(January 23, 2020) Is it possible to teach students about gerrymandering and the Four-Color Theorem by asking questions and not lecturing students about anything?* My plan was to ask the students just four questions:

1) What’s the difference between the House and the Senate?

2) If your state has a population of 50 people, 30 affiliated with the blue party and 20 affiliated with the red party, and you’re electing 5 representatives to the House, what’s the most fair way to draw district lines?

3) Could the red party win a majority of your state’s seats in the House?

4) What’s the minimum number of colors needed to color the Pennsylvania Legislative Map?

My hope was that when the students started discussing and asking questions, that my answers provided any direction they need.

**VOTING ON MILK**

When I posed the first question above, it turned out that the students weren’t totally sure what the House and the Senate are. One student tentatively posited that they are part of the government.

“What is the government for?” I asked, as I wanted to give our lesson some context. The students knew that governments engage in foreign policy and make laws. So we had two branches of the U.S. government covered. I wanted students to know that there is another branch, but I didn’t want to just come right out and tell them about the Judicial branch. I didn’t have a question planned, so I just blurted out the first thing that jumped into my mind:

“Suppose the Legislative Branch passed a law stipulating that no citizen is allowed to keep milk in your refrigerator for longer than 21 days. What are some possible problems with that law?” Students came up with ideas, but I was hoping to get them to think of the challenge of interpretation. (An important skill in mathematics is defining your terms, so I was hoping students might ask “How does the government define milk? How do they define refrigerator?” Those are the questions a mathematician would ask. The students did not ask these questions, so I asked “What if some people drank soy milk and wanted to keep it for longer? Would that be okay?”

“That would definitely be the problem in my house!” announced M. Then followed discussion that generated the idea of courts and the Judicial branch and also the House versus the Senate. (I almost wrote here, “We were finally ready to get into some math,” but we were already into some math with the issues of definitions and precision.)

**DOES PROPORTIONAL MEAN FAIR?**

I drew on the board 5 rows of 10 dots: the top two rows red and the bottom three blue. “Suppose your job is to decide where the boundaries of your 5 districts will be. How would you draw the lines?”

F immediately saw a flaw in this model. “Why do they live in nice little rows?” I acknowledged that, as we sometimes do in mathematics, we’re oversimplifying to make a point and real life is messier.

No one had an idea how to draw the lines. “If we imagine each row is a street, what would the election results be if each street was a separate district?” Students calculated: 3 blue representatives and 2 red.

“Are those reasonable places to draw the lines?” The students said yes. I told them of the belief that legislative districts be “compact.” “What does compact mean?” Students talked about it for about 30 seconds and were done. I must tell you that I spent a very long time before class wrestling with and researching the idea of what compact might mean. It seems that no one can come up with a good definition of this term. I find that incredibly interesting, probably the most interesting concept in this entire discipline. But the students did not find this interesting at all, so in the spirit of IBL, I moved on.

What the students really wanted to talk about was whether this district division was fair. I told them that the conventional wisdom is that proportional representation is fair. M brought up the point that if the red party only has 40% of the representatives, it will always be outvoted and its interests will never become law. At this point, we had moved into the context of our milk-law discussion from earlier. The students wondered: if 40% of the citizens want to legally keep soy milk for more than 21 days, but 60% of the citizens want soy milk to get discarded at the same time as the dairy milk, how is that fair for soy-milk drinkers? In other words, how is proportional representation fair if 40% of the people don’t get their legislative desires satisfied? (This is something I had never thought about. I am so happy to be exposed to a new idea from students!)

At this point, A took the marker from my hand and demonstrated on the board what might be more fair: 5 people from the blue party switch their allegiance to the red party. Of course, A explained, we would have to know what the positions of each party are so that these 5 people really want to switch. At this point, we’d have 50% from each party. “If your district is entitled to 5 representatives,” I countered, “then what would the election results be?”

Students realized here that things are more complicated than they seem on the surface. They came up with a few ideas of their own:

- Instead of having a national government with a Congress comprised of people from each state, let states govern themselves and just have one nationwide annual meeting.
- Let’s have neutral representatives! Why do they have to be connected to one specific party?

**MAKING LEGISLATIVE DISTRICTS
MORE COMPACT**

“Suppose people on one end of the street didn’t even know their neighbors ten houses down and decided that the districts would be more compact if grouped vertically, with their neighbors behind them. Would the election results change?” Students calculated and immediately agreed that this districting is more compact but less proportional and less fair, as the blue party gets all of the seats. By optimizing one variable, we weakened another.

“When are we getting to gerrymandering?” asked F presciently.

