While spending years studying mathematics in university — more years than I was supposed to — I had plenty of time to make a series of reflections on the subject itself, how its perception changes from high school to higher education, how it is viewed by the general public and its actual purpose. Some of the conclusions would be out of reach to anyone who doesn’t have either the privilege or the dedication to study it in-depth, so I thought I’d share them.
The best way to describe a layman’s perspective of mathematics is complete obscurity. Obscurity in the sense of a secret art unavailable to them, a magical process that spits out numerical results and the workings of which are too time-consuming to understand. This view is as ingrained as an outright prejudice, to the point where it led to the creation of professions and degrees that only deal with the inputs and outputs of mathematics but dread what they would find if they took the machine apart. And the mathematicians responded in kind, holing themselves up in an ivory tower of obscure language and giving up on explaining themselves altogether. Do and then you shall understand. The stereotype of the obsessive recluse living in a system of inapplicable abstractions detached from reality is now (at least statistically) motivated.
I concede that merely explaining is sometimes not enough, and that very often one needs to do mathematics; solve heaps of exercises, build simulations, correct errors, hammer in concepts by repetition. It takes a long time. But it only takes too long if one has no motivation to do it and feels as though they’re wasting those hours instead of investing them. What are repetitive tasks to us? We do domestic chores, work jobs we don’t care about and commute, all repetitive tasks we accept to do because they have a purpose. Even in our entertainment, we’re capable of doing the same grueling thing over and over, to solve a crossword puzzle, reach a videogame achievement or follow a story to the season finale. Fortunately our environment’s subtleties are still rich enough not to drive us to terminal boredom (though they get very close to it at times). The problem is that people feel like mathematics is pointless. To put it more precisely: they feel as though the average person doesn’t need to study the process when they can easily just receive an output for practical purposes.
To quote a parody song from my childhood (of all things, but it stuck with me), “Why would I study if calculators exist?” What looks at first glance like an ignorant opinion from the lazy is, in reality, a perfectly legitimate question. Learning long division and the quadratic formula on paper is exactly as idiotic as it feels, to the ends of reaching university-level mathematics. Learning to do the specific task isn’t useful. What is useful, and ultimately the ideal reason students should do things like long division on paper, is a combination of two things: a) knowing why and when two numbers should be divided b) an intuitive grasp of how numbers are divided by interacting with them. These two points can’t even be neatly separated as they often imply each other indirectly. By knowing why you perform a computation you may get to the how, and vice versa; the how carries a hidden implication to the why. Going one way or another depends purely on cognitive style and current high school education usually obfuscates one, if not both, of these methods depending on the task.
This is where we stand now, with a general population who has learned a few basic mathematical rules with no explanation of where they come from and sometimes without even knowing their most common applications, which they would surely encounter if they were just one degree of abstraction below mathematics: in physics, biology, engineering, architecture or economics. There are countless examples so I’ll bring up one of the most frustrating ones; trigonometry. Sine and cosine are just how much the length of a segment shortens when you project it onto the axes. This is how they’re used for forces in physics, the length of shadows, whenever circular motion is translated into horizontal motion, etc. It’s obvious how universally useful this notion is as we project lengths onto surfaces all the time. We project by doing so much as looking at something from different angles. Yet, not a single time in my high school education was this fact concisely stated and I doubt most 18-year-olds even now would know it when asked. Going by the curricula they would say something about triangles and dividing their sides by each other and wonder why we would even do such a thing. (We do not.)
The answer to the question is something along these lines; You study because otherwise you wouldn’t know what to type into the calculator. This issue is especially urgent now that we have compact, portable computers instead of just calculators, with access to an endless stream of knowledge at almost all times. “But I don’t need a math education to type a division into a calculator!” You already have a math education: you have enough of an intuitive grasp on division to know what you need it for.
This brings me to my main point. Mathematics is within the grasp of everyone. I don’t say this to parrot clichés, but instead as somebody who grasped it by paying the cost. In one’s formative years a notion of ‘mathematical genius’ quickly arises in high school contexts; a small group of people in possession of particular talents that facilitate mathematics. One or two kids in each class who can calculate much faster and are maybe better at organizing information — abilities that in the end are equivalent to a machine’s higher computational power, or a better short-term memory, to be more humanizing about it. After some time everyone except these individuals is strongly disincentivized from educating themselves on mathematics, or even worse, the entire subject becomes a selection process for future university math students. I don’t come from a place of mystical talent for mathematics. I wasn’t one of those kids; I’m still slow at multiplying in my head and have to put extensive effort into studying.
So why would or should anyone spend time on mathematics, even without talent?
Mathematics is ubiquitous. It’s the set of fundamental rules by which our reality operates. Aside from the entire physical world these laws cover thought, perception, causality, similarity, the concept of a law itself. Take any law, generalize it as much as possible, and the end point will inevitably be an existing mathematical concept. This isn’t an arrogant pretense of knowing everything about the universe; mathematics is based on reality and in service of it. It is the study of how to develop tools to use for other purposes and making those tools as widely applicable as possible; sort of like manufacturing screwdrivers, except with interchangeable ends.
