A few years ago, I was in school on A-level results day when a parent approached me. Jane had just found out her grades. I knew already that they were excellent: a couple of A*s and an A in my subject, maths. She’d done enough to be able to take up her place at Oxford. So, as the parent approached, I was anticipating a nice conversation about how well Jane had done.

But no. The parent was coming to talk to me because Jane was upset. She’d not got that little star next to her grade in maths. Could I arrange a remark?

Jane was fantastic, super conscientious: the kind of kid who, if you cracked a joke in class, would want to know if it was coming up on the test. She’d achieved her goal of getting into Oxford. But - to her mind - because she’d not got A*s across the board, she’d done it in an imperfect way.

This is an extreme example, but the desire for everything to be perfect is all too common in schools. It manifests itself more frequently in students wanting to tear pages out of books if they make a small mistake, or at least make a grab for the correction fluid.

In my subject, this is a particular problem because students understand that there is an objectively “right” answer that they are aiming for. This has its attractions - they know when they have got something correct that it is unimpeachably so - but it also creates anxiety because small errors have big consequences. A stray fleck of paint might be no catastrophe. A quote in an essay pushed slightly beyond its supportable meaning can be coped with. But in mathematics, that iota of error cascades down and collapses the result: it is now wrong. Because perfection is achievable, imperfection becomes intolerable.

Out come the scissors and glue. While I’m not looking, a careful rectangle has been scalpelled from the back of their book, a perfect cut along the lines of the squared paper. By the time I’ve circled round to their desk again, the deed has been done. The error erased, buried under a graft of fresh squares, the correct solution written on top. There. No mistakes. They can relax.

It’s at this point that we have to talk together as a class about the principle of learning from our mistakes. “It’s fine to get things wrong,” I remind them. “We can learn something from that.” And they nod in agreement. But turn it around and there is more discomfort: “Unless we make mistakes, we are not learning.”

What if the only way to learn, to make progress, is to get things wrong? This is not so easy to grasp. Why on earth would it be a good thing to get something wrong? Surely perfection is the perfect way to be? How could it be more perfect to experience some imperfection?

It’s at this point that the class usually wants me to shut up and let them get on with their work but they’ve learned, over time, to expect occasional detours. The situation I pose to them here might go something like this: imagine two great leaders, identical in every way. Both have gained sufficient wisdom to know what the right course of action is. But from whom would you rather take counsel: one who had never set a toe wrong, or one who has gained their wisdom with a few mistakes along the way?

The general consensus is usually the same: even if the two leaders know the same number of “right” things, the one who has also made some mistakes - got some things wrong along the way - will know more.

In other words, imperfection is expansive and instructive. Perfection is a straitjacket. It has to move cautiously, desperate not to blot its copybook. Imperfection is exploratory, inquisitive, interesting. Two people who have arrived at a fixed point might be in the same place, but the one who took wrong turns on the way to finding the place will know more about their surroundings than the one who took the “right” route there. In the same way, the student who has got to the right answer via a comprehensive romp through the ground of the wrong methods that lie close by will be more in tune with the landscape next time around.

The practical application of this understanding is that the last thing students should be doing is covering over their mistakes in their books. Doing so is erasing the learning that they have done, the experience that they have gained. If their books become no more than lists of exclusively correct answers, then they have become sterilised. To push that (perhaps timely) biomedical metaphor further, an organism that has never fought off an infection is weaker than one that has. The struggle for immunity is a messy one; sterility is not a sign of strength but fragility.

Drawing this back to learning, students need help understanding that what they are creating in their books is not a final work of art but the story of their learning journey. When they come to revising, a book in which mistakes have been made, identified, reflected on and learned from is far more useful than a clean list of perfect solutions.

This idea of creating a learning story in their maths books is one that students have found helpful. It liberates them to introduce something of their own voice into their work, rather than feeling that they have to speak with pure mathematical grammar.

