The Talking Stick Blog

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ESCHER #6: Exploration

Math circle 2.24.2014

(February 24, 2015)  Our last session.  One parent asked - as everyone was walking out at the end - “Did you come to a conclusion in your math circle?”  The answer was a resounding no.  This course was more of an exploration.  We often begin a math circle course with an overarching question.  This time, we had nothing other than “What’s up with Escher?”


I did, of course, want to have some sort of closure/conclusion to the course.  My Inner Math Teacher (who is not the same person as my Inner Wise Teacher, to borrow a phrase from my daughter’s yoga teacher) screamed out “summarize summarize summarize!”  I listened to that IMT at the start of today’s class and tried to elicit a summary of symmetry types from the group.  No one was into that; it was like pulling teeth.  (The IWT was being quiet at this point.)  Eventually it got a bit interesting when the kids reverted to categorizing letters of the alphabet.  Students were tentative about classification, however, after last week’s big can of worms over whether translation was a form of symmetry, and whether you’re allowed to declare that, in your own math world, it’s not at form of symmetry.  In retrospect, I think the kids were unsure of whether we were in Conventional Math Land, or Invent-Your-Own-Math Land.

This ABC symmetry exercise had come from James Tanton’s pamphlet “The Mathematics of Symmetry.”  The kids had great fun with it in our first session nearly 2 months ago.   I reminded the group today about how we had argued about whether N was symmetrical.  X, who had been absent that day, asked excitedly, “Were you yelling?” The kids assured her that it had been a civil debate.

“But do mathematicians ever yell at each other in debate?” I couldn’t help asking.

“Oh yes,” said X, “and sometimes they undermine each other’s work!”  I reminded the group that within every profession, it’s just a bunch of humans, most of whom are kind and decent, but of course not everyone is.

Now my IMT told me to keep going with Tanton’s pamphlet, since nothing I have ever done in math circle that was written by Tanton has ever fallen flat.  I decided to use his example of zooming in on a map on a smart phone:  what is happening here mathematically, and is this thing symmetry?  After making some big-time assumptions (such as the phone itself magically enlarging so the entire original image is preserved), students came up with the idea of dilation.  Students generally felt that dilation was not symmetry because the size changed, but still people were very tentative in conjectures.  (Note:  several students were absent.  Those who were absent are pretty vocal mathematical risk takers.  Would the discussion have played out differently with the entire group present?  And if so, who is doing most of the talking in class?  Time to pay more attention to that!  A big goal of math circle is to foster contributions, particularly risky ones, by everyone.)

Next I passed out frieze patterns copied from the pamphlet.  This definitely would have fascinated kids on Day One, but now, again, they weren’t biting.  We talked a bit more about types of symmetry and what types of it they could find in the patterns in the handouts.  I had hoped to get into kids interested in making their own frieze patterns, with challenges issued by me (“Can you make one that has rotation without reflection?” etc etc etc.)  Surprise surprise, no one seemed interested.

I wonder whether the kids in this group might have been feeling a bit sad that today was their last math circle.  Looking back, I recall other final sessions in which energy and enthusiasm were low.  Especially in courses that were more exploratory, versus those with a big question.  I wonder:  is it better to have a natural conclusion versus a petering out?  I don’t know.  It’s not that kids were non-participatory. Some were coming up to the board on their own to demonstrate their conjectures, and others were using the manipulatives* to design tessellations and the like.  But still, the energy today was different.  Something to ponder.

Anyway, it was clearly time for something radically different.



I’ve been wanting to do symmetry kinesthetically during this course, but have been racking my brain for how to do it in a non-didactic way.  Finally today, about 10 minutes before class, I realized that I could just say something open-ended.  So in class, I asked the kids to stand up in a row, facing me.  (“Oh no,” someone groaned.)  “I’ve heard,”  I mentioned, “that you can teach little kids about symmetry just by using your bodies.  Maybe you guys could help me out here, since I lead math circles for little kids too.  How might you do that?”

“Like this,” said J.  She held out her hands, and put them together, palm facing palm.

“Or like this,” said L.  He held his hands up to face me, gesturing for me to place my palms against his.  We discussed what was different about J’s and L’s ideas, and also whether your hands are really symmetrical at all.  The kids knew that (1) the sizes etc. are actually a bit different, and that (2) the hands are chiral – they are alike but oriented oppositely.  (I did not use that fancy science word with them, but probably should have.)

