Why calculus isn’t the pinnacle of mathematics
And why I feel a need to write an article with that title
Not too long ago, my friend asked me if I could find or generate an illustrative example that would help calculus click for their mom. It got me thinking about why calculus has the reputation that it does, particularly among nonmathematicians. A lot of the mystique is because it’s usually the last math class taken in high school, and so if you study anything else at university, it’s likely to be the last or one of the last math classes you take. Certainly it’s the most advanced math class that you can reliably expect anyone to have heard about, under a reasonable definition of “advanced math class.” It’s very reasonable to see calculus as the apex of mathematical education when you’ve never heard of any of what comes beyond. Trust me, dear reader, when I say that there is much to learn and to study beyond calculus, but if you’ve never heard of it, it’s difficult to imagine. That having been said, there are some genuine ways in which calculus is uniquely challenging. As I see it, math education in primary and secondary school alternates between two different modes.
The first mode is in learning how to do something. You learn how to count to 100, how to execute long division, how to solve for x, how to compute a derivative. There is some procedure or concept that you don’t know until you’re told. The second mode is the extension of that knowledge. You learn to first count to 10, then to 100, then how to count to 1,000 just in case you’re ever extraordinarily bored. You learn to multiply 1-digit numbers, then 2-digit, then numbers with 3, 4, or however many digits you like. You go from solving equations like 2x=4 into more complex ones (literally) like x²+2x-1=4x-3. From computing derivatives to computing antiderivatives.
As much as I’d love to wax on about the importance of each of these modes in math education, I’m in fact going to throw them out entirely, and go on to describe the exception to the rule I’ve just made. In mathematical education there are, I think five major paradigm shifts when one is first learning math.
Numbers
Imagine meeting a small child at some social function. Perhaps a family reunion, or take your child to work day, or anything else. You ask them how old they are, and they confidently tell you “This many! 🖐️☝️” Rather than saying “six,” this (hypothetical) child holds up one finger for each year they’ve been alive. This is charming for sure, but few of us ever stop to think about why they choose to communicate like this. After all, it’s more syllables and more modes of communication. Why say your age so inefficiently?
The answer is that children at this age aren’t familiar with numbers. They have a sense of quantities, this many 🖐️ is more than this many ✌️, and they might even know the words “one, two, three, four, five,” and so on. But children are not born with the innate knowledge that “five” represents the number of fingers on one hand, the number of points on a typical starfish, the number of Platonic solids, or indeed even the idea that each of these sets of objects have anything fundamentally in common. “This many” is an easy concept to understand. It’s just that: as many things as this. To abstract that not only into quantities of things, but even to an abstract notion of “five” takes a lot of mental brainwork. I can ask you “What’s 5+2,” and that question (hopefully) makes sense even if I don’t actually present you with five or two of anything. Learning to cope with that when you yourself are five years old is genuinely difficult, and marks our first paradigm shift.
By now, hopefully numbers feel so second-nature to you that the previous two paragraphs are hard to understand. It’s difficult to empathize with someone who doesn’t have a concept of what numbers are. But I also hope this is encouraging. You’ve conquered at least the first, and perhaps the hardest, major paradigm shift in learning math. And indeed, maybe more.
Arithmetic
After one has mastered numbers, or at least gotten a decent handle on them, it’s time to move on to arithmetic. We have 3 and 5, but what about 3+5? Again, it may feel strange that addition is somehow a gulf of education, but again I want you to try to emphasize with our “this many🖐️☝️”-year-old. When asking “what’s seven plus four,” she’s likely to count on her fingers. But then you ask her “what’s 342 + 788?” She can’t count that on her fingers. Likewise, she might be able to count by 4’s to answer 4×7, but this technique is hopeless for something like 23×42.
To do arithmetic with larger numbers, we have to break the operations into smaller steps, and glue them together with slightly opaque algorithms. To add 343+788,we need to “carry a 1” into the 10’s and 100’s place
Whatever efforts we make to explain why this process makes sense, most students are likely to treat it as a black box that gets them the right answer. Likewise with subtraction and multiplication
And the epitome of opaque computation algorithms is certainly long division.
