How do children actually learn maths?
By building from an innate but crude sense of quantity towards exact, symbolic number — and the research picture is rather different from the practice-until-fluent model most adults were taught under.
What children start with. Infants can distinguish quantities approximately — noticing that one group is larger — long before they can count. This approximate number system is present across cultures and in other animals, and is imprecise: it can tell 8 from 16 but not 8 from 9.
What has to be learnt:
The counting principles. Reciting numbers is not counting. A child must grasp that each object gets exactly one number word, that the order is fixed, that the last word said represents the whole set — the cardinality principle, which is a genuine conceptual leap — and that any collection can be counted.
Subitising, instantly recognising small quantities without counting, which underpins fluency.
Number as a line, with a sense of relative size and distance.
Part-whole relationships — that 7 can be 5 and 2, or 4 and 3 — which is what makes flexible calculation possible and is more important than memorised facts.
Place value, the hardest and most consequential idea in primary maths, and the source of most later difficulty.
What the evidence supports:
Concrete before abstract. Physical objects, then pictures, then symbols.
Talking about number in ordinary life — sharing, cooking, shopping, comparing — which predicts later attainment well.
Fluency with facts genuinely matters, because working memory is limited and a child calculating 7+8 laboriously has nothing left for the actual problem. This is not in conflict with understanding; it enables it.
Games with a linear board, which improve number-line sense measurably.
What is unhelpful: maths anxiety, which is transmitted from adults strikingly effectively — a parent saying "I was never any good at maths" communicates that it is a fixed trait; speed pressure, which drives anxiety and is unrelated to ability; and treating errors as failures rather than as information.