There are two clear assumptions that underpin most recent government policy decisions on numeracy. These are that: * basic calculation skills are missing from the working population, and that * these are what increasingly matters for national prosperity.
Over time, it seems that the failure to explore these assumptions has played a significant role in removing much that is mathematical from school mathematics, and has led to an intolerably low expectation of what children can learn.
The world of work has been transformed. The computer has changed employment practices and appears to have radically decreased the need for people to think about the mathematics which underpins what they do. The checkout operator, the bank clerk, and thousands of other jobs have certainly been de-skilled, and many employees spend their working lives in a Fordist nightmare, a modern variant of the automobile magnate's system of controlled and authoritarian production which was so chillingly captured in Chaplin's Modern Times.
Central to Ford's vision was the idea that effective management of the labour process demands the separation of conception from execution, the removal of human intellect from the working process, and the fragmentation and gradual removal of skills and craft knowledge. Anyone who has ever phoned or walked into a large corporation to find that there is nobody who seems to understand anything has experienced its implications.
Yet at the heart of industrial and commercial corporations, the tide is beginning to turn. In many working situations, basic numeracy is far from enough; that kind of mathematics is already handled at lightning speed by the computer systems. What matters increasingly is an ability to think in a mathematical way.
Celia Hoyles, Stefano Pozzi and myself have recently been studying how different groups of employees - including clerical and technical workers in an investment bank and paediatric nurses - make use of maths in their work.
We find that in some jobs, human beings are no longer viewed as mere adjuncts of the computer system. On the contrary, management is coming to accept that employees must be in a position to make sense of the models which underpin the systems they use.
In banking, for example, the problem is not that the operators are unable to do arithmetic. It is that they have little understanding of what the computers do with their inputs, and how to make sense of the output the computer gives them. Employers are finding that their entire financial networks are becoming vulnerable to input errors made by unskilled operators and output errors of misinterpretation. The problem is not that employees cannot "add up", but that they have no mental model of the system to enable them to interpret data.
From an educational perspective, this means that the personal and social needs of individuals can sometimes converge, and that job satisfaction and personal empowerment are not necessarily antithetical to efficiency.
More surprising still, it is the computer which points to this possibility: it is possible for computer systems to treat humans as partners rather than inconveniently expensive appendages. Computer systems which leave no room for human intervention can be both alienating for individuals and inefficient for the purposes for which they were designed. Judgment and calculation, so often conceived as opposed, can become two sides of a single coin - with positive consequences both for the individual and the system.
Increasingly, workers will need to understand the principles of the systems they use. As information technology proceeds to define workplace systems, its complexity has already reached the point where only their designers fully understand them.
This poses particular challenges for those who work with such systems. Increasingly, they will need to represent what is happening, particularly whenever the situation becomes in some way non-routine. They will need to engage with the mathematical knowledge that has been buried beneath the surface of their computer screens.
The implications for our teaching of numeracies are manifold. They point to the need to build new educational cultures in which individuals have the means to construct and make sense of appropriate models, and the means to express them mathematically. More generally, changes in working practices will increasingly lead to a redefinition of the boundaries of what needs to be understood as a whole, rather than as isolated skills.
There are few signs in the United Kingdom that successive governments or their advisers understand the need to consider what kinds of new mathematical knowledge are required, rather than simply more "effective" ways to transmit old knowledge. There is no sign that any of the myriad task forces understand that grasping the little bits and pieces of numerical facts is not enough, and that such knowledge is only a very small part of what mathematics is about, and what it is for.
There is no sense of vision, of what will be necessary for informed citizens to play their part in the next century. And there is no sense of urgency in helping children to see the potential in their own lives - and the life of the nation - for seeing the broad mathematical picture.
Failing to address this issue will leave us unable to explain how, for example, the same "trendy" teachers who allegedly failed students in numeracy in the recent international tests of mathematical achievement, succeeded in taking them near the top in science. (A recent study shows conclusively that the amount of whole-class teaching in Year 5 is not a predictor of performance in either mathematics or science.) Neither will it clarify why English Year 5 children came top of the league in geometry and why Year 9 children did relatively poorly.
Ignoring the issue of knowledge (not skills or competencies or teaching style) will leave us oblivious to the kinds of understanding our children will need in their adult lives. It will divert us from asking whether "basic numeracy" provides an appropriate foundation for mathematical appreciation.
It will further depress the professionalism of teachers by prescribing sequences of numerical trees rather than showing children the mathematical wood. And it will fail to equip children to lead full lives both at work and in leisure.
Richard Noss is professor of mathematics education at the Institute of Education, London. A full text of this article based on his inaugural lecture at the Institute, can be obtained from the Institute Bookshop, or from http:www.ioe.ac.ukrnossinaugural.html