One can explain why borrowing from larger place values is necessary, why you need to add zeroes to later lines of multiplication, and so on. Even long division has a method to its madness, rooted in some very convenient properties of the integers. Yet, the experience of the student is nevertheless to push the details under the rug and chug along, especially in the case of something like long division. When dealing with small numbers, it’s relatively easy to simply count, or think directly about quantities, to get the answer. But here we need to push further – doing arithmetic with very large numbers requires an abstraction of what’s going on, and working with that abstraction is the second major hurdle of math education.
But eventually we push through. We learn how to compute arithmetic operations. And if we’re lucky, we learn why those algorithms work in the first place. We’re now aged somewhere between 10 and 13, and we’re going to tackle the next major paradigm shift of mathematical education.
Algebra
Our third hurdle is algebra, and this is the first one where people really tend to stumble. “I was good at math until they started adding letters do it” is a cliché, and that’s for a pretty good reason. At this point in our mathematical journey, we’re comfortable with numbers, and we’re comfortable doing computations. We can be given a Problem and then we Show Our Work until we find The Answer. We have combined two or more numbers into one number, which we write down and get a gold star from the teacher. Algebra throws that out the window.
Now, rather than being told “you need to do this computation,” it becomes a weird game. x+2=10 doesn’t tell us what to do, and now there are letters. The letter x is a number, but we don’t know what number, so how can we possibly do math with it?
The answer, of course, is that we need to work in reverse. If x+2=10, then x is the number that, when added to 2, yields 10. We then subtract 2 and find that x=8. If 3x-7=17, then we started with x, multiplied it by 3, subtracted 7, and got 17. In reverse, we start at 17, add 7 to get 24, and divide by 3 to get x=8.
And these even are relatively simple examples. Once also needs to learn how to solve things like x²+x-10=5-x. For this, we have to take this “do the same thing to both sides” concept even further – we’re not just following steps in reverse, we’re pushing the expression around to take a nice form (namely x²+2x-15=0) and then factoring the result – treating x less as an unknown number and more as an object in its own right.
This is the first major hurdle of mathematics that people really seem to get stuck up on. Nobody says “I was great at counting until they got these weird ‘number’ things involved,” but they will say “I was good at math until they added letters.” In algebra, one must learn to treat variables as though they’re numbers, keeping track of them and accepting long and complex expressions as complete answers that don’t need to be evaluated, and to problem-solve creatively rather than following a prescribed algorithm. To solve a typical algebra problem, one pushes an expression into a more convenient form, even if it’s not clear how that gets you to “the answer.” All of this is hard to reckon with, it feels very different to what math used to be.
But still, we push through. We learn many different ways to solve for x, or y or whatever variable you like. We even learn how to “solve” expressions that are entirely of variables, by pushing almost everything to one side of the equation. And once we can do that, it’s time to be whollopped in the face by yet another paradigm shift.
Functions
This shift is a subtler one than the last, but I think more impactful. It really comes into full swing late into one’s algebra experience, especially in any course called “precalculus.” Up until algebra, math has fundamentally been about numbers. Even when “they add letters to math,” those letters still represent numbers. Most problems are variations on “solve for x,” and your goal is to find the numerical value (or values) that x can take. Now that changes. We don’t look at individual numbers anymore; now we have functions.
It starts off rather tame. We’re given “the equation of a line” in one of a few forms. Perhaps y=mx+b, perhaps Ax + By = C, or whatever else, then asked questions about the line. This is difficult to reckon with – after all you’re given and x and a y and yet you’re not supposed to solve for them? Scandalous. But then it gets even more abstract. We’re introduced to logarithms, to polynomials, to trigonometric functions. About the domain and range of a function. We learn about addition and composition of functions. We graph functions, and ask questions about the functions based on their graphs, and likewise questions about the graph based on the function. We take a handful of base functions and we shift and warp them into new, related functions, and how this affects their graphs.
