The Implication Of Piaget's Work to Mathematics Education

By Marsigit
Especially attractive to workers in mathematics education were Piaget's conceptions that children's intellectual development progresses through well-defined stages, that children develop their concepts through interaction with the environment, and that for most of the primary years most children are in the stage of concrete operations. Further, she stated that in mathematics education, it was a natural consequence of belief in Piaget's theory about the central role of interaction with objects that, when they learn mathematics, children should be expected to work practically, alone with their apparatus, and to work out mathematical concepts for themselves. Piaget proposes that children create their knowledge of the world; however, he also argued that in the creation and unfolding of their knowledge, children are constrained by absolute conceptual structures, especially those of mathematics and logic; thus, Piaget accepts an absolutist view of knowledge, especially mathematics (ibid, p.185). For primary children (7 to 12 years) in which Piaget called they are in the stage of concrete operation, mental action occurs in a structure with its counteraction - adding goes with its reverse operation subtracting, combining with separating, identity with negation (Becker, 1975); thinking shows many characteristics of mature logic, but it is restricted to dealing with the real; an eight-year old, for example, has no trouble ordering a set of sticks according to height, but might fail to solve the problem.
In order to draw out the explicit implications of Piaget's work for mathematics teaching in primary school for the children who are at concrete-operational stage, it is useful to divide this stage into three stages : early concrete-operational (7-9 years), middle concrete-operational (10-12 years) and late concrete-operational (13-15 years). The children at early concrete-operational and middle concrete-operational will be discuss in the following. The child at early concrete-operational stage is confined to operations upon immediately observable physical phenomena; therefore, he states that the implications of the teaching mathematics may be translated as : (1) both the elements and operations of ordinary arithmetics must be related directly to physically available elements and operations, (2) there should be no more than two elements connected by one operation even with the restriction and the result must be actually closed to avoid the problem of any doubt about the uniqueness of the result, (3) the only notion of inverse is physical, (4) there is no basis for seeking a consistency in relationships with a system of elements selected two at a time and connected by an operation.
The child at middle concrete-operational tends to work with qualitative correspondences, e.g. the closer, the bigger; and thus is still reality bound and not capable of setting up a reliable system based on measurement. For these reason he outlined that its implications for teaching mathematics in the primary school are that : (1) Children begin to work with operations as such but only where uniqueness of result is guaranteed by their experience both with the operations and the elements operated upon; this in effect means two operations closed in sequence with small numbers or one familiar operation using numbers beyond his verified range, e.g. the child can cope with items involving the following types of combinations, (3+8+5) and (475+234); (2) The developing notion of the inverse of an operation tends to be qualitative; children regard substracting as destroying an effect of addition without specifically value of 'y' in y+4=7, they regard 'y' as a unique number to which '4' has been added, substracting '4' happens to destroy the effect of the original addition; (3) a basis exists for the development of a notion of consistency as being a necessary condition for a system of operations but the child tend to recognize the need without being able to give a logical reason for it. Preadolescent child makes typical errors of thinking that are characteristics of his stage of mental growth; the teacher should try to understand these error; and, besides knowing what errors the child usually makes, the teacher should also try to find out why he makes them (Adler, 1968). For these reason, further he suggests that an answer or an action that seems illogical from the teacher point of view on the basis of teacher's extensive experience may seem perfectly logical from the child's point of view on the basis of his limited experience.The teacher can help the child overcome the errors in his thinking by providing him with experiences that expose them as errors and point the way to the correction of the errors.
The child in the pre-operational stage tends to fix his attention on one variable to the neglect of others; to help him overcome this error, provide him with many situations (Adler, 1968). Due to the fact that a child's thinking is more flexible when it is based on reversible operations, the teacher should teach them pairs of inverse operations in arithmetic together, and teach that subtraction and addition nullify each other, and multiplication and division nullify each other (ibid, p.58). As Piaget summed up in Copeland (1979), 'numerical addition and subtraction become operations only when they can be composed in the reversible construction which is the additive group of integers, apart from which there can be nothing but unstable intuition'. Adler (1968) also suggested that physical action is one of the basis of learning; to learn effectively, the child must be a participant in events; to develop his concepts of numbers and space, for example, he needs to touch things, move them, turn them, put together or take them apart. Children should have many experiences in sorting common classroom materials, working with concrete shapes and sizes and colors, and discussing all sorts of relationship; this activities provides a basis for determining in a clearly defined way what progress children are making in their ability to realize, as well as copy, basic spatial distinction; and, most children will be ready at first-grade level to learn the basic shapes.
Since there is a lag between perception and the formation of a mental image, the teacher needs to reinforce the developing mental image with frequent use of perceptual data, for example, let him see the addition once more as a succession of motions on the number line when the child falters in the addition of integers (Adler, 1968). What the mind 'represents' may be and often is different from what is 'seen' or 'felt' by small children (Copeland, 1979); the teachers need to know the stages through which children go in developing the ability to consider geometric ideas. In order that the students are ready to learn a new concept, the teacher should examine the mastering of student's prerequisites concept; Piaget's theories suggested that learning was based on intellectual development and occured when the child had available the cognitive structures necessary for assimilating new information (Leder, 1992). In his teaching, the teacher needs to look at the way pupils go about their work and not just at the products; he also needs to listen to pupil's ideas and try to understand their reasoning and discuss the problems so that pupils reveal their ways of thinking. These activities are actually in the framework of teacher's method of assessing students' thinking.
Piaget developed his 'clinical method' as a way of exploring the development of children's understanding, and employed observation along with interview as a means of accessing children's views of the world (Conner, 1991). He is the pioneers who advocated using observations of children in real situation; the observation, that is more than just looking, serves a useful assessment purpose involves : looking at the way pupils go about their work and not just at the products, listening to pupils' ideas and trying to understand their reasoning, and discussing problems so that pupils reveal their ways of thinking (ibid, pp. 50-51). The observation were used to support the hypothesis that the children, at a certain stage, were discriminating between 'means' and 'ends' (Becker, et al.,1975). In this interview, the answers of students at various ages are then analyzed to see how properties of 'mental structures' change with age (ibid, p. 218). The justification of whether the Piaget's idea of assessment is practical or not in the classroom practice depends much on : the philosophy of mathematics education in which we start to do so, the characteristics of his paradigm of cognitive development or student's competence, the capability of the teacher ; and, in general, this is dependent on its interpretation.
References:
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.
Piaget, J. and Inhelder, B., 1969, The psychology of the child, London : Routledge & Kegan Paul.
Vygotsky, L.S, 1966, 'Genesis of the higher mental functions' in Light, P. et al. , 1991, Learning to Think : Child Development in Social Context 2, London : Routledge.
Tharp, R. and Gallimore, R., 1988, 'A theory of teaching as assisted performance' in Light, P. et al. , 1991, Learning to Think : Child Development in Social Context 2, London : Routledge.