# No one had any clue what the rule might be.

We started last week’s Math Circle with 2 rounds of “Tens Concentration.” This arithmetic game is also a mindfulness practice, focusing and sharpening attention. You could have heard a pin drop as the kids tried to remember the location of that other card that would create a sum of ten. I integrated the story of our sorcerer challenge. “Once three sorcerers advance beyond the Wisdom Challenges, Tens Concentration would be part of the Character Challenges. The sorcerer that gets the job of the kingdom’s advisor would not only be able to devise a collaborative version of Tens Concentration, but also would be able to actually *play* it collaboratively. Progress can’t be made if each person is for herself only.”

As the kids played, we discussed why cooperation would be a valuable trait in an advisor. The game naturally evolved into a cooperative pursuit as they played. “I wanna help everybody!” said E. Cooperative problem solving is a huge goal in Math Circles, so I metaphorically patted myself on the back (only to have to unpat myself this week).

This week, I spent a longer time “setting up” before class as students arrived. The deck of cards was on the table, in its box. L arrived first, opened up the deck, then told me, “I took out the face cards.” J and V then put the cards on the floor and started setting up Tens Concentration. A, who had missed last week’s class, arrived and sat at the table. The kids on the floor invited her over and explained the game, and, as others came in, started to play. So far, so good. I kept “setting up the room,” announcing that Math Circle would start in 5 minutes. After 5 minutes, P said, “Guess what, Rodi! We’re playing competitively!”

“You are?!” I replied, aghast. A few kids snickered, reveling in the joy of doing it their own way. While the natural cooperation from last week was gone, some important things were happening. Kids were having fun doing (1) their own math (2) by choice (3) in a mindful way. “So let’s start,” I said. On with our story…

“The Sorcerer of the Triangular Hat arrived this week to take the third challenge.” The kids interrupted me to tell A about last week’s challenger, “The Sorcerer of the Long Stick Hat.”

Last week’s challenge was a continuation of the prior week’s: the bunny question. In version two the sorcerer (female – despite objections that “there’s no such thing as a female sorcerer”) had to determine the number of bunnies after 12 months, not just 6. The kids astonished me with their ability to symbolize rabbits on the board when the drawing of bunny faces had become laborious. They suggested lines, squiggles, and other symbols to first represent individual rabbits, and then squares to represent pairs of rabbits.

Most of the kids were able to determine that the sorcerer’s conjecture was wrong, but did not progress far enough into the sequence to determine the correct answer. A few kids had tuned out toward the end as the arithmetic had become overwhelming. Interestingly, my assistant R, who had been taking notes, jotted down the observation that “not knowing your times tables can help you when forming conjectures about Fibonacci numbers.” She commented later that the people who did not automatically resort to algorithms experienced more success finding the pattern. At one point in the discussion, E revised her conjecture, so P changed his (“I’ll change my conjecture too because she’s good at math!”).

In this week’s session, I had planned to continue with the Fibonacci sequence from different directions: function machines and combinatorics of spatial arrangements. The function machines took much time, though, as the kids who were familiar with them became exuberant in both their descriptions and in their desire to participate. We were sitting on the floor when this exuberance got out of hand. (I think that next week we’ll stay at the table, despite the insufficient board space there.) Another distraction was that two of the kids had been fighting earlier in the day, and their disagreement bled into our circle. (Another reason to stay at the table.) My assistant R is usually a calming influence in these situations, but she was absent this week.

For the most part, those various energies were refocused on some challenging function machines. People were at ease with basic adding and subtracting, so I put up a function that produced 200 from an input of 2. The kids who had been fighting were competing with bigger and bigger numbers. “2,003,800!”

“5,500,170!”

“99,099,099,999!”

No one had any clue what the rule might be. The input and output numbers looked scary and incomprehensible. “I have a good idea,” said A enthusiastically, “Why don’t we break up into teams and try to figure it out in small groups?” I agreed that this was a good idea, but unfortunately an idea we didn’t have time for today.

I asked the kids if they thought they could solve the function if they keep putting in such huge unfamiliar numbers. “Let’s try 1,” suggested D. When 1 produced 100, people were still mystified.

“So, when you put in 1, you got 100. With 2 you got 200. With 3, you got 300. What if you put in 4?” I asked.

“You get 400,” answered everyone, but no one could name what was happening.

“What if you put in 11?”

This was met by silence. I started to suggest another number when P said, “Wait a minute. I think I know. You would get one thousand one hundred.” That was correct, but still not helpful. I asked if anyone knew another name for that number. No. I asked them to help me count by hundreds: one hundred, two hundred, etc, etc, nine hundred, ten hundred….

“Oh!” shouted someone, “It’s eleven hundred! The function machine makes hundreds!” Everyone agreed.

I asked if anyone knew how to make hundreds mathematically, and no one did. We spent a brief time considering it, but didn’t pursue it. (I would have reviewed it if it seemed that people had seen multiplication before. I could have explored it in an inquiry based way if we had time. But, no, and no.) Everyone was very happy to declare this function machine a “Hundreds Maker,” which it certainly was.

Our next function machines produced lists of numbers where order mattered. One produced the Fibonacci sequence, which a number of the kids recognized from the bunny question. The function machines were part of story about challenges the sorcerers faced. I told them that the Sorcerer of the Triangular Hat answered this question with a memorized list of the first 100 numbers in the Fibonacci sequence. Should he pass that challenge if he didn’t know the rule? We debated whether understanding and describing the rule was important. M said proudly, “Well, *we* figured it out. *He* didn’t.” The consensus was that he could advance to the next level, but hopefully someone wiser would come along.

With 4 minutes left in Math Circle, I reported that the royals would only let him advance if he could correctly answer one more question. “What’s the question?” several kids screamed excitedly.

“If 3 people enter an empty room and then 5 walk out, how many have to come back in for the room to be empty again.”

We probed this question a bit, but it’s mostly food for thought until next week. Thanks for sharing your children with me!

— Rodi

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