By Nick Hamshaw
Every year, I meet children who tell me they love mathematics. Their parents usually tell me much the same thing. Sometimes it is because mathematics is their favourite subject. Sometimes because they seem to finish every worksheet before anyone else. Occasionally it is because they have always been fascinated by numbers, patterns or puzzles. Those are always lovely conversations to have.
Over the years, however, I have found myself becoming interested in something rather different. Looking back over the thousands of students I have taught, I have gradually come to realise that the young people who eventually immerse themselves most deeply in mathematics often share a characteristic that is surprisingly difficult to describe and even harder to measure. They are curious in a distinctly mathematical way. That curiosity does not necessarily reveal itself through exceptional examination results or extraordinary speed, although it sometimes does. Instead, it appears in the kinds of questions they ask, the things they notice and, perhaps most tellingly, the things that they simply cannot leave alone.
One particular student comes back to me remarkably often. A simple example came about when I was teaching one of my sixth form classes several years ago. We had been exploring divisibility proofs using induction and had developed a perfectly good method that could be applied to a whole family of questions. Most of the class had completed the example and moved on. One student, however, remained staring at the page. He was not writing anything and, after a while, I wandered over to ask whether everything was all right. “It is,” he replied. “I just don’t think this is the best way.”
His answer intrigued me. He was not saying that the proof was wrong, nor was he trying to be awkward. He understood the argument completely and could have reproduced it without any difficulty. What interested him was something else entirely. He wanted to find a proof that reflected the underlying structure of the mathematics more naturally. His reasoning was that, if he ever encountered a less familiar problem in the future, recognising the mathematical structure would be far more useful than remembering a particular technique. At the time, I remember thinking that it was an unusually mature observation for a seventeen-year-old. Looking back now, I think it was much more than that.
A week later we had moved on to differential equations. Once again, I noticed him staring at the page. This time, the examination question finished with the word “Hence”. His concern was not how to answer it. He wanted to know whether the question would have been more interesting had it ended “Hence or otherwise”. Once again, he was asking himself whether the expected solution was really the most revealing one.
I no longer remember which proof he eventually preferred, nor can I recall the examination question itself. What I remember is the way he was thinking. Over the years I have had versions of that same conversation many dozens of times. Different schools, different pupils and different branches of mathematics, yet the underlying instinct has been remarkably consistent. The students who eventually become absorbed by mathematics are often interested not simply in whether a method works, but in why this method works, whether another might reveal something different and what the mathematics is trying to tell them underneath. It is one of those observations that sounds almost obvious once you have seen it often enough. The difficulty is that it can be remarkably easy to miss.
Schools are, quite understandably, very good at recognising mathematical attainment. They have to be. Public examinations require us to assess what students know and what they can do, and those outcomes matter enormously. They open doors, create opportunities and provide an important measure of progress. I would never wish to diminish their importance. What I am less certain about is whether they always help us recognise mathematical potential. This is a different challenge for a teacher altogether.
The qualities I have gradually come to associate with future mathematicians are not always the easiest to capture in a written examination. Curiosity, persistence, a willingness to remain with uncertainty for a while, the habit of asking better questions rather than simply producing quicker answers and the ability to recognise underlying structure are all characteristics that seem to emerge repeatedly. Yet none of them sits particularly comfortably inside a mark scheme.
One of the things I have slowly come to realise is that genuinely mathematical children are often surprisingly tolerant of not knowing the answer. Indeed, some seem almost disappointed when a problem yields too quickly. That does not mean they enjoy being confused or never become frustrated. Rather, they seem to recognise, almost instinctively, that the most interesting mathematics often lies just beyond the first successful solution.
Looking back over my own teaching career, I suspect I spent too much of my early years congratulating students for being quick. Speed certainly has its place. Fluency matters and confidence matters, yet these days I often find myself much more interested in the student who asks a question that nobody else in the room had thought to ask. A quick answer tells us something about what a student can already do. An unexpected question may tell us something about the mathematician they could become.
This distinction has become increasingly important in recent years, not least because the world around us is changing so rapidly. Artificial intelligence is already transforming the way mathematics is used in many professions. Procedures that once demanded considerable expertise can now be carried out almost instantaneously by a machine. That should not make us value mathematics less. If anything, I believe it should make us think more carefully about what aspects of mathematics are uniquely human.
As machines become increasingly capable of carrying out algorithms, our contribution lies ever more clearly in asking worthwhile questions, recognising unexpected patterns, making connections between seemingly unrelated ideas and constructing convincing arguments. Those qualities do not suddenly appear when a student begins a university degree. They develop gradually over many years, often beginning with the simple habit of asking, “Why should this always be true?”
It seems to me that this brings us to one of the most important educational questions currently facing us: what is school mathematics actually trying to achieve?
Part of the answer seems fairly straightforward. Every young person deserves a mathematics education that equips them to participate confidently in modern society. Financial decisions, statistics in the media, technology, science and the workplace all demand mathematical understanding and I do not think anybody would seriously argue that these things are unimportant. Another part of the answer comes from the technical requirements of many careers, including engineering, pharmaceuticals and many others.