“I’ve also heard that you can do this with your feet.  How about you each stand on one foot and see if we can figure out how that might work?”  They each stood on one foot.  I said things like, “How would you demonstrate rotation?” etc etc etc.  Yawn yawn yawn.  The kids did the tasks, sometimes in unexpected ways, but boy was this boring for them and for me.  “Let’s do this by playing Simon Says,” I suggested, not sure how that would go with tweens and young teens.  Fortunately, it was a vast improvement.  Now they had a reason to do the task.  Some collaboration began as students looked at each other’s feet to brainstorm.

“Simon says do a translation!... Simon says do a rotation!...  Now do another rotation!... I didn’t say ‘Simon says’!  Simon says do a rotation!  Now Simon says do both a translation and a reflection simultaneously….”  I listed on the board the moves the kids did to demonstrate each symmetry type.  (see photos for list) The game ended when X directed “Put your hands on your head!” and most of us did.

Then some of the kids wanted to go out in the snow and do footprint symmetries in the snow, but not everyone was dressed for the weather, so I suggested doing it at home.



Finally, my Inner Wise Teacher spoke up.   The IWT said that the closure/conclusion we really needed in this course was to do more about Escher.  When I originally planned this course, I pictured us doing a good amount of art, and discussing this artist.  Last year, I co-taught a math circle and henna body art course with Gina Gruenberg.  In that course, often the kids seemed more interested in doing the art than debating over the geometry.  Therefore I expected that the kids would want to spend a lot of time on the art projects I presented.  But other than paper and pencil tessellations, the kids didn’t really latch on to any of them.  Of course, I’m not Gina.  I’ve done more than my fair share of art in my life, but still, I’m a math person.  With me teaching alone now, the kids latched on to the math in this new math/art course.  We needed to get back to the art.

I told the kids that for our last half hour, they could do whatever they wanted with their hands (draw,  build, etc) while I read more of The Life and Works of Escher.  I mentioned this to you before, but I just can’t say enough about how well this book is done.  Miranda Fellows tells the story of Escher’s life through his works, with a brief paragraph of biographical info next to each work displayed.  Fellows also points out a few of the mathematical concepts in the works.  Here are a few highlights of the conversation triggered by these works and anecdotes:

  • Hand With Reflecting Globe: What did Escher mean about the ego when he said “The ego is the unshakable core of his world?”  (p13, Fellows)  “Your view is the center of the world to you,” posited someone.  Was he talking about his own view of the world?  Other people’s views of the world?  Everyone’s view of the world?
  • Still Life with a Street: Why, at a certain point in his career, did he “feel compelled to withdraw from the direct and more-or-less true-to-life illustrating of my surroundings?”  (p21, Fellows)  X was adamant that you need to stay grounded in reality to even use your imagination.
  • Fish (1941 woodcut in three colors): Why did he use fish so often in his explorations of infinity?  (p34, Fellows) (L drew and help up for everyone to see his quick sketch of “The Infinity Fish” – see photo gallery from class)
  • Eye: Was Escher part of the Illuminati? I was unfamiliar with this term, so A explained it.  I told the class that all of my reading on Escher gave me the impression that he was adamantly against people reading mystical intent into his work, that his attitude was more like “hey I’m just an artist who likes to play with math.”  Fellows (p46) explains the purpose of this work as an experimentation with a certain printing process and with using a concave shaving mirror – sounds like math and art to me.  The more of his works I showed from the latter part of Escher’s career, though, the more the kids thought that there were deeper messages.  Or at least they saw why other people would think that.  There were definitely messages in Escher’s works (such as anti-fascism, as we discussed in class), but the messages were not mystical in any way, as far as I knew.

We were almost out of time, and I wasn’t even halfway through the book.  I decided to show the students images of Escher’s most famous works, his most representative works, and some from later in his career.  And of course, we discussed his death, which was from illness in a hospital.  The students found this interesting since they posited that society seems to expect people like Escher to die dramatic deaths.

Finally, I sent kids home, suggesting that next time they’re bored, to get out that pencil and paper and tessellate!

Thanks for sending your kids to me for this course.  It would help me with future planning if you could email me some feedback from your kids on how the course was for them.  Let me know what they and you thought went well, and not so well.

Best to all of you,


*polydrons and pentominoes

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