Burn Math Class: And Reinvent Mathematics for Yourself (2016)

Good fiction’s job [is] to comfort the disturbed and disturb the comfortable.

—David Foster Wallace, interview with Larry McCaffery

Fiction and nonfiction are not so easily divided.

—Yann Martel, Beatrice and Virgil

Burn Math Class

Okay, don’t really burn math class. Or anything. Arson is mean, and extremely illegal. I. . . No, nevermind. I don’t want to start the book like that.

(Author thinks for a moment.)

Alright, I think I’ve got it. Sorry about that.

(Ahem.)

We should all be very angry. Something beautiful has been stolen from us, but we’ve never felt its absence because the theft happened long before we were born. Imagine that by some massive historical accident, we had all been convinced that music was a dull, tedious, rigid enterprise to be avoided except when absolutely necessary. Suppose that we had all attended music classes for more than a decade during our youth, and that through some feat of brilliant sadism among the instructors, we all left these classes with a firm belief that music is at most a means to an end. We might agree that everyone should have a basic level of familiarity with the subject, but only for reasons of practicality: you need music because it might — rarely — help you with other things. But the consensus view would hold that music more closely resembles plumbing than an art form.

The world would still be full of artists, of course. Just as it’s full of them now. By artists, I don’t necessarily mean art-school students, or professional artists, or the guy who wrote some stuff on a toilet and put it in a museum. I mean the people who pull things into existence out of nothingness; who refuse not to be themselves; who have their own way of fracturing reality that’s so authentic you can feel it in your nerve endings; who enter the world on fire and too often die young. “Music is not for them,” we’d agree. “Music is for the accountants among us, and it’s best if we leave it to them.” As farfetched as such a situation might seem, this is exactly what has happened to mathematics. Mathematics has been stolen from us, and it is time we take it back.

With this book, I am advocating a process of conceptual arson. The state of mathematics education all over the world has degenerated to a point where it no longer makes sense to do anything but burn it all down and start over. We begin by doing just that. In this book, mathematics is not approached as a preexisting subject that was created without you and must now be explained to you. Beginning on the first page, mathematics does not exist. We invent the subject for ourselves, from the ground up, free from the historical baggage of arcane notation and pretentious terminology that haunts every mathematics textbook. The orthodox terminology is mentioned throughout, and used when it makes sense to do so, but the mathematical universe we create is entirely our own, and existing conventions are not allowed in unless we explicitly choose to invite them.

The result is an approach that requires zero memorization, encourages experimentation and failure, never asks the reader to accept anything we have not created ourselves, avoids fancy names that hide the simplicity of the ideas, and presents mathematics like the adventure it is, in a conversational form that could easily be read as if it were a novel. While the primary goal of our journey is hedonism rather than practicality, we are lucky to find no conflict between the two. You will actually learn the subject — a lot of it — and learn it well.

When one attempts to construct a narrative through mathematics that does not ask the reader to accept facts established elsewhere, it is impossible not to notice what I think is the fundamental tragedy of existing mathematical pedagogy — a tragedy that is never mentioned, even in the harshest criticisms of orthodox educational practice:

We have been teaching the subject backwards.

Let me explain what I mean with a story. I got a C in basic algebra. All I learned was to hate the word “polynomial.” I got a C in trigonometry. All I learned was to hate the words “sine,” “cosine,” and “hypotenuse.” Mathematics had never been anything but memorization, boredom, and arbitrary authority — these are a few of my least favorite things. By my senior year of high school, I had completed all of the required mathematics courses, and I can’t describe how happy I was that I could now die without ever setting foot in a mathematics classroom again. Free at last.

One night during my senior year, I was hanging around in a bookstore, as I often did, and I saw a book on calculus. I had always heard that calculus was pretty difficult, but I had never taken a course in it, and I would never have to. . . a relaxing thought. This lack of any obligation to learn the subject somehow made the book seem more appealing, so I thought I would flip through it for a few seconds. I expected to see some scary symbols, think “Yep, that sure looks difficult,” and then put the book down and be done with it forever. But when I opened the book, it wasn’t just the usual garbage. In completely honest and unpretentious language, the author was saying something like this: Straight things are easier to deal with than curvy things, but if you zoom in far enough, then each tiny piece of a curvy thing almost looks like a straight thing. So whenever you have a curvy problem, just imagine zooming in until things look straight, solve the problem down there at the microscopic level where it’s easy, and then zoom out. You’ve solved the problem. You’re done.

