Math in Society, 2nd Edition: Tools for decision making

In this chapter we are going to study some of the math used in the United States government today. In doing this, it is very important to acknowledge the Native American genocide and race-based chattel slavery which are often left out of math textbooks.

We are including this history to show how racist policies, laws and practices like segregation, racial profiling, police brutality, mass incarceration, redlining and voter suppression continue to this day to limit the freedoms, rights and economic prospects of African Americans. With the COVID-19 pandemic and the death of George Floyd by a police officer, there is greater awareness of systemic racism than ever before.

Scholars estimate that people have lived in the Americas for over 20,000 years (Worrall, 2018). By the time Columbus arrived, more than 50 million people were living in the Americas and around 10 million Indegenous people were living in the area that would become the United States (History.com Editors, 2018).

The first colonists arrived in America in 1607 and African people who had been violently separated from their families and homeland were brought to Jamestown in 1619 as slaves. Slave labor was used on tobacco and cotton plantations, for domestic work and even as part of the labor for building the White House. The system of slavery was an integral part of the United States economic and political system (How Slavery Helped Build a World Economy, 2003).

At the same time, white settlements kept encroaching on the homeland of Native Americans. Many treaties were signed and then violated. For example, in 1785, the Treaty of Hopewell was signed with the Cherokees in Georgia, which set a boundary on white settlement, but there were already colonists past that border.

The U.S. government was officially formed in 1787. During the Constitutional Convention, the framers were tasked with making decisions on the three branches of government and how people (or those considered to be people) would be represented. At that time only white men with property were allowed to vote. Women were also considered to be property of their husband at the time.

In creating the Legislative Branch of government, it was decided that Congress would be made up of the Senate and the House of Representatives. Representation for the Senate would be equal with two senators per state. To account for the different sizes of the states, the representation in the House of Representatives would be proportional to the state’s population, but there was a controversy over how to count the population.

During the Constitutional Convention, the northern and southern states were at odds on whom to count. The South wanted more representation in Congress which would also increase their power, so they argued to count slaves as part of their population. The southern states did not want to count slaves for taxation, however, because that would result in them paying more taxes.

Slaves were considered property, not humans, and they were not citizens. Therefore, they did not have the right to vote or participate in government. The anti-slavery North wanted to count only free people, which included free Blacks. This led to the Three-Fifths Compromise that determined three out of every five slaves would be counted toward a state’s population and taxation (Clayton, 2015).

It is also important to note that there are about 4 million U.S. Citizens who live in the territories of Guam, the Virgin Islands, the Northern Mariana Islands and Puerto Rico who do not have representatives or senators. They pay federal taxes like Social Security and Medicare but not Federal Income tax.

The number of representatives each state gets is based on its population, so once the framers decided how to count the population, they had to figure out how to divide up the representatives. This math problem is called apportionment.

Apportionment is the problem of dividing up a fixed number of things among groups of different sizes. In the United States, there is a certain number of representatives as stated in the constitution, currently 435, and they need to be divided fairly among the 50 states. Since the states are different sizes, and we cannot use fractions of people, this is not an easy task. In addition, the population in each state may change over time. Every 10 years after the census is taken, the representatives are reapportioned.

The apportionment problem comes up in a variety of non-political areas too. Here are the rules for apportionment in general.

We will look at four ways of solving the apportionment problem, developed by different people: Hamilton, Jefferson, Webster, and the Huntington-Hill method that is used today. We will continue to look at U.S. history throughout this section including Hamilton, Jefferson and Webster’s relationship to slavery.

Alexander Hamilton (1755 - 1804) was raised in St. Croix in the U.S. Virgin Islands by a poor family. He was motivated by his low socioeconomic status to work himself into a higher social standing. He eventually came to the colonies and worked himself into circles of wealth and influence. While Hamilton wasn’t pro-slavery and considered himself an abolitionist, when choosing between his societal status and moral obligation, he chose the former.

