I have heard that the materials draw upon theories of learning. I thought I left all that behind years ago!
There are two main sources of psychological research and theory which underpin the CAME approach: the Swiss psychologist Jean Piaget's qualitative description of children's thinking as it develops; and the Russian psychologist Lev Vygotsky's description of how powerful in children's development is their interaction amongst the immediate social environment of their peers-children of the same age in school, or brothers and sisters and friends in their own homes.
Piaget was interested in describing the development of intelligent activity in children and adults. He wanted to describe (and not number, as per a standard IQ test method) the development of intelligent activity in the human, from birth onwards. He spent many years observing children's language and thoughts, their judgement and reasoning (often using his own children). These detailed observations and discussions enabled him to establish a theory of cognitive development which mapped the stages of cognitive development as:
Most pre-school children work at sensorimotor stage, and most primary age children will be working at preoperational level. Just a very few of the most able Year 6 children may have started to use a higher level of thinking again, formal operations, although that is more typical of 15 year olds.
These tables (Maths, Science, Literacy - need Appendix One from Handbook - page 1.141 - page a) gives just a few examples of the types of thinking characteristic of some of the stages, the development of which is the big mental event throughout the school years. We call these tables Curriculum Analysis Taxonomies because it is a way of classifying everything that appears in the school curriculum in terms of the type of thinking that is required.
Piaget suggested that experiences which are puzzling to a child, and which cannot be easily explained, or cause 'cognitive conflict', can stimulate the development of intelligence. This means that if we continually make our demands on children easier and easier, we are actually doing them a disservice. Rather, we should be devising challenging activities for them, and then helping them to meet these challenges.
Vygotsky was fascinated by the way that children absorb the culture of the society around them. He examined the process by which human beings, as social animals, grow up together watching and listening, trying things out in speech and action, looking for the effects on others and so learning from each other. From these observations, he developed the idea of 'social construction'. Early learning mainly takes place between parent and baby, and sometimes between baby and siblings. Much learning from age four onwards takes place between children in a group, but this process needs to be well-managed by an adult. Vygotsky described the difference between what children can do unaided and what they can achieve 'with a little help from their friends' as the Zone of Proximal Development.
In the CAME project we believe that the work of Piaget provides helpful descriptors of what pupils can do, in thinking terms, and it is this perspective that enables us to devise appropriate and challenging mathematical tasks. From Vygotsky comes the social side of the learning process. It is this focus that provides the guidance for the effective orchestration of classroom interactions.
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. But, for a pupil entering 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.
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.
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.