Mathematics must be creative, else it ain’t mathematics

Why school maths could do with more mash-ups

I recently discovered the talent that is Tokio Myers. If you haven’t yet come across his performances on Britain’s Got Talent, head to the video below. Thank me by coming back.

You don’t need to be a pompous judge to see why Myers won the show. It takes courage to mash up Debussy with Ed Sheeran. We are impressed because we are surprised to see classical music fused with contemporary pop, and to see it done so well that it sounds and feels inevitable. I recall telling a friend earlier this year that my musical tastes are as fickle as they are fleeting, and that I had no way of connecting my love for Hans Zimmer scores with my appreciation of Rag‘n’Bone man’s Human. That was before I saw Tokio’s performance in the BGT final.

When people ask, often cynically, what creativity looks like, it is surely this: the ability to join seemingly disparate ideas to form new expressions of thought and emotion.

By this definition, mathematics must be considered a creative pursuit. The mathematical world is governed by patterns and symmetries, some of them known and most of them awaiting our discovery.

There are no topics in mathematics; only artificial barriers that we have erected to help organise the curriculum. At school, we study topics in discrete chunks and come to understand them as separate islands of knowledge. Yet the most powerful and interesting mathematics arises when we cut through these barriers.

We rarely stop to consider the interplay between arithmetic and geometry. When we are asked to sum the first 100 positive odd integers, say, we invoke narrow methods of computation; perhaps even a formula. We do not pause to draw those numbers, to see in them shape and symmetry. When we take that daring leap to marry geometry with number, our bravery is rewarded with the discovery of new and unexpected patterns that transform unwieldy sums into beauty.

Students recoil from algebra as if it descended from Mars; who could blame them? Studied in isolation, algebra is ugly and utterly confusing. But when we lift its veil of abstraction and link algebra to its close relative, co-ordinate geometry, we arrive at a whole new plane of understanding. The idea of representing every point in a plane using just two numbers — what we now know as the x and y co-ordinates — was Descartes’ own nod to creativity.

We marvel at the simplicity with which Fermat’s Last Theorem is stated, and the complexity that must fill the 100 pages of Andrew Wiles’s proof. Did we stop to marvel at his creativity, and of those who preceded him, in linking together previously unlinked concepts? Modular elliptic curves is the mathematical mash-up of our time, unifying concepts in algebraic geometry and number theory and giving rise to new branches of research. This is what mathematicians do: they create and recreate their subject, seeking connections even where none had any right to exist.

An elliptic curve, waiting to be mashed up

Knowledge has its place; you cannot connect or create what you don’t know. Tokio Myers is a trained pianist; his thousands of hours of deliberate practice are the foundation of his creativity. But he is more than a pianist (he’s a BGT winner!), and that is because he dared to defy the the conventional norms of music. Wiles is similarly the world class mathematician that he is because his field of vision is not restricted to any one topic.

If only students were encouraged to transcend their study of individual topics. When a GCSE exam question dared to combine a quadratic equation with basic probability, the students roared with disapproval. Among them was my niece, who defiantly proclaimed that this isn’t how she were taught. Quadratics, fine. Probability, no problem. But a question that requires both? Call the press; it’s time to create another headline about a fiendish maths problem.

The tyranny of school maths lies in the false promise that stuffing oneself with facts and procedures prepares you for creativity. The act of creativity is deferred to an unspecified time — presumably it is for older, more knowledgeable people. It’s as if Tokio was instructed to learn his scales but never put hand to piano. No mathematician I have ever met learned their craft this way. They dived deep into topics for sure, but they habitually sought to join up concepts and apply their knowledge in novel ways to create entirely new understandings of mathematics. Young age is no barrier — my most impressive students are also the shortest; it only takes a well-crafted maths problem to unleash their innate creativity.

It helps to organise mathematical knowledge. There are obvious benefits to going deep in a particular area and I always offer a gracious nod to the fluency and fundamentals of mathematics. But mathematics at its most fundamental is an integrated body of ideas, replete in patterns. The patterns and connections are what makes it mathematics. Let that be your next headline.

I am a research mathematician turned educator working at the nexus of mathematics, education and innovation.

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