He wed Elizabeth Schuyler, who was from a prominent family who owned slaves. He was involved with the transactions of slaves for his in-laws which further muddled his anti-slavery stance. Furthermore, Hamilton also traded and sold slaves as part of his duties for the Continental Army.

Alexander Hamilton proposed the method that now bears his name. His method was approved by Congress in 1791 but was vetoed by President Washington. It was later adopted in 1852 and used through 1910. He begins by determining, to several decimal places, how many people each representative should represent (the divisor).

Let’s see how this works in an example.

Hamilton’s method obeys something called the Quota Rule. The Quota Rule isn’t a law, but an idea that some people think is a good one.

After using Hamilton’s method for many years, three paradoxes happened, on separate occasions, where unfair things happened in new apportionments. This led to other methods being needed.

The Alabama Paradox is named for an incident that happened during the apportionment that took place after the 1880 census. (A similar incident happened ten years earlier involving the state of Rhode Island, but the paradox is named after Alabama.) The post-1880 apportionment had been completed, using Hamilton’s method and the new population numbers from the census. Then it was decided that because of the country’s growing population, the House of Representatives should be made larger. That meant that the apportionment would need to be done again, still using Hamilton’s method and the same 1880 census numbers, but with more representatives. The assumption was that some states would gain another representative and others would stay with the same number they already had (since there weren’t enough new representatives being added to give one more to every state). The paradox is that Alabama ended up losing a representative in the process, even though no populations were changed, and the total number of representatives increased.

The New States Paradox happened when Oklahoma became a state in 1907. Oklahoma had enough population to qualify for five representatives in Congress. Those five representatives would need to come from somewhere, though, so five states, presumably, would lose one representative each. That happened, but another thing also happened: Maine gained a representative (from New York).

The Population Paradox happened between the apportionments after the census of 1900 and of 1910. In those ten years, Virginia’s population grew at an average annual rate of 1.07%, while Maine’s grew at an average annual rate of 0.67%. Virginia started with more people, grew at a faster rate, grew by more people, and ended up with more people than Maine. By itself, that doesn’t mean that Virginia should gain representatives or Maine shouldn’t, because there are lots of other states involved. But Virginia ended up losing a representative to Maine.

“All men are created equal,” are words penned by our third president and Founding Father, Thomas Jefferson (1743 - 1826), in the preamble to the Constitution. However, over the course of his life, Jefferson owned around 600 slaves. Among these hundreds of slaves, Jefferson fathered at least six children with one of his slaves, Sally Hennings.

Over his lifetime and his writings, Jefferson wrestled with his conscience regarding slavery, which can be seen in documents such as drafts of the Constitution before its final version. In the end, his personal and monetary gain surpassed that of his concern for the enslaved.

Thomas Jefferson proposed a different method for apportionment. After Washington vetoed Hamilton’s method, Jefferson’s method was adopted, and used in Congress from 1791 through 1842. Jefferson, of course, had political reasons for wanting his method to be used rather than Hamilton’s. Primarily, his method favors larger states, and his own home state of Virginia was the largest at the time. He would also argue that it’s the ratio of people to representatives that is the critical thing, and apportionment methods should be based on that.

The first three steps of Jefferson’s method are the same as Hamilton’s. He found the same divisor and the same quota and cut off the decimal parts in the same way, giving the same initial apportionment that is less than the required total.

What changes is how Jefferson assigned the remaining representatives. He said that since we ended up with an answer that is too small, our divisor must have been too big. He changed the divisor by making it smaller and looked at the new total. He would raise or lower the divisor until he found one that produced the required total. This is a trial and error process that takes some patience.

Daniel Webster (1782-1852) was a lawyer, congressman, Senator of Massachusetts and also served as the U.S. Secretary of State. He was from the North and did not own slaves. He was an opponent of slavery extension and he spoke against annexing Texas and against going to war with Mexico. He argued, however, that no law was needed to prevent the further extension of slavery in new states and he supported the Compromise of 1850, which disappointed his abolitionist supporters (History.com Editors, 2018b).

