# Integrals and Equity

## A math lesson prompts new awareness for prep school students-and their teacher

Illustrator: David McLimans

An AP calculus class at a prestigious boarding school doesn’t seem a likely venue for student reflection on privilege and wealth. But when I taught a group of academically inclined math enthusiasts several years ago, I made some important discoveries, and I believe I helped my students view their world with a wider lens.

It was my sixth year of teaching math and I loved it—the intellectual challenge, the complex puzzle that would suddenly sort itself when that key piece was put in place, and the interconnectedness that appeared among seemingly disparate ideas.

I took a new job at a small private boarding school in central Massa-chusetts. I was impressed with the way the school guided its 300 students to become moral, thoughtful adults. There was a school wide work program, so everyone pitched in to maintain the buildings and grounds. They had an extensive community-service program, where students volunteered at soup kitchens, nursing homes, and periodically undertook larger projects such

as building homes for Habitat for Humanity.

The academics, of course, were rigorous and nearly 10 percent of that year’s graduating class would find themselves with Harvard admission letters. The students were privileged, and the school knew it was preparing future leaders.

### The Lesson Begins

My AP class had a weekly “lab period,” which provided a luxurious 90 minutes of class time compared to the standard 42 minutes. The class had this extra time to allow for AP exam test prep, but it also afforded the opportunity to conduct some more exploratory lessons, which I tried to do as often as possible.

As with my other lessons, I planned this one with objectives that were decidedly mathematical. We had been working with integrals, finding areas under various curves and calculating infinite sums, and I wanted to introduce my students to a real-world application of this mathematical concept by examining the GINI Index.

The GINI index is a measure of income distribution (or dispersion) across a population—in this instance, we considered households in the United States.

In a single number ranging from 0 to 1, the index provides an indication of the equitability of this distribution. A GINI index of 0 corresponds to perfect economic equity—every household has the same amount of income. A GINI index of 1 corresponds to extreme inequity, where one household has all (100 percent) of the income and all others have none. These are theoretical extremes; in reality, the number lies somewhere in between.

The index is calculated by first dividing the population into income groups—for example, by quintiles (fifths) or percentiles (hundredths)—and finding the percent of total income “owned” by that group of households. The group that represents the lowest quintile, for example, will hold less than a fifth of the total income (the accumulation of all households). The top quintile will have more than 20 percent of the total income.

Here is a graph of income distribution from 1996.

The accumulated percentage of income is plotted on the vertical axis against the percent of households (horizontal axis). The point (20, 4) means that the lowest 20 percent of households held 4 percent of the total income. The point (40, 13) means that the lowest 40 percent (the two lowest quintiles together) of households held 13 percent of the total income.

The GINI index is twice the area between the 45 degree line (representing perfect equality) and the income curve. Thus it measures the “gap” between perfect equality and the state of affairs described by the income curve. Note that the curve always includes the points (0,0) to (1,1). Its exact shape is what changes, depending on the distribution of income.

### Estimating Income

After a brief introduction to the idea of income distribution, we began talking about income in the United States. Perhaps not surprisingly, my students made reasonable estimates when I asked them to guess what percent of total income the top 20 percent of the population held. And they made somewhat reasonable estimates for the other quintiles. They didn’t seem troubled by the fact that the top 20 percent of households held 47 percent of the wealth. (I assumed they attributed the disparity to super-rich individuals like Bill Gates and Michael Jordan.) But the most meaningful conversation arose when I asked them to consider the median income of the various quintiles. The median represents the middle value of the set of incomes in the quintile, often close to the average. I was perplexed by their lack of reaction to the percentage distribution and I was curious what they thought the median incomes might be.

“Median income of the lowest quintile—what do you think?” I asked.

“Eighteen thousand,” someone guessed.

“Twenty-two thousand,” another said.

I recorded their estimates on the board. I didn’t ask for any justification, leaving their reasoning private. Looking back, this is somewhat surprising as I was always asking my students “Why?” and “How do you know?”

I then asked them to guess the median income of the top quintile.

