I had the pleasure of presenting at the Telegraph Festival of Education on June 22, 2017. The transcript and slides follow.
Thank you, it’s a pleasure to be here and thank you to the organisers for the invitation. I love this event, it’s my second time here, first time speaking. It’s a great place to celebrate rich, diverse perspectives on education. When I saw that STEAM was one of the chosen themes for speakers, I knew what I had to do.
You see, I’m a former mathematician, although these days you might describe me as a recovering mathematician; you never really get over it. And now my work lies in education, working with students, parents, teachers and schools to raise their confidence and success in maths. And that term STEAM is interesting to me because it embraces the importance of the arts alongside STEM. And my main goal today is to go further and persuade you that maths and the arts have far more in common than we perhaps realise.
Over the next half hour or so, we’ll look at the influence of mathematics in art. And then we’ll get inside the minds of mathematicians and I’ll show you that mathematicians are the ultimate artists. And lastly, we’ll reflect on what this means for how we learn and teach mathematics. I believe we’ve created a false separation between maths and the arts, and bridging that divide is more urgent now than ever before.
The mathematician Carl Fiedrich Gauss once remarked that maths is the queen of the sciences. Today, we’ll hear from a whole host of mathematicians, philosophers, poets, painters and writers. And we’re going to see maths, and experience it, in a new light: as the queen of the arts.
Sound like a plan? Great, let’s begin.
We often take the visual arts for granted, and as a layperson, I’m definitely guilty of this. We overlook the technical craft of artists and yet the more we dig in, the more we can uncover the influence of maths on these great works, because maths can provide a powerful toolkit for artists.
So take the rules of linear perspective, which helps us to see 3-D objects in a 2-D plane, underpinned by principles of geometry and convergence.
These equations, which mean little to most of us…
…yet they govern vanishing points, which are nothing more than the intersection of projected parallel lines.
And how about multiple dimensions, which mathematicians routinely roam around in. The challenge of visualising four-dimensional space inspired the cubist and surrealist movements of the early 20th century. Abstract art owes much to this idea that we can express paintings mathematically.
More recently, art and algorithms have come together to produce fractal art. So fractals are generated by applying iterative methods to solving non-linear equations, and then these calculations are represented as still images.
It may not sound pretty, but it sure looks it – the concept of mathematical beauty is at the core of fractal art. And it’s the inspiration behind Islamic geometric patterns, the kind you’ll find here at the Selimiye Mosque in Turkey.
In fact, maths is everywhere; we just don’t always realise it. I live in Oxford, the city of dreaming spires. You just need to look up to appreciate its beauty. What’s perhaps hidden from view is the influence of maths on the architecture. Marcus du Sautoy has even developed a walking tour of Oxford that showcases the role of maths in shaping this wonderful city.
Along the way you’ll see the curiously shaped Sackler Library. The architect Robert Adam chose a rotunda for a strictly mathematical reason: he was looking to get the greatest volume with the least impact. And the circular form is the result of his mathematical reasoning.
Here’s a simple beaded pattern that wraps around the columns of the Ashmolean museum. It’s a celebration of translational symmetry. The inside of the Ashmolean is decorated with more of these so-called frieze patterns that go back centuries.
And there’s the famous Sheldonian theatre, where, among other things, thousands of students mark their official entrance and graduation from the university every year.
Back in the 17th century, Christopher Wren was tasked with designing the building, and he was told by officials to maximise the space inside, to allow for all the dancing that was to take place, if you can believe that. So the supporting columns were not allowed to take up too much space, and this amounted to designing the world’s largest unsupported roof at the time. It seemed impossible and yet, an ingenious mathematical solution involving interlocking beams made it happen.
If you venture to the mathematical institute you’ll be greeted by this novel paved entrance – many people comment on its prettiness and who could blame them?
If you look closely, you’ll see there are just two shapes of tiles here. And what makes this tiling so special is that it’s non-periodic. That means that there’s no way of creating the whole tiling by repeating any one part of it. They’re called Penrose tiles after Roger Penrose, who established the underlying geometry behind these curious patterns.
And no trip to Oxford is complete without a visit to Christ Church College, home to the Harry Potter dining hall. Christ Church was also the college of Lewis Caroll; one of the most creative minds to ever grace the university.
