The Mathematics of How CAMI Got Its Name

What does it mean for a decision to be democratic? What role can math play? We explore 7 different voting methods to find out (and name our math teacher circle).

Facilitator(s): Eric
Date of Meeting: March 13, 2015
Further Reading: pdf


So before our last meeting (Feb. 2015), we sent out an electronic survey to choose a name for our group. Teachers were asked to put the 12 choices in their order of preference from most preferred to least preferred.

Mark presented the results from the electronic survey to name our group. Here are the Results from Online Survey he created. Rather than announce a winner, Mark asked everyone to take a few minutes to look over the report. The report shows a few different ways of presenting the data. Here are some things people noticed:

  1. MELT had the most occurrences in the top three with 5. TIPS was second with 4
  2. MELT had the most occurrences in the number one spot with 3. “Continue search was second with 2.
  3. If we assigned 3 points for each first place vote, two points for each 2nd place vote and 1 point for every third place vote, the top three choices were MELT (11 points), TIPS (9 points) and CAMI (7 points).
  4. Looking at the orders of preference for all 12 choices, from all 9 respondents:
    • MELT and CAMI had the lowest total (lower is better) with a 40 and 41 respectively.
    • CAMI had a smaller range than MELT (8 and 11 respectively)

Though MELT seemed to be the winner, it seemed like there might be a deeper story if we explored more data and asked more questions. Usha especially had a lot of great ideas right off the bat about some mathematically alternate election models: (1) Using a point system, where folks get a certain number of points to assign, where you could give your top choices more points (or even put them all in one “basket”), (2) Voting in rounds, narrowing it down to the top 5 and then down to the top two, (3) Mapping out “If this name was put against this name, which would win?” – maybe even doing this will all the possible pairings, maybe visually with post-it notes.

We decided to do another online survey, this time using a wider variety of preference measures. Voters had to:

  • Choose their favorite name
  • Use a Likert scale (Strongly agree to Strongly disagree) for each name
  • Rank the six choices from 1-6
  • Choose all of the names that were acceptable
  • Imagine that you have 5 points to distribute between your favorite names. You can give all 5 points to one name or you can distribute points among your favorite names. Your votes must add up to five.

Fourteen CAMI members voted.

Today’s meeting was about using that data to explore different voting methods and the idea of fairness so we could come up with a name for our group.

Naming Our Math Teacher Circle

These were the names that people voted on:

    • MEL (Math Educators Learning)
    • MICK (Math Investigations Community Knowledge)
    • MIC (Math Instructors Community)
    • MELT (Math Educators Learning Together)
    • CAMI (Community of Adult Math Instructors)
    • TIPS (Teachers Investigating Problem-Solving)

To frame the meeting, Eric started us off with one-question interviews. We were each given slips of paper and a short amount of time to respond to our question with a partner.

  1. Is the “will of the people” important? What does it mean to you?
  2. What is the purpose of voting?
  3. What does it mean for a process to be democratic?
  4. Do you vote?

Some ideas that came out:

  • The purpose of voting is group decision making
  • Voting is to reach some kind of consensus
  • Is consensus process the same as voting?
  • We vote so it is not just one person or small group voting for only their interests
  • Does the “will of the people” always represent what is right? Or put another way, what is more important – what is right or what is representative?
  • Voting can sometimes feel meaningless, especially in regions where there is an overwhelming majority and because third party candidates are usually not likely to win

Next, Eric gave out a data packet with an anonymous results for each of the 14 votes submitted.

Launch: The Plurality Flaw

Eric introduced what is known as “the plurality flaw” by sharing the following results of the 1970 New York US Senate election. James Buckley was the winner, having received a plurality of the votes. But far more people voted against him than voted for him. Also, the voters for both of his opponents preferred the other candidate to Buckley. And yet, Buckley won.

Is this fair? Is this representative of the will of the people? Are there other ways of voting that allow for more depth in understanding the will of the people?

Click here for more on Election Math and the Plurality Flaw

To get started, Eric broke us up into groups, assigned each group a voting method, and gave a description of the method along with the following instructions:

In your small group…

Choose a facilitator (someone to keep the group moving, help resolve disagreements)

Choose a reporter (someone to take notes on the group’s process, questions/concerns that came up, agreements)

Be prepared to present your method/s and results to the large group

The Borda Count (Kevin and Mark)

“An alternative system that avoids the kind of outcomes of the 1998 Minnesota Governor’s race is the Borda count, named after Jean-Charles de Borda, who devised it in 1781. Again, the idea is to try to take account of each voter’s overall preferences among all the candidates. As with the single transferable vote, in this system, when the poll takes place, each voter ranks all the candidates. If there are n candidates, then when the votes are tallied, the candidate receives n points for each first-place ranking, n-1 points for each second place ranking, n-2 points for each third place ranking, down to just 1 point for each last place ranking. The candidate with the greatest total number of points is then declared the winner.”

Single Transferable Vote (Ramon and Allison)

“For instance, several countries, among them Australia, the Irish Republic, and Northern Ireland, use a system called single transferable vote. Introduced by Thomas Hare in England in the 1850s, this system takes account of the entire range of preferences each voter has for the candidates. All electors rank all the candidates in order of preference. When the votes are tallied, the candidates are first ranked based on the number of first-place votes each received. The candidate who comes out last is dropped from the list. This, of course, effectively “disenfranchises” all those voters who picked that candidate. So, their vote is automatically transferred to their second choice of candidate — which means that their vote still counts. Then the process is repeated: the candidates are ranked a second time, according to the new distribution of votes. Again, the candidate who comes out last is dropped from the list. With just three candidates, this leaves one candidate, who is declared the winner. In a contest with more than three candidates, the process is repeated one or more additional times until only one candidate remains, with that individual winning the election. Since each voter ranks all the candidates in order, this method ensures that at every stage, every voter’s preferences among the remaining candidates is taken into account.”

