities. There were high school students from the Riverbend Community Math Center. All of us participated in morning sessions led by Bob, Ellen, Amanda, and Leo on various mathematical topics, as well as regarding how to set up and run a Math Circle in keeping with how the Kaplans have been doing it. In the afternoons, we planned teaching sessions and then worked with students whose attendance Amanda arranged: some were in elementary school, while the eldest were in high school or getting ready to start college in the fall. I worked with three groups, doing a problem from Computer Science Unplugged on finding minimal spanning trees with a group of 2nd and 3rd graders on Tuesday, then trying a modified version and the original problem with 4th and 5th graders on Friday. The first group struggled a lot with both the language of the problem and with the complexity of the diagram. However, I finally came up with a very simple version on the spot that they were able to work with. I'd say that with proper modifications, they could have gotten the original problem, and indeed several students from this group came by later in the week to show me their correct solutions. The second group was somewhat "bi-modal" in that four students blew through both the modified and original problems: I left them to consider how many unique minimal spanning trees could be found for the given graph. One student was able, with help, to get through the modified problem. One student, who may have been cognitively impaired, seemed thoroughly out of his depth and likely would need to learn some requisite skills before tackling these problems, along with simplified examples and language.

On Thursday, I worked with a mixed-age group of students on several questions I'd been interested in going back to November 2009 surrounding something called "number bracelets." Specifically, after the students worked through some of the basic questions that arise when working the problem modulo 10, I posed the following to them: a) for a given base b, how many disjoint orbits will there be? and b) for a given base b, what will the lengths of those disjoint orbits be?

This proved to be a deeply intrigued problem for all six students, regardless of age. One of the older students, who was perhaps 14 or 15, really sank his teeth into it. On Friday, he returned with a partially developed solution that could potentially answer part or all of each of my questions. He has promised to follow up with me as he continues to work on them.

We also had the opportunity to observe Bob, Ellen, Leo, and Amanda work with these students on several lovely problems on Monday, and to observe some of our peers teaching when we ourselves were not so engaged. I had originally planned to teach only the Friday group, so I lost out on opportunities to do more observing of others' styles and interests, to my regret. But since we debriefed collectively for about 90 minutes every day after the one-hour classes, there was ample opportunity to hear feedback from those who did observe and comments from my fellow student-teachers.

I cannot overly praise Bob and Ellen Kaplan's work in every aspect of this institute, as well as that of Leo and Amanda. And my "classmates" were all bright, dedicated, and highly-motivated to talk seriously about and work on mathematics, mathematics teaching, their experiences with the students we worked with, and much else. The week was enormously rich in terms of math content, thinking about proof (we spent one morning with Ellen leading us through some of the early section of Lakatos' PROOFS AND REFUTATIONS, not directly, but through exploring the Euler Formula for polyhedra, Cauchy's well-known attempt at proving it, and problems with his method that are raised by Lakatos), pedagogy, and various notions about what it means to teach mathematics effectively.

2010 has been one of my favorite years, and the week of July 4th has certainly been the most memorable thus far.

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