If you want to divide 8 objects equally among 3 recipients, then each recipient should receive \(\frac{8}{3}\) of an object. That’s easy if the objects are measured by area, volume, weight, or any other continuous quantity. But what if they’re animals? Or books? Or congresspersons? The apportionment problem requires that we assign an integral number of objects to each recipient.
There are a number of solutions to the apportionment problem; some of these (and their history) are presented in “Apportioning Representatives in the United States Congress” [Caulfield, 2010]. However, most treatments of the apportionment problem focus on the paradoxes. While these are interesting to mathematicians, politicians are wellpracticed at supporting antinomies, and voters are welltrained to accept them.
Instead, we might approach the congressional apportionment problem by considering its goal: that, based on its population, each state gets its fair share of congresspersons. In this activity, geared towards students in a liberal arts mathematics course, we consider what this means, and how different methods of apportionment might have altered the course of American history.
Using the 1790 US Census Data as an example, the project presents the Founding Fathers’ first solution to the congressional apportionment problem, prompts students to explore the shameful 3/5 constitutional compromise (according to which slaves were counted as 3/5 of a person), and then examines the different ideas for measuring and redressing disparities in representation that culminated in the HillHuntington Method of Apportionment used today.

Portraits of Founding Fathers Alexander Hamilton (right) and Thomas Jefferson (left), both of whom proposed congressional apportionment plans based on the results of the first census. Hamilton's plan prevailed in Congress, but was vetoed by thenpresident George Washington—the first Presidential veto in US history. Jefferson’s proposal was approved by both Congress and Washington. John Trumbull and Rembrandt Peale, respectively, painted the portraits, which are owned by the White House and in the public domain.


The studentready activity “Apportionment: What's Your Fair Share” (pdf) is fully selfcontained, and provides links to government documents and Supreme Court decisions that can be used by students and instructors who wish to explore the history of apportionment in the US via primary sources. In addition to its intended audience of liberal arts students, the earlier sections of this activity in particular would make an excellent interdisciplinary mathematics/social studies project for high school students.
About the Author
Jeff Suzuki is an associate professor of mathematics at Brooklyn College who specializes in the history of mathematics with an emphasis on the period between 1500 and 1800; in mathematics education; and in the legal, political and constitutional applications of mathematics. He grew up in southern California with an inability to decide what he really wanted to do, so he earned a bachelor's in mathematics (with a physics concentration) and history from California State University, Fullerton. He went on to earn his M.A. and Ph.D. from Boston University, with a dissertation on the history of a topic in mathematical physics. In addition to his book Constitutional Calculus, Dr. Suzuki is the author of Patently Mathematical: Picking Partners, Passwords, and Careers by the Numbers and the history of mathematics textbook Mathematics in Historical Context. In 2009, his paper "A Brief History of Impossibility" was awarded the MAA's Allendoerfer Award for expository excellence published in Mathematics Magazine.
References
Caulfield, Michael. "Apportioning Representatives in the United States Congress.” Convergence, November 2010.
Suzuki, Jeff. Constitutional Calculus. Johns Hopkins University Press, 2015.
U.S. Census Bureau. Results of the 1790 Census.