Why Your Child Is Good at Familiar Problems and Freezes on Unfamiliar Ones
There is a moment, somewhere in the middle of secondary school, that many parents will recognise.
The child has been doing well. Top quartile, sometimes top of the class. Worksheets come back ticked. Examinations are passed without drama. The parent feels a quiet, tentative pride — the system, whatever else may be wrong with it, seems to be working.
And then, sometime around Sec 3, something changes. The marks don’t fall off a cliff, exactly — but they wobble. A test comes back lower than expected. A homework problem that should be routine takes the child an unreasonable amount of time. The child, who used to glide through mathematics, now sits at the table and stares at the page. When asked what’s wrong, he says he doesn’t know how to start.
Most parents, faced with this, conclude that the work has simply gotten harder. They are half right. The work has gotten harder. But the deeper truth is that the harder work has revealed something that was already there — something that had been quietly hidden by easier material.
What was hidden is this: the child had been trained to recognise problems, not to solve them.
This essay is about what that means, why it happens, and — more importantly — what can be done about it.
The trick of recognition
Almost all of school mathematics, up to about Sec 2 in the Singapore system, can be passed by a sufficiently disciplined student through a single skill: recognition.
The student looks at a problem. He recognises its type. He retrieves the procedure that fits that type. He executes the procedure. He gets the answer. He gets it right.
This works because the problems on examinations, at this level, almost always belong to a small finite set of recognisable types. A student who has practised enough examples of each type can — without exaggeration — pass without ever having genuinely understood the underlying mathematics. He is matching, not thinking.
From the outside, matching looks identical to thinking. The marks are the same. The answer is the same. The face during the examination is the same. The only difference is what is happening inside the student’s head — and from the outside, nobody can see that.
So the parent, watching from outside, sees a child who is doing well at mathematics. The teacher, marking the worksheets, sees a child who is doing well at mathematics. The child himself, getting questions right, also believes that he is doing well at mathematics.
All three are wrong. He is doing well at recognising mathematics.
Why this stops working
It stops working because at some point, usually around Sec 3, the examinations begin to include problems whose type cannot be quickly recognised.
These are the problems that require the student to read the situation carefully, build a representation, decide what mathematical structure applies, and reason from first principles. They are, in other words, the problems that require thinking — actual thinking, not retrieval.
A student who has been doing well by matching has no equipment for these problems. He looks at the question. He doesn’t recognise it. He waits, expecting recognition to arrive. It doesn’t. He reads the question again. Still nothing. He has, in a real sense, never been here before — never been in a place where mathematical problems are not solved by recognition. He doesn’t know what one is supposed to do.
What he does next is informative. Most students, in this situation, do one of two things.
Some panic. They mark the question and skip it. They tell themselves, afterwards, that they “didn’t have time,” but the truth is they couldn’t begin.
Others go through the motions. They write something — anything — that looks like the kind of thing that worked on previous problems. They apply a procedure that doesn’t fit. They produce an answer that’s wrong. They lose the marks.
Either way, the parent sees the result and is bewildered. He’s been doing so well. What happened?
What happened is that he ran out of room. The trick of recognition works for as long as the problems are recognisable. The moment they aren’t, he has nothing else.
Why this happens
It would be easy to blame the schools, or the tutors, or the culture of relentless drilling. All of these play a role. But the deeper cause is more subtle, and it begins much earlier than most parents imagine.
It begins with the very first time a child is taught a procedure before he has been allowed to wonder about the problem the procedure solves.
Consider how multiplication is usually taught. A young child meets the multiplication table at six or seven. He memorises it. He drills it. He gets faster. By the time he is eight, he can produce 7 × 8 = 56 in less than a second. Everyone is pleased.
But ask him what 7 × 8 means, and you will, surprisingly often, get a blank look. He knows it equals 56. He doesn’t know it means seven groups of eight, or eight groups of seven, or the area of a rectangle with sides 7 and 8, or any of the other things that multiplication actually is. He has the answer without having the idea.
