I think it started at University. There I was, eighteen years old, hundreds of miles from home, and surrounded by thousands of complete strangers. In such an environment everyone soon refined the art of the conversational opening gambit down to three key pieces of information: what’s your name, where are you from, and what are you studying. It was almost a script that everyone knew and no one knew how to avoid. You could meet someone who introduced themself as John, from Leeds, and before you knew it you’d be asking what they were studying, smiling apologetically but unable to avoid finding out that third little trivium.
For me it was the “where are you from” question that usually broke the ice and opened up the conversation. I’m from Boston, which would usually be met with exclamations of “Really? But you don’t sound American.” And then I could say “Well it’s funny you should say that…” and the conversation could go from there. Every now and then, though, they’d pick up on my course instead. I studied maths, and by far the most common response to this was “Oh, I was rubbish at maths at school.”
At the time I didn’t think too much of it. I got my degree, then my PhD, and now get paid actual money to sit around doing maths. It’s rather incredible. And yet as time has passed I’ve found that whenever anybody discovers I’m a mathematician, whether they be my optician, my landlady, a barista, someone sat next to me on a plane, or anyone, they are almost guaranteed to respond “Oh, I was rubbish at maths at school.” Some people seem to feel obliged to tell me they never saw the point of the subject, which always strikes me as odd. Like meeting an author and telling them you don’t see the point of books. But having heard the sentiment so many times I got to wondering why it’s so common. Why do so many people recall being bad at mathematics at school to the point where it’s almost a point of pride? I was talking to two young men while waiting for a pizza late one evening, and when they found out what I did for a living they got into a genuinely competitive debate over which of them had been worse at maths at school. I wondered what would happen if I was an author, how many times would I witness people debating which of them was the most illiterate? I guessed never.
So why is this? Why do people hear “mathematics” and think “Oh yeah, I did that at school. I was rubbish at it.” Maybe before I ponder that I should ask a more fundamental question.
What is mathematics? It shouldn’t be a difficult question to answer. We all studied the subject at school after all. And yet, try to put into words just what maths is, and you may find it more difficult than you expected. Maybe the answer is to ask a mathematician. So I did. I asked a handful of my colleagues how they would define mathematics. Three of them gave the same answer, which was ìUhmî followed by an awkwardly long pause. The fourth person I asked mixed things up a little by saying “I don’t… erm,” and then came the long pause. Ultimately, two of them said they would let me know when they thought of an answer (they still haven’t); one of them told me he wasn’t sure what maths is, but he could tell me what it isn’t; and the fourth one told me that she didn’t think you could define maths, you just got to know what it was by doing it.
Why is this so difficult? We all studied history at school and I’m pretty sure most of us could define that: it’s the study of the past. I mean it’s right there in the word; history can also mean “the past”, so studying history is literally “studying the past”. So what of mathematics? It’s not really synonymous with anything the way history is, but etymologically it comes from the Greek word máthema, which means “knowledge”. So that’s not so useful, especially when you find out that the word science comes from the Latin scientia, also meaning “knowledge”. And history comes from the Greek historia, which means, you guessed it, “knowledge”. (Apologies to any linguists currently frothing at the mouth over my barbaric translations of ancient languages.)
Maybe my colleague was right, if we want to understand what mathematics is we need to roll up our sleeves, pick up our calculator, and just do it. Fortunately we don’t have to do it right now, because as I said earlier, we all did maths at school. What we learn in maths lessons at school is not the totality of mathematics, but it is hopefully a representative sample. So what do we learn in school? Well, we learn to count. And then we learn to solve problems. What is five times nine? What is eighteen divided by three? Solve this equation for x. What’s the area of this triangle? Differentiate this function.
Oftentimes these mathematical questions are just that: purely mathematical questions. But sometimes they have some flavour text, a little real-world dressing. You know the kind of thing, “Farmer Bob has a field that’s a right angled triangle. How long is the perimeter if the shorter sides are 300m and 400m long?” And this is good, because it teaches abstraction.