“One-hundred seventy-five thousand.”

“Yeah, around there.”

“One sixty.” “Two hundred.”

And others offered thoughts.

“And how about the next quintile down—this 60 to 80 percent group?” And we went through the rest. Once we had their set of values on the board, it was my turn to tell.

“Median income of the lowest quintile is $8,600,” I reported.

The response was “Whoa,” followed by some silence.

“That means half the households are below that—10 percent of the people.” Although this fact seemed to be mind-blowing for many, they did not challenge it.

I continued. “The next quintile up is $21,097. The third quintile is $35,486. The fourth quintile is actually $54,922, and the top quintile is $115,514.”

My students expressed disbelief. “No way!” “Is that right?” I think the median incomes of the fourth and fifth quintile surprised them the most: They had thought that the fourth quintile would be around $90,000 and the top quintile would have a median income closer to $170,000.

Given their privileged backgrounds, perhaps I should not have been shocked that the students’ conjectures were so far off. And perhaps I should not have been taken aback at their disbelief upon learning the median incomes for each quintile. Though I don’t know the actual incomes of these students’ families, this group was firmly situated in the top quintile, and most the top decile. These data were news to them—top headlines.

But I was surprised, perhaps most surprised, by the fact that the students had so little sense of their economic positions relative to the rest of society—this was despite all their community service, attention to issues of race and class in their history and literature classes, and interactions with classmates who were there on scholarships. Even with these experiences, the students inhabited a world where they could think of their families as more or less “average.” In their social context there was very little evidence that would have led them to believe that only one in 10 people was as economically advantaged as they were. I don’t know that this group of students had ever been asked to look at the “cold, hard numbers” before and pinpoint their positions relative to others. They knew they were from privileged backgrounds, and had “more,” but this quantification of how much more they had—orders of magnitude—was quite a surprise.

### Worlds Apart: Math and Social Justice

The experience of seeing these students gain awareness of their positions in society helped me along in my evolution as a teacher. I had gone into teaching because I loved math and loved supporting students in their journeys to become good people living in a challenging world. I also wanted students to be attentive to their communities and others around them, to understand that the world does not offer all an equal playing field. Ideally, I hoped they would use these understandings to change the world in ways that helped address some of the gross disparities that exist. But these were separate aspects of my “teaching.” Math took place in my classroom. It existed a world apart from what took place in the dorm, on the soccer field or basketball court, during the van ride to the soup kitchen, or over an ice cream at the favorite local shop.

In retrospect, it’s odd that these were separate worlds for me. I had been a math major at a progressive and politically active university. I had participated extensively in volunteer activities and lived in the Public Service House, a dormitory whose members were committed to community service. But math and being a good person in the world were separate for me there as well. My aspirations for social justice had not had a place in the math classrooms at the university and I didn’t bring them with me to the mathematical part of my teaching. These worlds were not in opposition, but I didn’t see them as natural bedfellows. Math was about the intellect and individual power. Being a teacher for me was about giving to society and connecting with individuals in ways that might make transformation possible.

It became clear to me that there was a large gap here, for me and my students. Despite my prep school students’ general awareness and participation in activities, they remained unaware of their positions in that landscape. The hard and fast numbers brought this into relief.

I would like to say that I followed up on this initial lesson right away and implored students to examine other data that helped them understand themselves anew. We could have pursued the difference between men’s and women’s incomes, or the incomes of different racial groups. We could have compared the United States to other countries. We could have looked at historical trends in the GINI index, and analyzed the factors that contribute to the greatest and least equity in income distribution over time. Regrettably, I did not pursue any of these.

Nonetheless, I was left with a powerful lesson. Through this activity I realized that those parallel pursuits could serve one another—where mathematical learning goals, even on a high level such as AP calculus, could be pursued at the same time as goals aimed at personal growth and social awareness.

#### Reference

The Office of Social Justice website, www.osjspm.org/101_income.htm, is a good teaching resource.

This article will appear in the forthcoming book Rethinking Mathematics: Teaching Social Justice by the Numbers.