(1/16/2020) Our overarching Math Circle goal is for students to invent and discover math for themselves. In this course, the plan/hope is that I’ll ask a bunch of questions and the students will invent their own voting methods before discovering what voting methods people have created throughout history.

Pet of the World

 “What would happen if you got to vote for Pet of the World, you had to choose between Dog and Cat, that there were exactly 100 people in the world, and you were the only people who showed up to vote?” After I posed this question, the students voted and Dog won with 75% of the vote.

“Why did Dog win?” I asked.

“Dog had more votes!”

“So whoever has the biggest number of votes wins, right?”


“What would happen if all of those people who didn’t vote actually wanted Cat to be named Pet of the World?” Now the game was afoot, the game being to shake up our assumptions about math. This question incited a big discussion about fairness and problems with voting systems and methods, so we began a list on the board and added to it every time someone raised an issue.

Plurality versus Majority

Of course students wanted to talk about the current US election. One goal for this course is to let students experience Emily Rhiel’s wonderful presentation “The Mathematics of Social Choice.”  I’m using her presentation as an outline, turning her statements into questions, and letting the group take it from there. (In other words, we’re taking an inquiry-based approach.) *

“Can you name every candidate who was running for the democratic nomination this past summer?” The students brainstormed, with help from me and Ellen (my helper). We got a full list on the board and then I asked students to cross off a few so that we had only 20 to make the calculations easier. **

“What would happen if all 100 people in the country voted and Cory*** got 6 votes, Seth got 4, and everyone else got 5 apiece?” Great hesitation from the students now, not that they didn’t know the answer, but the results were disturbing. Then an even more disturbing question: “What if Wayne dropped out and all 5 of his supporters preferred Seth to Cory?”****

After some discussion, the students suggested aggregating votes to fewer candidates. They dictated who should drop out and to whom those votes should go. Each time one candidate dropped out, I asked, “Now does any one person have enough support to be sure of winning?” The answer was no every time. Things got really interesting when the students manipulated the voting so that Pete had 34, Bernie had 41, and Elizabeth had 25. Students soon stated that a single candidate had to have one more than half of the votes to be sure that the real preference won. We discussed plurality versus majority.

“Is majority the best and fairest method?”

“It’s a no-brainer,” said A. Everyone agreed. After a little more discussion, J pointed out that it depends upon how many candidates there are. Majority is fine for a two-candidate election, but harder to achieve with more than two. M brought up another issue: gerrymandering. Then F brought up yet another issue: the electoral college. And the students already knew about strategic voting. By now our list of issues/problems with voting systems was getting longer.

Student list (so far) of Problems with Voting Methods

  • People not voting
  • Most preferred candidate not winning
  • Electoral College (is this a problem or a solution?)
  • Gerrymandering
  • Miscounts
  • Majority could produce a bad leader
  • Strategic voting (is this a problem or a solution?)
  • Uninformed voters
  • Too many candidates (is this a problem?)

We didn’t have one single discussion of problems during our 90 minutes. Instead, this seemed to be the theme that every discussion throughout the session circled back to. The “problem” discussion generated a new list: “ideas.”

Student Ideas

We kept a running list on the board of ideas as they occurred:

  • A point system whereby voters list multiple preferences and candidates get different numbers of points. (This is what I mean by students inventing and discovering math for themselves! Mathematicians have developed ideas like point systems for hundreds of years. We’ll talk more about that method during the course.)
  • Ruler is appointed by the predecessor but can be deposed by a majority of citizen votes. (This idea prompted a discussion on whether we were assuming a democracy in this course, and whether this system is democratic. Again, prescience here with the idea – there is a precedent for voters stating whom they disapprove of – we’ll get to that soon.)
  • Add in an element of chance to the election. (I will research this!)
  • Write-in candidates. (Once again, students had not previously heard of something that is done in real life but thought of it themselves.)

Vote for Two Method

“What democrats are left in the election as of today?” We got the list of 12 on the board. “Let’s suppose that there are only 6 left. Of these 6, suppose every voter is allowed to vote for two. What would happen?” The students winnowed the list down to 6 candidates.  I pulled a puppet out of my bag and said to M, “If this puppet was allowed one vote, whom would she vote for? And if she had a second choice, what would that choice be?” I circled around the room several times getting students to say whom each of the 26 puppets would vote for in a vote-for-one and a vote-for two election. We then compared the results of the two methods. The students agreed that different methods can produce different results and that different methods might be better for different situations. The students had no conjectures yet about what those situations might be.

Other questions

As we worked, students asked other questions. Ellen acted as our Math Circle’s real-time fact-checker. So some of these questions got answered on the spot:

  • When is the last date a candidate can join an election?
  • How does the Iowa Caucus work?
  • Can people write in anyone?
  • Is there an age minimum for write-ins?
  • What if nobody votes?

Getting away from numbers

At this point, students’ brains were tired. I abandoned my plan to introduce yet another voting scenario and instead mentioned a topic in which politicians are involved: utilities. I posed the famous “Gas Water Electricity” Problem.

“Three houses in a row need to be connected to three utilities (gas, water, and electricity), the sources of which are also in a row. Can it be done with no lines crossing?”

M immediately noted that this problem is similar to a problem we did with names a year or two ago. I explained that it’s similar since it’s in the field of graph theory. As soon as I said “graph theory,” one student reacted excitedly with “yes!” The students spent the rest of the session with paper and pencils trying to solve this problem.

Follow-up at home

There is no homework in Math Circle. But for those of you who have asked, here are some ways to extend what we did here today. If you chose to do that, try having the students create the scenarios.

  • Explore methods for how to tally up large sets of data (i.e. if you had 100 people voting for Dog versus Cat, how to count quickly?).
  • Make a fraction from a ratio (i.e. if 65 voted for Dog and 35 for cat, what fraction of people voted for Dog?).
  • Convert from a fraction to percent.
  • Research how utilities are related to politics.
  • Explore the name problem that M mentioned. It’s an unsolved problem posed in 1977 by Krishnamoorthy and Deo. Visit https://mathpickle.com/unsolved-k-12/ and click on the video at the top right in the array of videos.
  • Work on the Gas Water Electricity problem.

A big goal for this course is to expose the students to as many voting methods as possible and for them to evaluate them at the end to see whether any truly is a no-brainer.


‘* I plan to do this with the guideline that we only talk about the mathematics of voting theory and not about policy or candidate preferences. Using puppets helps to keep the conversation non-partisan.

**One student asked “Why just talk about democrats?” I explained that it makes the math more interesting since there more candidates.

*** The students helped to formulate the questions. This writing seems to insinuate that I had a script of questions and was just reading them. I didn’t. I only came in with a printout of the slides from Dr. Rhiel’s presentation. The students not only asked a lot of questions, but they also came up with the names and numbers when our candidates dropped out or their supporters switched preferences.

**** In the first situation, a candidate won with only 6% of the votes. In the second situation, the last-place candidate became the first-place candidate when one person dropped out.

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