A child’s non-commutative model of multiplication
I was playing math with a niece and nephew a few weeks ago. Really, just playing games. And challenges.
We played who can get to 12 (by adding 1s or 2s). I wasn’t going for a rule, but my niece was close, so we (sister-in-law helped) got her to discover that 9 was a good number. And so was 3. And my nephew (younger) wasn’t going to discover it, but once his sister announced it, he kind of sort of followed.
We played puppies and kittens (game I learned from Sue Van Hattum. Adopt as many puppies as you like. Or as many kittens. Or an equal number of each. And – here’s a twist – whoever adopts the last furry animal loses). The two kids played with each other (I watched), and while the girl discovered some strategy, it was not a complete solution, and the two seemed to enjoy it.
I broke out some wonderful dice that my games mentor gifted me. Blue dice have the numbers 5 – 10 on the sides. Red dice have 0 – 5. 2cm, wood. I gave my niece one red, and I rolled one blue, and we saw who got higher. I won two or three rounds before she called me on it. Then I gave her two red dice, and him one blue one, and they rolled against each other, sum of the red against the blue. And then I gave her five red dice, and him 2 blue dice, and they both had some quick adding to do. They played for almost fifteen minutes, and needed to be stopped. Completely engaged. (And no, not a fair game. I didn’t calculate the probability, but the expected value favored the younger child. Intentionally, to help maintain interest).
I pulled out some graph paper (1/2 inch) and some crayons. Here I didn’t involve my nephew (I asked him to draw me something), but I drew a rectangle for my niece, 4×3, horizontally oriented. I counted the boxes (12) and the lines on the outside (14). I asked her if she could make another 12 box rectangle. She copied mine. I asked her if she could make a different 12 box rectangle. She drew a 2 unit high rectangle, and counted, and closed it at 6 wide. I asked her to draw another 12 box rectangle. She drew a 6×2, but this one vertically oriented. I asked her to count lines for each rectangle – 14, 14, and 16.
My sister-in-law asked my niece if she could write a multiplication for each rectangle. Not where I was headed. But I understand that it is not obvious to non-teachers that not every encounter with mathematics needs to reach “fruition.” And it was fine, the girl knows a little bit about multiplication, so I sat back, and watched.
Next to the 4×3 she wrote 4 x 3 = 12. Next to the next 4×3 she wrote 4 x 3 = 12. Next to the 6×2 she wrote 6 x 2 = 12. Next to the vertical 6×2 she wrote 6 x 2 = 12, and started to cross it out. My sister-in-law started to speak, to interrupt the process, but I motioned to let my niece continue, and she did. And after crossing out 6 x 2 = 12, my niece wrote 2 x 6 = 12.
I was delighted. My sister-in-law was concerned. She wanted her daughter to see that 4 x 3 and 3 x 4 were the same thing. I did not. I thought the girl was in a good place, was developing a strong sense of multiplication, and would transition nicely, later. So I intervened to assuage her mother’s concerns while only denting, not exploding, her non-commutative model. I turned the paper, and let her conclude that a 4 x 3 could be a 3 x 4 if you looked at it differently. And I asked if 3×4 had the same number of boxes as 4×3. She answered without pausing. And 2×6 and 6×2? Ditto. Right, 4×3 and 3×4 in her mind were different things, with the same answer, and that’s ok.
I dragged out the rectangles challenges by asking if there was a different rectangle with 12 boxes with even more lines. She was stuck, so I drew a 1×12. She carefully counted. 26 boxes. I asked if that was the most, she was not sure, I began to draw 1/2 by something, counting half boxes with her along the way (she was good at counting by halves!), and she was certain that there were more lines. She counted anyhow. My brother, who had only watched part of this, asked if we would ever be done (with the most lines) and she articulated nicely a “keep cutting in half” approach.
Then I taught them Set (or rather, what makes a set. I turn teaching someone how to play Set into an enjoyable game itself. I’ve done this with high school and middle school students for years. The first day we never play). And then I sent them some turn-taking rules that I thought would be better for adults playing with little kids, and kids of different ages playing together. (I played these rules on Thanksgiving, 2 math teachers and a 2nd grader, fun for all).
I never wrote about going to the Math Circle 2013 summer conference a few months ago, at Notre Dame. But I believe my experiences there had some influence on this story.