The other day, I presented some students with the following game:
Last cookie is a game for two players. In this game, a number of cookies are divided between two cookie jars, and each jar has at least one cookie. Each player can take cookie in one of two ways:
- By taking any number they like from just one jar or
- By taking the same amount from both jar.
The winner is the person who takes the last cookie(s).
Some of you may recognize this as Wythoff’s game; on the NRICH website, which has a great online version of the game, it’s called Last Biscuit (I modified the name for a Canadian audience). If you aren’t familiar with the problem, I highly encourage you to give it a go – very easy to understand and play, but the optimal strategy is not particularly obvious.
I assigned this game as homeplay (get it?? because it’s a game… terrible, I know). Then, the other day, we picked it up again after about a week of rumination.
To my surprise, one of the students had come up with the optimal strategy, which (spoiler alert) revolves around “cold piles” and “hot piles” (credit to Wikipedia for the terms): when the piles are “cold” (e.g., 1 cookie and 2 cookies), the player whose turn it is to draw will lose with best play; conversely, when the piles are “hot” (e.g., 1 cookie and 3 cookies), the player whose turn it is to draw will win with best play.
I let the students play the game for a bit in pairs to test and refine their strategies; however, there seemed to be a roadblock that prevented productive exploration: because one of the students had already found the optimal strategy (and was very vocal about it), the students seemed to feel that the work was done – even though this strategy was not universally understood. Why were the pairs 1 and 2, 3 and 5, and 4 and 7 special? Because a (peer) authority said they were. Case closed.
Luckily, I had something in my back pocket. Having done some research about Wythoff’s game, I came across this article, which shows that it is isomorphic to a came called “Cornering the Queen”. (This game is played with a chessboard and the Queen piece. The Queen is initially placed in the far right column or in the top row of the chessboard, and can only move south, west, or southwest. The player who gets the queen to the lower left corner is the winner.) Not mentioning the connection, I gave the students some paper chessboards and let them play in pairs, keeping an eye out for optimal strategies.
This turned out to be far more productive – and if you compare the intermediate work of one group below to the images in the article cited above, you will see that the students were hitting the main ideas. Eventually, all of the groups found the first few “cold positions” (actually, they called them “dead zones”) and were able to make the connection between Cornering the Queen and Last Cookie. They were even able to extend the chessboard to find more dead zones. Brilliant! A happy ending! Except…
Even when everyone else had found the pairs, the student who had found them first through Last Cookie (and was therefore quickly able to apply them to Cornering the Queen) still felt that he “owned” the solution: “They’re my numbers!” he repeated – jokingly, but with a slight air of resentment. He found them first, after all. “Nobody owns numbers,” I replied politely, but this did not seem to help. I think that some of the students left feeling that they somehow had less of a claim on the solution, even though they had all worked so hard to find it.
But, in today’s day and age, don’t many teachers want students to “own” their knowledge? I think that many of us want to get away from the scenario where it is the teacher who owns the knowledge, doling it out to passive recipients who are expected to gulp it all down, no questions asked. I certainly do. So what was wrong with this situation? I’m trying to pin it down, and I think that perhaps, what we may actually be looking for is a sense of community ownership of knowledge among students in our math classrooms. This was certainly missing in the situation I described. So how do we build it?
I think I’ll need to refer back to Strength In Numbers, Ilana Horn’s excellent book on community learning, for some ideas. However, I would also love to hear what you think. Have you had similar experiences? Do you strive to foster individual or community ownership of knowledge in your math classroom? Perhaps you have a different perspective on the issue here? Feedback would be very much appreciated. (In other words: help!)