Webster proposed a method similar to Jefferson’s in 1832. It was adopted by Congress in 1842 but replaced by Hamilton’s method in 1852. It was then adopted again in 1911. The difference is that Webster rounded the quotas to the nearest whole number rather than dropping the decimal parts. If that didn’t produce the exact number of representatives, he adjusted the divisor like in Jefferson’s method. (In Jefferson’s case, the first adjustment will always be to make the divisor smaller. That is not always the case with Webster’s method because some numbers may be rounded up.)

Let’s see how Webster’s method works in this example:

Like Jefferson’s method, Webster’s method carries a bias in favor of larger states but rounding the quotas to the nearest whole number greatly reduces this bias. Notice that Bates County, the largest, is the one that gets a representative trimmed because of the increased quota.

Also, like Jefferson’s method, Webster’s method does not always follow the quota rule, but it follows the quota rule much more often than Jefferson’s method does. In fact, if Webster’s method had been applied to every apportionment of Congress in all of U.S. history, it would have followed the quota rule every single time.

In 1830 the Indian Removal Act was signed by President Jackson. The Act relocated Native Americans to land west of the Mississippi and the forced journey is known as the Trail of Tears. To learn more, here is a link to an online museum exhibit with a timeline of Native American history and artifacts

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http://recordsofrights.org/themes/4/rights-of-native-americans

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Toward the end of the Civil War, slaves were freed by Abraham Lincoln’s Emancipation Proclamation on January 1st, 1863, but it was not until June 19th, 1865 that the last group of slaves in Texas learned they had been freed. This is now celebrated as Juneteenth

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juneteenth.com/history.htm

. You can read more about the timeline of slavery in this Jim Crow Museum of Racist Memorabilia

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https://www.ferris.edu/htmls/news/jimcrow/timeline/slavery.htm

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Although they were technically free, Black people still faced unequal and unjust treatment. After Black men were granted the right to vote in 1869, voter suppression laws like poll taxes and literacy tests were implemented barriers to voting (American Civil Liberties Union, 2020).

In 1887, the Dawes Act was passed, which allotted Native American land to individuals and 60 million acres were taken by the government or non-Indian homesteaders (Indian Land Trust Tenure Foundation, 2020). This act also provided citizenship under certain restrictions which theoretically allowed native men to vote. All women were granted the right to vote in 1920, but the barriers enacted for men of color applied to women of color as well. You can read more about voting rights and suppression in this online timeline

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https://www.carnegie.org/topics/topic-articles/voting-rights/voting-rights-timeline/

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In 1920, no new apportionment was done, because Congress couldn’t agree on the method to be used. They appointed a committee of mathematicians to investigate, and they recommended the Huntington-Hill Method. They continued to use Webster’s method in 1931, but after a second report recommending Huntington-Hill, it was adopted in 1941 and is the method of apportionment still used today.

The Huntington-Hill Method is similar to Webster’s method, but attempts to minimize the difference in the percentage of how many people each representative will represent.

Let’s look at why Hill and Huntington decided to use the geometric mean for rounding instead of the arithmetic mean of 0.5, which is halfway between two numbers. By calculating some geometric means and looking at them in a table we can see a pattern:

Notice that each geometric mean is between \(n\) and \(n+1\text{.}\) For smaller numbers, the decimal part is less than 0.5 and as n gets larger, the decimal part gets closer and closer to 0.5. This gives smaller states a chance to round up before larger states. That’s how the Hill-Huntington method reduces the bias toward larger states and makes it more fair for smaller states.

Now let’s see how to use the geometric mean in an example.

In 1980, two mathematicians, Peyton Young and Mike Balinski, proved what we now call the Balinski-Young Impossibility Theorem.

In this section we have learned four different methods of apportionment used in U.S. history, intertwined with the U.S. treatment of women, Native Americans and Black people. In the next section we will continue with our brief history of voting rights and look at different methods used for voting.