Alice’s adventures have secured his legacy, but few people know that Lewis Caroll was actually a pseudonym. His real name was Charles Dodgson. And Charles Dodgson was a lecturer in…mathematics. Before he earned fame as a writer, he was a geometer and an algebraist. And Charles Dodgson, or Lewis Carroll, is a wonderful example of how these two seemingly disparate disciplines – maths and the arts – fuse together.
And I’m here today to make the case that every mathematician, by the very nature of the subject, are artists. It’s not just that maths influences art, it goes both ways: art is deeply engrained in mathematics.
We’re going to look at five concrete traits that define the artist – and I don’t think anyone would dispute them. They are Beauty, Creativity, Curiosity, Knowledge and Courage. Not all terms we would immediately associate with maths. Yet the examples will, I think, speak for themselves. Some are inspired by Will Gompertz’s book, Think Like an Artist. When I read it I was struck by how one could just as easily substitute the word mathematician for artist. I’ll let you be the judge.
Let’s start with beauty.
We’ve seen the obvious beauty that results when maths is fused with the visual arts. But maths is beautiful in its own right. We so often tend to think of maths in terms of its apparent dullness and technicality. But that is not the maths that mathematicians fall in love with.
Here’s Bertrand Russell, the mathematician and philosopher – he says:
“Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere…sublimely pure, and capable of a stern perfection such as only the greatest art can show.”
A beauty cold and austere. What does he mean? One of Russell’s contemporaries at Cambridge was G H Hardy, a mathematician who had this to say in his famous Apology:
“Beauty is the first test. There is no permanent place in the world for ugly mathematics.”
Now it’s important to know that Hardy was not your typical romantic. He was the purest of mathematicians. He studied maths for its own sake. He celebrated maths as the ‘one true science’ that is free from the real world. And yet when pressed, he chose beauty as the defining characteristic of his subject.
Hardy picked out two pieces of mathematics as his favourite. And they say a lot about how he understood mathematical ‘beauty’.
The first is the proof that the square root of 2 is irrational, which the Greek mathematician Euclid first produced. What that means is that this number, whose square is two, cannot be written as a fraction. Now what makes this beautiful to Hardy is not the result itself, but the proof – the argument that shows no such fraction exists. Now I won’t go through the details but here is the proof in all its glory.
It doesn’t strike you as beautiful, does it? But to those that have studied it, beauty shines through. There is an elegance to this argument. It starts by supposing that root 2 can be written as a fraction. And it then goes on to show a contradiction. This is the technique of reductio ad absurdum, proof by contradiction. And when used effectively, it can establish difficult truths with an amazing succinctness. In this case, it shows that none of the infinitely many fractions out there will ever equal root 2.
And Hardy’s other favourite is the proof that there are infinitely many primes, again from Euclid. So we know that primes are integers with two distinct factors – 2, 3, 5 and so on. Ever wonder how many there are? Euclid did, and he argued, again using proof by contradiction, that there are infinitely many. Here’s the proof.
Again, the details can be deferred but savour this: in a few short lines, Euclid has proven the existence of an inexhaustible supply of these strange objects. He has grasped at the infinite through flawless mathematical reasoning, and such deft moves like adding the one over here. It’s this kind of elegance, this simplicity with which we establish irrefutable eternal truths, that makes maths beautiful.
Hardy’s complete criteria for beauty was “seriousness, depth, generality, unexpectedness, inevitability, and economy” – all found in those two marvellous proofs. To Hardy, and to all mathematicians, these arguments are aesthetically pleasing.
Mathematicians seek beauty. Hardy said: “A mathematician, like a painter or a poet, is a maker of patterns.”
Artistic expression lies in the ability to synthesise our knowledge, experiences, intuitions, feelings, into a single, unified entity. Picasso showed us that art is often about reducing, taking things away, not adding them. Ernest Hemingway would spend hours writing a single sentence; not because he was looking to create the perfect sentence, but because he wanted that sentence to weave seamlessly into the whole text. To connect with the sentence that came before and after. The Apple products that we’re all addicted to today are governed by much the same ethos. And it’s this ethos of simplicity and elegance, and completeness, that mathematicians have adopted for centuries.
