Relationship of primary school children's ability to conserve number and quantity to their achievement in arithmetic and to certain other characteristics

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A present-day theorist of cognitive development in young children, particularly in reference to development of mathematical concepts, prominent among educators, is Jean Piaget, a Swiss biologist, psychologist, philosopher, logician, and mathematician. The most notable Piagetian theory relative to children's cognitive development pertains to of number and quantity. Piaget uses the phrase conservation of number and to indicate a mental process called self-regulation or equilibration, a process involving reversibility of thought.' According to Piaget, a child has attained of number when he continues to say that two sets of objects matched in a one-to-one relationship have the same number of objects regardless of the spatial transformation of one of the sets. He is a conserver of liquid quantity if he continues to say that equal amounts of liquid in two containers of the same size and shape will remain equal regardless of size and shape of a third container into which liquid from one of the original containers is poured. He is a conserver of solid quantity if he continues to say that equal amounts of clay, or similar material, in two identical forms will remain equal no matter what shape the material in one of the original forms assumes. At this stage of mental development, which Piaget calls the concrete operations stage, he contends that the child's decisions do not depend upon perception alone but in the main depend upon logical thinking. Piaget has labeled the concrete operations stage as the third one of the four stages through which he contends the child passes in the development of mental structures. Stage one, identified as the Sensori-Motor Stage, he describes as preverbal in nature and covers, in general, the first two years of life. Stage two, the Pre-operational Stage, marks the beginning of language in the form of words, of thought, and of symbol manipulation. Stage two lasts from about age two years to age six and one half years. The Concrete Operations Stage, the third stage, covering the years from about six or six and one half years to about eleven, constitutes a period of cognitive development when the child obtains ideas from operations on concrete objects. At this third stage, or invariance is fully de-