It’s important to make sacrifices for your craft. Today, for instance, I had to pony up $12 to the Wall Street Journal so that I could read about why calculus is so last century. (You will have to do the same.) The article’s title (I confess that at the writing of this opening paragraph I haven’t read past the paywall yet) reminded me that I have a whole soapbox about why everyone and their grandmother seems to take calculus, despite the large swath of its content being useless to all but engineers, physicists, math majors, and the occasional business or biology major. Let’s get started!
Now I have read the article. I’ll quote some of it here, which I’m sure is fair use on a two-post blog that has me about thirty-four dollars in the hole.
Was that quote too short? I’m sorry, that’s a consequence of the fact that I just paid for a nine-paragraph article! I may have the attention span of a millennial goldfish, but I make exceptions for the first time ever I bother paying my way past a paywall.
We’re not saying calculus shouldn’t be taught. — But the singular drive toward calculus in high school and college displaces other topics more important for today’s economy and society. Statistics, linear algebra and algorithmic thinking are not just useful for data scientists in Silicon Valley or researchers for the Human Genome Project. They are becoming vital to the way we think about manufacturing, finance, public health, politics and even journalism.
Linear algebra holds a privileged place in my heart, probably because it is the first time a student sees the “map that preserves an important quality” theme of mathematics. This is such an important feature of mathematical thinking that an adjunct instructor at a southern community college with a crappy blog might be willing to call it the most important feature, but he doesn’t want to get yelled at.
But why is it so important, Dr. Li and Professor Bishop (the former of whom runs a data science training institute)? I squirmed through a master’s degree without really learning statistics, and some attempts at learning topological data analysis (or even just regular data analysis) were thwarted by my 50-hour work week. It’s not obvious to me what relevance my appreciation for linear algebra as to do with understanding large data sets. (Otherwise I would be able to understand them.) (I realize I can and should educate myself, but I just read a persuasive article presuming to discuss the topic.)
My first point is this: unless you were already jonesing for a subscription to the WSJ, don’t pay for a nine-paragraph advertisement for learning data science.
I have a second point, and I think it’s much more interesting.
The WSJ article’s main point was that calculus was a critical body of knowledge in the space-and-missle-ridden 1900’s, but that in the information age it is less useful to students. This thesis supposes, as so many do but so few question, that the purpose of education is to get you a job.
Maybe it is. Education costs so much, whether in money, opportunity, or sanity, that it is not unreasonable for a student to expect a steady paycheck as a result of their hard work.
Maybe it isn’t. Maybe education, or at least the compulsory twelve years we all shuffle through in the United States, should be a public good designed to produce more well-rounded, more knowledgeable Americans.
In either case: who in God’s name needs to know how to evaluate an integral by trigonometric substitution? The answer is whoever tells WolframAlpha how to do it. Does the robot know how to do it now? Good, that first person can forget.
“But, idiot,” you say: “What if the computer forgets? What if all technology is destroyed in a freak EMP burst?” Right, and what if an airplane falls on me in front of this hookah bar? I should not have left my house today. We’d have bigger problems in such an unlikely scenario than losing a calculus trick, and if we still had the Pythagorean theorem, it would be fairly easily recovered anyway. (Exercise: Do you know how to explain trig substitution with just the Pythagorean theorem? It’s pretty neat.)
Look, I’m not saying that there’s not a class of people well-suited to preserving the ancient and mysterious art of crunching integrals and derivatives. I’m just saying that those people are not Every 18-Year-Old.
I believe calculus is incredible, and I believe as a subject it has something to offer both the utilitarian and hippy-dippy views of education I put forward at the beginning of this section. I do not believe—even if you are taking calculus to get an engineering job—that your life is improved by knowing the quotient rule.
Here are some interesting things that you could learn about in a calculus class. You could learn about how the limit was devised to answer a paradox posed by the Greek Zeno regarding motion, and how it can be applied to allow humans to use arbitrary small and large quantities. Without the limit, you’d try to invent a derivative that divides by zero and Newtonian physics wouldn’t exist. Instead, we take these arbitrary quantities that could as well be imaginary and apply them to solve real-world problems ranging from chemical decay to population growth to compound interest and joke’s on you, dear reader, those are the same thing anyway!
Calculus is a stroke of human genius, and we have relegated it to a required class in which the student comes away thinking that calculus is for slipping ladders and estimating how far away something is when you’re driving. (In the differential calculus course I TA’d as a graduate student, I asked my students for a freebie at the end to list some applications of calculus. One of my favorite students answered with the preceding. We are giving them the wrong problems.)
You can’t beat Something with Nothing, so goes the theorem of Republican game theory, and I am thinking of Something. This is where my article will sputter out and be ultimately unsatisfying, but at least I didn’t charge you $12.
Something here is a class, maybe called General calculus or Calculus: history and application. It consists of two pieces: whether these pieces are one, then a midterm, then the other or interwoven is currently unknown to me. The first act is the cultural and historical context in which calculus was birthed, all the times it was birthed: not just Newton and Leibniz, but Zeno and Weierstrass too. The second act is what calculus does for us now: e.g. optimization, arc length (required to be paired with a nod to Descartes’ claim we could never do it), and an extremely healthy understanding of differential equations. A student leaving this class would know that a differential equation is an equation whose unknown quantities are functions. They may even know how to solve a few. The A+ student would know how to grind one out by hand. The D student would know that chemical decay, population growth, and compound interest are all modeled by , and why.
My next move is to start poking around on Twitter to see if such a class exists, where, and how it’s doing. In the meantime, I want to ask a question: Why teach calculus? Because it is the cash cow on which all mathematics research rides around the pasture, waving its ten-gallon hat around and raving about the zeroes of something called the Riemann-Zeta function. Other than that, what’s the point? What do our students get out of it?
They will never die on a slipping ladder.