(January 20, 2013) “What’s this for?” asked the kids as I handed everyone a piece of triangular graph paper. I explained that you can make fancy tessellations on this kind of paper.
“What’s wrong with squares?” asked A, referring to traditional graph paper. I then handed out square grid paper for anyone who wanted to try it on squares. M commented that it’s hard to tessellate shapes because when you copy the basic shape, if your copying is not perfect and you repeat it a bunch of times, the design is off. The kids wondered how Escher got his so perfect. Was it freehand? Did he use measuring aids?
“You’d have to have a computer to do it perfectly,” said someone. Others nodded and vocalized assent.
“Computers weren’t around in Escher’s time,” countered someone else. Most of the kids then concluded that he must have done it freehand. I was reminded of Vermeer, whose paintings were so accurate that they looked almost photographic. Vermeer was accused of “cheating” by using some sort of measuring/tracing apparatus. Some people still wonder about this. I relayed this anecdote to the kids. Then I asked,
“If Escher did use some kind mechanical aid, would you consider that cheating?” The consensus was an adamant no. I posited that in our age, with computers and all, that we’re more comfortable with technological help. I think the kids agreed.
Then I explained that I had been attempting Esher-esque tessellations at home, with problematic results: my shapes, as M predicted, were not matching up well. I hoped that if we all did tessellations together here, maybe the kids could help me improve mine. I showed them my flawed attempts, then started a new design on the board.
First I drew a triangular grid. Next I isolated just one triangle in the grid by tracing over it with a different color marker. Then I isolated a single side of that triangle and draw a curvy line over it with yet another color. (I quoted a math teacher from youtube1 who refers to this step as “funkifying” the side – not a mathematical term, but an effective one – everyone knew what I meant.) I funkified the other two sides of that triangle, and ta-da, had my shape that I was going to tessellate. I copied that shape to the right of the original, and then above these two, creating something in the middle. That something in the middle resembled an upside-down version of the original shape. (SEE PHOTOS) Then came the question that served to fill an hour of our time: “Is the shape in the middle the same as the other three?”
A big debate ensued. For clarification, I called the original shape “outie” because it seemed to have what resembled icicles outside of a trianglular window. I called the resulting shape in the middle “innie” because it seemed to have icicles inside a triangular window. (I wish I had explicitly mentioned the strategy of understanding concepts by relating them to real-world familiar things.) The kids all thought that innie and outie were the same. Identical. Rotations, for sure, but still the same shape. Many came up to the board to demonstrate or prove their point.
I was confused. I wasn’t seeing their point. Innie and outie looked different to me. The kids were trying to convince me that the differences were in my eyes, a matter of perception, like Escher’s birds/fish of last week where the conclusion was that the birds and fish were identical shapes. More kids came up to the board. I still didn’t understand/buy into their conjecture. Finally G accused me of purposely being dense in my perceptions in order to develop better mathematical communication skills in the students.
“Do all of you think that?” I asked the group? Yep, they all did. “Well, I do think that it is part of my task here to help you improve your explanatory skills, but I’m really not doing that. I really do disagree with you. I think the shapes are different.”
“Prove it,” demanded M, handing me a marker. 2
A Seeming Tangent
Now, while all of this debate was going on, many of the kids were working on their own tessellations at the same time. A had been trying to get my attention and a turn at the board for quite some time to posit her own question/conjecture. So we put the innie/outie question aside for a bit and examined her tessellation issue. She made a green triangular pattern on the board (see photos), with each triangular cell having funkified sides, a dot in its middle, and a curved line connecting the dot to one side. She explained that on her paper, when she continued the design inward, her drawing filled the space so that there was no negative space. Was this still a tessellation? Was it a fractal? Was it something with predictable structure, or something random?
This discussion gave rise to a question that has come up again and again here: can every shape be tessellated? Several students came up to the board to present their thoughts on this. Then we revisited innie and outie. It turns out that A’s question enabled the group to look at our scenario with new eyes.
Things Get Emotional
Before I had a chance to make a case for my conjecture, a few more kids came to the board with justifications for their position. At one point, in the midst of an explanation in front of the group, R sat down, saying, “Wait a minute.” Someone else went up to the board, then R returned. “I needed a minute to gather my thoughts,” she explained. She continued her presentation as a few students commented on my conjecture.
“How would you demonstrate that they’re different?” asked someone to me.
“I would do a rotation of one of the outie cells and compare it to an innie cell,” I explained somewhat vaguely. R was listening as she was putting things on the board, and wanted to try what I said. She hadn’t been to our other sessions, though, so wasn’t sure how to rotate a cell 180⁰. Her attempts didn’t end up proving one conjecture or the other. She sat down, doubt creeping into her mind.
Doubt, I think, was creeping into everyone’s mind. No students were at the board, so I used this time to demonstrate why I thought that the 2 shapes were not identical. Everyone was silent. Faces fell. A few kids were visibly shaken.
“Math can be a really emotional subject,” I told them. “Some people think that it is really dry, that you just add your numbers or whatever and know clearly whether you’re right or wrong. It’s not really like that at all. Most of the time, you don’t know whether you’re on a productive path or not. It feels really bad when you are attached to a conjecture, and then it’s disproven. It’s important to acknowledge your own feelings to yourself when this happens. It happens to me, it happens to you, it happens to everyone.” (The words might not be exact here, but that’s the general idea.)
“We would have seen that the shapes were the safe if you had…. Oh, never mind,” said someone, still clearly put out.
“No, don’t stop. Continue, please. How could I have presented this topic so that you all could have had an easier time of it? There’s always a better way.”
“You could have made your diagram larger. When you copy the shapes bigger, it’s easier to see.” At this, I moved to the other side of the room and looked at the board. It was harder to tell that the shapes were different when viewed from far away.
“You’re right. Thank you for telling me.”
The Tension Eases
We then moved on to other topics. The kids all showed the tessellations and other interesting mathematical art they had created. No one did end up using the grid paper to make square-based tessellations. I took pictures (see photos!). I started to demonstrate the tessellation project involving the “funkifying” of sides. We talked a little more about Escher. Then time was up, and everyone went home.
PS I found this game online, and thought it was kinda fun: http://www.sciencekids.co.nz/gamesactivities/math/transformation.html
1 Here’s a link to his video: https://www.youtube.com/watch?v=212XC1zfxXY. This activity will be harder than I thought, since many kids did not have protractors. And… no one knew how to use one, but that’s okay – that was kinda gonna the point of the activity as far as I was concerned.
2 We had a visitor this week – a neophyte math circle leader who hoped to hone her skills by observing other circles. All this relatively comfortable back and forth between me and the students brought to mind an analogy that Bob Kaplan made between math circles and Quaker Meetings: he said that math circles need a lot of time to mature so that everyone feels comfortable enjoying the freedom and creativity and risk-taking that mathematics allows, that a mature math circle is like a mature Quaker Meeting. Even though we have a few kids in the group who have done few or no prior math circles, many of these kids have been doing this with me for years. People are really comfortable testing limits, questioning authority, and taking risks. It makes me wonder how things would have gone had we discussed this topic on day one years ago?