Jerry King, another mathematician, proclaims that “the keys to mathematics are beauty and elegance”, and that this is the motivating force for mathematical research. I can tell you he’s right. A good many research mathematicians, who are paid to develop mathematical proofs, do what they do not for glory or material gain, but for the innate beauty and satisfaction that their otherwise obscure work brings.
The final word goes to Paul Erdos, the Hungarian mathematician, for whom mathematical beauty was so obvious.
“Why are numbers beautiful?” he asks. “It’s like asking why Beethoven’s Ninth Symphony is beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful.”
I don’t know that I agree entirely with Erdos, because I think we can learn to discover and understand the innate beauty of maths, and I hope you’re starting to be convinced by that to.
If maths is beautiful, it must demand something of our creative talents. Back to Hardy, who said:
“I am interested in mathematics only as a creative art. Like creative art, maths promotes and sustains a lofty habit of mind, increases happiness of mathematicians and other people.”
There are many ways to define creativity, let’s say it’s what emerges when our brain combines two or more seemingly random elements in a new way. That seems like a reasonable definition. And Einstein said that this “combinatory play is the essential feature in productive thought.”
It’s why I love playing word association with my nephews and nieces; it can reveal their most creative tendencies, and sometimes just the outright absurd. And creative thinking, combining ideas, is an exercise in problem solving, the lifeblood of maths. So to show you that maths is fundamentally a creative pursuit, we’re going to solve a maths problem. And here it is:
We’re going to sum the first 100 odd integers. So 1,3,5…all the way to around the 200 mark.
Now, the solution isn’t immediately obvious. But that’s okay, mathematicians, like artists, regularly confront the unknown. So what do we do?
Well, a common problem solving strategy is to reduce the problem – we can’t swallow all these calculations in one go, but maybe we’ll sum them, a few at a time. Let’s do that, and we’ll stay alert to any interesting patterns.
So if we just took the first odd integer we get 1, obviously.
And if we summed the first two odd integers, we get 4.
When we add the first three integers we get 9.
We’re just playing here, having a go and staying alert to anything that might look familiar.
Adding the first four gives 16.
Huh: 1,4,9,16. That is familiar – they’re the square numbers. Maybe that’s the pattern.
But why? That’s the most important question a mathematician can ask. Why does summing the odd integers give squares? And will it always be the case?
Well, since we have squares, maybe we can visualise the problem. We’ll draw a picture. And here we are.
There’s 1, 4, 9, 16. The squares. And now we can sense this connection between adding odds and squares. It’s revealing itself because we’ve made the problem visual. Let’s draw it again, with some colour this time. 1,4,9,16.
Now we can see it, right? Adding the next odd integer gives us the next layer of the square. So we can solve the problem – adding the first 100 odd integers gives a 100×100 square whose area is 10,000. So that’s the answer: 10,000.
But the answer is not what’s important. Because think about what we’ve just achieved. We started with an arithmetic problem, summing those integers, and we ended up in the realm of geometry, drawing squares. We actually discovered the geometry of odd numbers – we found a link between two different topics. That, in a nutshell, is creativity. And it all came from our willingness to explore and our ability to detect patterns.
And you know, creativity isn’t just this thing that magically springs up from nowhere. It is an iterative process. Thomas Edison was one of the most creative innovators of all time, and he knew this. It took him over 10,000 attempts to invent the light bulb. And Edison was clear that
“I have not failed 10,000 times. I have not failed once. I have succeeded in proving those 10,000 ways will not work.”
Now on a good day, a mathematician won’t quite need 10,000 attempts to solve a problem. But problem solving has always been an iterative process. It’s captured rather well by the Hungarian mathematician George Polya, who laid out a 4-step cyclical approach to problem solving.
We always start by understanding the problem at hand and we then map out a plan, a line of attack. We have a go, carry out the plan, and we review. If we have the solution, great. If not, that’s okay too, we reflect on where we might have gone wrong, or how far off the solution we now are. Rinse lather repeat. And mathematical creativity is like all other forms of creativity – it takes a lot of experimentation and quite a few mistakes.
And one of the real drivers of creativity is curiosity. Mathematicians are, almost by default, curious beings. Their job is to ask questions.