Anyone can understand this idea, and it has nothing to do with mathematics. If you’ve got a hard problem, then break it up into a bunch of easy problems, solve those, and put them back together. The idea had a feel of elegance and necessity that I had never experienced in a math class. I flipped through the book some more, and when I saw that it had a section where the author complained about the way mathematics is usually taught, I knew this guy was my kind of person.

So I bought the book, and I started reading it whenever I didn’t have anything else to do. I liked the way the author wrote. It gave me an odd sense of justification for having always disliked mathematics in school, while at the same time convincing me that I had been entirely wrong about the subject. I wasn’t planning to learn calculus, and I didn’t remember any of the prerequisites from high school, so I didn’t even know how to solve the “easy problems” down at the microscopic level. But that didn’t matter, because I was free from all the restrictions of formal education, and there was no one to punish me for doing things wrong.

Thus began my strange journey of learning calculus before I knew algebra, trigonometry, what a “logarithm” is, or any of the other stuff they say you have to learn before calculus. I bought a notebook and started playing around. Whenever I didn’t understand something, I’d draw a picture and try to convince myself that it was true. I usually didn’t succeed in doing this.

Weirdly, the calculus concepts were by far the simplest parts of the book. Much harder were the so-called “prerequisites” to calculus: the algebra, trigonometry, and other conceptual packing peanuts with which modern high school courses are filled. All the stuff about zooming in made sense to me: derivatives and integrals were not only simple computationally but easy to understand from first principles. From their pre-mathematical motivation to their definitions to the methods of computing them, there was a coherent and well-motivated narrative tying everything together. But once in a while the author would make use of things that were supposed to be more “basic” — things I couldn’t understand at all, though I vaguely remembered hearing them from some teacher in a quiet, boring classroom. I couldn’t for the life of me figure out where all those supposedly simple things came from, like the area of a circle, or the pile of unexplained “trig identities.”

Fortunately, there was no one to force me to memorize any of it, so I kept learning the calculus bits without learning the algebra and trigonometry bits. I’d be reading something about calculus in the book, understanding it just fine, and then get lost because I didn’t remember how to add fractions. Occasionally, in cases like this, staring at the confusing step for a while was enough to eventually realize, “Oh, they’re just multiplying by 1 twice. It’s like they’re lying to make the problem easier, and then correcting for the lie so that they don’t get the wrong answer. Interesting . . .” Other things weren’t so easy to figure out by staring at them, and they continued to stump me. Logarithms, sine and cosine, the “quadratic formula,” and “completing the square” were not part of my vocabulary, and the terms themselves felt like a hangover from all the negative experiences I’d had with the subject in school.

After learning a bit more calculus, I still didn’t understand the “prerequisites,” but I started noticing some interesting things. I noticed that the derivative of a sphere’s volume was its surface area, and the derivative of a circle’s area was the distance around it. I still had no idea where the area and volume formulas themselves came from, but this weird “zooming in” operation suggested that they were all related somehow. This was my first exposure to a strange fact about mathematics: we may be completely defeated by two different questions, unable to make progress on either in isolation, and yet nevertheless manage to demonstrate with certainty that they have the same answer, all while remaining completely ignorant of what that answer is. This fact, which appears at first to be some sort of black magic, turns out to be a fundamentally important feature of abstract mathematics at all levels. This was clearly not the dull, authoritarian field I had been exposed to in school.

When it came time to start my first year of college, I did the unthinkable: I decided to take a calculus class. Having always hated mathematics with every fiber of my being, because of a freak accident in a bookstore I found myself taking calculus 1 for fun. Then calculus 2. Then my calculus 2 professor suggested that I take a graduate-level math course during my second year. I reminded him that I didn’t know anything and that he was insane. I took it anyway, and got the highest grade. By my senior year of college the department gave me one of those plaque thingies that said something like “Congrats on being the best math major we have.” I want to stress that I have absolutely no innate mathematical talent whatsoever, and nothing in my thirteen years of pre-university mathematics education suggested I’d find any enjoyment in the subject either. In any education system where the above series of events is possible, something has gone horribly wrong.

