Ask an adult what a child needs to learn about number and they will say counting. Counting looks basic. It is not. It sits on top of several things children already have, and it requires a conceptual step that is genuinely difficult and almost invisible from outside.
Two systems that are not counting
The research points fairly consistently to two capacities present very early.
The first handles very small sets precisely. Presented with one, two or three objects, infants respond to changes in number, and the responses are sharp. This appears to work by tracking individual objects rather than by counting them, and it runs out at around three or four.
The second handles larger quantities approximately. Infants discriminate between sets when the ratio between them is large enough, and the precision improves through childhood. This is a magnitude sense rather than a number sense in the exact meaning: it tells you that one collection is bigger, not by how much.
Both are found in other species, which suggests neither depends on language or culture. Neither, crucially, gives you exact number beyond the smallest sets. You cannot get from approximate magnitude to knowing there are exactly seventeen.
Counting is a technology
Exact number for larger sets requires counting, and counting is an invention rather than an endowment. Languages differ in their number systems, and some have very limited exact number vocabulary. Speakers of such languages perform well on approximate tasks and struggle with exact ones, which is about as clean a demonstration as this area offers that exact number depends on a cultural tool.
The tool has several parts that must be coordinated. The number words in order. One word per object, no more and no less. Each object counted once. And the principle that the last word said gives the size of the whole set.
That last one is the conceptual step, and it is the one children take last. A child can produce the words in order, point at things, and still not grasp that the process answers a question about the set.
How researchers tell the difference
The standard method is disarmingly simple: ask a child to give you a certain number of objects from a pile.
Children go through a sequence. Some hand over a handful regardless. Some can give one correctly but not two. Then two, then three, each apparently learned separately. And then, at some point, the child can give any requested number, having grasped that counting is what produces the answer.
That transition is the interesting one. It is not the addition of another number to a list; it is understanding what the list is for.
The mental number line and how it changes
Another well studied phenomenon involves asking children to place a number on a line with the ends labelled.
Younger children space small numbers widely and squash large ones together, a pattern resembling the approximate magnitude system, where the difference between two and three feels large and the difference between eighty and ninety feels small. With experience, placements become more evenly spaced.
The interpretation is contested. Some read it as a change in an underlying representation, others as the acquisition of a strategy for the specific task. This is a good example of a robust finding whose meaning is unsettled, and it is worth knowing that the robustness of a result and the correctness of its usual interpretation are separate matters.
Why arithmetic sits on all of this
Once counting is understood, arithmetic can begin, and the same automaticity story applies as in reading.
Early addition is often performed by counting. Counting is slow and consumes the workspace described in working memory in plain words. A child spending everything on the arithmetic has nothing left for the structure of the problem, which is why children who understand a situation perfectly can fail a word problem about it.
As facts become retrievable rather than calculated, capacity is released. This is the argument for fluency in number facts, and it is the same argument as for fluent decoding: not speed for its own sake, but freeing the space that thinking needs.
The thing adults transmit without meaning to
One finding in this area is worth stating carefully, because it is easy to turn into blame.
Adults' own discomfort with mathematics appears to be associated with children's attitudes and, in some studies, with attainment, particularly where an anxious adult helps with mathematics homework. The proposed mechanism is not mysterious: a person who finds a subject unpleasant transmits that, and help delivered under strain is not good help.
The correct response to this is not for anxious parents to feel worse. It is to notice that saying you were never any good at maths, which is said constantly and warmly, is a sentence with content. It is not a moral failing to have said it. It is simply worth knowing that it is heard.
What this changes
Mainly what counts as progress. A child reciting to fifty may have learned a song. A child who can give you four objects has understood something. The visible achievement and the conceptual one come apart, and the visible one is the one that gets praised.
It also lowers the stakes on early formal maths. Approximate magnitude and small exact number arrive without teaching. Counting is a tool that will be taught. The interesting question is not how early a child counts but whether they know what counting answers, and that question is better explored over a pile of buttons than over a worksheet.
For the parallel story about a different cultural invention, see how reading is learned.