# Course PoliciesMath 217, Mathematics, Voting, & Democracy, Spring 2022

Over the last two centuries, certain questions involving the design of political structures have arisen that at first glance, appear fairly straightforward to answer. However, they actually require some deceptively subtle, and sophisticated, mathematical analysis. These questions have contributed to the development of the academic discipline of Social Choice Theory, which has attracted attention from a variety of fields including economics, political science, mathematics, geography, and psychology.

In this course, we will focus on three major questions:

1. Every ten years, a census is taken in the United States. How are the seats in the House of Representatives assigned to the states? This seems very straightforward, but the fundamental problem is that you cannot assign a fraction of a seat to any state. In addition to studying the very thorough and beautiful mathematical analysis of apportionment laid out by Balinski and Young, we will also examine the interesting, and sometimes entertaining, politics that have come into play.
2. The term gerrymandering refers to the practice of one political party drawing the legislative district boundaries to give them a durable systemic advantage over their opponents. How can you identify, and quantify, a gerrymander? Some gerrymandered districts are apparent from their odd, meandering shapes, but it is also possible to tip the scales with more compact looking districts. This is fundamentally a problem of political geography - you need to know the political distribution of the population within the state to analyze any districting plan. We'll use some techniques that have been developed in the last five years to analyze districting plans based off the 2010 census, as well as to analyze plans based off the 2020 census that have been released in the several months!
3. If there are more than two candidates in an election, how does a voter indicate her preferences, and what method is used to determine the winner? We will approach this through the very nice geometric framework that Saari has developed to understand why different voting methods give different outcomes, even if no voter changes her vote. One of our goals is to understand why different procedures behave as they do so that we can identify when we have inadvertently made a bad decision. This has applications not just in political elections but in any group decision process.

This is going to be a really fun semester.

The plan is that this will be close to a "normal" semester, but there are still a lot of unknowns about exactly how things will play out, both for the campus overall and for each of us individually. We'll need to be flexible, and we might have to make some adjustments as the semester goes along.

Let's all be kind to each other, and we'll figure it out.

## Goals for a 200-level Mathematics Course

There are several primary objectives of any 200-level math course at Wheaton. By the end of this semester you should:

• Be able to formulate a concise, precise mathematical argument, including recognizing when it is complete or when further justification is needed
• Be willing to approach a problem even if you do not know whether or not your approach will be successful. If it doesn't work out, try something else!
• Appreciate the necessity of rigorous mathematical arguments
• Continue in your development from being a consumer of mathematics to being a producer of mathematics

## Goals Specific to Mathematics, Voting, & Democracy

You should gain a deeper understanding of:

• The history of reapportionment in the United States
• The mathematics underlying different apportionment methods and why the methods may give different results
• The multitude of factors involved in building a redistricting plan
• Modern techniques for identifying gerrymandering
• The structure of elections with three candidates based on a geometric model of ℝ6
• How to use this model to explain why different voting methods may give different outcomes for the same set of voters' preferences
• How these models extend to higher dimensions when there are more than three candidates

## Expectations

Mathematics is a very active discipline that is best learned by doing rather than by observing. One of the features that makes your Wheaton education so special is that we have time in small classes to explore material together. The class meetings are not intended to be a complete encapsulation of the course material, but instead will focus on the major concepts from the Pre-Class Assignments and clarifying the more subtle ideas in the course.

You should expect to put in approximately 3 hours outside of class for each scheduled hour of class. In other words, expect to spend a roughly 9 hours per week on Math 217 outside of the scheduled class meetings. There will be some weeks where you spend more time, and there may be some weeks where you spend slightly less.

## The Honor Code

We operate under the Wheaton Honor Code for all of your academic work at Wheaton. This carries certain freedoms and responsibilities for both you as a student and me as a professor. I take this quite seriously.

Most likely, no Honor Code issues will arise this semester. If you are uncertain about whether a particular situation falls under the Honor Code, then please consult with me. However, if an Honor Code issue does come up, I will assume that you are prepared for the full consequences. Remember that you should write out, and sign, the following statement on all course work:

"I have abided by the Wheaton College Honor Code in this work."

