Math Classes Should Be Less Useful
"If you want to build a ship, don't drum up people to collect wood and don't assign them tasks and work, but rather teach them to long for the endless immensity of the sea." - Antoine de Saint-Exupery
One of the most common complaints about math classes is "when am I ever going to use this?" Often people respond with something along the lines of "All of modern science and technology depends on math" or "If you become an engineer or scientist these skills will be crucial to you." Complaints of "I'm not going to be an engineer or scientist" are met with "well, math teaches you how to think logically about problems, so even if you don't end up directly using trigonometry it's still useful for you to learn math."
And while all of these arguemnts are true, they're not particularly convincing. Instead, we should embrace that lots of math is "useless." Note that I do love math; don't take this as a criticism of math, but instead as praise. We should focus on the beauty of math, not its utility.
The goal of math classes is to teach people math. Math classes would be more effective if we focused on useless concepts in math, rather than so-called "useful" concepts. Effective here means that they would accomplish their goal better; that is, more people would understand math better.
Who cares what's useful?
Let's first say what useful means, at least for this article. When people complain about the uselessness of math classes, they mean that they won't personally use it in their future jobs. Naturally, something is useful if they will use it in their future jobs. As a remark, most people are right: they won't use trigonometry in their daily jobs.
People don't dislike things because they're useless. Naturally, we can tell this is true by exhibiting a concept that people like, despite being useless.
The same people who complain about math being useless in high school are happy to spend hours listening to music, reading books, watching T.V., etc. Are these activites "useful"? Obviously not! And yet, people spend hours and hours doing them anyway. There's nothing wrong with that, of course, but what happened to the aversion to spending time doing useless things?
Sometimes "This is useless, so I shouldn't have to learn it" is just an argument to avoid learning things you don't like. If you don't like math, it may be hard to argue "I shouldn't have to learn this because I don't like it", and it is much easier to argue "I shouldn't have to learn this because it's useless."
Interesting vs. Useful
The answer to the question "Why spend time doing 'useless' things?" is actually quite obvious: because useless things are fun, or rather, because they're interesting.
People like interesting things.
For example, I think we can all agree that some pieces of knowledge are interesting, some are useful, some are both, and some are neither. But of these groups, I would most like to hear about interesting things, to the exclusion of things that are useful.
For example, the fact that the Halting Problem is undecidable is interesting, though not very useful. The fact that \(pi\) is approximately \(3.14159\) is quite useful, but not interesting. The fact that vaccines prevent diseases is both interesting and tremendously useful. And the fact that a 40kg watermelon rolled down a (frictionless, of course) 10m tall hill at an angle of 30 degrees will be going about 14 meters per second when it reaches the bottom is neither interesting nor useful.
Which of the above would you like to learn more about? Personally, I'd like to hear about the Halting Problem, or possibly about vaccines if that were the only option, but it'd be a sad day before I'd choose to go to a lecture about the other two. Obviously people have their own personal preferences for what's interesting, but if you think there's nothing interesting in math you probably haven't explored it enough.
You also may argue that the Halting Problem being undecidable is useful—if that's the case for you, probably one of the following applies to you: either (a) you're a researcher in some mathematical field; or (b) it's not actually useful to you, at least by the definition above.
So by the above, being useful doesn't make something interesting, though there's probably some correlation there. So rather than trying to make math seem "useful", we should make it "interesting".
Now we have made some progress towards the main goal of this article. If math classes taught interesting concepts, more people would enjoy math class. This on its own implies that teaching interesting concepts would make math classes more effective.
Of course, this relies on the assumption that math classes can accomplish their goal of teaching math while teaching interesting concepts.
Math is Interesting
However, this assumption actually poses no problems at all.
Almost every topic in math is interesting or is necessary to understand some other interesting topic. Often these tangential developments spawn their own fields of mathematics. For example, basic arithmetic, while not particularly interesting, is the basis for so much math that to understand math you simply must understand it. That is, even focusing on so-called useless mathematical concepts will still involve teaching basic arithmetic.
But just because an uninteresting topic must be taught doesn't mean that it should be taught via meaningless drills.
The simple act of playing with numbers often gives rise to surprisingly deep questions. There are numerous examples of this: the Graceful Tree Conjecture, the Collatz Conjecture, the sum of three cubes problem, and many, many others. All of these are fascinating questions. Naturally in the course of exploring all of these questions, one will acquire an understanding of arithmetic.
Perhaps not all of these problems are of immediate interest to your average 8 year old, but there are plenty of similarly instructive problems that are. One such collection of problems can be found on MathPickle, a website collecting unsolved problems suitable for teaching, organized by grade level.
