The open secret of math is that the hardest thing to do in it is count. -Andrei Zelevinsky
Last night I had the extraordinary privilege of teaching my first Math Circle class, to a group of 9- to 12-year-olds at Harvard. I think they had a blast. I KNOW I had a blast. The theme is counting, and we started out by discovering that all our names turned out to be four letters long. We then counted up the squares in this figure,
and once they settled that, we spent the rest of the hour following a diagonal. We came up with several good conjectures on the diagonal problem, and I’m very excited for next week; I expect they’ll nail it. There were a lot of great victories along the way—the youngest in the room really wanted to use fractions, and found a way when we had to enumerate different sizes of square—but the one that pointed the way forward for the rest of the course was when they confidently told me that the diagonal of a 1 million-by-1 million square would pass through 1 million grid cells. I pointed out that they had just effectively counted 1 million things in a couple of seconds. I thought that was cool. I hope they thought it was cool. I think they did.
I’m also excited for what this might teach me about my high school students. In this little laboratory where I’ve controlled out teenage apathy and uneven experience with algebra, I’ll be interested to see how natural it is (or isn’t) to start using abstractions and symbols. Great fun!
Sounds like a lot of fun! How did you present the “follow the diagonal” problem? I’m curious because of the interesting discussion on Dan Meyer’s blog about the relative merits of different formulations of this problem.
Aaron,
Yes, Dan’s page on this problem got a lot of interest!
Let’s see: I think I drew a 3×4 on the board, drew the diagonal, and asked how many squares it went through. Then I outlined a 2×4 within the 3×4 and asked about that. Then I put a table up, dimensions in one column (taught them what “dimensions” means) and number of squares in the other, passed out graph paper and asked whether they could find a pattern. They all started drawing—next time I’ll remember rulers!—and nearly immediately we got a conjecture of m+n-1 (which fit both so far) and m*n/2 (which fit the first but not the second). Then, I think, they gave me some more data points. Shortly after that someone noticed that m+n-1 doesn’t work for perfect squares. At the end of the hour we were 99% confident that it’s n for nxn: kids were sure, but didn’t know if we’d proved it. They thought m+n-1 was probably right for non-squares but weren’t sure. I put a 4×6 on the board right as they were leaving. I have no idea if they will progress or regress or what on their own before next week.
I’m actually trying to find a team of 5 to apply for the Math Circle workshop this June in Palo Alto, CA. There is a strong network of math circles in the Bay area and in Los Angeles, but really nothing in between — where I live. Sounds like everyone had a great time doing math, and I love what the kid said about arguing. Thanks for sharing!