“I’ve never used any of that trigonometry we had to learn in school - what was the point in learning it, when we could have been taught something useful?” This question, and questions like it, are put to all of us who work in education from time to time. And, all too often, we undermine the true worth of what we do by framing our answer in the values of the question.

If there is a battle to be fought here, then we have already lost it if we allow ourselves to be drawn into fighting it on the questioner’s terms. It doesn’t matter how confidently we retort, “Well, if you were to find yourself a known distance from a building you need to calculate the height of…”

Quite understandably, this contrived approach to justifying a mathematical education by imagining increasingly surreal scenarios into which we can crowbar bits of the curriculum does not persuade anybody. And nor should it.

The truth is that if we want to teach people the maths they would actually need to use in their adult lives, for the vast majority of learners, we would teach them arithmetic until somewhere around the end of Year 4, and then call it a day.

The direction of travel in mathematics pedagogy in recent years has championed the ideas of US psychologist Jerome Bruner and cognate theories of learning, which consciously withhold abstraction until the latter stages of learning. There is, of course, enormous value in teachers considering that learners need to be properly prepared before being presented with abstruse symbolic representations. But we must not fall into the trap of thinking that the abstract is to be feared or avoided. Mathematics is abstract; that’s the point.

Indeed, we must be cautious that the desire to anchor all learning in the concrete, and to emphasise real-world contexts for the maths we are teaching, does not obscure the true value of mathematics and mathematics education: that it is inherently abstract and that what we are actually teaching is a capacity for abstract thought.

To put it bluntly, we do not teach children to calculate the area of composite shapes in case they one day have to tile an odd-shaped bathroom floor. Lurking behind the stated learning objective for the lesson is always the same latent goal: teaching the application of reason to solve a problem of logic. Teaching children to calculate is a side endeavour; we are teaching children how to think.

This principle is perhaps at its most acute when discussing maths, but maths is certainly not the only place we find it. Education in any and every area of study must fight back against the existential threat posed by those who seek to diminish it by interrogating how directly applicable it is in the fabled “real world”.

The questioners themselves are nothing new. As Bertrand Russell wrote in his 1945 work, A History of Western Philosophy, framing the views of Aristotle from more than two millennia earlier: “They must, of course, learn to read and write in spite of the usefulness of these arts. But the purpose of education is ‘virtue’, not usefulness.”

Fertile battleground

The reason the threat is on the rise, however, is that we have lost our defence. There have always been those who place no inherent value in the intentions and the soul of the subjects we hold dear. But, too often, educationalists of today pander to these worldviews in vain attempts to win the holders round. It is no exaggeration to say that this is a battle for the heart of education.

English is another fertile battleground for these arguments, and a subject whose true worth is ripe for restating. It’s disheartening to hear so many teachers mumbling resigned renunciations that “they’ll never need to understand the subjunctive mood, anyway” or parents proudly crowing, “Well, I never learned about this passive-voice thing and I’m doing just fine.”

Certainly, one needn’t be able to explain many of the finer (and more interesting) points of the English language curriculum in order to hold a conversation, bluff one’s way through a job interview or even write an article for Tes. Pointing to moments when someone might need to recognise the past perfect is a challenging - and, more importantly, fruitless - task.

Instead, we should be pointing out that the true virtue of teaching children to notice language, to speak about language and to think about language is that the observer will always see more of the game. To notice patterns in the way those around us - and we ourselves - are using language is to notice the way people think. Metalinguistic understanding is a vital vehicle to metacognition. If mathematics education is really about teaching people how to think, then linguistics education allows them to reflect and comment upon that thinking.

Of course, English also contains the study of literature. If any area of study has come under fire for not focusing enough on utility more than mathematics, then surely the study of literature is it. There have always been those who eschew matters of culture, decrying the idea of pupils discussing poetry as an indulgence.

So, how do we present its value? By listing parts of the job market where content writing might be a valued skill? No. We do it by standing our ground on the value of culture, the power of the metaphor and the unique invitation that literature offers to reflect on the self and the human condition. Children don’t learn empathy in a didactic PSHE lesson about sharing but by exploring what it is to be human and to recognise humans - and this is nothing less than what is happening when children are guided through explorations of literature.

The arts have long suffered the damage of sullied defence: making the case that theatre matters because of what the West End does for the London economy has created an environment of very precarious footing for the sector. Education in the arts, including the performing arts, is no different. It has always had to fend off interrogation about its value.

Here, again, the true danger lies in the misappropriation of curriculum content to contrive vulgar defences about presentation and public speaking. If we wished to teach public speaking, then that is what we would teach - not Brecht. The value of arts lessons is about creativity, expression and the deeper experiences of thought. It does not need translating into the narrow, the parochial or the functional.

Creativity brings us back to maths. The late author and educationist Ken Robinson defined creativity as “the process of having original ideas that have value”. Mathematics seeks to foster this capacity by stripping away all but the essential known parameters of a problem. The long-standing puzzle of the Bridges of Königsberg, for example, saw people attempt to navigate the islands of the city, crossing each of seven bridges only once. When the great Swiss mathematician, Leonhard Euler, set out to prove, once and for all, whether it was possible to do so, he did not put on his coat and wander the streets. Instead, he removed the problem from anything that remotely resembled the real world and created an entirely new way of thinking about location, inventing what we now know as graph theory in the process. This is mathematics at its truest: unfastened from the crudities and limitations of the real world, home to thinking at its purest and most divergent, and unapologetically abstract.

Galileo wrote in The Assayer that the universe “cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles and other geometric figures, without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth.”

This ability to shine a light on the real world is what we need children to understand about maths. If we hide it from them by obsessively holding them in the relative comfort of the graspable and easily manipulatable world around them, then we deprive them of the very thing we are trying to provide them with: the deductive reasoning and systematic thought strategies that are the core tools of the mathematician.

Thoughtful pedagogies

Clearly, it is neither feasible nor desirable that we should create an education system that dogmatically presents subjects only in their purest form. The teaching of a discipline is a different endeavour from the practice of that discipline. For this reason, we must, of course, ensure that we have thoughtful pedagogies, which create navigable routes of access for our pupils.

We must, however, protect the integrity of our subjects, not apologise for them. Perhaps every subject needs a clear mission statement: a set of guiding principles detailing core purposes and intentions for the field of study. This could both inform our pedagogy and allow all teachers to sing from the same hymn sheet as we eulogise about the non-concrete reasoning children are learning in maths lessons or the metalinguistic cognition at the heart of a lesson about grammar.

If not, we will continually have to answer to the real-world contexts brigade, with their incessant focus on the practical and the utilitarian. We must proudly nail our colours to the unrepentant mast of maths as formal logic, or be doomed to spending years trying to explain how simultaneous equations may one day help pupils to work out which is the best route to cycle to the train station.

Thomas Kent is a middle leader at a primary school in Southend-on-Sea

This article originally appeared in the 12 March 2021 issue under the headline “The noble art of self-defence”