In the end, the Mathematics Department, my sworn nemesis in high school, ended up being the department where I felt most at home.¹ After college I went on to enter a Ph.D. program in mathematical physics at the University of Alberta. In the summer after my first year, following a lifelong pattern of doing everything except what I’m supposed to be doing, I became obsessed with psychology and neuroscience. I eventually applied to some Ph.D. programs in the field, somehow got accepted, left my mathematical physics program with a master’s degree, and I’m now living in Santa Barbara, California, studying the brain and behavior using mathematics. During my first year of graduate school in the Department of Psychological and Brain Sciences, I met tons of extremely smart students who had the same unjustified fear of mathematics that I always had in high school. Every time I see the flash of fear in someone’s eyes when advanced mathematics is mentioned, I want to tell them that their entire experience of the subject is a lie. The perceived difficulty of mathematics is entirely the fault of how we teach it, and I hold myself to that standard as well. If there is anything in this book that you’ve repeatedly attempted to understand, but failed to do so, that is my fault, not yours. The underlying ideas are extremely simple. All of them. I promise.

I was lucky to have amazing mathematics teachers in college, and I should mention (R/y+V)icky Klima, Eric Marland, and Jeff Hirst. I had many other wonderful teachers, but these four deserve special mention for being unbelievably helpful, and always putting up with me storming into their offices with bizarre questions unrelated to any course.

Throughout my first year in Santa Barbara, I couldn’t help but think that research in every area of science could be accelerated significantly if only everyone in the various fields of science knew more mathematics. By “knew more mathematics,” I don’t mean “had more mathematical facts in their heads.” I mean “had been explicitly trained in abstract reasoning.” What’s worse, I’m fairly certain that nine out of ten people have more “innate skill” at mathematics than I do (whatever that means). The only reason I happen to know more of it than my fellow graduate students is because of a random accident in a bookstore that led me toward a subject I never thought I’d love.

It’s summer now, and I’m writing this book for everyone who ever hated mathematics. Not only for the young and disenchanted, but also for the many scientists who secretly regard mathematics as a distasteful but necessary professional requirement, and have dutifully endured it but have never felt the fire, the anarchy, and the hedonistic pleasure of the subject. Unless I have failed miserably, we’ll have a lot of fun along the way.² However, I should stress that this is not another one of those tired attempts to “make math fun,” which usually translates to spreading a thin layer of silly faces and bad puns on top of the same old approach. While for some this might be a minor improvement over the standard textbooks, such books never present the subject in the way I always wished it would be presented: clearly pointing out everything that is arbitrary, everything that only looks the way it looks because someone is trying to sound fancy (consciously or not), separating historical accidents from timeless processes of reasoning, acknowledging the well-justified contempt that most students in most math classes feel most of the time by poking fun at the way that the subject is typically taught, and most importantly: backwards.

The above sentence was (of course) written by the author of this book, a heavily biased source whose opinions of his own work should not be trusted. However, the same principle applies to that last sentence as well, suggesting that the aforementioned distrust should itself be distrusted. We appear to have reached an impasse. Think what you will.

The subject as it is usually presented in modern educational institutions is something that no creative, independent thinker should be able to stand, and books that try to remedy this fundamental flaw with chapter titles like “Funky Functions and Their Groovy Graphs” are missing a large part of why so many students find the subject so alienating.³ But mathematics itself, when stripped of everything unnecessary, everything pretentious, and presented in as honest and human a manner as possible, is clearly one of the most beautiful things our species has discovered. It is a scientific art form that doesn’t need to justify itself by being “useful,” though learning it is one of the most useful things you can do.

To be fair, this is a chapter title from a very well-written book. Mark Ryan, I’d love to meet you in person someday. You’re an incredible teacher.

