Piaget's theory of intellectual development focuses on two central aspects of the progressive view of childhood; first, on the centrality of children's experience, especially physical interaction with the world; second, on the unfolding logic of children's thought, which differs from that of the adult (Ernest, 1991). Piaget proposed four major stages of intellectual development: (1). the sensori-motor stage (birth to 1 1/2 to 2 year), (2). the pre-operational stage (2 to 7 years), (3). the concrete operational stage (7 to 12 years), (4). the formal operational stage (12 to 15 years and up). The characteristic in which a remarkably smooth succession of stages, until the moment when the acquired behaviour presents seems to be recognizes as 'intelligence' (Piaget and Inhelder, 1969); there is a continuous progression from spontaneous movements and reflexes to acquired habits and from the latter to intelligence. They further stated that this mechanism is one of association, a cumulative process by which conditionings are added to reflexes and many other acquisitions to the conditioning themselves. They then regarded that every acquisition, from the simplest to the most complex, is a response to external stimuli, that is a response whose associative character expresses a complete control of development by external connections. They described that this mechanism consists in assimilation, that reality data are treated or modified in such a way as to become incorporated into structure of subject.
In moving from the sensory-motor stage to operational thought, several things must occur during the preoperational period (Becker, et al., 1975); there must be a speeding in thought or actions, there must be an expansion of the contents and scope of what can be thought; and there must be concern not only with the results of action but also with understanding the processes by which a result is achieved. Piaget and Inhelder (1969) outlined that there are three levels in the transition from action to operation; at the ages of two or three there is a sensory-motor level of direct action upon reality; after seven or eight there is the level of the operations in which concern transformations of reality by means of internalized actions that are grouped into coherent and reversible systems; and between these two level there is another level obviously represents an advance over direct action in which the actions are internalized by means of the semiotic function and characterized by new and serious obstacles. Toward the end of the preoperational stage, the basis for logico-mathematical thinking has been laid in the use of language, but the child is still far from reaching operational thought (Becker, et al., 1975).
During the years between two and seven the child learns much about the physical world; some of this is spontaneous, while other is deliberately taught by parents and teachers; despite the many intellectual feats of the period, children do not reason in a logical or a fully mathematical way. Children's thinking in the pre-operational period is characterized by what Piaget called moral realism as well as animism and egocentrism (ibid, p.102). Animism is the failure to adopt one stance towards inanimate objects and another towards oneself; moral realism is the consequence of viewing morality in one sense only; egocentrism is the consequence of the child's taking only one perspective; and the child achieves the next stage of intellectual development when at last he can consider a situation from several different aspects - in other words, he can de-centre (ibid, p.103). After many experiments, Piaget and his colleagues concluded that there is a sequence of development for each of the conservations; each experience requires that the child must judge whether the two things are still the same or are different when the entities is transformed in appearance by being changed in shape or transferred to another receptacle (Sutherland, 1992). It has been shown that children in the period of concrete operation can perform the mental operation of reversibility and can attend to several aspects of a situation at once (ibid, p.109).
For Piaget, an operation is a mental action (Becker, et al., 1975) that usually occurs in a structure with its counteraction - adding goes with its reverse operation subtracting, combining with separating, identity with negation; an operation is said to be concrete if it can be used only with concrete referent rather than hypothetical referents. The first obstacle to operations (Piaget and Inhelder, 1969), then, is the problem of mentally representing what has already been absorbed on the level of action. In the concrete operational stage, thinking shows many characteristics of mature logic, but it is restricted to dealing with the 'real' (Becker, et al., 1975). The second obstacle to this stage is that on the level of representation (Piaget and Inhelder, 1969); achieving this systematic mental representation involves constructive processes analogous to those which take place during infancy; the transition from an initial state in which everything is centered on the child's own body and actions to a decentered state. The third obstacle is related to the complexity of the using of language and the semiotic function involving more than one participant.
Formal operations involve thinking in terms of the formal propositions of symbolic logic and mathematics or in terms of principles of physics (Becker, 1975); one can deal with the hypothetical and one can deal with operations on operations. Piaget studied the development of logical thinking in adolescence and reflective abstraction, that very human capacity to be aware of one's own thoughts and strategies. Piaget assert that the basis of all learning is the child's own activity as he interacts with his physical and social environment; the child's mental activity is organized into structures and related to each other and grouped together in the pattern of behaviour (Adler, 1968). Piaget also asserts that mental activity is a process of adaptation to the environment which consists of two opposed but inseparable processes, assimilation and accommodation (ibid, p. 46). The child does not interact with his physical environment as an isolated individual but as part of a social group; as he progresses from infancy to maturity, his characteristic ways of acting and thinking are changed several times as new mental structure emerge out of the old ones modified by accumulated accommodations (ibid, p.46).
Piaget found that there is a time lag between the development of a child's ability to perceive a thing and the development of this ability to form a mental image of that thing when it is not perceptually present (Adler, 1968). The development of the child's concepts of space, topological notions, such as proximity, separation, order, enclosure, and continuity, arise first; projective and Euclidean notions arise later; and his grasp of order relation and cardinal number grow hand in hand in the concept of numbers (ibid, p.51). Piaget also asserts that a child progresses through the four major stages of mental growth is fixed; but, his rate of progress is not fixed; and, the transition from one stage to the next can be hastened by enriched experience and good teaching (ibid, p.53). Based on all the above propositions, some of their implications for the mathematics teaching in the primary school can be asserted. Piaget maintained that internal organization determines how people respond to external stimuli and that this determines man's unique 'model of functioning' which is invariant or unchangeable (Turner, 1984); a person attempts to make sense the environmental stimulus by using his existing structure or by assimilating or accommodating it.
The structure and their component schemes were said to change over time through the process of equilibration; if a subject finds that her present schemes are inadequate to cope with a new situation which has arisen in the environment so that she cannot assimilate the new information, she will be drawn, cognitively, into disequilibrium (ibid, p.8). Given these fundamental postulates of Piaget's theory : internal organization, invariant functions, variant structures, equilibration and organism/environment interaction; what then are the implications for mathematics education in the primary mathematics school ?
Piaget, J. and Inhelder, B., 1969, The psychology of the child, London : Routledge & Kegan Paul.
Adler, I., 1968, Mathematics and Mental Growth, London : Dennis Dobson.
Becker, W., et al., 1975, Teaching 2: Cognitive Learning and Instruction, Chicago : Science Research Associates.