# Section Two: The thinking behind Let’s Think Maths

“ *We are looking to move children’s thinking forward, and not all children are **going to get to the same point at the end. That’s not what it’s about. We are **trying to raise awareness, so that the children will continue to think about the problem even when the session is over.*

“ *That’s what’s nice. You can leave it in the air. It’s the talk.*

Primary teachers talking about Let’s Think Maths

Mathematics is a shared activity and is one way in which children seek to make sense of the world. If children are to become successful, mature thinkers and learners, they need to develop ever more general and abstract mathematical frameworks incorporating mathematics’ ‘big ideas’. However, simply telling children about ‘big ideas’ does not work. Children’s need to understand often leads them to develop misconceptions and misunderstandings of key mathematical concepts and procedures, which they will need to address.

The Let’s Think approach to teaching makes use of children’s thought processes to enable them to develop greater mathematical understandings. To do this, the original CAME project draws upon two theoretical approaches to children’s learning: Piagetian theories about the individual development of mental powers and Vygotskian and social constructivist theories about the social interaction that drives this individual development.

The Piagetian contribution

Piaget described children’s thinking in terms of qualitatively distinct stages. The broad stages that concern us in Years 5 and 6 are concrete operational thinking and, to a lesser extent, formal operational thinking, where a child has begun to think through logical propositions.

Concrete operations are thought processes based on a person’s perceptions. The images and ideas to be worked on may arise from practical activities or from something read, seen, heard or even imagined. The crucial characteristic of concrete operational thinking is that children have the ability to describe situations but not to explain them beyond simple cause and effect. By contrast, formal operational thinking allows people to step back from a problem and to create or use general explanatory models of events and to test how accurate those models are.

Piagetian ideas are particularly powerful in describing the underlying difficulty of mathematical learning tasks in the same terms as those used to describe children’s thinking. Piaget’s thinking stages are very broad. Subsequent research has sub-divided into sub-stages or thinking levels. Concrete operational thinking is sub-divided into early, middle, mature and concrete generalisation. Formal operational thinking is sub-divided into early formal and mature formal.

These new sub-stages provide an enriched description of children’s thinking in a Let’s Think Maths lesson.

Formal operational thinking allows people to generate an idea (model) about events already described through concrete operations, and then test how well the idea connects the events.

*F**igure 1: Graph of plant growth*

For example, if a simple graph was plotted on the height of a growing plant against the number of weeks it has been measures (Figure 1), a child using concrete operations can easily answer questions such as: ‘Between which weeks did the plant grow fastest?’. This is because there is a direct relation between the steepness of the graph and the rate of plant growth. The one describes the other.

*Figure 2: Graphs describing journeys*

Suppose the task is to examine the three graphs in Figure 2 and then to answer questions such as ‘Which of these best describes a lift rising from the ground to the fifth floor and stopping twice on the way up?’ or ‘Point to a part of a graph that represents an impossible journey’. A child using descriptive models only (concrete operations) is apt to choose a or b as the answer to the first question and to offer no answer to the second. The ability required is to step back from the problem, and to say to oneself: ‘This is about the relationship between the height axis and the time axis’, then ‘If the graph is going vertically, this would mean the lift is travelling up in zero time’ which is impossible, and finally, ‘The graph with the lift movement shown at an angle must represent its travel in time’. To do this requires formal operational thinking.

All but the most exceptionally able Year 5 and 6 children will be concrete operational thinkers. A few children may still be pre-operational thinkers – able to recognise and name things, but are unable to think through the results of actions without actually carrying them out. Let’s Think Maths lessons aim to help children become more mature and more successful concrete operational thinkers across the range of mathematical ‘big ideas’, preparing the way for their development as formal operational thinkers. The potential of this approach is demonstrated by the significant results that have been achieved in the Leverhulme Numeracy Research Programme Focus 5 Primary CAME project research, particularly in terms of Piagetian measures. Significant results have also been demonstrated through Piagetian and Mathematical Reasoning achievement test results in Year 8 at the secondary level and on national tests at the end of Key Stage 3.Property of Let’s Think Forum – not to be copied or reproduced without permission

The hypothesized Piagetian level of thinking associated with the different conceptions and understandings within the lesson is shown on the left-hand side of each lesson abstract, but these are only indicators of thinking difficulty. They are not intended to give a full picture of every child’s thinking during the lesson.

Vygotsky and children’s development

Mathematics is a social activity. Mathematical concepts have been developed over time by people working together. Similarly, children construct and discuss mathematical understandings together. It is through participation in the mathematics classroom and discussion of mathematical meanings that children learn to become mathematical thinkers.

In this respect, Piaget’s description of development as an interaction between an individual and their environment is inadequate on its own. Those involved in the development of Primary CAME see Vygotskian theories of learning as providing the necessary social complement to Piaget’s ideas.

Vygotsky emphasised that children’s learning is mediated by other people. In early childhood, the child’s immediate world is framed, simplified and thus mediated by older family members and adults in general. However, as children become older and engage with tasks in and outside school, their main mediators are their peers. Children learn through interacting with and listening to their peers. Although they still do some of the work in developing their thinking for themselves, more usually, they see or hear another pupil showing an idea which is just beyond the competence level they are at already. They then immediately make that idea their own.

The CAME approach uses Vygotsky’s work on development and Piaget’s work on children’s thinking to put children in a position where, in collaboration with their peers, they must construct key mathematical reasoning patterns for themselves.