At each point of our journey, I will focus on the ideas that I consider to be of largest conceptual importance, whether or not they’re typically presented together. Although we’ll start at an extremely basic level, we’ll eventually start learning some things that aren’t usually taught until the latter years of a four-year degree in mathematics. If there has ever been a book that goes from addition and multiplication to calculus in infinite-dimensional spaces, I’ve never found it. If you continue reading, I hope to show that this approach is not nearly as delusional as it seems.

I try at every stage to run the ideas through a conceptual centrifuge before I present them. What is presented in courses on any subject is usually a cloudy, confusing mixture of the essential with the historically contingent, a mixture that hides the simplicity of the underlying ideas from even the most attentive students. I’ve always wished that academics would spend much more time attempting to separate this mixture into its component parts before writing their books or giving their lectures. I’ve attempted to do this throughout the book, but for an example of what I mean, see the first few pages of Chapter 4, “On Circles and Giving Up.” Also notice that circles first enter the story long after we’ve invented calculus. And they should; they’re extremely confusing before then.

Here’s an example of how we do things differently. One of the few things I remember hearing about in high school mathematics was the “Pythagorean theorem,” but I didn’t know why it was true, I didn’t know why we should care, and I didn’t like the unnecessarily fancy name. We’ll avoid all three of these problems like this: I’ll use the term “shortcut distance” instead of “hypotenuse,” I’ll think of a more descriptive name than “Pythagorean theorem,” I’ll offer the simplest explanation of why it’s true that I’m aware of (it takes about thirty seconds to explain), and once we’ve invented it for ourselves, I’ll show you a simple derivation of the fact that time slows down when you move.⁴ This fact comes from Einstein’s theory of special relativity, but the explanation uses no mathematical ideas more complicated than the “Pythagorean theorem,” so at that point you’ll be able to completely understand the argument. The conclusion will still seem surprising, though. It seems surprising to anyone with a human mind, no matter how long you’ve known it! Given the fact that this argument is perfectly comprehensible once we’ve invented the formula for shortcut distances (formerly known as the Pythagorean theorem), it’s a tragedy that this short argument isn’t a mandatory part of every high school geometry class. They should ring a bell, throw confetti, and start explaining it to you five seconds after they teach you the Pythagorean theorem. But they don’t. We will.⁵

To be a bit more precise, whenever two objects are moving in different directions or at different speeds, their “clocks” start moving at different rates. But it’s not just a fact about clocks. It’s a physical property of time itself. The universe is crazy. More on that later!

You’ll have to provide the bell and confetti. Not that I’m unwilling to provide them, but I’m probably not where you are at the moment.

Burn Math Class breaks a lot of conventions and a lot of rules, probably too many for its own good. No method of learning works for everyone, and I certainly don’t claim that this book is a universal cure for the ailments of mathematics education, nor do I claim that it is guaranteed to be suited to everyone’s learning style. If this book’s approach doesn’t work for you, please stop reading it and find one that does. Your time is valuable, and you shouldn’t waste it trying to trudge through a book that isn’t to your taste. This book was written as a labor of love, entirely for fun, not as part of a job. Ideally, it should be read for the same reasons.

Whether or not this experiment contributes anything of lasting value, radical changes simply are needed in education. As it stands, our educational institutions at all levels — from grade school to grad school to the style requirements of academic journals — appear to have been optimally designed to induce a kind of reverse Stockholm syndrome, causing us to revile subjects we might have otherwise loved. Students are graduating from these institutions deeply bored by the most phenomenally mind-blowing things our species has discovered. If they think mathematics, physics, evolutionary biology, molecular biology, neuroscience, computer science, psychology, economics, and other such fields of inquiry are dull and uninteresting, it isn’t their fault. It’s the fault of an education system that is brilliantly engineered to punish creativity; a system in which they are taught the spellings of words, but not how to think without deceiving themselves, as we all inevitably do; a system in which the laws of nature and arbitrary fiats like the prohibition of prepositions at the end of sentences are presented on equal footing, as if they were both equally valid descriptions of The Way Things Are; a system in which they are legally obligated to spend the majority of their young lives. For them, and for any of you who have ever had a similar experience, this book is my note of apology.