All we know for sure is that math is art.
I had hoped that we could quickly focus attention by starting with a game of “How Are They Different?” I had thought that we could eliminate the distraction of inequitable turn-taking by saying, “We’ll go from youngest to oldest.” I was wrong. It was easy to know that N goes first and V second, but then came the seven-year-olds. All five of them compared birthdates and expressed surprise about their places in the age ranking. While it wasn’t a quiet game of “How Are They Different,” some useful mathematical concepts emerged: a method to rank ages by month and date, and also the fairness criteria that no one should point out more than one difference on his turn, so that O, the oldest (by weeks), would have a fighting chance as the person who goes last. While everyone had come to class simmering with physical energy this day after Halloween, all became silent and focused when O made the final observation.
Before class began, I had asked M and A to help me finish making my Mathematician Cards. The kids in the room helped me glue pictures onto backgrounds as the rest of the group arrived. All were asking questions: “Why do you need Mathematician Cards?” “Why are you making your own instead of buying them at a store, like baseball cards?” “Why is there a big question mark glued onto this one?” P noticed the x-y axis drawn on the board and asked “Why is there a measuring stick on the board?” He then stood up against it to measure his height. It was at that point that we regrouped on the floor to play “How Are They Different” with pictures of two unicorns.
After that activity, I told them about some of the people on the Mathematician Cards. First we remembered what we had accomplished in our map-coloring session. Then I told the story of how the Four-Color Conjecture had become the Four-Color Theorem. I showed cards with Kenneth Appel, Wolfgang Haken, and an IBM 360 Mainframe (that one had the question mark).* The kids made guesses regarding what that thing was. P suggested old-fashioned movie projectors. J said copy machines. A proposed those cases that protect statues in museums. We talked about how the proof that Appel and Haken developed with computer aid was not initially taken seriously because computers were so new. The kids were fascinated that mathematicians are still trying to come up with a four-color-theorem proof that humans can do and understand. At this point I sloppily shuffled through all my cards and I heard M say “Oooooh! Women!” I showed them Emily Peters, who recently presented a paper on the theorem at MIT. Then I showed them Martin Gardner.
The year before Appel and Hakim published their proof, Gardner had published a map that he claimed required five colors, therefore supposedly debunking the Four-Color Theorem that had been around since 1852. I gave each child a copy of this map, and told them it was published on April 1. “So was it a joke?” Some thought yes, some no. They voted unanimously that I should tell them whether it was a joke, so I relented and said yes. N and P immediately started coloring and testing their maps, while the others asked that we get back to the dying unicorn story.
The kids once again attempted to solve the problem mathematically, but many were jumping and rolling around the room within a minute. So I told them, “You’re all filled with extra energy from Halloween, so let’s take a break from the unicorn and talk about Halloween. Raise your hand if you trick-or-treated last night.” Immediately, all were in their seats quietly with their hands up. “How many pounds of candy would you estimate that you collected?” A few called out numbers, a few looked puzzled, and then P said, “It would be impossible to tell because at our house had an all-you-can-eat buffet last night so the candy is all inside me now and impossible to weigh or even imagine.”
“So let’s rephrase the question,” I said. “N, how many pieces of candy did you eat?”
So I wrote “PIECES” on the x-axis on the board, counted up three units to the three, and put an N in the graph. I asked the same question to all of the children, and marked their initials on the graph at the relevant point. Soon we had a scatter plot of Halloween consumption by person and were able to look for patterns. All agreed that most people had eaten somewhere around three candies. V pointed out another pattern: “3 (people) ate 3 and 1 (person) ate 1. “
Then P made the claim that this graph was not necessarily realistic because his number (13) was so much higher than the rest only because he ate his candy all at once instead of rationing it out over time. Might it be possible that most people in the world do it this way, and our graph with his number as the outlier is misrepresentative? (Of course, I am paraphrasing here, but we then used this language in our extensive discussion of his question.) Someone suggested that another way our graph might be misrepresentative is that we all ate candy, while in the rest of the world not all people even celebrate Halloween. I asked how many people we would have to survey to be sure our results were accurate if we want to generalize about this whole country? Everyone in the class firmly agreed that we would have to ask every person, or at least one person from each family. Surely, statistics would be a good topic for a future math circle with this group.
With fewer than 10 minutes remaining, I asked the whether they wanted to do more statistics, or return to saving the unicorn. The consensus was the unicorn. Some in the class were actually hoping that the unicorn would die, and wanted to prove that saving it was impossible. I told the anti-unicorn faction that in another world, a unicorn was living, and would only die if they could get to it within 17 minutes. Finally we were all doing the same problem, sort of. This time, with little coaching from me (other than helping them remember their failed calculations) and a lot of mental effort from themselves, the group collectively figured out how to combine the numbers to get 17. “The unicorn lived!” shouted M, as she and A started dancing around. “The unicorn died!” shouted O and P. J, N, and V stayed at the table, but I think that everyone was feeling satisfied that this math question posed many weeks ago was finally answered.
We snacked and took photos, and as usual no one wanted to leave. Once they did, A came running back in. “I forgot my Gardner’s map!”
On the drive home from Math Circle, J and I saw a deer lying on the road with open eyes, moving its head. Another deer was standing over it, nudging it with hoof and nose. We called 911 and were told that someone would be there as soon as possible to try to save the deer’s life. We drove past an hour later and both deer were gone. Was this a case of life imitating art? Art imitating life? All we know for sure is that math is art.
*Read this article for the fascinating details of the story.