CAME aims to contribute to the teaching of mathematics at specific times in primary and secondary schooling, where youngsters have a 'window of opportunity' for rapid intellectual development in the context of mathematics with the help of specialist teachers. This development is to be achieved through cognitive stimulation using carefully selected challenging classroom tasks over two-year periods. The emphasis in these tasks is on 'big ideas' or organising conceptual strands in mathematics, rather than on procedures and algorithms.

The CAME approach helps to underpin the notations and algorithms stipulated in the mathematics syllabus with real understanding through pupils effectively reconstructing the underlying concepts for themselves. Without the weight of exam revision on their shoulders these young pupils will be relatively open to challenges that require deep thinking and conjecturing. This complements the other ingredients in students' mathematical experience: instruction and practice and/or investigations. This balanced diet of mathematical experience, delivered with forethought and co-ordination, significantly raises the whole of the thinking capacity of the students, with a lasting effect that creates a stable basis for higher achievement in later school years, whether for exams or for further learning. The aim, therefore, is cognitive development, permanent and general, as well as meaningful learning of mathematics here and now.

The Thinking Maths approach

The CAME approach to teaching can be described as: From each according to his ability: to each according to his need. Rightly, normal mathematical learning experiences comprise on the one hand instruction and practice, and investigation and problem-solving on the other. How do Thinking Maths (TM) lessons differ from both of these?

Instruction and practice focus on a given objective and assume the ability of the whole class to benefit: TM recognises a range of levels of ability in any class in any one context, of which only one may properly process the lesson objective. Hence TM proposes agenda to be addressed rather than objectives to be reached by all. On the other hand, investigation and open-ended problem-solving allow each pupil to proceed in their own direction within their own capability: TM corrals the pupils within the mathematical context, and asks all to collaborate, from wherever they are, in constructing insight at all levels of understanding which are implicit in the concepts within.

The structure of each TM lesson is based on an analysis of the different levels of difficulty of all the concepts that may be implicit in the mathematics featured: in the next Section this will be called the Piagetian element. The conduct of each lesson is based on the principle that collaborative learning, well managed, allows each to contribute from where they are to the collective insight achieved, and allows each to draw from that source what will move their thinking forward: in the next Section this will be called the Vygotksian element. The CAME art is to marry these two.

Using Vygotsky and Piaget in tandem

For some pupils the combination of adequate mediated learning in the home and social environment, and satisfactory primary school experience has led to cognitive development in a natural unconscious way (Feuerstein et al. 1980, chapter 1). But, for a pupil entering secondary school already below potential a more conscious intervention in his or her development is required . Some deliberate strategy is needed for teachers to accelerate development in their pupils which would not otherwise take place.

Vygotsky described the qualitative changes in mediation as children get older. Well before they reach adolescence their main mediators have become their peers. Although they still do some of the work of developing their thinking for themselves, on their own, more usually they see or hear a fellow pupil showing a completed skill which is just beyond their own competence level. They then immediately make it their own. This is because each child possesses, in addition to the assured competencies which enable immediate solution of problems such as test items, a whole spectrum of skills in the process of partial construction. The usual mini-steps of development are from outside in the social space adolescents share with their peers, to inside as their own possession. And even this view of the process is too individualistic: all children - or indeed, any learners - contribute to the interaction that results in the production and expression of insight when learning is truly collaborative. Just as no child does all the work of accommodation on his own, so too the pupil in which the new insight has crystallised has been assisted there by the efforts - even the doubts or difficulties expressed - of the other pupils.

The teacher must be able to look ahead on behalf of the pupil. The aim is a very long-term one in which the pupil cannot see more than the immediate task, but the teacher frames the specifics of each task so that 'the road ahead' leads in the right direction. As mediator the teacher has to realise that the role is less to act as a model for the pupils, and far more as a manager who directs their small-group learning and whole-class discussion in such a way that for each pupil, the probability that they will witness in some other pupil just that next step in thinking they are ready for, is increased far above the usual chance level.

This view of cognitive development underlies much of the style in which the CAME lessons are conducted. Teachers need to take a Piagetian view of what is implicit in the mathematics, but only if, in addition, they conduct the lesson on a Vygotskian view of psychological development, will they be successful. Both views are necessary, and need to be integrated in their teaching skills.

"It is helping me to learn alongside the children in an open environment where the questioning and challenging of ideas is encouraged (and celebrated). I am learning to respond more effectively to children's ideas during the lesson."