You never know which idea in a math circle will take root in someone’s imagination.
We had a few visitors at Math Circle today, so we started with a collective recap of our voting scenario from last week. Kids immediately started suggesting still more voting/counting methods. M, who has asked if we could get back to the Dying Unicorn story (we will!), asked what we will do if we get “the answer” really soon. I reminded them that the question is whether there is a system for voting that is fair. O suggested multiplying votes in different positions (first, second, etc.) by different numbers. I briefly told them about Borda Counts, which count votes in such a way. P suggested some sort of competition among the candidates to determine who should win. Some kids were getting distractingly physical with some of the math manipulatives on the table, so we engaged in an attention- focusing activity: the Bobble-Head doll.
The Bobble-Head doll (who is “a distant relative of the man who owns the zoo”) sat in the middle of the table. I tapped his head, which is on a spring, and told the kids that they had to watch until it stopped moving, then put their own heads down. The doll never stopped; with every fidget (and possibly truck passing outside) the bobbling/vibrations increased. At this point, attentions were sharpened, and we decided to put him away and try him on the floor next time. I told the kids that the doll is somehow related to math, and with that, we were ready to return to our story.
The kids were curious to hear what the man decided to do, and I said that he did a second election where people indicated their likes and dislikes (“approval voting,” as invented last week), and I showed the results on the boards. The kids immediately called out “The monkey did win! Hooray!” I told them that the anteater’s supporters were pretty mad about this, and had extracted a promise from the man to explore other voting methods before choosing a new animal the next time one dies, hence keeping the door open for more play with voting.
I told them that since the possum’s pen had been split into two (cemetery and anteater) that a new map was needed, but since the man was still struggling for money, he needed to save money on printing costs. He wanted to distribute a colored map to visitors that had each animal’s region a different color from its adjacent animal’s, but that used the minimum number of colors. We looked at a world map that used many colors. I passed out maps of the zoo and colored paper squares . I noticed aloud how N got right to work placing the squares onto the regions of the map, and my comment to her got the rest trying it out. I said,”You can work with a partner if you want,” but no one did. This was the first time the members of our circle worked independently on something.
J, who prefers cooperative work, immediately left her seat with the comment, “I don’t want to do this.” I said, “That’s fine.” She spent the next 10 minutes approaching the other members and their work, and in a gentle and cooperative manner, pointing out their mistakes. Fortunately, no one minded. N was “tiling,” as opposed to “map-coloring”, hers, so I told her that what she was doing was also something mathematicians work on (“tessellations”). I let her know that the goal of tiling is that no pieces overlap or have gaps. I wondered whether she’d continue with that approach, but when I walked back to her side of the table, she had switched to map-coloring by category (yellow for the animals, red for the man’s personal regions, etc), prompting me to tell the group about a map of Rome I have at home that is colored by categories: the city regions are all one color, the suburbs another, the countryside yet another. As I looked at each person’s completely colored map, I asked, “Now can you do it in fewer colors?”
Soon P announced that it could not be done in fewer than 4 colors. The rest were still industriously working, so I gave him a map with more regions (including the neighbors properties too, increasing the number of regions from 15 to 29). He very quickly colored that in with 4 colors also. A and V had colored theirs in with 5 and were staring at them. M and O were still rearranging. At this point a few people looked frustrated. I asked a few questions that didn’t always help, so we talked about how mathematicians handle frustration. (P asked if any mathematician had every gotten so frustrated as to cut off his own arm? I don’t know, but to anyone who’s reading this and knows the answer: please let me know!) We also talked about the origins of the map-coloring problem, the later-disproved multiple attempts at proofs over 150 years, and how long it often takes to prove something in mathematics. How helpful it is to hear about unsuccessful attempts pushing mathematics forward.
We decided that anyone who was frustrated could come to the other side of the room and play function machines. Most of the group did so for the last 5 minutes, and 3 kids bounced back and forth between the activities. The kids really enjoyed trying very big numbers in the function machine. As we dismissed, some of the kids wanted to take their maps and coloring squares home to play with some more, and I gave out a United States map to color for all those who were interested. I told them to email me if they wanted something even more challenging because “I have two more very interesting maps that we are not going to do in the group.” I told them that we’ll talk more about their discoveries next week, after anyone who wants to gets to play more with them at home. “But of course,” I said, “there’s no homework in math circles.”
You never know which idea in a math circle will take root in someone’s imagination. Hours later that same day, J was playing with something like an old-fashioned whirligig, and said to me, “This has something to do with math, doesn’t it?” “Yes; how could you tell?” “It just seems like it would.”