The Chessboard

I introduced the Polya Problem solving steps to the class and we talked about how we would implement such a framework.  As a class, we recollected our golf ball math problem, and applied these steps to the learning that took place during that problem.  Once we had a good idea of how it could help us, I introduced a new problem for the day, and asked them to keep this framework in the back of their mind while they worked through it.  At this point in our classroom growth, this is not something that I am going to force on them, but something that I will bring up and talk about after and before we embark on  complex math puzzles.

Today, we had a great problem that I picked up in The Elephant in the Classroom, by Jo Boaler.  The problem is simple, and the introduction is easy.

How many squares on a chessboard?

At first glance, this is easy.  64, you say!  Well, what is a square?  The board is a square.  2x2 squares.  3x3 squares.  And so on.  Also, they overlap!  This simple counting activity just got complex.  To solve it, you need some serious critical thinking, most importantly (and this is why I love it), organization.

Students were making tables, using unix cubes to plot out the center points of each square.  Finding patterns in their tables and making assumptions about what would happen next.  All in all, it was great learning, and a great problem that had they absorbed for almost an entire hour. I actually had to psychically carry one boy out of the classroom for lunch.  He didn't want to give it up!

The power of good problems, problems that are interesting and seemingly simple, is something that cannot be under-estimated in a math class.  This problem seemed so easy, but was so difficult because of the complexity.  This follows the video game model of learning that I read somewhere but cannot recall where.  Kids love video games because they are simple, yet they get incrementally harder as you progress.  Designing math activities that are similar to this model of understanding is something that I intend to play with over this year.

At the end of the problem, we went back to the problem solving framework and reflected on how it helped (or didn't), and how we used it to solve the problem (or didn't use it).  All in all, the framework was successful and got the kids thinking about the how and why of solving problems, but this will need to be something that is done and refereed to very regularly in order for it to become part of the classroom understanding.


  1. Lowell Cross5.11.12

    Your chessboard near the top of this page needs to be rotated 90º (either clockwise or counterclockwise will do) -- a bit unfortunate considering your topic.

  2. Thanks! I do appreciate accuracy! Fortunately for us, that detail has no impact on the mathematics behind the problem. Thanks for your well trained eye...


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