In fact, we know that people, even those who do not consider themselves to be "math people" (itself an unfortunate classification), will willingly pursue mathematical knowledge when they are interested. Somewhat famously (at least, as far as this sort of thing is concerned), an anime (The Melancholy of Haruhi Suzumiya) inspired a 4chan user to prove a new result about superpermutations. You can see the paper which is based on the work of the 4chan user, though not written by them here. On their own, superpermutations are exactly the sort of thing that people would complain about being useless. But once interested, people study them without regard to their utility.
It's also common for video games to spark interest in math: exploring questions such as "is this part of the game fair?" or "what's the optimal strategy?" A few weeks ago, I told someone about the binomial distribution, much to their joy, because it helped them answer a question about fairness in a video game they played. The point being, people willingly do math when it's interesting to them, regardless of whether they can cure cancer or even make a living off of it.
Of course, let's not suggest that the only point of studying math is to use it to do other things, but that often is the lure that takes people in.
Why do some people like math?
So I've argued that we should teach interesting concepts in math class, which is hardly a revolutionary idea. But the real point is that trying to pick topics to teach by using their usefulness as a proxy for their interestingness is a bad idea: though there may be some correlation, it's just not that strong.
Thus far, I've largely presented this choice as a dichotomy: either we teach useful concepts, or we teach interesting concepts. But didn't I admit there are concepts that are interesting and useful earlier? Surely we could teach those, to get the best of both worlds.
Well, not quite. Let's ask the question, "Why do some people like math?" If we could emulate the conditions that made this happen for them, perhaps more people would be interested in math, or at least, fewer people would hate it.
What sort of person likes math? Mathematicians, of course. Obviously there's other kinds of people who also like math, but we'll just focus on mathematicians.
Personally, I like math because it's really cool. I don't like math because it consistently produces useful knowledge, though that may be true. In general, I think this is also true of many mathematicians (though I don't claim to be one myeslf).
As an anecdote, I've spent some time doing research in both computer science and in math. In computer science I was working more on the engineering side, rather than the theoretical side. Once, some fellow students and I were giving a presentation about my work in math (I'll post a link here once I have something written), during a session of math presentations by students. Not once during my presentation, or anyone else's, did someone ask "Why is this useful?" In contrast, in computer science talks, that's usually among the first questions if the presenter does not immediately address it to the audience's satisfaction.
This difference isn't because mathematicians have a more innate sense for when something will be useful than computer scientists, so they just don't need to ask. So why do mathematicians study these problems, if not because they're useful? It must be because they're interesting: there's no reason to study something it's neither useful nor interesting. So if we want people to be more enthusiatic in their study of math, we should follow suit and focus on interesting, but not necessarily useful, concepts. A result of this is that we shouldn't try to find concepts that are both interesting and useful, because such concepts are generally "less" interesting than those which are simply interesting. Everyone wants a little escape from the real world, every now and then.
As a side note, the above anecdote isn't to say that computer science isn't interesting, of course, but it's a difference in culture.
Math doesn't have to be useful
You may object, "How will students learn necessary skills to succeed in life?"
Here are some responses:
- Basic math skills are necessary to explore cool math problems. So if people are motivated to do the exploration, they will either know how to do basic arithmetic and algebra, or they will want to learn so they can explore.
- The math most people use in their daily lives is not very advanced. As I said before, most people are right that most of math class is useless. If you need more than what "Algebra 1" teaches, then you probably have a technical job. Even many software developers, who people would generally say have a technical job, rarely use more advanced math.
- Finally, let's respond by asking the question, "Why should math be useful?"
We teach art in school—most schools require it in some form or another. Is art useful? I think we can all agree, just as very few people will need to know how to calculate derivatives in their future jobs, very few people will be painting or playing an instrument in their future jobs. In fact, we can go further: very few people will need to analyze Fahrenheit 451, to calculate the charge in a capacitor, or to say why or when any particular event in history occurred.
Does that mean that all of the major subjects are useless? Maybe it does, but most people would agree that education is not useless. So it seems that even though any particular topic (or even every topic) is, on its own, "useless", as a whole, school is not useless. So we should not disqualify math which is, on its own, "useless".
In fact, I, and many others, would argue that math is art. And I don't mean that you can draw pretty pictures of fractals. I mean that math is a creative pursuit of beauty, like any other art.
Math is an exploration of fundamental truth that transcends the laws of the universe—it builds monuments that last forever. A mathematical truth cannot be torn down or concealed, or worn away by the passing of time. Along with the other arts, mathematical truths are the greatest expression of imagination and creativity. As Paul Erdős said, "Why are numbers beautiful? It's like asking why is Beethoven's Ninth Symphony beautiful...I know numbers are beautiful."
"Phyllis explained to him, trying to give of her deeper self, 'Don't you find it so beautiful, math? Like an endless sheet of gold chains, each link locked into the one before it, the theorems and functions, one thing making the next inevitable. It's music, hanging there in the middle of space, meaning nothing but itself, and so moving...'" - Village, John Updike