2 The theory behind Thinking Maths

The Piagetian contribution

Some time between the ages of about 12 and 18, many people’s thinking shifts up a gear. Using Piaget’s labels, in early teens the pupil is still using concrete operational thinking, whereas later adolescents are also able to use formal operational thinking.

Concrete operations Formal operations
  • Thinking processes using logic and based on perceptions and memories of these perceptions, not necessarily linked directly concrete ‘hands-on’ experiences.


  • Describing situations, including simple causal relations, but not explaining them.
  • Most daily transactions (for people of all ages) require only concrete operations.
  • Generating an idea (model) about the perceptions and memories of these underlying connections in events described perceptions, testing how well the idea explains the events.


  • Thinking abstractly and drawing conclusions from the information available.


  • Much of the agenda of secondary mathematics requires formal operational thinking.

Through plotting a simple graph of the height of a growing plant against time in weeks

(Figure 1), a person using concrete operations can answer questions such as:’Between which weeks was the plant growing fastest?’

Because there is a direct relation between the steepness of the graph and the rate of plant.

growth, the graph functions as a concrete model, almost a simplified picture, where the values for time in weeks and height of plant are in one-to-one correspondence with perceptions.The one describes the other.





‘Which of the graphs in Figure 2 best describes a lift rising from the ground to the fifth floor, and stopping twice on the way up?’

“Point to a part of a graph which represents an impossible journey”

A person using concrete operations is likely to choose (a) or (b) as the answer to the first question, and offer no answer to the second.

To answer the questions requires the following formal operational thinking:


  • the questions are about the relation between the height axis and the time axis
  • a vertical graph would mean the lift is travelling up in zero time, which is impossible
  • the graph with the lift movement shown at an angle must represent its travel in time


In a 1974/75 survey a representative sample of 14,000 pupils between the ages of 10 and 16 were given

Figure 3 shows the findings for 14-year-olds (Year 9) (Shayer, Kiichemann & Wylam, 1976). 1974/75 CSMS survey of 14,000 children between the ages of 10 and 16 data for 14-year-olds

Barely 20% of the population showed even Early Formal thinking (3A). But the more striking finding was the very wide range of thinking levels, including even some thinking at the level of the average 6-year-old. In the National Curriculum for Mathematics formal operations are generally required from about level 6 upwards.

The Key Stage 3 national statistics show the same general pattern:the majority of pupils are below level 6.

Key Stage 3 National Statistics for Mathematics 2000: all 14-year-old


Figure 4

1 Concepts in Secondary Mathematics and Science (1974-80). Research programme funded by the SSRC at Chelsea College, University of London.

Unless pupils are at level 6 in maths by the end of Year 9 they are very unlikely to achieve a GCSE pass at B grade or above in mathematics. In order to improve the maths achievement of pupils in the population the proportion of pupils using formal operational thinking by the end of Year 9 needs to be increased. The general reasoning patterns (Piaget called them schemata) that are characteristic of formal operations include:

• control of variables, and exclusion of irrelevant variables;

• ratio and proportionality;

• probability and correlation;

• the use of abstract models to explain and predict;

and are particularly relevant to mathematics.

Thinking Maths encourages the development of thinking from concrete to formal operations, through lessons that use these reasoning patterns in topics across the National Curriculum for Mathematics (see section 4).

The Vygotskian contribution

Piaget described development as an interaction between the individual and the environment, either with new stimuli assimilated into existing thought processes or cognitive structures, or the cognitive structure accommodating to stimuli which could not be simply assimilated. But this explanation is inadequate, since it implies that the development of formal operations is universal.Vygotsky argued that Piaget’s theory is only a part of the story. From birth the child’s immediate world is framed or simplified by the parent and other family members in such a way that learning is facilitated, and the child’s competencies in perceiving, walking, talking, etc. are developed.This process, called mediation, becomes part of the stimuli which the child either assimilates or to which his cognitive structure must accommodate. When the mediated learning experience of the child is adequate, the child develops confidence in his/her own ability to learn directly from the environment: development is ‘normal’ and is as Piaget described.

Vygotsky realised that many children’s experience of mediation is inadequate for them to realise their potential. He argued that school should not just present children with learning at the level of understanding they can cope with:’… instruction is good only when it proceeds ahead of development,

when it awakens and rouses to life those functions which are in the process of maturing’ (Shayer, 2003). Well before children 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. When learning is truly collaborative all children – or indeed any learners – contribute to the interaction that results in the production and expression of insight.The pupil in which the new insight has crystallised has been assisted there by the efforts – even the doubts or difficulties – of the other pupils.

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 leads 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. Thinking Maths lessons take a Piagetian view of what is implicit in the mathematics and incorporate this into lessons conducted according to a Vygotskian view of psychological development.The activities are framed so that road ahead’ leads in the right direction but the pupil can only see the immediate task. The teacher as mediator directs small group learning and whole class discussion: ways that increase the probability for each pupil that they will witness in some other pupil the next step in thinking that they are ready for.


2 Recent evidence (Shayer, Ginsburg & Coe, 2006, and Education Guardian, 24/1/2006, p.3) has suggested that children’s development has deteriorated, with the average child of 11 showing by 2003 only the level of the average 8/9-year-old in 1976 shown in p,3. This questions government statistics that appear to show continual improvement.


Thinking Maths Copyright © by Michael Shayer and Mundher Adhami. All Rights Reserved.

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