At first, there were two different results to this method of voting… on the one hand, it was a 7 to 7 tie between CAMI and MELT. On the other hand, it was a 8 to 6 victory for MELT. The complicating factor was due to inconsistencies in the voting measures on the part of two voters. When asked to choose their favorite name and to rank their choices from 1 to 6, two voters had a different name representing their top preference. In other words, when asked to pick a single, top preference, two people then choose a different name for their number 1 ranking (when ranking 1-6). Allison and Ramon decided to use the top preference as indicated by a number 1 ranking. The results of looking at the data in this way result in a 7-7 tie.

Approval Voting (Ramon and Allison)

“Yet another system that avoids the Jesse Ventura phenomenon is approval voting. Here the philosophy is to try to ensure that the process does not lead to the election of someone whom the majority opposes. Each voter is allowed to vote for all those candidates of whom he or she approves, and the candidate who gets the most votes wins the election. This is the method used to elect the officers of both the American Mathematical Society and the Mathematical Association of America.”

Point Distribution  (Ramon and Allison)

No model found. Make up your own. From the survey: “Imagine that you have 5 points to distribute between your favorite names. You can give all 5 points to one name or you can distribute points among your favorite names. The name with the most points from all voters would be the preferred choice, given this method of voting. (Your votes must add up to 5.)”

Condorcet Method (Parvoneh, Kevin and Mark)

Preference Schedule

group of 18 group of 12 group of 10 group of 9 group of 4 group of 2
Killians 5 1 2 4 2 4
Molson 1 5 5 5 5 5
Samuel Adams 2 3 4 1 3 3
Guinness 4 4 1 2 4 2
Meister Brau 3 2 3 3 1 1

The Condorcet method is the final method for computing the winner. First, for each pair of candidates determine which candidate is preferred by the most voters.

For example, here is a comparison between Samuel Adams and Guinness (the number of supporters in the first row represents the number of voters who prefer Samuel Adams to Guinness, and vice-versa for the second row):

# of supporters
Samuel Adams 43 (18 + 12 + 9 + 4)
Guinness 12 (10 + 2)

Winner for this pair is Samuel Adams

If there is a candidate who ‘wins’ EVERY comparison with all other candidates, then this candidate is the winner. If there is no such candidate, then there is no Condorcet winner.

Note: you can define a “winning” candidate as that candidate having a number of preferential votes which is greater than or equal to the number of preferential votes of all other candidates when the candidates are compared pairwise. There isn’t always a Condorcet winner. If no candidate satisfies this condition for winning, then there is no Condorcet winner.

Summary of Results

Once the results were in, the discussion and interpretation began.

  • MELT won the plurality vote.
  • CAMI won the Borda Vote, the Approval Vote, the Likert Scale Vote and the Point Distribution Vote.
  • The Single Transferable Vote was a tie between CAMI and MELT.
  • Neither won the Condorcet Vote.

But, what do any of the results actually mean? Does MELT win because it won the plurality? Does CAMI win because it won the votes of four different voting methods? Does CAMI win because one of the four is more democratic than all of the others?

We took turns going around one at a time and weighing in. After several rounds, through a consensus process, and with no major objections, we decided that choosing CAMI as the winner represented the “will of the group” as best as it could be determined by the data at hand. A few arguments kept coming up to support this decision. Look at the Approval Method and consider the disapproval voiced. 13 voters approved of CAMI and 10 voters approved MELT. Put another way, 1 voter disapproved of CAMI and 4 voters (out of 14) disapproved of MELT. Also, two people ranked MELT last, while CAMI was in everyone’s top four. MELT received more first place votes (6 out of 14), but CAMI received more top two votes (10 out of 14).

Were the group’s ideas of fairness and representation correct? Is any one voting method more democratic than another? Is there some way to combine the data resulting from each voting method? If there had been a different set of CAMI members could the decision have gone another way?

Extending this Work in the Classroom

If the US used some of these methods, how might US history be different? It would be interesting to find some historical data and run some alternate endings to some key elections in US history. If the US had a single transferable vote would enough of Ralph Nader’s votes have gone to Al Gore to make Gore v. Bush unnecessary? Which voting method would have kept Abraham Lincoln from winning the election of 1860, when there were 4 Presidential candidates and more people who voted against Lincoln than voted for him?

 Additional Resources

Usha and Tyler suggested materials for teaching the mathematics of choice and elections, both from COMAP (the Consortium for Mathematics and Its Applications)


In attendance: Alison, Eric, Kevin, Mark, Parvoneh, Ramon

Programs represented: Brooklyn Public Library, CUNY Start, Fortune Society, CUNY Adult Literacy PD Team

Location: CUNY, 16 Court Street, Brooklyn, 17th floor.

Respectfully submitted by Eric & Mark

1 thought on “The Mathematics of How CAMI Got Its Name”

  1. Check out this New York Times article discussing the role of the plurality flaw in the rise of Donald Trump in the 2016 Republican Primary. They compare the plurality vote (which Trump wins) to a ranking method that has Kasich winning.

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