"The Locker Problem"

 I first became familiar with the so-called locker problem in the first year of my undergraduate studies. I had the opportunity to revisit it in my final year, this time from the perspective of a teacher professional development day that I assisted with. The leader of the workshop was Richard Hoshino, of whom I am an undying fan. He used a pack of playing cards to simulate the first twelve lockers, which I think is a good idea for two pedagogical reasons. First, it allows the problem to be visualized and even walked through by doing the exercise of turning cards. Second, it suggests to students the value of taking a toy version of a large problem. Even just in this small decision, there's a lot to be learned.

Beyond that, he actively engaged the group of teachers by assigning them a number. We walked through the problem, each flipping our relevant cards. At the end, we could all clearly see that the square numbers (1,4,9) were closed. In this case, that was enough for us all to observe the pattern. However, we still hadn't actually solved the problem to a relational learning standard.

To push us further, we looked at the specific histories of a few lockers. Who opened them? Who closed them? We then observed that we would of course touch a locker if our student number was a factor of the locker number. Again, this was a group of teachers, so that observation came easily, but I can imagine that it might require some teasing out in a classroom setting. 

As a final step to reach an understanding of why the square number lockers were closed, we once again did an activity to involve everyone. We picked an open locker and everyone who had touched it raised their hand. Then, since we knew that we were factors, we paired up with the other part of our factor pair. Naturally, for a non-square, everyone had a partner. When we did a square number, somebody was left all by themselves. Why was that, we asked? Of course, it was because the other part of the factor pair was them! They were, as Emma Watson would put it, self-partnered.

Witnessing all of this engagement and discussion on a problem that could easily have turned into a dozen people sitting at notebooks working individually was very inspiring. It's something I'd be interested to try in a classroom myself.