### The Kids Take Over

(February 5, 2013) What would you do if you were settled around the table with your students, about to engage in a civilized discussion about math history, when suddenly the students started chanting in loud rhythmic unison “Puppets! Puppets! Puppets!”?

That’s exactly what happened in our Math Circle this week. First, the kids introduced themselves to S, our new member. Then we easily finished off the John Lasagna question from last week when V gave a convincing explanation of how a dead person could attend a party. Interestingly, not every student was convinced that we had finished that problem. “Is that THE answer?” wondered a few. They were surprised that some questions have multiple possible solutions. Moreover, I think they were surprised that they themselves had the power to debunk an apparent contradiction.

When several kids and I started to fill in S on Dodgon’s biography so far, a quite murmer of “puppets!” could be heard. It quickly crescendoed. Why not? We shelved Dodson, moved to the other side of the room, and got out the puppets for a game of Knights and Liars. We reviewed the categories as I hung signs for each: Knights, Liars, Normals, Negators, and Blenders. We argued over whether Negators were a subcategory of Normals. The kids adamantly said no, so I hung the Negators sign far from the Normals sign. We began.

- First, E introduced Lammy, her lamb from home. She asked the group to give statements to which Lammy would respond. The class quickly deduced that Lammy was a Knight, so Lammy and Ella moved to sit under the “Knight” sign.

- H then introduced Leopard the turtle, who also joined the Knights after replying to student statements.

- C was so excited about winning the opportunity to use Waggy the fox that he immediately sat down with the knights. (The students who had not brought their own puppets guessed numbers to determine order and preference. C won the puppet jackpot with Waggy.)

- Next, S ended up with Koko the gorilla. By now, the kids were directing the game. They told each other what to do and how to do it. I sat back comfortably with my legs stretched out, only assisting when asked. I quietly watched S and the group interact to determine that Koko was also a knight. This activity was not, by the way, my agenda* for today; I was ready to move into new material. I planned to make sure we explore enough new material (Progress! Progress! Progress!) to ensure that we can solve the course’s original mystery (“All Puddings are Nice”) before our six weeks are up. But what is the overarching agenda of a Math Circle in general? The primary agenda of this one is to give a group of kids ownership of mathematics. So what the kids were doing here was actually pretty awesome.

- L chose Rooney the raccoon, and announced that he might not necessarily be playing the same role as he did last week. Students raised their hands and gave statements, and L made Rooney negate each one. Each negation was truthful, however, so Rooney’s role was not clear. V raised his hand and said to Rooney, “You are a Negator.” Now L hesitated. The kids thought about how to help her.

M gave her some advice: “Say you are not a Negator if you are one.” Not everyone agreed, though, so debate ensued.

Finally L asked me for help. She whispered in my ear that she wanted Rooney to be a Negator. I whispered back that she should make him say that he is not one, so she did. Through the ensuing debate, the kids realized that Negators are, in fact, a subcategory of Normals, so I hung the Negators sign under the Normals sign. The kids also realized that they needed to ask more questions to ascertain Rooney’s role. “The sun is shining,” said M.

“No it is not,” said Rooney. Everyone looked at the rain out the window and realized that Rooney was telling the truth. Therefore he could *still* be a Knight, a Normal, or a Negator. So people thought some more.

“The sun is not shining,” said M. L wasn’t sure how to negate this one, so came to me for advice. I suggested that she have Rooney tell group that the sun* is* shining. She thought that she couldn’t say this because she didn’t realize that a negation does not have to include the word *not*. Despite my assurance that it could, she wanted to use not in her reply, so I had her say “The sun is NOT not shining.”

Most students were now convinced that Rooney was a Negator. Two students, however, were not. Some people wanted to move forward anyway. I explained that we try to reach consensus when we can in a Math Circle. I asked the group if I could give Rooney a statement. Some students thought that I shouldn’t – that only the kids could participate in this. I asked them to give a statement that Rooney would respond to in a way that would make his role obvious to everyone. No one wanted to do this, so they granted me permission. “Rooney,” I said, “you are a raccoon!”

“I am not a raccoon!” responded Rooney through Lila. Now everyone was convinced and time was just about up. Four students hadn’t played yet. Two of them chose to rush their turns to get them in. Two decided to wait until next week. A few others grumbled about the unequal amount of time each student had to make their puppet talk. The remaining few students explained how the duration of each turn can’t be controlled and we moved on.

- J did something different with her puppet. The puppet quickly walked up to each person in the group and gave an obviously false statement (“You have green skin,” etc.). “Liar!” called out everyone. J was appalled at how short her turn was and in an amazing example of thinking on her feet, said “You are all wrong!” She then had her puppet approach each person with an obviously true statement (“You’re wearing a blue shirt,” etc.). Now folks agreed that her puppet was a Normal. I quickly mentioned that you can’t always trust a pattern in mathematics, and wished we had more time to discuss this principle. But we were in overtime with only one minute left and it was D’s turn.

- D brought his elephant Buttons from home. (I had asked him to go last because at the start of class, he had announced that his puppet was a new category.) He asked the class for a statement, and had buttons respond with the exact same statement but for the order of the words. “Are you a Scrambler?” asked V?“That’s really close,” said Buttons through D. “But I am actually called a Backwardser.”

My displaced mathematical agenda for the day had been to move into the converse and inverse. The kids, though, had arrived at the Backwardser. It looks like we took a different route but ended up at the same destination anyway. We took a child-led detour and emerged in a stronger position. Today’s session has deepened the kids’ understanding of the concepts of true, false, contradiction, negation, and most importantly, how to frame questions effectively. Their exploration has naturally led us to where we need to be. To get there, I had to allow the kids to take control. I’m so glad I did.

Rodi

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* I just read the Newberry novel Walk Two Moons. The book’s recurring quote “everyone has his own agenda” inspired this reflection on my own agenda.