Date of Degree
David J. Bearison
Geoffrey B. Saxe
This investigation was designed to examine the development of the child's capacity to make numerical and quantitative comparisons. It was hypothesized that the nature of the child's understanding of correspondences and one-way functions informs his capacity to compare arrays of decals and quantities of liquid. The relation of Piagetian operative level to the child's capacity to use crystallized skills, or solution aids (Cattell, 1963; Horn, 1968), in comparing arrays was also investigated.
A total of 171 children who ranged in age from four to seven years were administered numerical and liquid comparison tasks. The numerical tasks included paired arrays of green and red decals represented as the candies of puppets named Bert and Ernie. Some paired arrays were related by injective and/or surjective correspondences. Other paired arrays were equal in length but different in density. In a different set of tasks, the green and red arrays each comprised two subarrays. In these tasks, one puppet got more candy in one green-red subarray comparison and again in the second subarray comparison, or a different puppet got more candy in each of the two green-red subarray comparisons. In all subarray tasks the child was asked to determine the relative numerosity of the total amount of green and red decals. In the liquid tasks, paired green and red liquids were represented as Bert and Ernie's juice. Tasks included paired quantities of juice contained in transparent cylinders that were either the same or different in diameter. In the latter task, paired quantities had the same height. In other liquid tasks, each quantity of green and red liquid comprised two subquantities. In these tasks, one puppet got more juice in one green-red subquantity comparison and again in the second subquantity comparison, or a different puppet got more juice in each of the two green-red subquantity comparisons. In all subquantity tasks, the child was asked to determine the relative quantity of the total amount of green and red liquid.
Results indicate that: (1) Preoperational children possess comparison-making capabilities reflecting a rudimentary understanding of injective and surjective correspondences, one-way function based mappings of height on to quantity, and one-way compositions of same-directional subquantity comparisons. (2) Concrete operational children develop powerful comparison-making capabilities based on the capacity to coordinate countervailing subquantity and correspondence relations; they also begin to judge quantities on joint bases such as density and length, or diameter and height. (3) Comparison-making capabilities found in the preoperational subperiod become more accurate with the development of the concrete operations. (4) The child's capacity to use solution aids in making accurate numerical comparisons is structured by operative level.
The educational implications of correspondence and function based knowledge were discussed.
Schonfeld, Irvin Sam, "The Early Development of Quantitative Cognition: Correspondences and One-Way Functions" (1980). CUNY Academic Works.