With the Presidential Election days away, and challenges to voting integrity already simmering, juniors and seniors in Jess Zeldes’ "Topics in Advanced Math" class spent the last several weeks testing out a variety of voting theories.

“I jokingly said to the students: This is a class about how democracy ‘doesn’t work’ because what we find very frequently when we look at the mathematics of voting is that there are all kinds of deep paradoxes that lie at the heart of voting theory,” said Zeldes. 

 

It turns out that there is no voting system that is perfectly fair and this was proven by economist Kenneth Arrow in 1949 through his Arrow’s Impossibility Theorem, one of many topics explored by students in this unit focused on the election. 

The class, which explores a variety of themes that aren’t typically taught in high school, uses ideas from discrete math. They have looked at plurality voting systems (the U.S. system), which they found, might seem to make sense–whoever gets the most votes wins–but is actually susceptible to a lot of problems, explained Zeldes. And they are planning to examine gerrymandering and redistricting, and the impact of all of the possible permutations. 

“We can mathematically quantify these problems in any situation where there are more than two candidates,” said Zeldes. “So in a plurality system, oftentimes if there are more candidates who are closer to people's views, it could actually lead to the candidates who are farther from people's views to win.” 

The 2000 election is a good illustration of this phenomenon. The candidates in that election cycle were Al Gore, George Bush, and Ralph Nader, and Zeldes asked students to establish and assess a preference order for each. The data showed how Ralph Nader took votes away from Al Gore, and that those who voted for Nader actually caused their third choice candidate to be elected. 

Another problem Zeldes shows students is how five different voting methods involving five candidates could each yield different results. Using real data and real events drove home the mathematical connections students were able to make. 

The big takeaway is that it’s important to investigate voting systems because it helps voters determine what aspects of a system they value, and what they prioritize, such as fairness, accuracy, and simplicity. 

“Math doesn't just exist in the ether,” said Zeldes. “Math is this beautiful world of abstraction and creation, but it is always connected to reality, which means we need to take our own implicit value systems into account. That's something I'm really trying to emphasize in the class: when you're doing math, even when you're doing pure abstraction, there are choices you're making and those choices have meaning.”

Zeldes believes it’s valuable to keep taking math classes and continue into the higher level courses. The work helps students build mathematical maturity, or the ability to grapple with abstract and complex concepts. 

“Politics can be a really cool entryway to advanced math because it is both very concrete in that we are looking at real world examples, but it also introduces a lot of the fundamental ideas of abstraction,” they said.  

In the eight-day cycle used for Upper School courses, Zeldes’ class happens to be meeting on Election Day and they expect a robust discussion. 

“It's a super politically active, historically active, and mathematically active group of kids,” they said. “I mediate the conversation, make sure everyone's voice is heard, and sort of move things along. These kids can talk about this kind of stuff all day.”