Abstraction is at the heart of mathematical thinking. I’m not going to say mathematics is abstraction, because that might suggest that I think Jackson Pollock was pouring maths all over his canvasses. And I’m pretty sure he used paint. Nevertheless, if maths is a toolkit for solving problems (spoiler alert: it is) then abstraction is the most important tool, the one that gets used first on every job. This toolkit metaphor is already creaking under the strain, I’m not really sure which actual tool abstraction would be compared to. Maybe the screwdriver you have to use to jimmy open the toolkit because you bought it second-hand and it just won’t open without some gentle screwdriver-based persuasion.
Less metaphorically, abstraction is the technique that lets you take any old problem and turn it into a mathematical problem which you can hope to solve. I have a friend who teaches maths at a secondary school. One year he was teaching a year nine class about the formula used to solve quadratic equations. Part way through the lesson a young man raised his hand and said those magic words, variations of which have been uttered in maths lessons since time immemorial. “This is pointless, I’m never going to need to know this in real life. When are we going to learn useful stuff, like how much change I get in a shop.” (I believe in the US the request is to learn how to do taxes, whatever that means.) Of course the student did know how much change he should get because he’d learned subtraction many years earlier. He’d just never made the connection between this rote-learned process of abstractly taking one number away from another number, and the real-world process of seeing how many pennies one should be handed after buying a £3.19 coffee with a £5 note.
Misguided as his particular complaint may have been, the complainer did perhaps have a point. Does everyone need to know how to do all the maths taught in school, even calculus? Does learning how to differentiate trigonometric functions benefit you if you’ve no intention of becoming an engineer, scientist, or mathematician? Maybe people recall being bad at maths at school because they had already known they wouldn’t need to know how to do calculus in order to become a restaurant owner, so they stopped trying. And now they own a restaurant they can feel vindicated that all that maths was a waste of time. The obvious problem with that is that most people in their mid-twenties have no idea what kind of career to follow, never mind people in their mid-teens. A guy in my year at both primary and secondary school never showed much interest in maths. He now has a PhD in nuclear physics and spends his days using rather high-brow mathematics to run computer simulations on prototype nuclear-powered engines. He’s as surprised as anyone that this happened. And that friend who’s a maths teacher from the previous paragraph? He was definitely vocal at school about the inapplicability of what we learned in maths class. Of course it can go both ways. One of my housemates when I was an undergraduate was doing a degree in maths and theoretical physics. He was regularly impressed at how important maths was when it comes to describing the Universe around us. And now he’s in charge of IT support at a University in London. Suffice it to say that his mathematical skills now seem less important than his management skills and ability to ask people if they’ve tried turning it off then back on again.
The solution to the “this maths stuff is pointless” problem isn’t, I think, not to teach this maths stuff. The years I spent in compulsory Art classes or German classes have proved wholly pointless in my chosen career as a mathematician. (Don’t get me wrong, I appreciate art, I’ve just never needed to make it; and I’ve given talks in Germany, but academics in Germany speak in English, sorry.) I don’t think the solution would be to let eleven-year-old me decide that he’ll never need Art or German and thus not study them. After all, there might be a parallel Universe version of me out there writing about his new installation in the Guggenheim. Who knows? No, I don’t think the problem lies in what is taught, so much as how it’s taught.
I hasten to add I don’t mean maths teachers are rubbish. I think maths teachers are brilliant. I quite literally would not be where I am today without a succession of the most wonderful people teaching me through my school life. A few minutes spent online will reveal hundreds of stories about inspirational teachers using ever-more imaginative methods to convey often difficult ideas. But ultimately their job is to make sure most of their pupils pass the next exam, or face the possibility of finding a new job. And the exam will be a series of abstract mathematical problems. “Solve this arithmetic problem.” “Solve this algebra problem.” “Solve this calculus problem.” The teacher is faced with the task of making sure as many as possible of their students can answer those questions before exam time. Inevitably then, some pupils are left twiddling their thumbs wanting to learn more but being fenced in by the curriculum; and other students will lag, unmotivated either because they don’t see the point or because they’ve already convinced themselves they’re rubbish at maths.
