I walked into a neat building in downtown Holland, MI (the kids were at a science camp at Hope College) that now houses a 5/3 bank. The facade is beautiful, but I'd never been inside. It was just as beautiful. One of the tellers was talking about how they just filmed a scene at this bank for Ed Harris and Jennifer Connelly's coming movie "What's Wrong with Virginia?". The ceiling has a fantastic pattern (idealized above) with regular hexagons and rhombi. One of the things I've promised myself I'll think about some day is tesselations of more than one tile, and how to do Escher tesselations.
I sat down and started sketching the pattern trying to think about alterations. These really boil down to what's the fundamental region and how are you going to move the region about the plane (or what subgroup the tessellation represents if you're more algebraic). The fundamental region needs to be at least a hexagon and a rhomb.
That led to noticing the obvious translation tessellation, but there's also a rotation tessellation possible. If you do two rotations and a translation, it becomes really an altered quadrilateral. (Or, what I actually notice from making this picture is that it's a pentagon with an additional rotation on the side between R1 and T2!) I also noticed that two of them made a hexagon tiling. So I thought about making more interesting tilings by dividing up standard tilings into congruent pieces. Which got me thinking about which shapes can be divided into congruent pieces. That got me thinking about rep-tiling alterations which is another thing I've promised myself I'll think about someday. I really got lost in the ceiling for 15-20 min when all I needed to do was make a deposit.
Why don't students play with mathematical objects?
At this point, take 5 minutes to read this Jonah Lehrer column at science blogs about video games and our interactions with them.
Video games used to appeal to a limited audience, like math. More people than in math, because it's easier to get engaged. The people who got into them could get really into them, like math. But the Wii has changed things. A much broader audience of people enjoy playing it. Why? Lehrer relates some of the neuroscience, which actually goes all the way back to William James. (Sometimes I feel like if we had just taken a century to figure out why he, Dewey and Piaget were really saying, things would be better now.) Someone at Emory's Education Division has assembled some great William James info. In particular on interest and his talks to teachers (pdf). He thought a lot about engagement and connections.
"The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal." - William James (available on a mug)
I think that games help some people see the playful nature of math, but are still not broadly accessible. Is physicality the key? How do we broaden the physicality of math? This is probably a multiple intelligences question. We played this Prism Power Game, and I shared with the preservice teachers how I had considered having them draw isometric views or just keep data for the game, instead of using blocks. They unanimously thought it was obvious that the blocks made it more fun. (I agreed.)
I'm very curious what other teachers do that relates to this idea of physicality, or if you think it is important, too. Please share your thoughts!
PS> the game!
Prism Power Game
PPS> Math at my university actually started in William James College. Those were the days!
PPPS> Had to make a geogebra sketch of this pentagon tiling. It's pretty cool and flexible. Be fun to Escherize.