The Locker Problem and Why the why is important

Today I had one of my classes solve the locker problem.  The basic premise of this problem is that a group of students is running up and down a hall in the school opening and shutting one hundred locker doors.  The first student opens all the lockers, the second closes every other one, the third opens or closes every third one and so on.  The question that then comes up is which lockers will be open after all one hundred kids get done.

A couple years ago, I was struck by the idea of making the problem more three dimensional instead of just a story on paper.  (Thanks to reading Dan Meyer of course.)  I also thought this might lead to the students generating the question instead of me putting it out there.  So I created this video and Google site.  I showed the video today as the kickoff to the problem and for the first time ever I had a student basically solve the problem out loud at the end of the video.  He looked at my video ending showing lockers 1, 4, and 9 open and said well all the square numbers are going to be open!  Now this is one of the better students I have ever had and fortunately most of the other kids were more interested in seeing what exactly was going on instead of listening to him.  More importantly, I was able to catch him from blurting out the more important "why"were only square numbered lockers open.  While the other students got to work on figuring out what lockers would be open, I pulled him aside and asked him why he thought it was square numbers, which he quickly explained to me (again not surprising, he's a good thinker) and so I set him off to another task and told him not to help anyone else. 

The rest of the kids were working in groups and many came up and told me square numbers would be the lockers that were open.  At this point, I was wondering if I would have to come up with something else to do with the remaining class time.  In year's past, kids took at least the whole period to get even to this point.

So as I often do when we are problem solving, I replied that their answer was great but why? And here is where the rest of the kids got stumped.  For the most part, they spent the remaining class period trying to figure out the "why" of the problem.  They watched the video time and again, they went back to their notebook items and made great connections (such as the people opening and shutting lockers were representing factors, the lockers were representing multiples.)  Still they had a lot of trouble with that question "why."  I even had one student tell me that I wasn't allowed to ask "why" for the rest of the week.  (I was most proud of that, apparently I've been doing my job and actually making her think!)

In the end, I had to give some hints and talk some parts through as they watched the video, but most of them were able to explain the "why" by the end of the period.  Tomorrow we will see how they did on their "Locker Problem Home Enjoyment."  I wanted to know some more "whys!"