## Evaluation

 Class Engagement/Participation 15% Problem Sets + Final Assignment 30% Group Presentations 20% Two Take-home Exams 35%

### Class Engagement/Participation

A significant part of the class meetings will be devoted to working in small groups on problems that delve more deeply into the content introduced in the Pre-Class Assignments and discussed at the beginning of class. A substantial amount of your learning will happen during these collaborative sessions by bouncing ideas off of other students and seeing how other groups approach the problems. I will determine your Engagement/Participation grade for each class meeting using a binary scale: You were present and engaged with your peers or you weren't.

However, I also know that there may be times when you have a valid reason for missing class, especially this semester when there is so much uncertainty and we're taking extra precautions to keep each other safe. I'll be really flexible, so if you need to miss class, please let me know. Let's just keep the lines of communication open.

### Problem Sets + Final Assignment

You will have a Problem Set due many Thursdays at midnight. I firmly believe that one of the best ways to build your understanding of mathematics is to explore the ideas with other students. You can discuss the problem sets with other students in the class, but you must write up your solutions independently. There are more details about the logistics and expectations for your write-ups on the Guidelines for Problem Sets page.

The Final Assignment will include some problems similar to those on the Problem Sets plus an end-of-semester essay.

### Group Presentations

You will give two group presentations during the semester: A short one in early March where your group will describe the current redistricting process in a particular state, and a longer one at the end of the semester where your group will teach the rest of us about a topic that we haven't had the opportunity to discuss during the semester. You will form groups of two or three students of your own choosing. I will give more details about the Group Presentation during the semester.

### Take-home Exams

The purpose of the exams is for you to demonstrate your understanding of the course material and, just as importantly, to give you feedback on where your understanding is strong and where you may need more work. The exams will be open-note take-home exams where you will have several days to work on them. See the Tentative Daily Syllabus for dates of the exams. I will provide more details about the structure of the exams as the time gets closer.

## Getting Help

Please come see me during my drop-in office hours! No appointment necessary! If you have a conflict and cannot make my office hours, please email me and we can set up an appointment for another time.

Remember that the goal of the course is to help you develop your mathematical thinking! If there's any point where you feel that the structure of the class isn't working for you, please come by and we can figure out some possible strategies.

## Having difficulty purchasing required materials?

Support is available for students having trouble purchasing required materials for classes. Students can contact Karen McCormack (mccormack_karen@wheatoncollege.edu) in the Office of the Provost for help finding support for required materials.

## Accessibility at Wheaton

Wheaton is committed to ensuring equitable access to programs and services and to prohibit discrimination in the recruitment, admission, and education of students with disabilities. Individuals with disabilities requiring accommodations or information on accessibility should contact Autumn Grant - Associate Director for Accessibility Services at the Filene Center for Academic Advising and Career Services. ~ accessibility@wheatoncollege.edu or (508) 286-8215

## Wheaton Student Support & Wellness Resources

• The Counseling Center is the confidential and free mental health resource on campus for all students and will be offering in person and telehealth care this year. To learn about services, check out the website, or give the office a call at 508-286-3905. Additionally, the Mental Health Support Line is available anytime the CC is not, by calling the front desk 508-286-3905 and following voicemail prompts to be connected to a clinician (24/7, available in languages other than English, and accessible from anywhere you are in the world).
• The Filene Center strives to support your learning pathway by fostering successful academic, career, and personal development. The academic advising staff will work collaboratively with you, faculty and campus resources to ensure that you have the access and guidance to become a confident and reflective learner at Wheaton and beyond. Contact us at advising@wheatoncollege.edu.
• Many other offices on campus can also help support the holistic wellness of students. For students who identify as low-income, first-gen, LGBTQ+, or have a faith or spiritual practice they adhere to, the Center for Social Justice and Community Impact and Center for Religious and Spiritual Life (the Base) are good places for support and engagement. The Marshall Center for Intercultural Learning supports BIPOC students and those working towards breaking down barriers across differences, and the Center for Global Education supports international students, and students seeking educational opportunities abroad. The SMART and Title IX Office support students through sexual and gender based misconduct, and the Bias Incident Response Team supports individuals through a wide variety of bias events. We encourage you to reach out to any and all of these offices for support.
• Health Services through Norton Medical Center is available to support students with a variety of physical health needs including specialty support for GYN and STI care. Contact the office at 508-286-4500 to make an appointment for care. There is no copay for visits and most services are free, with select procedures and labs billed to insurance.