So if there’s a problem, what’s a solution? Well… I don’t know. I mean I’m a researcher, not a teacher. Every now and then I give talks to school kids to encourage them to study maths at University. Honestly, I find it much more nerve-wracking than standing up in front of a bunch of professors and talking about my research. I’ve no idea how people do it several times a day, every day. Luckily I do know a disproportionate number of maths teachers. So I asked them if there was a better way to teach maths. The consensus seemed to be “I’ve no idea.” There are simple physical limitations when one teaches maths: class sizes, student ability, curriculum, and so on. Given these limitations teachers do a fantastic job. But what about without these restrictions? I probed my maths-teacher-friends further. In an ideal world, could maths be taught better? Again there was a consensus, this time wholly positive. It’d be great, they agreed, if the fastest students could be taught in smaller groups. These students regularly want to know more, explore farther into the world of maths, but the teacher can either go off-piste with these students or help the ones struggling. Obviously they choose the latter, not being able to do more than suggest some topics for the faster students to investigate in their own time. Similarly the slower students would benefit inordinately from small groups or even one-on-one tutelage. Even teaching at University I see students nod and smile their way through a class only for their assignments to reveal they weren’t following the material. Luckily I do have time to help them out one-on-one, and once they get over the stigma of saying “I don’t understand, could we go over that again?” they come along in leaps and bounds. And one of my friends had a bee in his bonnet on the “it’s pointless” complaint. “I just want to Google ‘applications of maths’, stick it up on the smartboard, and spend the next hour saying ‘Look! Maths!’” he explained. “Most of the students think it’s only good for figuring out how much is left in their bank account. And they’re so wrong I can’t even explain just how wrong they are, it’d take too much time.”
And he’s right, maths is everywhere. I study number theory, an area of pure mathematics that until the early twentieth century prided itself upon not being applicable in the real world. The English mathematician G.H. Hardy wrote in 1941 that “No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years.” Unfortunately he was already wrong. The famous e=mc² consequence of special relativity was at the time being turned into the world’s most powerful bomb in Los Alamos. Meanwhile in Europe a secret war was going on between Nazi code-makers and British code-breakers, much of it based on number theory. And now, seventy years later, virtually all of internet security is based on the theory of numbers. Indeed, if there’s one thing mathematics consistently manages it is to become ever more abstract yet still find applications. Knot theory has found applications in studying how DNA replicates, helping in the study of cancer. Information theory led to the development of codes that allow your DVD to keep playing even after being scratched (up to a point). Game theory has changed our understanding of the global economy. Modern robotics uses deep ideas from algebraic topology and Lie theory. Partial differential equations give the best models for studying biological phenomena including the spread of diseases and the distribution of endangered species. And then there’s group theory, which has applications to chemistry, cryptography, and even the leading candidates for the so-called “theory of everything” in physics. Oh, and it gives the quickest ways to solve a Rubik’s cube.
Knowledge begets questions of course, especially when a person’s curiosity hasn’t been quashed yet. Knowing all these applications of mathematics only raises other questions, like “What on Earth is algebraic topology?” or “Why’s it called Lie theory*? Is that, like, the theory of untruths?” But that’s fine because questions are a good thing. The landscape of mathematics is vast and often difficult to traverse. But it’s full of wonderful sights, like infinities of different sizes, ways of cutting a ball into pieces and reassembling them to get two balls back, and numbers that transcend algebra itself. It has its quirky sights too, like the ham sandwich theorem, the hairy ball theorem, or theorems concerning optimal pizza eating strategy. Every question asked is another step into this weird and wonderful landscape, and curiosity is the most powerful incentive for taking these steps. And nobody is more curious than a child. Why not let them roam? Help them learn maths because it answers their questions, not because it answers yours. Hold their hand, but don’t hold them back.
* (For the curious, algebraic topology involves using fairly advanced algebra to study topology. Topology? That’s basically geometry except you do it with bits of play-doh instead of bits of stone. Lie theory is a deep area of maths connecting ideas from algebra, geometry, and calculus. It’s named after the Norwegian mathematician Sophus Lie. His surname rhymes with Lee, as all the best names do.)