Everything has something to do with math.

“What do pick-up sticks have to do with math?” asked one of the kids.  We were playing this attention-focusing game as people arrived for Math Circle this week.

“Everything has something to do with math,” I posited.

“What does a butterfly have to do with math?” countered M.  Once everyone had arrived and was seated, I threw these questions back at the group.  M and A both mentioned that you could keep score in pick-up sticks.  I agreed, and then asked the kids how they had to use their minds in that game.
“You have to think very hard.”

“And how could that help with math?”

That question elicited the idea that pick-up sticks could help focus your mind so that you can give the math the attention that it needs.  But what about the butterfly?  What does that have to do with math?

A said that you can measure how fast its wings are flapping.  I agreed, and attempted to draw a butterfly on the board.  I asked if its shape made them think of anything mathematical.

“A dog bone!” said someone.  (I’ll have to work on my drawing skills.)

“Oh, I know,” called out V excitedly, “it’s symmetrical!”  (The others agreed.)

Then we returned to our sorcerer story.  P had a question.  Before asking his question, he pointed out that when he asks a question in math circle, the discussion about the answer often takes 15 minutes, delaying the progress of the story.  Can’t this happen faster, with a yes/no answer?  “Let’s just get on with it!”

We talked about how Math Circles are for deeply exploring a question – how interesting math questions usually don’t just have a yes/no answer.  Then he asked his math question, and we happily spent way more than a minute answering it.  I’m always glad when the topic of pedagogy comes up with students.  It’s empowering for them to know what instructional methods are being used with (hopefully not “on”) them.  My conjecture is that if meta-pedagogy is not empowering, then the instructional methods may not be sound.  But we’re here to talk about math, so let’s get on with it:

“A path of lapis lazuli bricks will be built in front of the new sorcerer/advisor’s home.  The space available is 2 feet by 10 feet.  The bricks are 1 foot by 2 feet.  How many ways are there to build the path if exactly 1 brick is used? …then exactly 2? …then exactly 3, …and so on up to 10?”

This question is so involved that the kids didn’t even realize that they didn’t understand it. We spent 10 minutes trying to answer and clarify before it was apparent to all that the answer is a list of numbers, not a single number.  Many kids (P, D, V, A, L) came to the board to diagram their conjectures.  Several points needed to be clarified in order to form conjectures:

• Spacing doesn’t affect the number of arrangements, but positioning does. (P)
• The path must be 2 feet wide so that the sorcerer will not have to do a weird walk out of his or her door.  It can’t be only 1 foot wide, nor can it be wider than 2 feet. (A)
• The bricks must be flat, not standing on their sides, to count as a different arrangement. (L)

And speaking of “his or her,” this week’s sorcerer challenger is female:  The Sorcerer of the Square Hat.  That announcement was met by cries of “Male! Male!” from the male members of the class, and by cheers from the rest.  The kids looked for a pattern in the sex of the challengers, and predicted that the challenger on the last day of math circle would be female.  Once again, the natural human instinct for finding structure in our world emerged.  Once again, too, the issue of social justice has emerged in our Math Circle.  This issue also emerged at the recent Math Circles on the Road Workshop (detailed in my 4/17 blog entry), where Math Circle leaders from around the country discussed what we can do to make the field of mathematics more representative of our population at large.

Today, in addition to kids coming up to the board to contemplate the path-building question, one student worked on paper, and others used their hands.  At one point, I couldn’t understand A’s conjecture, so I got out some dominoes to make sense of it.  And it did make sense.  Isn’t it ironic that I am the one with the goal of moving kids from the physical to the abstract, yet I was the one who needed manipulatives to figure out what was going on?  I was stuck “in the box” of narrow-minded grown-up thinking.  How refreshing it is to explore interesting questions with  second graders.

The dominoes triggered someone to form a new conjecture.  Once stated, the rest of the group was silent, until V lamented, “Oh no, now everyone understands but me.”  This was a great opportunity for me (and a few of the other group members) to point out that actually, probably only one person in the group understands right now, and that silence does not necessarily indicate comprehension.  We also talked about how the Math Circle works as a collaborative:  no one here could work this problem alone, but each time one person said or asked something, it triggered an idea in someone else, who shared that, and so on.

At this point, the group had figured out the first four numbers on the list (0,1,2,3) and had four active conjectures about the problem going:

• The list is the consecutive counting numbers (the sorcerer’s conjecture).
• The list is not the consecutive counting numbers (H’s conjecture).
• The list is the Fibonacci numbers (P’s conjecture).
• No conjecture (“I don’t know.”)

Next time, I’ll work harder on eliciting a conjecture from everyone.  Even if it’s ridiculous (“Ten trillion, twenty trillion…”), a working hypothesis gives you something to test against.  None of the above four positions was dominant – each had about 2 supporters.   When we were just about out of time, V asked that I email the question home to parents so that kids could work on the problem at home.  “I don’t think I’ll be able to explain it to them,” she said.

E protested that she’d be too busy to work on it at home.  I announced that there are no homework assignments in Math Circle, but that if a question moves you deeply enough to continue exploring it after class, that’s fine and have fun.

At the end of class today, a few kids were so hungry that they ran out of class to get some food.  The remaining kids asked if we could resume our fall practice of having snacks in the last 5 minutes.  I told them why I had stopped:  (1) no matter what I brought, someone didn’t like it, (2) a lot of foods would spoil their dinner appetites, and (3) we need a mess-free food as we were using the classroom immediately after it had been cleaned for the next day’s classes.  Several students took on the challenge of figuring out a good snack.  I promised to bring some apple slices, carrot sticks, and raisins next week, unless any parent objects. (Please email me if this is a problem).

Finally, a lot of interesting mathematical vocabulary came up in class today, so I’d like to share it with you:  bird’s-eye view; combinatorics; arrangement; spacing; positioning; length; and width.

— Rodi