Paul Lockhart is a mathematician and maths teacher who gives a blistering account of his subject in ‘A mathematician’s lament’. Here’s what he has to say:
“The mathematician’s art [is] asking simple and elegant questions about our imaginary creations, and crafting satisfying and beautiful explanations.”
There we are again, maths as art. Note too the ‘satisfying and beautiful explanations’ he seeks – another mathematician who cannot escape the beauty of maths. And according to Lockhart, the art of maths is in asking questions.
Now, this ability to ask questions is how artists produce their art. Edgar Allen Poe once described how he produced his poem, ‘The Raven’. He said it came about through a series of questions he asked himself. It’s tempting to think he was hit with divine inspiration but he debunks this entirely. In fact, he said he “proceeded to its completion with the precision and rigid consequence of a mathematical problem.”
A mathematical problem – I don’t think his analogy was accidental.
This use of questioning to generate ideas is hardly new. The Socratic method goes back thousands of years. And for Socrates, asking questions is an enlightening process. He believed our greatest insights come when we approach ideas with a natural scepticism – when we query, probe and reason logically, accounting for our assumptions. Well this is precisely how mathematical thinking unfolds.
There is an honesty to all art in that it doesn’t come for free. We are not born great artists, though we are born with potential. And one of the things we need to develop that potential, something that is often overlooked when we speak about the arts, is knowledge.
All of those traits – beauty, creativity, curiosity – are driven by a sound knowledge base. Knowledge sometimes gets a bad reputation, it’s often associated with the mindless drilling of Thomas Gradgrind in Hard Times. But knowledge is power to the artist because it enables so much else.
John Steinbeck put it best when he said:
“Ideas are like rabbits. You get a couple and learn how to handle them, and pretty soon you have a dozen.”
By the way, I’m not sure how familiar Steinbeck was with Fibonacci numbers but he definitely missed a trick here because they can be modelled using a strikingly similar metaphor.
Anyway, what Steinbeck was getting at was that the more knowledge we have, the exponentially more tools we have at our disposal. In his recent book Originals, Adam Grant describes the characteristics of so-called ‘creatives’ and his research is clear: the most creative people are the ones with the most ideas. Artists hedge their bets.
And a key aspect of knowledge is that so much of it is out there already, it’s established. Artists have long known that their first job is to mimic those who came before – art is apprenticeship. Or as Voltaire said, “originality is nothing but judicious imitation.”
And going back to Picasso, however revolutionary Cubism was, it was simply the next link in the chain of artists who came before. His work followed from Cezanne’s dual-perspective of the late nineteenth century. And Cezanne likewise followed his own predecessors.
Knowledge is the foundation upon which students’ creativity, curiosity, artistry will emerge. This is the justification for most of the maths we learn in school. But knowledge also has to be presented in ways that enrich students’ understanding and let them connect deeply with their subject matter. The way in which we represent knowledge makes all the difference.
I’ve never really understood music, never played an instrument. This is what music always looked like to me, a sprawling mess of symbols.
But one day, in graduate school, a professor played a clip of classical music that switched my perspective in an instant. This is Bach, as you’ve never seen him, represented by Stephen Malinowski:
For someone like me, who never understood music, this was magic. Just by changing the representation, I felt like I could connect with Bach, and see his inner workings, for the first time. That’s the power of strong mental representations – they reveal truths to us in new ways. They make us feel smart.
We don’t always find these representations in the maths curriculum. But with some imagination, we can easily get them.
Take times tables. Should you learn them, shouldn’t you? Well, if we just see them as a lump of disconnected facts, there’s little value either way. But I want to show you times tables as you might not have seen them before.
So here’s the standard multiplication grid, a showpiece of classrooms and home studies the world over. Nothing special there – in fact, it’s quite dull. It doesn’t excite and it sure doesn’t enliven the rich structures of multiplication. It’s a pretty weak representation.
Here’s a slightly better one, where we see times tables as nested within one another. If you stare at this long enough, it will reveal familiar results in novel ways. You can find the solution of summing the odd integers if you look hard enough.
And here’s another multiplication grid – courtesy of @the_chalkface. It’s the same as before, except now the numbers are drawn to scale.
With this simple tweak, the grid is beginning to speak to us. It conveys size and proportion rather than just numerical outputs. It connects otherwise disconnected topics, binding together number with geometry; multiplication with area.
