Math Circle Explores Systems of Voting and Counting

We began this week’s math circle sitting in an ogre – not a circle, not an oval, but an ogre, as suggested by the kids. This used the math skill of mentally shifting from the concrete to the abstract.

Once situated, we played a version of the shell game aimed at increasing our attention spans and developing the Jeffersonian hand required for extended mathematical inquiry. (Next time we will do several focusing activities at different points to help contain enthusiasm just a smidge.) Then we moved to our task at hand: to create a scenario that would involve land plots and voting so that we can explore the topic of our semester, “maps, voting, and counting.”

I told the story of a man who collected animals and got his near neighbors to vote on a new animal to include in his zoo. Of course, the children greatly improved the story, hence attaining ownership of the problem. The problem is how to fairly set up a voting system and fairly count the votes. L wondered aloud whether this was a problem with a single answer that I knew the answer to and was coaching him to figure out. I answered that we don’t know whether there is a single answer.

We discussed how in some places and times, certain classes of people were not allowed to vote, an idea that shocked most of the participants. P said “that’s like saying that you can’t vote if you’re wearing a purple jacket!” L said “that’s not fair to people who live in hotels!” I told them about Marquis de Condorcet and Madame Sophie de Condorcet, figures from the French Revolution. He was a mathematician who worked on fairness criteria in voting and also was a feminist. Sophie was a highly educated woman who stood up to Napoleon on behalf of a woman’s right to an education. Only V had heard of Napoleon and told the group “he was a guy who thought he owned the world.”

Then the group came up with its animal voting plan for the above-mentioned man’s neighbors: each person would (confidentially!) write on a slip of paper which animal she preferred, and then on the board a check would go under each animal’s name for each vote. The plan was to count the checks and whichever animal had the most would win. Notice that the group (not “the participants”) came up with this plan (not “plans”). The group is already moving nicely into a practice of collective inquiry. One child had came into the class saying “I am the best at math here because I know my times tables up to 12,” but that competitiveness that so often mars math classrooms for people was totally absent once we got to work.

I reported to the group that the neighbors had used this exact system, and the result was this: tamandua (anteater) 6, clouded leopard 5, and emporer tamarin (monkey) 4. (Note: these are real animals that can be found online and in the London Zoo.) There was excitement that the tamandua won! But then I broke the news that all of those who voted for the leopard or the tamarin were totally opposed to the tamandua because of the possibility of unpleasant odors. Excitement immediately evaporated. V suggested combining the votes of the defeated animals. L reported that 9 are against the tamandua. P called out, “then the leopard should win!” Most cheered this suggestion until M wondered whether the tamandua’s supporters might actually prefer the tamarin over the leopard, and hence would generate 10 votes combined. (Sighs.)

M and J pointed out that this occurred because “we only counted the likes, not the dislikes.” Excitement filled the room as several alternate voting methods were suggested. I listed these on the board and our group voted on which one should be used to be fair. I mentioned that I was bothered that we ended up with 7 voters and 8 votes (“someone voted twice – who was it?”), but since most voted for the same option, this incongruity was dismissed.

I told the group that they invented something on their own that is actually used in real life: approval voting. (We’ll talk a bit more about this next time.) I asked whether they thought there was one system of voting and counting that is always fair. No one was sure; we will revisit this. Circle ended as folks raced up to the chalk board to vote for their own personal preferences.