Have you ever had the following mathematical experience? You persevered through a math problem, found a solution, saw the related mathematics in a completely new way, and your brain felt “So good! So good! I got you! Whoa!” That type of experience is what I am referring to when I talk about the James Brown brain.
Unfortunately, the first time I can remember experiencing my brain feeling “so good” from solving a math problem occurred after I had already been teaching middle school math for eight years. Sadly, I now realize that if my brain was not experiencing any wonderful feel-good mathematical moments, I am confident my students’ brains were unaware of how great a math problem could be.
The math problem that changed my attitude towards the teaching of and learning about mathematics was so monumental that I simply refer to it as THE Square Tile problem. I’ve seen this problem in a variety of forms, but below is a version that is similar to the one I encountered during a professional development opportunity in the summer of 1990. (Were you even born yet?)
The presenter posed the following task:
Create the shape below using square tiles. What’s the area of this shape? What’s the perimeter of this shape?
After counting square tiles and edges (several times), I was satisfied that the area was 8 in2 and the perimeter was 18 in. Not entirely interesting to solve, but it was fun trying to build the irregular shape.
The next task posed by the presenter was this: “If possible, add more square tiles to the area without changing the perimeter.”
The phrase “if possible” did not even motivate me to pick up another square tile. I knew that if I the area increased, the perimeter would obviously increase. After all, I had been teaching students for the last eight years about how to calculate area and perimeter. What else was there to know?
I sat there with my arms across my chest waiting for the other participants to join me on realizing the obvious conclusion. After listening to the sounds of square tiles moving around on tabletops, I finally heard one of the teachers say, “I found a way to add one more square tile without changing the perimeter.” Then someone else said, “I can increase the area by 2 and still have a perimeter of 18.”
What the heck? Now they had my attention. My arms were off my chest, and my hands were back on the square tiles. How could the area increase while the perimeter stayed the same? I placed a blue square on the irregular shape (see figure below) knowing it would change the area but wondering how it would change the perimeter.
While I was adding more blue square tiles to the shape, the presenter posed one more question by asking, “What’s the maximum number of square tiles you can add to the area of the original shape without changing the perimeter?”
It was a great moment for my brain when I figured out how and why you could add square tiles without changing the perimeter. But the real James Brown so-good mathematical moment came when I finally saw the maximum number of square tiles that could be added without changing the perimeter….I didn’t even see that solution coming.
Not wanting to rob any reader from experiencing their own James Brown brain moment, I am not revealing any of the answers to this problem. Get out your square tiles, and let your brain enjoy the pursuit.
How about you? Tell us about a math problem you have solved that made your brain feel “So good! So good! I got you! Whoa!”