This is the seed. From here, the same pattern repeats — formula after formula, technique after technique, year after year. The student accumulates a vast library of procedures, each one attached to a recognisable problem type, none of them connected to a structural understanding of why they work. By Sec 2, he has thousands of these. He is, by any reasonable measure, very well prepared.
He is also, by any reasonable measure, in serious trouble. He just doesn’t know it yet.
The role of being good
There is one more layer to this, and it is the most uncomfortable one.
The students who get hit hardest by the wall in Sec 3 are very often the good students — the careful ones, the conscientious ones, the ones who never gave their parents trouble. The ones who, in the quiet language we use in Chinese, are 乖 — well-behaved.
This is not an accident.
A child who is gently rewarded, year after year, for getting things right and gently discouraged from getting things wrong develops a particular relationship to learning. He learns to stay close to what he knows works. He learns to avoid the discomfort of trying something he isn’t sure of. He learns, slowly and without anyone meaning to teach him this, that the safe thing is to recognise, retrieve, and execute — and the unsafe thing is to wonder, to wander, to risk being wrong.
By the time he is fourteen, this has hardened into a temperament. He is excellent at the kind of mathematics that rewards staying close to known territory. He is, almost paradoxically, quite bad at the kind of mathematics that requires venturing out.
When the examination begins to test that second kind, he discovers — for the first time, in many cases — that his entire training has been preparation for a different test.
What can be done
I want to be honest: this is not a problem that resolves in three weeks. A student who has spent eight years building a recognition-based mathematics has built something real. It can be rebuilt — but the rebuilding is slow, and the student often resists at first, because the new way of working feels less productive than the old way. He used to get answers in two minutes. Now you’re asking him to spend twenty minutes thinking about a single problem. The early weeks are often the hardest.
But the rebuilding is possible, and it follows a fairly clear shape.
It begins by slowing down. The student is asked to spend more time on each problem than he is used to — not to solve more, but to understand each one more deeply. Why does this method work? Where does this formula come from? What would happen if one of the conditions changed?
It continues by allowing the student to be wrong. Not occasionally — frequently. The student is given problems that are slightly outside what he has practised, and is invited to try, fail, try again, and slowly find his footing. The teacher’s job during this period is to resist solving the problem for him. (This is harder than it sounds. Most teachers cannot do it.)
It deepens by rebuilding the foundations. Topics that the student thought he knew turn out to have been understood only superficially. He is taken back to those topics — not for review, but for genuine first-time understanding. This is uncomfortable for parents, who paid for him to learn this material the first time. We try to be honest about why it’s necessary anyway.
And it stabilises when the student begins, on his own, to enjoy unfamiliar problems. This is the moment we are working toward. It usually arrives without warning, somewhere in the third or fourth month. The student looks at a problem he has never seen, hesitates, and then says — with something like surprise — “let me try.” That moment is the moment a real mathematics student is born.
After it, the wall in Sec 3 is no longer a problem. The student has become someone for whom unfamiliar problems are interesting rather than terrifying. There will still be hard days, and there will still be examinations he finds difficult. But he will not freeze. He will think.
What to look for at home
If your child is younger and you’d like to avoid this trajectory, the most useful thing you can do is also the simplest. Pay less attention to whether he gets answers right, and more attention to whether he can explain why they are right.
A child who can produce 7 × 8 = 56 instantly but cannot tell you what it means is a child for whom the wall is being built, slowly, in the background. A child who is a little slower but can show you what is going on — seven groups of eight things, look, like this — is a child whose foundation is sound.
The difference, at age eight, looks like a difference in speed. It isn’t. It is a difference in what is being built.
By Sec 3, the slower child is overtaking the faster one, and the faster one’s parents cannot understand why.
The deepest training a child receives is not the content of his lessons. It is the unspoken instruction about what learning is for. A child taught that learning is the production of correct answers will, for as long as that strategy works, look like he is winning. A child taught that learning is the building of understanding will look, for some years, like he is behind. The first child hits a wall around fourteen. The second child does not. Almost everything depends, quietly, on which one we are raising.