What We Get Wrong About Mistakes
When a student gets a problem wrong, the most common thing that happens next is that an adult tells him the right answer.
This is so universal, and so apparently helpful, that it can be hard to see it as a choice at all. A child has erred. The kind thing to do is to correct him. So we correct him. We point to the mistake, we explain why it’s a mistake, we show the right method, and we move on. Sometimes, if we are particularly attentive, we ask him to redo the problem to consolidate the correction. Then everyone moves on.
I want to suggest, in this essay, that this entire sequence — well-meaning, ubiquitous, almost reflexive — is one of the deepest pedagogical mistakes we make. Not because correction is wrong, but because of when it happens, and what it forecloses.
What it forecloses is the most important thing in education.
The signal hidden in a wrong answer
A correct answer, considered carefully, tells us almost nothing.
The student may have understood deeply. He may have matched a memorised pattern without understanding. He may have guessed. He may have copied. From the teacher’s seat across the table, all four of these look identical: the right number sits on the page, the box is ticked, the lesson moves on. The correct answer is opaque. It hides what is happening inside the student’s mind.
A wrong answer, by contrast, is luminous.
A wrong answer is a precise, specific, often-beautiful indication of what the student was actually thinking. It tells us — with a clarity that no correct answer ever provides — exactly which step of his mental model is faulty. Was he confused about the operation? Did he misread the question? Did he apply a rule that doesn’t apply here? Did he carry a misconception from an earlier topic? The wrong answer points, like a finger, at the place where teaching is needed.
If the teacher rushes to correct, this signal is lost. The student is told the right answer; he updates his memory; the rule is filed; the box is ticked. The teacher has taught a fact. He has not taught the student.
This is the difference between treating mistakes as failures and treating them as data. One way of teaching erases the very information that real teaching depends on.
What is supposed to happen at the moment of error
When a student makes a mistake, something is supposed to happen inside him.
He is supposed to feel the small, characteristic flicker of wait, that doesn’t seem right. He is supposed to look back at his work. He is supposed to find the place where his thinking diverged from what would have worked. He is supposed to ask himself — even if only for a moment — why did I think that was right? And in answering that question, he is supposed to revise the small piece of his mental model that produced the error.
This entire sequence — flicker, look, find, ask, revise — is what learning is, in any meaningful sense. It is the moment a structure inside the student’s mind is rebuilt. It cannot be transferred. It cannot be shortcut. It can only be lived.
When an adult intervenes — “no, like this” — the entire sequence is bypassed. The mental model is not revised; it is simply overwritten with a new fact. The student walks away with the right answer. He does not walk away having actually understood anything.
Repeat this enough, across enough subjects and enough years, and you produce a particular kind of student: one who can be told things, but who cannot, on his own, work them out. He is dependent. He is dependent because the moment when independent thinking should have happened — that moment, again and again — was taken from him by someone who meant well.
Why we do it
The reason adults rush to correct is rarely cruelty. Almost always, it is something more sympathetic and harder to fight.
It is the discomfort of watching someone you love struggle.
A child sitting at the table, pencil hovering, slowly working out that his approach isn’t working — this is a hard thing to watch. He looks lost. He looks like he’s wasting time. He may also look slightly upset. The adult’s body, watching, registers all of this and produces an almost involuntary urge to step in. The urge is felt as care. Let me help him. He’s having a hard time.
But what the urge is, underneath, is a desire to end the adult’s discomfort. The student’s struggle is making the adult anxious — about time, about the child’s intelligence, about whether the parenting is working — and the fastest way to relieve that anxiety is to give the answer. The child stops struggling. The adult stops feeling anxious. Both feel better. Almost nothing is learned.
It takes a particular kind of discipline to resist this. Not patience with the child — most parents have plenty of that. Patience with one’s own discomfort. The willingness to let a few minutes of ambiguity pass through the adult’s nervous system without intervening. This is harder than it sounds. It is, in our experience, the single most difficult skill a parent can develop, and the one that produces the greatest difference in a child’s intellectual life.
What to do at the moment of error, if not correct
When a student gets something wrong, the question we have learned to ask — and almost the only one we ask — is some version of:
What made you think it was that?
This question does several things at once.
It does not tell the student he’s wrong. The student usually knows. (Or, if he doesn’t yet, he is about to figure it out — which is better than being told.) It does not give the right answer. It does not preempt the work he is supposed to do. What it does is invite him back into his own thinking, where the error actually lives.
The student, asked this question, almost always begins to walk us through his reasoning. Well, I thought because the question said X, and I remembered Y, so I… And somewhere in this walking-through, very often, the student himself notices the place where the reasoning went off. Oh. Oh wait. That’s not — hmm.
This is the moment we are after. Not the moment of correction. The moment of self-correction — when the student, looking at his own thinking, finds the flaw himself. That moment is worth, educationally, more than a hundred correct answers. Because in that moment, a structure has been rebuilt from the inside. The student didn’t receive a fact. He revised a model. The next time a similar question appears, he won’t need us. He has the equipment.
When this happens consistently, over months, the student begins to do it on his own. He gets something wrong, pauses, frowns, looks again, finds the error. He doesn’t need the prompt anymore. The voice that used to come from outside has moved inside. This is, in the end, what we are trying to build.
A note on what mistakes are not
I want to be careful, before closing, not to be misunderstood. I am not saying mistakes should be celebrated. They shouldn’t. The goal of mathematics is not to make beautiful errors. The goal is to think well, which usually produces correct answers.
What I am saying is more specific: mistakes, when they occur, are not the failure of the lesson. They are the lesson. The work of teaching is not to prevent them, and not to erase them when they occur, but to use them — to follow them carefully back to the place inside the student’s mind where the misunderstanding lives, and to do the slow, real work of helping him see it.
A child who is allowed to err, and then walked back through his errors with patience and curiosity, becomes a different kind of thinker than a child who is rushed past every error toward the right answer. The first child develops the capacity to find his own footing. The second develops the capacity to follow instructions accurately.
Both are useful. Only one is enough for the world he is going to grow up into.
Education does not advance through correctness. It advances through the moments when a student looks at his own thinking and notices that something doesn’t fit. We can help that happen, or we can foreclose it with a well-meaning correction. The decision we make in those small moments, repeated across thousands of lessons over many years, becomes the difference between a child who can be told things and a child who can think.