This requires some flexibility on my part. Obviously, it is far, far easier for me to mark work that has been presented in a highly structured, controlled way that I have mandated. But this, again, is desire on my part for clean perfection and is not necessarily focused on giving them the best chance to develop their own learning story. So I encourage students to find their voice with colours and highlighters, call-out boxes or emoji. This does not mean that chaos rules or that anything goes. But it does mean that mistakes should be celebrated and embraced, rather than covered over and erased.

The Nigerian writer and poet, Ben Okri, puts it well in his 1996 essay, Birds of Heaven, where he says that “where perfection lies, there is no more story to tell”. If we insist on clinical perfection in work, then there is nothing more that can be said. The problem with this is that this perfect formulation, because it has no story, is not necessarily memorable. It offers little learning.

What lends any story this essential quality of being memorable is the drama. In John Yorke’s seminal work on the art of storytelling, Into the Woods, he explores why the classic five-act structure of Shakespeare’s plays (and almost every film you’ve ever seen) is rooted in the arc of human experience. It is about how our protagonist learns. Though the book is essentially a manual (and manifesto) for screenwriters, it parallels beautifully the process of teaching. The learning arc that Yorke describes for characters in a drama is pretty much what we see in classrooms every day. In one analysis of the archetypal five-act structure, Yorke describes is as follows:

Act 1: our hero begins in ignorance of a problem, quickly encounters the problem and receives a call to adventure to solve this problem.

Act 2: our hero shows an initial reluctance to heed this call because they refuse to acknowledge the power of the problem, but - through a misstep - a mentor figure shows them how important and powerful the problem is, and helps them cross the threshold, even though the hero has no idea how the problem will be resolved.

Act 3: our hero engages directly with the problem and has initial success. They are exultant: this hasn’t been as hard as they thought it would be.

Act 4: disaster strikes. Our hero realises that the problem is bigger and deeper than they’d realised. They experience failure and are brought right to the edge. They experience anxiety and fear as full knowledge of the scale of the problem is understood. They feel that this problem is going to kill them and they again turn to giving up, wishing that they’d never started out and accepted this quest.

Act 5: our hero makes a choice to turn and face the problem one final time. There is a climactic battle and our hero finally overcomes the problem.

This is Frodo’s journey in The Lord of the Rings (and sometimes Year 9 lessons can feel like traipsing through Mordor), but isn’t this also the dramatic arc of a good lesson? Our students as heroic little hobbits struggling with their quest … and us as wizards who, quite frankly, could sort the thing in a flash - but that would leave the hobbits having learned nothing.

Yorke’s book is brilliant because it shows just how universal and fundamentally human this five-act arc is. But, importantly, embedded in this structure are key moments where our protagonists fail. They do not remain perfect - they cannot remain perfect - because to do so would mean that nothing was learned. They do something wrong in Act 2, and this is mirrored by a more terrible failure in Act 4. But it is precisely these moments that teach them about themselves and thus empower them to be able to master the problem at the end of Act 5.

In Okri’s formulation of “where perfection lies, there is no more story to tell”, we see that, if Frodo only acted in the perfect way all the way through The Lord of the Rings, there would be no story, no drama. He might have destroyed the ring but he would have learned nothing - would have mastered nothing - because he never made any mistakes.

Too often, my students want a magic formula that will just get the ring of power from The Shire to the fire in a flash. They want me to show them how to wizard it there. But our job as teachers is to show them the power of struggle and failure.

If education is a quest - one that sees far beyond mechanical numbered grades and out into the drama of real life - we should celebrate the failures and setbacks that lead to wisdom, not elevate automata who never err. This release from the pressure always to be right often comes as a relief to my students. “Now that you don’t have to be perfect,” I tell them, quoting John Steinbeck at the end of East of Eden, “you can be good.”

To teach students to be good. Yes. That should be enough.

Kester Brewin teaches maths in south-east London. While working as a teacher, he has been a consultant for BBC Education, and is the author of a number of books on culture and religion. He tweets @kesterbrewin

This article originally appeared in the 10 April 2020 issue under the headline “Failure should be an option”