The secret life of numbers

How do you spark off an interest in maths when the curriculum seems dreary? It's all about mystery, big stories and journeys to infinity and beyond, says Marcus du Sautoy [via guardian]


Maths can come alive when people learn how it runs seductively below the surface in art and music. Photograph: Daryl Benson/Getty

My son is 13. In his English lessons, he spends time learning the grammar and vocabulary of the language - basic necessities for anyone leaving school. But he has also been exposed to some of the great works of literature that have been created using these building blocks. He has already read Richard III and George Eliot's Silas Marner. He probably didn't understand the intricate complexities and subtleties of these great works, but he was excited by the contact with such stimulating literature.

In mathematics, he has also been learning the basic grammar and vocabulary of the world of numbers. Percentages, long division, some basic algebra and geometry. Techniques that are also regarded as core skills that every child should leave school with. But the curriculum has not exposed him yet to the creative possibilities of mastering these tools. And nor is the curriculum likely to, even as he advances through the school system.

The teachers are required to teach a utilitarian and unadventurous curriculum that leaves them no room to explore the creative side of the subject. Indeed, most people are utterly surprised to discover that there is any creativity in mathematics.

When I was 13, I hadn't caught the mathematical bug yet. I wasn't particularly interested in mathematical computations. But then my mathematics teacher took me aside after one lesson and recommended a few books that he thought might interest me. He conspiratorially intimated that the maths we were doing in the classroom wasn't really what maths was about. It was something much more exciting, creative, imaginative. Those books provided me with a key to the secret garden of mathematics.

In that garden I discovered that mathematics also has great stories. Unsolved mysteries like the enigma of prime numbers. Magical mathematical machines that could help you see in four dimensions. Mathematicians who had journeyed to infinity and beyond, discovering that there are many sorts of infinity, some bigger than others. Like my son reading Shakespeare, I certainly didn't understand everything I read, but it inspired me to want to navigate this world, to put in the hard graft to master the language and grammar of maths so that I could read and one day create my own mathematical stories.

One of the books my teacher recommended was GH Hardy's A Mathematician's Apology. At the time, I was very interested in music, I was learning the trumpet, hanging out with the arty crowd, doing plays and singing in choirs. Science hadn't really captured my imagination. But I also had a desire for things that made logical sense, for solving puzzles, for a rational perspective on the world. A Mathematician's Apology suddenly opened up a bridge between these two competing desires, these two cultures.

As I read Hardy's book, there were sentences which revealed to me that mathematics shared a lot in common with the creative arts. It seemed to be compatible with things I loved doing: languages, music, literature. Here for example is Hardy writing about being a mathematician: "A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas." Later he writes: "The mathematician's patterns, like the painter's or the poet's, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics."

For Hardy, mathematics seemed to be a subject with a sense of aesthetics. His book contained two proofs. Like playing a delicate mathematical minuet, he explained the ancient Greeks' discovery that there are infinitely many primes. It was a revelation that one could prove with such a simple piece of logical reasoning that these indivisible numbers with no discernible pattern spiral off to infinity. That our finite minds could master the infinite was inspiring. Here was the power of analytical thinking to get you to new places, new discoveries, new knowledge.

The other proof he explained was the discovery that the square root of 2 cannot be written as a fraction, another proof for which the ancient Greeks were responsible. It led to the creation of a whole new sort of number called irrational numbers. Mathematics is full of these extraordinary moments of creativity and discovery, breakthroughs that have had an impact on understanding the world we live in.

The creation of a number whose square is -1 seems a moment of absurdity, but led to the maths that allows us to formulate quantum physics. Imagining new geometries that exist beyond our three-dimensional world and will never have a physical reality gave birth to the physics of relativity. Creating strange new symmetrical objects has inspired the invention of codes that are the basis of the telecommunications industry.