Fundamentally, somewhere between learning to solve for x and learning calculus, our fundamental object of study has changed. And this shift I think is generally harder to reckon with than algebra even, since it’s harder to see. “Adding letters to math” feels like a rather concrete event, but rarely are students told “Now we’re not actually studying numbers anymore, this is really about functions.” We’re expected to become comfortable on our own that these
aren’t problems that need to be solved, but complete answers in their own right. They’re not numbers, but we’re not studying numbers anymore. We’re studying functions. To be sure, conducting that study requires working with numbers quite a lot. The functions we’re dealing with are made of numbers, so we don’t totally eschew them. What’s changed though is that this is a tool in service of our main objective, rather than being that main objective.
For this reason, I think that this is one of the harder hurdles for students to conquer. Not because it’s inherently difficult, but because it sneaks up on you and nobody explains that it’s happening. Still though, we can move past it. And at this point, we’re ready to approach calculus.
Calculus
And then we have calculus. To be sure, I think the mental hurdle here is one of the milder ones. It’s hard, don’t get me wrong, but if you spent your precalculus study getting used to thinking about functions, then you’re in a good spot for studying calculus.
Fundamentally, calculus is about a couple of operations done to functions, and exploring what happens when we do those. The previous article was about convergence, and indeed the limit is one of those operations. You have some function, maybe as a sequence and maybe as a real-input function, and find the limit of that function as the input approaches infinity, or some finite value. Maybe we’re interested in what happens when we approach the value from the left, or the right, or both, or something else entirely.
There’s also the derivative, which basically takes a function and asks “what happens to the output when you change the input just a little bit.” And its inverse, the integral, which asks “What happens if you keep track of everything the function does over this interval and added it all together?”
This trio, the limit, the derivative, and the integral (the derivative and integral are just special cases of limits, mind you) are basically all there is to it. The rest of calculus is just figuring out techniques to compute them for particular functions, to unpack their relationships, see where they’re defined, and so on. Not unlike how primary school is basically just about the arithmetic operations: addition, subtraction, multiplication, division; the rest of it is learning and practicing how to compute them for specific inputs.
Again, even if I think this mental hurdle is less substantial than the rest, I don’t mean to imply that calculus is easy. After all, “learning how derivatives and integrals work” leads to results like this
But that’s hard because of implementation. It takes concerted study and a lot of effort to get that far, but it also takes concerted study and a lot of effort to multiply two ten-digit numbers. Computing a derivative is hard in the way that digging a ditch is hard. It takes a lot of time, effort, and energy, but it’s a straightforward process that will be over with enough time.
Final thoughts
If you’ve read this far, you clearly care at least a little bit about math (or if not, you at least care a lot about the person who sent it to you.) But I don’t think this phenomenon is really mathematical in nature – it’s true across all disciplines. The ways that we introduce concepts is almost never the most useful or complete way to think about them. This is true in math, science, the humanities, the arts, and everything else. There’s a fundamental tradeoff between a framework that’s flexible and useful enough to do complex work, and one that’s easy to understand while you’re still learning a topic. If I were more of a philosopher, I might say something about Plato’s allegory of the cave, but I’ll leave that to David, over at Words Without Knowledge. Instead, I’ll just say that whatever you like to do or think about, however you understand or think about it, you can almost certainly make it deeper or better.
In my wildest fantasy, someone reads this article, realizes that this is the missing piece that makes math make sense, dedicates a the next few years to study, and comes out a veritable mathematical whiz. If this describes you, please let me know. I can be reached at threeven@danemea.de. But to those of you who actually exist, I just hope that you think about this any time you’re learning something. If it seems like it just doesn’t make sense, try and view it from another angle. Ask your teachers, your peers, or field experts how they think about these problems, and don’t be too attached to what’s made sense before.