Sir Christopher Ball explains how children acquire mathematics skills
Numeracy is a complex skill. All sorts of adults and children suffer from "maths anxiety", which ranges from discomfort with numbers to a deep fear of them.
Why should this be? The fundamentals of numeracy are innate. Children, like some animals, are born with two specific skills - subitising and number sense.
Subitising is what we do when we immediately recognise a small number of objects without having to count them. Imagine two bowls with three apples in one and 13 in another. We recognise three without counting, but we have to count 13 apples.
Babies can subitise numbers up to three. They also possess number sense, the ability to distinguish a larger from a smaller set of objects. Consider your own ability to differentiate, as larger or smaller, two groups of people, say of 10 and 20, without counting them.
There is a difference between natural development and "artificial education". The first happens inevitably, the latter requires teaching. We all learn to walk and talk, unless we are severely disabled or maltreated.
But we are unlikely to learn Latin or calculus without instruction.
One of the key skills of parents and teachers, especially in the early years, is knowing when it is enough to provide a good example and a "good-enough" environment for natural development, and when we need to give instruction.
This distinction is especially important in the development of numeracy.
All the growing child needs to develop subitising and number sense are plenty of opportunities to practise them. In good-enough homes children are encouraged to develop naturally without pressure. Perhaps many who have not had these advantages are not ready for the school maths curriculum because their development is incomplete.
The next major step in the acquisition of numeracy is the mastery of language. This, too, develops naturally. As children learn to talk, they learn to count. Children enjoy counting but do not grasp the skill immediately.
Most children learn how to reproduce the sequence of numbers in the correct order from nought to 10 some time before they understand the rules of counting sets. These rules are so well embedded in the adult brain that we take them for granted.
Grown-up counting requires a one-to-one correspondence between the number words and the objects in the set, such that wherever you start, and whatever the order you choose to enumerate the objects, you reach the same answer.
While all this is obvious to adults, it is a revelation to the growing child, who is often surprised to find that there are five fingers on a hand whether you start counting with the thumb or the little finger.
Wise parents and teachers encourage and celebrate the process of learning that enables a child to switch from childish to grown-up counting. This step is momentous.
It is the first clear example (in the realm of numeracy) of the growing child moving from innate, developmental learning to the culturally-specific learning characteristic of "artificial education".
Grown-up counting enables the child to make a link between their innate skills of subitising and number sense and create a unified theory of number.
So important is this learning stage that I am almost inclined to dismiss the rest of mathematics as mere elaboration of this fundamental step. But not quite.
In a later article I shall consider three more key elements of learning which are required before the child can be said to have mastered the fundamentals of numeracy: reading numbers, understanding order (units, 10s, 100s and so on), and grasping the four arithmetical operations (addition, subtraction, multiplication and division).
Sir Christopher Ball is an adviser to the Organisation for Economic Co-operation and Development on numeracy. He is the chancellor of Derby university, chairman of The Talent Foundation and patron of the Campaign for Learning