# Developing collaborative problem-solving through the joyful art of mathematics.

Since Euler codified the concept of functions, we started with a function machine. It was challenging to get past the design phase to play with the math, but after applying a chimney to a fish, we did.

The first rule was quickly deduced: “It makes it half; it takes half of it.” (G)

We almost ran out of board space for the numbers we tried before solving the second machine, which had blue eye shadow added to the fish. All conjectures failed. We needed a new strategy. I drew a number line on the board and put the numbers on it. Finally, D said “if an odd goes in, you add one, and if an even goes in, you add two.” This did work for every example we had. I asked for the rule to be stated as a single rule that works for all numbers. Collaboratively, the group rephrased the rule: “the next larger even number.”

The function changed with the addition of a cupcake. After three numbers went into the revised machine, everyone had spotted a pattern. No one, however, wanted to say what it was. “It’s too easy.” “That can’t be it.” “It looks like you just add one to the number, but it couldn’t be.” I asked them for a strategy to test the validity of their unvoiced conjecture.

“Try a different kind of number!”

“How about minus 2?”

“Okay, I’ll put in minus 2,” I said.

“Negative 2,” corrected M.

“Yes, you’re right, negative 2,” I agreed. Normally, I use the mathematical term for everything, but slipped here, and the kids caught it. They want to be accurate, and it’s fun to know the correct term.

“Oooh, negatives!”

This didn’t help in their figuring out the function, so they tried other different types of numbers: evens versus odds, small versus big, whole numbers versus fractions. “Put the number line back!” Once the number line was back on the board, the group was able to collaborate in seemingly solving it with 2 rules: “Subtract 1 if it’s even and 2 if it’s odd.” But G was frowning. “What about 6 ½?” Wow, it didn’t work for that. The group suggested that I had miscalculated for 6 ½. After I insisted that I hadn’t, M suggested the rule “The next lower odd number.”

We talked about what it would take to break a function machine, and also what functions the kids want to create themselves for next week. (Their idea.) Then D asked, “What about solving the problem from last week?” This was the perfect time to move into that, since everyone had shown a clear understanding of even and odd numbers. Last week, N had stated that the Konigsburg Bridge problem was all about even and odd numbers. This week, we tested that. It turned that he was correct, but with a few conditions. About half the group participated in this proof. The other half worked on compass art or polyhedron construction while they listened. As my assistant R points out, doodling can actually enhance thinking.

I asked for votes about the frustrating Gas Water Electicity problem: (1) it can be solved in 2 dimensions; or (2) it can be solved in 3 dimensions; or (3) the solution is that it is impossible. Option 2 was the unanimous, and correct, result. I said that if you want to think about this some more at home, try to figure out what type of solid shape the solution requires.

Then I told a math joke – the one about the engineer, the chemist, and the mathematician on a camping trip. This broke the tension of the aforementioned problem, and drove home the point that one goal of mathematics is determining without any doubt whether a problem has a solution. (It’s not just about “How do you solve this?”)

And speaking of engineers, the group suggested names for J’s completed polydron construction. After J chose “Mr. Egg,” X asked, “Does it break easily?” We decided, with J’s permission, to end class with an attempt to answer that question. The students very politely applied various efforts into breaking it, noting that the solid Polydrons came apart much more readily than did the open “Frameworks” Polydrons, both of which were used to construct Mr. Egg. The group guessed that the frameworks were more stable because the hollow inside allowed for greater flexibility.

The announced topic of this course was “The Platonic Solids.” It looks like we won’t get to that this time. I’ve presented a number of jumping-off points that could have logically progressed into this topic. None of them grabbed the collective interest of the group. The things that have grabbed their interests have led to great discussions ripe with mathematical concepts. But it’s not the concepts themselves that make a Math Circle. The goal here is to develop collaborative problem-solving through the joyful art of mathematics.

So don’t expect the Platonic Solids next week, our final week for this session. This group is so into Euler that we will use him to discuss a key component of proof. Unless, of course, we do something else entirely.

— Rodi

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