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PARITY #5: Discovering Collaboration

(April 14, 2016)  The kids finally solved The Very Clever Prince problem.  My helper J and I put a list on the board entitles “What We Know.”  (We had never done this before.)  One student posited a conjecture which made someone else think of something.  That something made another student realize something else.  And so it went, as it has for the past five weeks.  Throughout the process today, I pointed out to the kids what was happening.  “This is real collaboration,”  I explained.  “Have you heard the word collaboration?”  I asked.  None of them had, so I explained it, and reminded them of it every time some new conjecture only could have been formed based upon another’s prior statement or question.

As the kids were getting closer and closer to the solution, excitement mounted.  Finally S announced the solution.  “You’re right,” I said to everyone.  They were surprised, even shocked.

“That’s it?  That’s really it?  We solved it?” someone asked.  When I confirmed, they got excited and proud.  “What can we do to celebrate?” someone said.

“Wait a minute,” said someone.  “We didn’t solve it.  S solved it.”

“Yeah,” said another, “S solved it, not me.”  The mood in the room darkened a bit.  I reminded them of the exact chain of statements that led to the solution.  S pointed out to the others that he would not have said what he said had they not said what they said.  I reminded them of the word collaboration, that they were working as mathematicians do.  This cheered them up, but they were still not as excited as they were initially.  If I had more time with them, I’d make collaboration to focus of many discussions in the next few sessions.  It was not enough to experience collaboration.  The kids needed to think about their own thought processes.  Talking about this thinking about thinking, called metacognition, can be very helpful in math (for confidence, frustration tolerance, conceptualization, problem solving, and more.)

 

PARITY OF SUMS

To celebrate, we played “Teacher’s Trick”* – that game we played before where one person turns his back while the others write numbers on the board, and the person has to guess the parity of the sum.  The kids had fun, made some conjectures about patterns, but had no definitive consensus about strategy.   This game is well worth doing at home, as it is the only activity we’ve done during the whole course where no one has posited a successful conjecture.  BTW the kids have no idea what it’s called.  I never referred to it by name.  The kids call it “that one where someone hides.”

 

CHEATING AT NIM

“We’re setting up the stones today.  It’s our turn,” I announced, referring to me and my assistant R.  We started counting out a few stones for a few piles.

“You have to use all the stones,” said S.  “No cheating.”

“Oh!” I said with eyebrows raised.  “It’s possible to cheat at nim?”

“Yes,” he explained.  “You’re cheating if you don’t set out all the stones.”

“I want to see how she wants to set them up,” said B to S.

“Me too,” said M.  So.  We had a situation where the kids on the team against me disagreed on the rules.  They huddled and whispered.  They announced that I could use however many piles and stones as I wanted, as long as they got to choose who goes first.

“Sounds fair,” I said.  I set up two piles of two stones.

“You go first,” they said to me.  As soon as I reached toward the stones, they said, “You lost!”

“I’m switching teams,” announced M.  She joined my team.  We played again.  I lost again, even with M.  “You lose,” said the other kids to us.

“That’s okay,” said M.  “I switched teams because I wanted to lose.”

“Ah,” I said.  “Did she really lose if her goal was to lose?  Does that mean she wins?”  The kids thought that it might, except that the kids who wanted to win did win.  Hmmm….

We played a few more rounds of various set ups under S’s condition – that we use every stone.  Most, but not all, lost interest in the round I set up of 35 piles of one stone each.  I pushed them too far on that one.  The good news is that after five weeks of experimenting with this games, each child is fully loaded with conjectures about strategy and set-up.

 

KIDS HELPING

You may have read in the reports of the past weeks about the activities I had some older kids facilitate.  We did a bit of that today, but it wasn’t a good day for that.  One student just didn’t have it together enough for collaboration or cooperation.  His behavior was more than helper J could handle when she was facilitating The Very Clever Prince, so she quit in frustration.  (I did spend time debriefing her later.)  Another student in the class was lying on the floor, adamant that he participate in math circle despite a painful injury.  (He did participate very well from the floor!)  These incidents had me managing the group much more tightly than usual.  Therefore, when R was facilitating the game of nim, I butted in too much, and the kids turned their attention to me instead of her. (Sorry, R.)

 

FLIPPING CUPS*

We wrapped up class revisiting the game where kids try to get 4 cups in the same direction by moving 2 at a time, starting with 3 up and 1 down.  They all remembered the progress they had made with this a few weeks ago, and asked a bunch of new questions.  That’s what I love to hear in math:  “What happens when you try it this way?”  They had answered the original question earlier, but then turned it into something new with their curiosity.

Thanks for sharing your kids for these weeks!
Rodi

*From my constant source of material for this circle – the “Playing with Parity” math circle from Julia Brodsky and her colleagues, http://www.mathcircles.org/content/playing-parity.  Remember, this link has videos and activity descriptions of almost every activity we did here, and I didn’t use all of them.  Enjoy them at home!

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