Fraction Knowledge in Preschool Children.

Much school mathematics is devoted to teaching concepts and procedures based on those units that form the core of whole number arithmetic (ones, tens, hundreds, etc.). But other topics such as fractions and decimals demand a new and extended understanding of units and their relationships. Behr, Wachsmuth, Post, and Lesh (1984) and Streefland (1984) have noted how children's whole number ideas interfere with their efforts to learn fractions. Hunting (1986) suggested that a reason children seem to have difficulty learning stable and appropriate meanings for fractions is that instruction on fractions, if delayed too long, allows whole number knowledge to become the predominant scheme to which fraction language and symbolism is then related. Alternatively, there is some evidence that suggests that children can solve fraction-related problems before the appropriate procedures are taught in school. Polkinghorne (1935) concluded from a study of 266 children in kindergarten, first, second, and third grade that considerable knowledge of fractions is held prior to formal instruction in this topic, and Gunderson and Gunderson (1957) demonstrated that second graders had concepts and ideas about fractions that could be developed subsequently. Children's concepts of fractional units have been studied by Piaget, Inhelder, and Szeminska (1960), when children of ages 4 to 7 were individually asked to cut a circular clay "cake" for different numbers of dolls. Streefland (1978) reported "phenomenological" sources of the fraction concept arising from a 12-month study of two young children. Pothier and Sawada (1983) have proposed a five-level theory of partitioning based on observations of children's attempts to subdivide continuous quantities represented by cakes of various dimensions and shapes. Whereas studies by Korbosky (1984) and Smith (1985) sought to clarify aspects of earlier work of Piaget et al. (1960), evidence that children use different strategies to subdivide discontinuous quantities was reported by Hiebert and Tonnessen (1978). Children aged 9 and 10 years have been found to possess cognitive structures for defining, relating, representing, and transforming fractional units (e.g., see Hunting, 1983a). However, little is known about the origins of such knowledge. It is possible that young children have sufficiently developed cognitive processes for dealing with problems involving subdivisions of quantities to allow fraction instruction to begin earlier. Both Kieren (1983) and Vergnaud (1983) have emphasized the primacy of partitioning for establishing rational number knowledge. A careful study of younger children's responses to fraction problems was designed to clarify what (if any) knowledge in regard to