Voting is not simple. The way ballots are designed and collected, district boundaries are drawn and votes are counted vary drastically between different democratic nations, states, and counties even though they all have the same goal: to represent the will of the people. In math teacher Bradley Stoll’s Discrete Math class, students explore myriad systems of democracy through the voting theory unit.
For national elections, the U.S. uses plurality voting: each member of the electoral college chooses one candidate, and the candidate with the most votes wins. However, the class breaks down several flaws of the system, like how third party candidates can split votes and lead to the election of an unpopular president, and explores alternative methods.
“I think we have to accept the fact that our voting system, whichever one we use, is not gonna be fair,” Stoll said. “It’s not really about if it’s fair or not anymore. It’s about if this is the best we got right now, and can we do better?”
A feature of the plurality voting system, which is used in most U.S. elections, is that it promotes a two-party system. Since Americans can only choose one candidate to support in a presidential election, only the two major parties have realistic chances of winning elections.
Discrete Math introduces students to alternative voting systems, like ranked choice voting — which is not used on the national stage, but has been implemented in local elections like the New York mayoral race. Ranked choice voting allows voters to rank candidates by preference instead of picking a single candidate.
“Implementing ranked choice voting would give third parties and independent candidates a fair chance,” Discrete Math student Victoria Ma (12) said. “A large number of people might support them, but not as much as Democrats or Republicans. It would also level out the playing field when it comes to the high monetary barriers of entry of American elections.”
Voting theory proves that the many voting methods all have advantages and pitfalls. Some, like the electoral college, might contradict the popular vote. Others might be unduly affected by the introduction of third-party candidates.
“The math itself is objective, but the definition of fairness is subjective,” Math Club Co-President Heather Wang (11) said. “In voting systems, there’s a lot of different definitions of fair that make sense to us, but it’s mathematically proven to be impossible to satisfy all of them.”
Discrete also dives into the subjective debates over apportionment, the mathematical process of assigning the number of U.S. House Representatives to states in proportion to their populations. The issue of apportionment seems straightforward, but each seat holds a lot of weight in Congress and the Electoral College, as illustrated in California’s divisive Prop 50. Political incentives created fierce competition over apportionment throughout American history.
“Apportionment was established by the U.S. Constitution, but it doesn’t specify exactly how to do this,” Heather said. “States fought over apportionment because different methods lead to different numbers of representatives per state — and that gives some states advantages over others.”
Students analyzed these different methods in a Mathematica project, calculating apportionment results based on current U.S. census data.
Beyond the mathematics of voting, the Discrete Math class connects students to direct applications of voting theory. It asks them to understand their local county elections as well as analyze how political power is distributed on a national level.
“Being good citizens of a country takes practice and education,” Victoria said. “We’re all going to become full-fledged adults who can vote in this country and understanding what our vote means in the context of our state, of our electoral college system, of different voting systems is going to benefit us, as young adults, to get our voices heard.”