We tend to imagine mathematics as a solitary pursuit. The mathematician sits alone at a desk, surrounded by paper, thinking deeply until a moment of inspiration produces a solution. It is a familiar and rather romantic image, but it tells only part of the story.
Mathematics is also a profoundly social activity. Mathematicians discuss ideas, challenge one another's reasoning, share conjectures and collaborate on problems that none of them may be able to solve alone. Mathematical discoveries are communicated, scrutinised and refined by other mathematicians. An argument is not accepted simply because its author believes it to be correct; it must withstand the questioning of others.
One of the most famous examples is Andrew Wiles's proof of Fermat's Last Theorem, a problem that had resisted mathematicians for more than 350 years. Wiles worked largely in isolation for seven years before announcing his extraordinary achievement in 1993. Yet when other mathematicians examined his work, a gap emerged in the argument. Wiles subsequently worked with his former student Richard Taylor, and in 1994 the difficulty was overcome. The proof was eventually published the following year.
The achievement remains overwhelmingly Wiles's, but its history illustrates something important. Even the most brilliant mathematical thinking benefits from being communicated, questioned and tested. Mathematics advances not simply through individual insight, but through the collective efforts of a mathematical community.
What happens when young mathematicians work together?
The same principle applies in a classroom.
Imagine four pupils tackling an unfamiliar mathematical problem. One notices a pattern and suggests a possible solution. Another immediately finds a counterexample. A third proposes a different approach, while the fourth asks whether they can prove that the new idea will always work.
None of the pupils necessarily knows more mathematics than the others. Yet their conversation has taken them somewhere that none might have reached independently.
This is one of the great pleasures of teaching mathematics. A pupil begins explaining an idea and, halfway through, realises that something is missing. Another offers an unexpected observation that changes everyone's understanding. Sometimes a discussion that begins with a relatively straightforward question develops into an exploration of something much deeper.
These experiences are not simply enjoyable additions to mathematical learning. They help young people develop the habits of thinking that mathematics requires: constructing convincing arguments, questioning assumptions, considering alternative approaches and recognising when an explanation is incomplete.
There is also a particular value in having to explain an idea to someone who does not immediately agree. A pupil may feel entirely confident about a solution until another asks, 'But why must that always be true?' Suddenly, an intuitive answer is no longer enough. They need a mathematical argument.
Why not simply learn individually?
We live in an age in which extraordinary mathematical resources are available to almost anyone with an internet connection. Online courses can introduce children to advanced topics, while individual tutoring can provide highly personalised teaching and allow pupils to progress at their own pace.
These are valuable opportunities, and for some young people they can be transformative.
But they do not necessarily provide everything that a mathematical education should offer.
A one-to-one tutor can challenge a pupil's thinking, but a group of mathematical peers introduces something different. Other young people bring their own ideas, misconceptions, questions and ways of approaching a problem. Their contributions are less predictable, and the discussion can move in directions that neither teacher nor pupil anticipated.
There is also an important difference between being taught mathematics and participating in a mathematical community. In the latter, young people are not simply recipients of knowledge. They become contributors to a shared intellectual endeavour.
Interestingly, mathematicians themselves have demonstrated that this kind of collaboration need not always happen in the same room. The Polymath Project, begun in 2009, brought mathematicians together through online discussions to tackle difficult research problems. Participants shared partial ideas, challenged one another's approaches and collectively developed new proofs.
The lesson is not that online mathematics is somehow inferior. It is that access to mathematical content is not the same as mathematical interaction. Technology can be an extraordinary means of collaboration, but working through an online course alone is a rather different experience.
Finding people who share your enthusiasm
There is another reason why learning mathematics socially matters, particularly for young people with a strong affinity for the subject.
A child who is fascinated by mathematics may not always find classmates who share that enthusiasm. They might enjoy spending an afternoon exploring a difficult problem, discussing an unusual pattern or trying to prove something they have noticed. But such interests are not necessarily shared by their immediate friends.
Finding other young people who enjoy mathematics in the same way can be enormously liberating.
Suddenly, asking an unusual question is entirely normal. Spending a long time thinking about a problem is something to enjoy together. Someone else might have a better idea, and that becomes a source of excitement rather than embarrassment.
This is not simply about improving mathematical attainment. It is about friendship, belonging and the pleasure of sharing an intellectual passion.
For some children, discovering a community of mathematical peers can change their relationship with the subject altogether.
The importance of thinking alone
None of this means that mathematics should always be collaborative. Quite the opposite.
Deep mathematical thinking often requires silence, concentration and prolonged independent effort. There are times when a pupil needs to wrestle with a problem alone, try different approaches and experience the satisfaction of making progress without assistance.
Indeed, collaboration becomes much more valuable when everyone has had the opportunity to think independently.
The richest mathematical education therefore combines both experiences. A pupil might begin by investigating a problem alone, share a conjecture with others, defend or revise their reasoning in response to questions and then return to their own work with a deeper understanding.
The aim is not to replace individual thought with group activity. It is to allow each to enrich the other.
Building a mathematical community
This belief is central to the educational approach at 1729 Maths School.
Our pupils spend considerably more time studying mathematics than they would in a conventional school, but the additional time is not simply devoted to learning more material. It allows them to explore unfamiliar problems, discuss ideas, construct arguments and work alongside others who share their enthusiasm. Teachers and research mathematicians contribute to that community, but so do the pupils themselves.
We want young people to become confident independent thinkers who are equally comfortable sharing their ideas, listening to others and recognising that a different approach may be better than their own.
Ultimately, mathematics is not simply a collection of techniques to be mastered or problems to be solved. It is a living intellectual discipline, developed over centuries through the exchange of ideas between people.
Learning to think independently is essential. Learning to think with others can take us somewhere entirely new.