I want my CSA sophomores to understand many sorts of visual representations of data — tables, charts, graphs, etc. — especially in the context of elections, since we’re applying mathematics to models of voting. This summer, of course, we have a wealth of material to choose from, both prospective (polling data) and retrospective (historical data). Perhaps the best source for both is FiveThirtyEight; Electoral Projections done right; the only downside is the complexity of their representations. Being a teacher, I try to turn this complexity into a teachable moment, but it’s still necessary to figure out how to help kids digest and understand the representations. Here are two examples:

- One day we looked at FiveThirtyEight’s Electoral History Charts. Here’s a small part of one chart, so you can see what I mean about complexity:

This is pretty overwhelming even for college students, so it certainly can’t simply be thrown without explanation at kids who’ve just finished their freshman year in high school. After digging into it, you realize that it’s a wonderful (and almost successful) attempt at representing several different dimensions of information in one two-dimensional table — but it does take some work:- It’s clear what’s going on horizontally. Each row represents one state.
- The vertical axis is almost as clear; one realizes from context that each column is a presidential election fro 1948 through 2004.
- By now we’re all familiar with the symbolism of red states and blue states, so of course the red cells show elections where the state voted for the Republican presidential candidate, and the blue cells where the state voted for the Democratic candidate.
- But there are different shades of red and blue: the darker and more saturated colors (such as Maine in 1956 or Massachusetts in 1996) represent landslides. The lighter or less saturated the color, the closer the election was (in that state); really pale colors, almost indistinguishable from white, show races that were so close that they were almost a tie.
- Finally, there’s a letter and a number in each cell. These show the winning party and the margin by which it won. For example, in 1984 the Republican candidate (Reagan) took Connecticut by a 22-point margin.

The class is already divided into six groups, each with five students. To get them to dig into the data

*and*to understand the interpretations, I assigned each group a region of the country and asked them four questions:- For each of your states, figure out the average margin by which the Republicans won those states in the past ten elections (1968–2004). One way to do this is to count the “r” numbers as positive, the “d” numbers as negative, and average them. Do this separately for each state in your region(s), and list the results.
- If your region(s) contained both positive and negative averages, find the state with the most positive average and the state with the most negative average. List them, along with their averages. If your region(s) contained only positive or only negative averages, find the states with the smallest and largest averages, and list them, along with their averages.
- In which year did your region(s) vote most heavily Republican? Find out who the presidential candidates were in that year.
- In which year did your region(s) vote most heavily Democratic? Find out who the presidential candidates were in that year.

This all took a long time, but it worked out well. Each group then reported their findings back to the whole class, after which we were able to make some horizontal and vertical generalizations: What happened (nationwide) in 1984, where we see a sea of red? What happened (chronologically) to Vermont? And so forth.

I hear you asking what the chart does when a state votes for a third-party candidate. Using the RGB principle, the site colors the cell green. For example:

This, of course, gave us an opportunity to talk about third-party candidates. Students were suitably amazed when they learned how a segregationist candidate (Strom Thurmond) could have won Alabama by a huge margin in 1948, despite what should have been a large number of black voters. - These charts are abstract and difficult, partly because they’re so textual. So what about a more visual representation, such as a map? Maps are a great way of presenting this sort of information, though they really have to be snapshots in time and are therefore not suitable for longitudinal historical data. It turned out that the maps we used had a great side benefit: students learned a bit about geography (which isn’t much taught these days), since most of the maps omit the names of the states. Here were two of the maps we used, the one on the left showing the current polls for the upcoming election, and the one on the right showing the results of the 2004 election:

These maps let me ask questions like these:

- Of the states that Kerry won in 2004, which ones are now likely to switch to the Republican candidate (McCain) according to the map? What is the total of their electoral votes?
- Of the states that Bush won in 2004, which ones are now likely to switch to the Democratic candidate (Obama)? What is the total of their electoral votes?

More on simulations later. Stay tuned…

Categories: Math, Teaching & Learning