A child brings home a page of subtraction. 62 − 28 = 46. 71 − 35 = 44. 84 − 19 = 75. Every answer is wrong, and every answer is wrong in exactly the same way: in each column, the smaller digit has been taken away from the larger one, whichever order they were written in. That page is not eight careless mistakes. It is one misunderstanding, repeated eight times — and that is much better news, because a single misunderstanding can be sorted out in a fortnight.
Most parents can tell when a child is struggling with maths. The hard part is working out where. Maths is unusually unforgiving in this respect: it stacks. A gap left at age six does not stay at age six. It sits underneath everything built on top of it and quietly makes the new work impossible, so a child looks like they cannot do long multiplication when the real problem is that they still cannot subtract across ten without counting on their fingers.
Patterned errors and random errors are completely different problems
Before you can find a gap, you need to know whether you are looking at one.
Random errors are scattered. The child gets 7 × 8 wrong on Tuesday and right on Thursday. They copy 46 from the top of the page and write 64 in the working. They skip a line. They are fine until question nine and then fall apart, because it is 7pm. Random errors are about tiredness, attention and handwriting. They cost marks, and they are not a gap.
Patterned errors are consistent. The same shape of mistake appears again and again, and if you gave the child ten more questions of the same type, you could predict most of the wrong answers before they wrote them. A patterned error means the child is not confused — they have a rule. It just happens to be the wrong rule.
The quickest way to tell them apart costs about thirty seconds. Point at a wrong answer and ask, with no tone in your voice at all, "How did you get that one?" A child making a random error shrugs, looks again, and often fixes it themselves. A child with a patterned error explains fluently and confidently, because from inside their rule the answer is obviously correct. That confident explanation is the thing you are hunting for. It tells you exactly what they believe.
Wrong rules survive for years because they work often enough. "Take the small digit from the big one" gets 68 − 25 right. It gets 62 − 28 wrong. A rule with a sixty per cent success rate feels, to a seven-year-old, like a rule that works.
The three places maths usually breaks
Across very different school systems, the same three walls come up.
Place value (roughly ages five to seven). The idea that the position of a digit is what gives it its size. A child who has not got this will write "three hundred and five" as 3005, because they are transcribing the words they hear rather than building a number. Ask what the 7 means in 7, in 70 and in 700. If the answer to all three is "seven", you have found something important.
Regrouping (roughly ages six to nine). Carrying and borrowing. This is where the subtraction page above comes from. It also produces answers like 47 + 25 = 612, where the child has added each column separately and written both results side by side. Regrouping depends entirely on place value, which is why it is so often the visible symptom of an invisible earlier gap.
Fractions (roughly ages eight to eleven) — the classic wall, and the one that ends a lot of children's confidence. Fractions demand that a child give up something that has been true for their whole mathematical life: that a number is a count of things. A fraction is a relationship between two counts. So 1/3 + 1/4 = 2/7 is not silliness; it is a child applying everything they have ever been taught about adding. Likewise, saying 1/8 is bigger than 1/5 is a child correctly noticing that eight is bigger than five.
Reading the error backwards
| What you see | The rule the child is using | Where the gap actually sits |
|---|---|---|
| 62 − 28 = 46 | Take the smaller digit from the bigger one | Regrouping, and number bonds to ten |
| "Three hundred and five" written 3005 | Every spoken part gets its own digits | Place value |
| 1/3 + 1/4 = 2/7 | Add the tops, add the bottoms | A fraction as one quantity, not two numbers |
| 1/8 called bigger than 1/5 | Bigger number means bigger amount | What the bottom number counts |
| 6 × 0 = 6 | Multiplying never makes things smaller | Multiplication as equal groups |
| 4 + __ = 11 answered 15 | The answer goes after the equals sign | Equals as balance, not as "do it" |
That last one is worth dwelling on. A great many children read "=" as an instruction meaning work it out now rather than as a statement that two sides balance. It causes no trouble until algebra, and then it causes nothing but trouble.
Tracing a Year 4 struggle back to a Year 2 gap
Here is the method, using a real shape of problem. A nine-year-old in Year 4 cannot do 400 − 137. She freezes at the zero. The instinct is to reteach column subtraction with zeros in it. Don't. Instead, halve the difficulty and ask again, and keep halving until she succeeds easily.
- 400 − 137 — she stalls completely.
- 40 − 13 — she gets there, but it takes nearly a minute and she uses her fingers.
- 14 − 6 — correct, in about eight seconds, counting back on her fingers.
- 10 − 6 — instant.
The floor is now visible. She knows her bonds to ten. She does not know how to cross ten. Every subtraction she meets that bridges a ten costs her eight seconds and her entire attention — which is why there is nothing left over for the structure of a three-digit sum, let alone a word problem. That skill is normally secure somewhere around ages six to seven. The Year 4 problem was a Year 2 problem wearing a bigger hat.
Notice what did the diagnosing: not the wrong answers, but the timing of a right one. Correct-but-slow is the most commonly missed signal in primary maths. Working memory is small and shared. If basic facts eat all of it, the child looks as though they cannot reason when in fact they cannot spare the capacity to.
Checking without turning it into a test
Ten minutes at the kitchen table is enough, if you run it well.
- Sit beside them, not opposite. Opposite is an exam.
- Use their own wrong answers as the material. You are not setting new work; you are asking about work that already exists.
- Let them finish before you say anything. If you correct mid-stream you will never see the whole rule.
- Ask "how did you get that?" for a right answer too, at least once. Otherwise the question itself becomes a signal that they are wrong.
- Time things quietly in your head. Under about three seconds is fluent; ten seconds is a gap forming.
- Stop after three questions at the point of failure. You are looking for the floor, not grinding at it.
Software that adapts to a child — Kid Genius World included — is genuinely useful for generating the fifty small repetitions a bridging-ten gap needs without you hand-writing them. It is much less good at the thirty seconds of "how did you get that?", and that conversation is where the gap is actually found.
What to do once you have found it
Name it small, and out loud. "You're fine at this. We just need to make 14 − 6 fast." A child who has decided they are bad at maths is carrying something much heavier than a missing skill, and being told the problem is one specific rule is an enormous relief.
Then: ten minutes a day for two or three weeks, on that one thing only. Practice spread across days sticks far better than the same total time crammed into a weekend, which is one of the most reliable patterns in how memory works. Do not fix two gaps at once. Do not move up until the skill is both correct and quick — correct-but-slow will collapse again the moment it has to share attention with something new.
After a fortnight, go back and try 400 − 137 again without teaching it first. Very often it has quietly repaired itself, because it was never the problem.
Finally, tell the teacher what you found. They have thirty children and very little diagnostic time, and "she's counting back on her fingers for anything that crosses ten" is far more useful to them than "she's struggling with maths."