And I’ll finish by showing you the standard 100-square in a not-very-standard format. Take a look at this, from Dan Finkel.
Let it speak to you. It may intrigue you. It may perplex you. And it should provoke some interesting questions. What do the colours mean? This grid will let you sink into the depths of multiplication, and if you stare long enough, you might stumble upon some old friends.
Since I want you to focus on the rest of this talk, I’ll give one spoiler and leave it to you to dwell on the rest later: this is the layout for a board game called Prime Climb.
Do you see what Hardy meant when he said mathematicians are painters? Makers of patterns? We took a construct as seemingly bland as times tables and brought it to life. And these mental representations – or schemas as the educationalists like to say – are so important to the artistry of mathematics. And for those of you who struggled with maths in school, they really make you think – that maybe you weren’t bad at maths; maybe you just weren’t looking at it the right way.
Art is a craft, but it’s also a mindset. Henri Matisse told us that ‘creativity takes courage’. We often think about artists in terms of their end result. I say Michaelangelo, you think the ceiling of the Sistine chapel: divine. But you know, even the great Michaelangelo was frightened to take on the challenge. His reputation, his status as an artist, his life’s work, was threatened. He was risking everything. Even artists have to face up to their vulnerabilities.
And when I think of the greatest feats in mathematics, that word courage keeps coming back. In 1994, Andrew Wiles issued his proof of Fermat’s Last Theorem, a problem that stretched back over 350 years and had defeated some of the greatest mathematical minds that had ever lived. The problem is captivating because it’s so simple to describe and so profoundly difficult to solve.
It says that this equation, x^n + y^n = z^n, has no whole-number solutions when n is greater than 2. Now when n is equal to 2 we know this equation does have whole-number solutions, because of Pythagoras’ theorem for right-angled triangles. It has infinitely many solutions. But Fermat reckoned that when you raise the power n, you no longer get any solutions. Quite a claim, and he even said he had a simple proof, but that the margin of his notebook was too narrow to contain it. Well, Fermat died, and the alleged proof went undiscovered. It took over 350 years for mathematicians to find a new proof, and it was finally achieved by that man there, Andrew Wiles.
I have a spurious claim to fame, by the way. Here goes: Andrew Wiles did maths at Oxford. His undergraduate tutorial partner at Oxford was Charles Batty. Charles Batty was my doctoral supervisor. So I think that makes me Wiles’s nephew, in academic terms, anyway. I did say it was spurious.
Now Wiles became obsessed with Fermat’s Last Theorem when he came across it in a library book at a young age, and he devoted his whole career to solving it. Imagine the audacity of taking on a problem many considered beyond reach. And Wiles locked himself away for seven years, working largely in isolation in search of a proof. He emerged to present his solution, only for a flaw to be revealed. It took a whole extra year for him to correct the error. The rest is mathematical folklore.
I mean, talk about courage. And I don’t think a mathematician can be without resilience, that ability to overcome doubt. And I think the same is true of all artists.
You know, when people tell me, as they all too often do, that they can’t do maths, that mathematical ability is innate, I despair. But then again I often felt the same way about art. I could never draw, never paint, never produce anything of visual significance. And so I found myself espousing Carol Dweck’s growth mindset in maths – the belief that our abilities are not fixed, and yet for some reason I refused to apply it to my own feared subject.
Well, maybe that could change. I came across a TED talk by Graham Shaw that aims to do for drawing what I and others have been championing for maths. He reckons that in 15 minutes he can get even a hopeless drawer like myself producing some pretty decent looking cartoon characters. And you know what…he’s not wrong. So here is the fruit of my labour.
Now I’ll grant you, I’m no Picasso…seriously. But this was far beyond anything I’d assumed I was capable of. That’s Einstein, by the way. And in those 15 minutes, I was persuaded, for the first time ever, that maybe I do have an innate talent for drawing. Maybe it just needs to be nurtured, with carefully guided instruction, with the freedom to explore and experiment. Without fear of being labelled a failure.
We are all consumed by doubt, and the richest art, and the best maths problems, will push us to the edge of our vulnerabilities. If we’re seeking courage, we just need to remember that artists, and mathematicians, are not just born, they’re also made.
And I have to say, as an educator that excites me.
