Math Circle 12.9.2014


(December 9, 2014)  We began our final session in this course with a few rounds of The Takeaway Game.  My goal was to focus attention for deep mathematical thinking, and to plant seeds of a solution strategy for The Very Clever Prince.  (We never did get time to return to the logic puzzle about the Prince; some of these kids have been searching for a mathematical/logical solution to this puzzle since last spring.)

Anyway, here’s how the game works:  You place a bunch of assorted objects on a tray, and identify them with the group.  Then the kids hide their eyes and you put a towel over the tray.  Surreptitiously remove an object.  The kids open their eyes and try to figure out which object is missing.  (See PHOTO to know which objects we used.)  The kids had a great time, could have spent all afternoon doing this, and after 10 minutes were quite attentive and calm.  They were also armed with the strategy of discerning a fact about the unseen based upon the seen.



It was a few days after Thanksgiving.  I got out my puppets and said to the kids,

“Isn’t one of the hardest things about Thanksgiving how you have to wait to eat until everyone has been served all of their food?  […nods of understanding…]  It’s hard, but many families do practice this mannerly custom.  Do any of you know anything about the manners of goats?  Do you know what they eat?  […brief discussion…]  Did you know that my puppets here are actors?  […assistants J and L, well acquainted with my puppets, chorus that fact with me…]  Today the puppets will be playing goats.  I would like to teach the goats some manners.  So, when I pass out the food, none may eat until everyone has been fully served their whole meal.  […each student chooses a role as either a puppet/actor/goat, or goat feeder…]”

The other goat feeders (L, J, and R) and I now began feeding the goats paper.  I continued talking.

“I’m tearing this piece of paper in half.  I’m putting half of it under my arm.  The other half I’ll divide equally among all of you.  […we feeders tore paper and distributed accordingly; no goats ate it…]  Is there any more food left?  […removed piece from under arm…]  Now I’ll take this piece, tear it in half, put one piece under my arm, and split the other half up so that all of you can have an equal share.  […did so, still no eating…]  Is there any more food left?”

This process of tearing and sharing went on for a while, until each goat had a pile of paper slips, and I had a very small piece left.  I asked whether the goats were allowed to eat yet.  No, they weren’t.

“When will they be allowed to eat?”

“Never,” said M.  “The pieces just get smaller and smaller.”  Not everyone agreed with her.  Some thought that I was on the verge of needing to give my last tiny piece to a goat – even though I was trying to give only half my stash to the goats each time, the piece would be too small to split.  We discussed hypothetical situations of splitting using precision scientific instruments.  A few more kids came around to M’s idea that the goats would never get to eat.  Not everyone agreed.  I didn’t push it any further, since the goal in this course has been to plant seeds of mathematical thinking.  Process over product.  IMHO it’s fine to not reach mathematical consensus sometime, and to not reach a solution within a given time limit.  I’ve seen time and time again that when the problem is interesting, kids just naturally mull over it on their own long after courses end.

Teaching Goats Manners, BTW, is a game I decided to play because the novelty of playing “Math Red Light Green Light” (RLGL)* had worn off for a few kids.  The goat game is isomorphic to the RLGL game, i.e. it mimics the structure/properties/relationships/sets of the original game.  Isomorphs are useful in that they can make a concept more accessible.  That sure did happen here.  Interestingly, I thought that RLGL would be more accessible because it makes the point with embodied mathematics.  I wonder what it was about the puppet game that made it more powerful: the narrative/drama?  …the use of physical props?  …the fact that we did it second, so the seeds were already planted? … something else?

Also interestingly, no one realized, at least no one mentioned, that the goat game is isomorphic to Math Red Light Green Light.  I have seen kids recognize isomorphism, although maybe not kids quite this young.  If you do decide to continue any of these games at home, please let me know if the insight does arise.

BTW we’ll be exploring isomorphism explicitly in a circle for 9-11 year olds in the spring (River Crossing Problems course beginning 4/21).  Also, we have a course for 7-8 year olds starting 2/24.



Next, with time running out, we picked up where we left off in the reading aloud of The Cat in Numberland.  The kids were pretty into chapter 4.  Then, we still had one chapter to go – chapter 5, which gets into fractions.  I was worried that the content would be so over these kids’ heads that the strength of the narrative wouldn’t be enough to hold their interest.  So I took an aside from reading.

I asked a question that’s a favorite of Bob and Ellen Kaplan’s:  Are there numbers between numbers?  We got into an interesting discussion because at first everyone said no, and then most said yes when I asked them how old they are.  The idea of half (e.g. age 5 ½) was clear.  But then – and here’s where the age range in the group really became apparent – only about half the class could get beyond that.  Some talked about ¾;  M even posited that there is an infinite number of numbers between numbers. But the younger kids were only able to grasp the ideas of whole numbers and halves.  Everyone demanded more of the book, so against my better judgment I continued reading.



As I read, interest waned.  All the kids tried to grasp onto the plot, but just couldn’t, since it was dependent upon advanced ideas such as numerator, denominator, and division.  And we were out of time.  Parents were waiting in the other room.  I gave kids a choice:  go to their parents, or stay for a few more minutes in hopes we could finish the book.  Very soon, the group dwindled down to only 2 students.  Then only M was left.  She really wanted to finish it.

“Let’s wait till next year,” she said, when she realized that she was getting a private reading.

“I could lend you the book,” I offered.

