I wrote a book. One major premise of the book is that changing the scale of a mathematical concept and/or exploration can help math learners make sense of the math at hand in new and helpful ways. This is a short story of how the blue tape I use to define the dancing space in Math in Your Feet brought unexpected new insights to me as both a teacher and a learner in a surprising, unexpected way.
I recently returned from Twitter Math Camp, a professional learning experience for math teachers, by math teachers. It’s tremendous. For my morning session (three days, two hours each morning) I chose Tessellation Nation (#tessnat). It was three days of open ended exploration and cross pollination of individual inquiry focused on making sense of a topic that often seems pretty straight forward but is actually NOT, in any way, shape, or form (ha!).
The first day I chose a random bag of shapes crafted by Christopher Danielson (of many fames) and see what might happen. Bryan joined me and we were completely engrossed in investigation for the entire morning. We both left for lunch inspired and looking forward to the next day.
— Joe Schwartz (@JSchwartz10a) July 16, 2016
The second day I decided to act on my wondering about how pentominoes would tile. I also decided to challenge myself to work solely on graph paper. I spent a good 30 minutes with the F. People kept coming by to see what I was doing and every time I would say: “Visualizing transformations is not my strong suit. I really want to see if I can do this on paper. I don’t want to cut out a manipulative.”
Someone convinced me (probably Max, who was spiraling some kind of shape next to me at the table by the open window) to try a different shape. I picked W. I was more successful at creating the W on graph paper, but was still really disoriented trying to tile one W with another visually.
Max, who has been my partner in crime on and off on matters of blue tape on the floor since TMC14 wondered if we might take this investigation to the lobby. We taped out the W pentomino and I immediately thought of naming the pathway using spatial terminology. It sounded like this: “Start, down, over, down, over” as pictured below.
I walked and talked the pathway a few times but my bigger goal was to “nest” the Ws, so we built a larger grid and Max took colored squares of paper to keep track of our individual pathways (I’m pink, Max is green).
I immediately felt a rush of adrenaline and ran to the graph paper to draw it out. It was like the sun had come out after a week of rain. Partly because I FINALLY UNDERSTOOD the structure of the W pentomino and how it could tile with itself, and partly because it was a spectacular example of what I know and have been writing about for the past couple years:
Changing the scale of a mathematical investigation has the potential to create new insights about previously understood and not-understood mathematics. In children AND adults.
Then Max and I tried standing on different corners of the grid to see what a double reflected or rotated W pathway might look and feel like. You can see the Start-Down-Over-Down-Over pathway with each of the colors. But it doesn’t end there.
The best part of putting blue tape on the floor is that we usually can only do it in open, common spaces which means folks walking by often get curious about what we’re doing.
— Sadie Estrella (@wahedahbug) July 17, 2016
During this time period one of the Tessellation Nation folks happened by (I’m sorry I don’t remember your name!! But I remember you!!) This person had very bravely, not half an hour before, shared that she really didn’t understand reflections. We showed her the W pathway and then, again I think it was Max, asked her if she could reflect it. She got a lot of hints about using opposite feet, and she had some productive struggle but I think you can see how elated she was when she finally GOT IT. We were all so very happy. High 5s and hugs all around!
And THAT is the tale of two different kinds of transformations that happened during the TMC16 Tessellation Nation morning session — at once both personal and mathematical. I’m feeling verklempt just thinking about it but I have enough poise to share this one final thought:
— Judy Larsen (@JudyLarsen3) July 17, 2016