What I wonder, however, is whether that is the only responsibility we have. Some young people will go on to become engineers, economists, computer scientists, physicists and mathematicians whose work depends upon mathematical thinking of extraordinary depth. If that is the case, should we simply assume that exactly the same curriculum, with exactly the same emphasis and trajectory, is necessarily the best preparation for every mathematical future? That is not because one group somehow deserves a richer education than another. Quite the opposite. Every young person deserves rich mathematics. The question, perhaps, is whether richness always takes exactly the same form.
One of the reasons I have found myself thinking more carefully about these questions over the past few years is that mathematics education seems to be entering a particularly interesting period. Artificial intelligence is changing the way we work. The Curriculum and Assessment Review has encouraged us to think again about what should be taught and why. Across the country there is renewed discussion about whether school mathematics has become too abstract, too procedural, too difficult or, depending on whom one listens to, perhaps not abstract enough. Those debates are important and I hope they continue. They have encouraged all of us who care about mathematics education to reflect on first principles rather than simply debating the detail of individual specifications.
Every generation contains a relatively small number of young people whose relationship with mathematics is rather different. Some of them will become engineers, economists, computer scientists or physicists. A very small number will become mathematicians whose work extends the subject itself. Others will make contributions in fields that have not yet been imagined but will nevertheless depend upon deep mathematical thinking. Those young people matter too. That is not because they are somehow more important than anyone else, nor because they deserve a better education than everybody else. It is because society has two responsibilities, not one: to ensure that every young person leaves school mathematically equipped for the life they choose and to ensure that exceptional mathematical potential is recognised, nurtured and allowed to flourish.
Those responsibilities are different. The difficulty, it seems to me, is that we often behave as though they must always be achieved through precisely the same curriculum, taught in precisely the same way and progressing at precisely the same pace. Perhaps they can and perhaps they cannot. I simply think the question deserves rather more discussion than it sometimes receives.
One of the things that has increasingly intrigued me is that many countries do not regard this as an especially controversial conversation. Romania, Bulgaria, Russia, Singapore and several others have long traditions of specialist mathematical education within their state systems. Those schools are not built on the assumption that some children are worth more than others. They exist because those societies have concluded that exceptional mathematical talent is itself something worth developing. Whether we should adopt similar approaches in the UK is, of course, open to debate. What surprises me is not that people reach different conclusions, but that we sometimes seem reluctant even to ask the question.
Part of the difficulty, I suspect, is that discussions about excellence in education can very quickly become discussions about fairness. Those are important conversations and they should never be dismissed lightly. We should always ask who benefits, who is excluded and whether opportunities are genuinely accessible. Yet mathematics has a rather awkward habit of refusing to conform to our educational philosophies. One of the things that distinguishes mathematics from many other subjects is that, in the end, it is remarkably indifferent to our educational philosophies or our political ideologies. We can and should debate the purposes of school mathematics. We should think carefully about what every young person needs and how best to engage them. Those are important questions. Mathematics itself, however, remains entirely unmoved by those debates.
A student who hopes one day to contribute to modern mathematics will eventually need to grapple with abstraction, proof, generalisation and increasingly sophisticated structures. Those demands do not arise because teachers enjoy making life difficult. They arise because that is the nature of the discipline itself. If we genuinely wish to develop future mathematicians, then at some point our curriculum must respond not simply to what young people find engaging, but also to what mathematics itself requires.
I have gradually come to believe that this is where many educational discussions become unnecessarily polarised. People sometimes assume that the alternative to a curriculum rooted in real-world applications is one dominated by abstract manipulation for its own sake. In my experience, neither picture is particularly accurate. The most memorable mathematical lessons I have ever taught have rarely been those in which students learned the greatest amount of new content. They have usually been the lessons in which students found themselves thinking differently. A simple counting problem suddenly became an exercise in symmetry. A geometry question revealed an unexpected invariant. An algebraic manipulation exposed a much deeper structure than anyone had anticipated at the beginning of the lesson.
When we began designing 1729 Maths School, these were the conversations we found ourselves returning to again and again. We were not asking how we might teach GCSE mathematics more quickly or how we might introduce A Level content at a younger age. Those seemed to us to be relatively small questions. The larger question was this: if our aim is genuinely to help young people become mathematicians, what experiences should they have between the ages of eleven and eighteen?
That question led us in some rather unexpected directions. It led us to think about curiosity before content, about discussion before explanation and about mathematical habits of mind as well as mathematical knowledge. It led us to create opportunities for students to work alongside research mathematicians, not because they need university mathematics at the age of eleven, but because they deserve to see what it actually means to live as a mathematician. It led us to distinguish between developing mathematical foundations and developing mathematical imagination, because both seemed essential and neither quite replaced the other.
Whether we have answered those questions well remains to be seen. New schools should be humble enough to recognise that they have much still to learn. What I do know is that the conversations themselves have changed the way I think about mathematics education. Looking back over twenty-five years of teaching, I no longer think the most important question I can ask a young mathematician is whether they can solve today’s problem. I am much more interested in whether today’s problem leaves them with a new question that they cannot wait to explore tomorrow.
Perhaps that is one of the simplest ways of recognising the difference between enjoying mathematics and beginning to think like a mathematician.