**RIGGING THE SYSTEM**

“Can we draw the lines so that red wins?” I asked. M had seen this example before, so we already knew the answer. So I changed the question: “How can we draw the lines so that red will win?” I gave each student a paper with a 5x10 grid of dots, a red pencil, a blue pencil, and a regular pencil. I instructed them to circle the dots in the top two rows red and the bottom three rows blue. “Use the regular pencil to draw your district lines.” The students got to work.

“Is this legal?!” asked someone incredulously.

“I can’t do it,” said F.

“Imagine it’s your job. Your boss has hired you to redistrict in favor of a certain party. If you can’t, you’ll get fired.” I hoped drama component would stimulate creativity. It didn’t.

“Good. I wouldn’t want that job,” said F adamantly. As the students worked, they were still holding on to some shock about the legality of this.

Soon, M had figured out a way. “Are your districts compact?” I asked. They were. “What happens sometimes is that districts get gerrymandered into shapes that are not compact, that are the opposite of compact. Can you do it that way?” No one was able to get five non-compact districts of the same population with red winning, but M came close. “Can you look at that weird-shaped district and see an animal shape?” She saw a rhinoceros-shaped district.

I passed around the legislative map carved out by Massachusetts Governor Elbridge Gerry in 1812 and asked students why they think it’s called gerrymandering. They figured it out. We also discussed the pronunciation of the word and name. The students helped to correct my pronunciation for the rest of the class.

“Are there any examples of this
happening in our times, or did it just happen in history?” asked someone. I passed
around some maps of current gerrymandered districts (Maryland’s 3^{rd }district,
North Carolina’s 12^{th}, Florida’s 5^{th}, and Texas’s 35^{th}).
Then one of the most famous gerrymandered district, our local former 7^{th}
District of PA.

**GERRYMANDERING IN PENNSYLVANIA**

I then gave students a handout comparing PA’s congressional districts from 1992 versus 2011. After students discussed differences, I gave them a handout showing how the districts changed from 2011 to 2018 after the PA Supreme Court declared gerrymandering illegal in our state. The students examined these maps and asked a lot of questions about the legality of gerrymandering. My helper Ellen did some in-class research and answered student questions about this.

**MAP COLORING**

I then gave everyone a map of the current congressional districts in PA. “What’s the minimum number of colors you need to color this map?”

“Can we try?” asked someone.

“First, in the spirit of mathematics, make a conjecture about how many colors it will take,” I responded. Conjectures ranged from 3 to 18. They then tried it. F realized it would be quicker to test conjectures using hashmarks to symbolize colors. After a short time of asking clarifying questions, making mistakes, and testing ideas, the consensus was that you need four colors to color the PA Legislative Map. I asked 2 follow-up questions:

- Is it possible to color this map with 3 colors if you try harder?
- Are there any maps that require more than 4 colors?

M had seen a map of the U.S. colored in four colors, so we knew that was possible. I added these questions to today's running list of questions on the board.

We were almost out of time when the students asked to return to our graph theory problem from last week.

**GAS WATER ELECTRICITY PROBLEM**

I asked whether anyone had thought about this problem since last week. A few had. No one had come up with a solution. “If you were allowed to change the rules to make it easier, what would you change?” The students agreed that allowing one pair of lines to cross would make a solution possible. As we were discussing this, F jumped up to the board with an idea.

“Never mind,” he said, partway through, and sat down.

**FOLLOW UP AT HOME**

- Come up with an actual mathematical definition for “compact” (if that interests you!).
- Do some research on different perspectives on whether proportional voting systems are fair.
- Find out whether gerrymandering happens outside of the U.S.
- Ask your student what PA’s former
7
^{th}district resembles. (Some call it Goofy Kicking Donald Duck, which I did not mention in class.)

**GRATITUDE**

This class would not have been possible without my helper Ellen or content suggestions from Haley Horton. Before leading this session, I posted to the Facebook group 1001 Circles a request for help connecting gerrymandering to the Four-Color Theorem. Haley came through, with the idea of using our local legislative map and also content from this Washington Post video on gerrymandering. The video is less than 3 minutes and covers a lot of what we did in class today.

Rodi

*This plan was different from a pure “Ask Don’t Tell” approach because I was prepared to give students information in answer to their questions. We don’t have enough time in our 90-minute sessions to allow for students to research all answers. So I planned to answer (with Ellen’s help) at least some student questions beyond the scope of mathematics. In a pure IBL (inquiry-based learning) class, students would do their own research and possibly go off on a tangent. I used a guided-inquiry approach here in the interest of time and also because I wanted the focus to stay on mathematics.