When I learned how to multiply matrices in 10th grade, my initial reaction was, “Why on earth would anyone ever want to do that?” Compared to addition and subtraction, the rules of matrix multiplication seemed arbitrary and meaningless. Alas, this perception lasted for years, as I’m sure it does for many college students, particularly non-math majors. Thus, in honor of Martin Luther King Day, I present here a method that will allow your students to see why matrix multiplication is intuitive and can have important applications for racial justice.
In February 2013, the Wall Street Journal Reported, “Prison sentences of black men were nearly 20% longer than those of white men for similar crimes in recent years, an analysis by the U.S. Sentencing Commission found.” Is this evidence of racism, intentional or subconscious, on the part of judges? We might initially think so; after all, the statistics have already been adjusted for type of crime committed, so the outcome shouldn’t be skewed by one race committing more serious crimes. However, “The commission, which is part of the judicial branch, was careful to avoid the implication of racism among federal judges, acknowledging that they ‘make sentencing decisions based on many legitimate considerations that are not or cannot be measured.’” That may be, but some factors can be measured, and that is what we will try to suss out here using matrix multiplication.
Perhaps what we are really seeing is the effects of class. (Aside from the original WSJ quote, the following percentages are made-up). Suppose for a particular crime low-income defendants are handed down a long sentence 30% of the time, middle-income defendants 20% of the time, and high-income defendants 10%. It could be due to the high cost of good lawyers, biases by judges, harsher laws in states that have more poor people, etc. Now suppose 85% of black defendants are low-income, 10% are middle-income, and 5% are high-income. What percent of blacks should we expect to be receiving long sentences if the reason is class alone? Clearly the answer is:
Note that what we’ve just done is to take the dot product of two vectors:
Which we can also write like this, for reasons that will soon become clear:
More complicated than addition? Sure. But as this example illustrates the definition of dot product is not some arbitrary construction of interest only to mathematicians. There are times in life when multiplying the individual components of something separately and then adding them all up is extremely useful. Now suppose the percentages of low, middle, and high income white defendants are 60%, 25%, and 15% respectively. We can do the same thing we did for blacks by changing the values in the second vector:
But since the first vector is the same in both cases, let’s see if we can avoid having to write it twice:
And tada! You have matrix multiplication. It’s no more scary or complicated than vector dot products; we’re just doing those dot products side-by-side to save space. Those savings become even more apparent when we add more rows and columns. Perhaps we want to include a third column for Asian-Americans:
Or a separate row for death sentences:
And now that we have these percentages, we can start to make inferences…carefully. If the number in the top left corner of our final matrix is lower than the actual number of blacks serving long sentences, than we have good reason to suspect that racial disparities in sentencing cannot be fully explained by class differences. Or more broadly, if any of the numbers in our final matrix do not reflect the numbers we actually see, and we determine the differences are statistically significant, than we know that class cannot be the sole cause of the racial sentencing gap. Of course, there are many other things to control for including criminal record, age, whether the defendants live in heavily-policed urban areas, etc. And one can still ask the broader question of what societal and historical forces have led higher percentages of blacks to end up in poverty in the first place. I’m not claiming that you can fully explain racial sentencing disparities in one lesson. What I am claiming is three things:
- Matrix multiplication seems less arbitrary when it is motivated by a real-world problem
- Matrix multiplication appears less complicated and overwhelming when you start with the dot product of vectors and then expand it one row at a time
- There is plenty of room in the curricula of introductory college math courses to tackle race, class, and social justice
And indeed, there is no excuse to stand on the sidelines in an age of such inequality and injustice. Many math, science, and engineering students badly need to be exposed to the reality of these inequities, and what better a place than in a course that they value, in a context they find engaging? Students from other disciplines merely trying to fulfill their quantitative requirement might suddenly find that math is important to the world they live in and the values they hold. Moreover, both types of students will learn how math can serve as a valuable tool for fighting injustice. Even if they never use that tool themselves, they might someday be faced with a problem like this and realize that hiring someone with mathematical expertise would prove useful. And if you’re still not convinced, recall that much of our research is funded, however indirectly, by the American taxpayers. It helps to have voters to whom math matters.
Palazzolo, Joe. “Racial Gap in Men’s Sentencing.” Wall Street Journal. Feb 14, 2013. http://www.wsj.com/articles/SB10001424127887324432004578304463789858002 (Accessed January 21, 2016).