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Between the steps on "The Mind's Staircase" : individual pathways to the development of young children's mathematical understanding Loewen, Susan Nedda

Abstract

This study explored how individual, 4- to 5-year-old children, who displayed average to above-average mathematical ability for their age responded to an instructional program designed to facilitate construction of the mental counting line. Case's (1996a) neo-Piagetian theory, Case's (1996b) model of the process of structural change, and Porath's (1991b) model of the intellectual development of academically advanced children provided the theoretical framework for the study. The microgenetic approach advocated by Catán (1986a) was used to explore change and variability in the developmental pathways of each of the children in the study. Three girls (aged 4.0, 4.1, 4.8 years) and 1 boy (aged 4.11 years), who had not yet constructed the mental counting line, participated in the study. An instructional program (7 weeks long) was used to stimulate the development of the mental counting line. The Quantitative Reasoning subtests of the Stanford-Binet Intelligence Scales, Fifth Edition (Roid, 2003a) and four measures of conceptual understanding (Number Knowledge test, Balance Beam, Money Knowledge and Birthday Party tasks) were administered prior to instruction. The measures of conceptual understanding were readministered following instruction. A qualitative analysis of the children's pretest and posttest scores, descriptive microgenetic quantitative and qualitative analyses of the children's responses to the instructional program, and a trend analysis of the children's performance were conducted. The results indicated the children progressed from the pretest to the posttest. Intra-individual and inter-individual differences in the rate and the pattern of construction of the mental counting line were apparent. The results provide evidence for individual pathways to development as children negotiate the critical transition between the predimensional and dimensional stages of Case's (1996a) theory. The results support Case's (1996a) neo-Piagetian theory, Case's (1996b) model of the process of structural change, and Porath's (1991b) model of the intellectual development of academically advanced children. The results are consistent with the results of studies from Case's (1996a) neo-Piagetian theoretical perspective, the results of studies from other neo-Piagetian theoretical perspectives (Case, 1996c; Siegler, 1996b), and the results of studies on the development of children's mathematical understanding (Abbott, Berninger, & Busse, 1996; Robinson, Abbott, Berninger, Busse, & Mukhopadhyay, 1997).

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