Children who can count like grown-ups are already mathematicians. Last week I explained that the fusion of our two innate skills of subitising (direct recognition of small quantities such as three apples) and number sense (differentiating between larger and smaller groups), along with the development of language, enable children to learn the rules of grown-up counting and create for themselves a unified theory of number.
This is a momentous learning achievement. But hard on its heels comes a second breakthrough that enables the child to master the fundamentals of numeracy - reading numbers.
In fact, there are at least three major challenges for children to face once they have learned how to count. They are reading numbers, understanding order (units, 10s, 100s, and so on), and grasping the four arithmetical operations (addition, subtraction, multiplication and division).
Although it is possible to make some progress with understanding order and the four operations without being able to read numbers, one cannot go very far. I think it is necessary - as well as desirable - to be literate in numbers to comprehend the relationship between millions and billions, for example, or to practise long division.
After the development of language and the unified theory of grown-up counting, the second fundamental stage of learning to be numerate is number literacy.
This may seem obvious, but it does raise some interesting issues. First, it raises the question again of whether maths anxiety may be, at least, in part, the result of failure to master with confidence the two key processes of grown-up counting and number literacy.
I believe that number literacy precedes, or should precede, word literacy.
The heart of literacy is the understanding of a triangular relationship between meanings, sounds and symbols.
That is what we mean by reading. It is no small achievement to learn to read. Children often embark on it before they have fully mastered the sound system of their first language.
They have to wrestle with the complexity of an alphabetical system of symbols that has only a partial fit with the distinctive sounds of English.
And they have to begin to map these systems on to the infinite realms of semantic space - all possible meanings. I sometimes think it is a wonder that we ever learn to read at all!
There is some limited and fragmentary evidence that those who start with number literacy find word literacy much easier to master. Why should this be, if indeed it is the case?
I think the answer may lie in the idea of numbers (one to 10) as a sort of simple initial practice alphabet, ideographic, related to well-known and often spoken terms, and referring to a limited and coherent subset of meanings.
For the learner, the task of reading numbers is simpler and makes better sense than reading words.
I have called this idea the Pembroke hypothesis, since it was derived from an interesting observation of children in Pembrokeshire, in Wales, who responded to a numeracy intervention in early primary education, designed to help them learn elementary mathematics, with literacy outcomes. They became better word-readers.
I am now promoting a new research project in a number of Jigsaw day nurseries to test this hypothesis.
Children who have mastered grown-up counting and number literacy have achieved the fundamentals of numeracy. From this foundation they can move forward to the mastery of order and the four arithmetical operations - and then tackle the more advanced tasks of the measurement of time, space and money, fractions and powers.
All these are in the realm of what I call "artificial education". They require good teaching.
Beyond these lie geometry, algebra, trigonometry and the curriculum of advanced mathematics. It would be both interesting and useful to know whether maths anxiety arises for some of us in these later stages of numeracy development or whether the roots of this anxiety lie in the early phases described above.
As ever, we need more research.
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