If one considers mathematics the fundamental laws of reality, it becomes more evident that every child going through mandatory education should have access to them, better yet that all conscious beings have an innate knowledge of them on some level. By merely existing, going through sensations, observing and reaching conclusions a creature utilizes mathematical principles, although the latter should more accurately be called principles which mathematics formalizes. These exist independently from our human perspective as something higher than us — nobody really knows why reality works the way it does — but unlike a deity we actively participate in it and interact with it every day.
Understanding your own reasoning is the first thing mathematics will help you with. Just elementary logic, covered in 3-4 university classes in algebra, is incredibly helpful to follow a more organized train of thought, to pinpoint exactly when you perform a deduction and which fact follows from which. The general principles of logic are also quite simple and can easily be rendered entertaining; not only is there no need to relegate them to higher education, they are adequate for even elementary school. Besides, being trained in logic would subsequently make studying mathematics (or the process of studying) much more efficient for the rest of mandatory education.
Then there is the benefit of another attitude one gets used to adopting: a control over generalization. A great deal of societal conflict and faulty reasoning is a consequence of not knowing how to properly generalize, not being aware when one does it, or doing it constantly and obsessively. Think of discrimination, conspiracy theories, sophism or any verbal slight of hand, really. As I already mentioned mathematics has the set goal of finding the general principle, of stripping away every unnecessary bias and looking for the actual characteristic a set of elements share. Better yet the defining characteristic, the one that applies for good reasons and not by accident. To achieve this, mathematicians put a hellish amount of work into testing and re-testing the established principles, looking for counterexamples, sometimes coming up with contrivances that require an outright artistic level of creativity. This might sound like hairsplitting so you must trust me when I say this: when a relevant counterexample is found what usually happens is that the entire paradigm collapses like a house of cards and the theory (and practice, through physics) moves forward in the newfound freedom.
While I personally don’t believe we can ever evolve to a “better state of the world”, or an ideal one, the way of progress is clear: a lot of susceptibility to dangerous rhetoric could be prevented by teaching the general population how to systematize their own reasoning. We already teach a whole multitude of subjects precisely because they’re considered to be basic life skills; literature and grammar to properly communicate with others, history to place ourselves in the world’s timeline, geography, biology and physics to know what the physical world around us is like, etc. In a sense we got as far as acknowledging that mathematics is a necessary skill, but I feel it’s not taken seriously. Mathematics is the business of clowns for laymen and high school students, clowns who will reside in their secluded circus and occasionally perform for the public. They’re paid as a sign of recognition but most people don’t want to be clowns.
The next step of this discussion is figuring out why mathematics arrived to this level of gatekeeping. And the answer is annoyingly simple; language. All of mathematics is written in a particular language that looks deceptively similar to the ordinary spoken word, but they are not the same. This needs to be made abundantly clear because many people are terrified and confused by mathematical terminology and notation — rightfully so! Trying to understand it without help is like deciphering a language one doesn’t speak; some of it may look familiar but just translating it is a great effort, let alone assimilating what those statements are trying to convey.
I could get into the details of basic math terminology but that’s a topic for another time that needs its own lengthy, dedicated section. What’s important is that the fundamental truth that you need to learn another language to properly understand mathematics and follow university classes is widely overlooked. It is important to consider the gatekeeping mathematicians’ side of this debate: “Terminology was developed to preserve logical rigor and ordinary language cannot be used for this purpose due to its ambiguities”. This is entirely legitimate, but considering that mathematical language is based on spoken language (which carries its own fat load of philosophical implications) we know for sure it can always be translated to phrasing understandable for a layman. This phrasing might take up ten times the space but the information would be conveyed nonetheless; the same thing can’t be said in reverse. That is to say, the nuances of human language in the overwhelming majority of times can’t be translated to mathematical language, which doesn’t come from a place of snarkiness as much as a need to highlight that they serve different purposes.
Overall: yes, the average person should learn mathematical notation. I have been talking about it as though it was a different language but, in the realm of practicality, it’s orders of magnitude simpler than any language you’d learn. (French comes to mind. It’s extremely, frustratingly irregular. Mathematical language is laughable in comparison.) I’m not qualified to make estimates but an introductory course of just translating back and forth between mathematical and ordinary language shouldn’t take more than a semester. It should take at least that much though, and be done before taking a single university class. I suppose the awkwardness of when such a course should be introduced is due to these circumstances; one normally enters higher education directly from high school — certainly without a semester to spare— and is then meant to follow several courses at once, all of which they’d need the linguistic preparation for. When should one acquire mathematical language? They don’t teach it in high school because it’s seen as futile, while in university one needs to start applying such language right away to different subjects.
The reality of the matter is that the metaphorical gate keeping ordinary people away from mathematics is very thin. It requires the acquisition of a skillset not more complicated than driving or plumbing. Though learning the latter is certainly recognized as an effort, we don’t regard people who achieve these as geniuses, and it ought to be the same for mathematics. Complex mathematics and mathematical creativity are definitely the realm of exceptional people, but ordinary and well-established branches aren’t. There are plenty of people already learning these and the success rate isn’t as narrow as they’d have you believe.