But mathematics doesn't always need to be linked to a technological or scientific breakthrough to accentuate its potency. Discovering that there were many different sorts of infinity (in fact infinitely many) was an exciting moment in my own mathematical career, as it was for the first mathematicians who discovered it at the end of the 19th century. It doesn't need to have an application to be exciting.

I'm a maths nerd. I love maths for its own sake. But for others the subject comes alive when they learn how mathematics is not an isolated subject, but runs seductively below the surface of many other subjects in the curriculum.

I've never understood why education is so compartmentalised. My son looks at his timetable: maths first lesson, history second lesson, music before lunch. The curriculum gives no hint at how integrated all these subjects are. To look at the historical evolution of mathematical ideas provides an invaluable perspective on why the mathematics was created in the first place.

The volume of a pyramid is a third the base times the height. Its discovery was first recorded in the Egyptian Rhind Papyrus. The Egyptians wanted to know how much stone would be needed to build the pyramids in Giza. A cosine is the adjacent divided by the hypotenuse. It was developed to allow ancient astronomers to measure distances in space without ever having to move from the comfort of their observatories.

In music, there are so many interesting mathematical themes and variations that can be investigated: Ghanaian and Indian rhythms exploit the indivisibility of the primes; the tension between fractions and irrational numbers like the square root of 2 is the key to problems of musical harmony; and modern music by Schoenberg and Messiaen is a musical expression of the mathematics of symmetry.

In art, the mathematics of symmetry is fundamental to understanding the beautiful designs of the Moorish artists in the Alhambra. Baroque art and science are both attempting to understand how to capture things in motion: one with paint, the other with calculus. Just like Riemann before them, the cubist painters are trying to see beyond our three-dimensional world into hyperspace. The architecture that adorns our modern skylines is the physical embodiment of many different themes from the mathematical palette.

In English and theatre, I've done workshops with maths and theatre teachers exploring ideas of infinity as part of my collaboration with the theatre company Complicité. The condition to join the workshop was that a theatre teacher had to bring a maths teacher with them. They went away wondering why they'd never talked to each other in the common room before. As Stoppard and Frayn discovered, mathematics is full of wonderful ideas to explore theatrically. Even Shakespeare's poetry is full of interesting mathematical structures. Mathematics could come alive for so many more people if it wasn't kept behind walls.

I am not an educationalist. I am a mathematician. But I know what turned me on to the subject. It was being shown what mathematics is really about. It was being exposed to the big stories, the Shakespeare of mathematics that inspired me.

Why are more children not given the key to this secret garden? Why can't we include the Shakespeare of maths in the curriculum? I admit it's not for everyone, just as Shakespeare doesn't work for every child. English has two GSCEs: literature and language. There is talk of having two GCSEs in maths. Why not dedicate the second GSCE to studying the great stories of maths? We are not frightened to throw Richard III at 13-year-olds. Let's be more brave and throw Riemann at them, too.

Marcus du Sautoy is professor of mathematics at the University of Oxford and Simonyi Professor for the Public Understanding of Science. He is author of The Music of the Primes and Finding Moonshine (both HarperPerennial)

5 puzzles for pupils

1 What's the next number in this sequence: 1,1,2,3,5,8,13...

2 What are the chances that two people in the class have the same birthday?

3 Is there more than one infinity?

4 Primes go on for ever, but what's the biggest prime so far discovered?

5 A one-dimensional line has two ends. A 2D square has 4 corners. A 3D cube has 8 corners. How many corners do you think a 4D cube has?

The Answers:

1 21. Each number is got by adding the two previous numbers together. These numbers are the key to mating rabbits, musical rhythms, growing shells and Le Corbusier's buildings.
2 More than 50:50 if the class has at least 23 children. The mathematics of probability is as counter-intuitive as it is useful in navigating the risks that surround us.
3 There are infinitely many infinities, some bigger than others.
4 The biggest prime is 243112609-1, which has nearly 13m digits.
5 It has 16 corners.

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