“No, that’s okay, I can wait.  Let’s finish it in the group next year.”  She was in last year’s math circle when we never finished The Very Clever Prince.  She knows that deep mathematical thinking takes time.  Ideas need to percolate.  And that’s okay.


After class, in a You-Blockhead-Charlie-Brown moment, I realized that I had never explicitly told the kids that throughout this course, they’ve been alpha testing/doing R&D on a new game.  Oh, I may have said once or twice that “I am inventing this game as we go along, and you guys are helping me find its faults and improve it,” but I never really explained the gravitas/significance of this.

What I wish I had said:  “You are doing something that some grownups do at their jobs…”

What you can say to your kids at home to jump on this teachable moment:  “I got an email from Rodi, and she is so grateful to you for helping to alpha test that new game.”

How I imagine your kids will respond:  What new game?

Once you remind them of “Math Red Light Green Light,” the conversation might go anywhere.



I seem to be able to connect any concept to math.  Many Math Circle kids have joshed/ribbed me about this over the years.  Here I go again.  On the Buddha Quote of the Day app, I saw a quote that applies to many things, including quite possibly (at least in my math-centric brain) math education:

“We are shaped by our thoughts; we become what we think.  When the mind is pure, joy follows like a shadow that never leaves.”

Did you see my advertisement for this course on the listservs?***  The little marketing blurb I wrote included this statement:

The reason that we offer a course for children so young is that it’s around this age when many kids make the shift from appreciating mathematics from a deep conceptual place to seeing math as a set of instructions to be followed.  Preschool-age children find natural joy in trying to understand the underlying structure of the universe.  Once kids hit kindergarten, though, somehow many get the idea that math is something to be “performed.”  Or even worse, “memorized.”  (Ask a child of this age, “Why does the number 6 come after 5?”, and you may get a surprising answer.)  Even kids who homeschool or whose schools use a more progressive approach to mathematics often develop this bias around this age.  One goal of this math circle is to prevent, stop, or at least slow this shift – to retain the joy, and to know that mathematics is more than just arithmetic.

This blurb relates to the quote in terms of our self-talk about math.  At some point, many kids (especially girls) start telling themselves “I’m not good at math” or “I don’t enjoy math” or something like this.  While I’m not going to directly coach kids this young on self-talk, I do think it’s important for us teachers/parents to notice and acknowledge it when we hear it.  I also have hope that joyful early math experiences can help kids arm themselves with positive self-talk (“I love math!”  “I enjoy math!”) so that when they delve into concepts that might require some stricter self-discipline to master, kids are motivated by an awareness of how these new concepts underlie or support what they’ve already enjoyed.


Back in the summer, we posted this description of the course for ages 11 and up that begins January 6:

In this math-meets-art circle, students will experiment with the four types of symmetry in a plane to create their own tessellations (tilings).  We’ll look at the work of MC Escher and that of the mathematician whose work inspired Escher, George Polya.  We’ll draw and draw and draw.  We’ll also attempt to determine which regular polygons can tessellate a plane, and then verify our answer with proof. 

As I’ve been preparing for it during the fall, I’ve added into the plan an exploration of symmetry inspired by James Tanton’s Pamphlet on Symmetry.  Expect this course to emphasize depth in mathematical thinking over breadth, to require rigorous conceptual analysis, but not to require any experience with algebra or geometry.

With gratitude to all of you for allowing your kids to go on this journey with me, and for reading and thoughtfully considering these reports,


PS  Two books in particular acted behind the scenes in this course, inspiring the activities:  Infinity and Me, by Kate Hosford, and Thinking Mathematics 6:  Calculus by James Tanton.  Thanks to the authors for writing these.



A few of the kids were VERY interested in continuing this game.  If you would like to do some more at home, I’d suggest each person doing their own, on construction paper, choosing their fractional amount to advance on each turn.  I would have done it this way if we only had one more week.  We still need suggestions for a better name for this game.  Email me if your kids have suggestions.

Also, for more information for yourself about infinity, here are a few interesting links: (informal and not scholarly but a fun short discussion) (I like this site because I can choose to focus on the comments that are at my own personal level of understanding, and also have my eyes opened by concepts I never even knew existed.) (both cool sites about the infinity symbol – the lemniscate, which surprisingly rhymes with “biscuit”)



“Why,” you may be wondering, “does Rodi need 2 assistants in a class with only 7 students?”  Or, you may have guessed that I don’t really need two assistants, or one, or any.  The point is that the assistants give me a chance to spread the Math Circle gospel.  You guessed it; the assistants deepen their mathematical thinking just as the students do.  Even though I forbid the assistants to make conjectures or ask mathematical questions during the class sessions, they always end up asking me a lot of math questions before and after class.  In this course, both assistants (J and L) whispered conjectures about infinity to each other during class sometimes, and then after class J posited those conjectures directly to me in private.  If you have or know a child who would be interested in a Math Circle internship, email me at


We typically market Math Circle courses in 3 ways:

(1) Post course description and details on the Talking Stick website

(2) Email the rationale for the course (coupled with a link to website) on area parenting and homeschooling listservs

(3)  Blog about the course by posting these parents’ reports on the website, on the Math Circle community listservs, and to my email distribution list

We’ve also experimented with flyers and postcards.  We’d like to do more, though.  We do not always fill the courses and would love a few more students each time.   Do you have marketing ideas?  If so, let us know.  Also, we’re looking for a volunteer or intern to help spread the word in the school math teacher community.

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