Because now we can ask: what does all this mean for how we educate our students? I don’t have a prescription for classroom teaching –that’s not my background. But I hope I’ve shared a vision of what maths can look like, and feel like, for both students and teachers. If we want to inspire students with a renewed vision for maths then we first have to be inspired ourselves. And we have to be honest; not everyone fell in love with maths at school. But that’s because so often, the maths we experience in school, the maths ordained by curriculum standards and assessments, is a world away from art.
Paul Lockhart was scathing when he said:
“No society would ever reduce such a beautiful and meaningful art form to something so mindless and trivial”.
It really makes you think what made us reduce maths to a jungle of symbols and abstract procedures. This is not the maths that mathematicians fall in love with. So we need to reconnect maths to its art form, and remove the artificial barrier that has separated the two.
Picasso once said that “every child is an artist, the problem is staying an artist when you grow up.” Well I think every child is a mathematician. And as educators, we need to keep them that way. We need to shower them with rich mental representations and let them loose in this wondrous landscape of ideas and logic, of patterns and theorems. At times, we just need to let them play. And of course, we need to support students with the foundations of knowledge, but we must never forget that facts and procedures and algorithms are a means to an end. They serve a higher purpose; to give students a way of thinking about the world. And a way of being. Mathematics is etched into my identity and I’m proud to say it; it’s something all students are entitled to.
And I don’t say this as an idealist. This artistic vision of maths is one that I think is so crucial in the world we now occupy.
We cannot assume that the maths we taught students fifty years ago, hell even twenty years ago, is as relevant today. Back then, I may have hired people for their computational skills. You know, that word ‘computer’ was originally coined for humans, whose job it was to calculate. And of course, now we’ve been overtaken in that function by silicon chips. We just don’t employ people to carry out masses of calculations. Mental maths helps, it may even be necessary, but it isn’t enough. We need students to possess an intimate relationship with the objects they study, a deep understanding.
As one example, in this age of Big Data, we need greater emphasis on topics like statistics and probability. We certainly don’t need number crunchers because we have computers for that. But we do need people, humans, who can provide insight and connect data to context, to find deeper meaning.
In the years to come, we’ll have to face up to the realities of automation. And we’re going to depend on these distinctly human skills that machines cannot replicate – social skills. Empathy. Creativity. It is the era in which artists will reign supreme. Art arises as the interaction of our intellect and emotion. Maths is at its purest when it fuses together those two things. As Francis Su said recently, maths is a medium for human flourishing.
And as technology takes a firmer grip on our world, the role of teachers will become more important than ever. The most creative acts are formed when humans solve their most pressing problems together. We cannot hope for students to become mathematician artists if they do not have exemplars to follow, mentors who inspire them, who guide them. Even Michaelangelo had Pope Julius II championing him on.
I’ve quoted Hardy several times today but his story is not complete without another mathematician, and one of my heroes, Srinivasa Ramanujan. And I’ll finish with this.
So Ramanujan was born in rural India in the late 19th century. He’s the one on the right, in case that wasn’t obvious, and there’s Hardy on the left. Now, Ramanujan was raised in poverty and had access to a substandard education. But he was a genius in the true sense of the word – he once stumbled across a simple mathematical text, the equivalent of an A Level textbook, and from that he was able to derive deep mathematical theorems, many of them unknown at the time. So he wrote up his findings and sent his manuscripts over to England. And they were ignored as the work of a crank, for the most part. But G H Hardy saw something that nobody else did – he probed those manuscripts and saw that their was genius buried within. Hardy arranged for Ramanujan to visit Cambridge, where the two men collaborated in the years leading up to the first world war around topics in number theory.
And it was the most fascinating collaboration: Hardy was an ardent atheist who found beauty in the formalities of rigour and proof. Ramanujan was a Hindu Brahmin, who believed the gods communicated to him through numbers. So you can imagine how well they got on. They were from different worlds, but ultimately they lifted up one another and were unified by their love for mathematics. And their talents combined together to produce some of the most creative, artistic work the mathematical community has ever witnessed. But where would Ramanujan be without his mentor Hardy, who believed in him, nurtured his talents, and led him towards greatness?
And so my final thought for you today is that every artist needs a champion, and the most profound responsibility we have as maths educators is to help our students discover the artists within them.
Thank you very much.