The development of the child's understanding of fullness as a ratio.

Abstract

One of the central problems in the understanding of the child's cognitive development is his acquisition and subsequent modification of concepts. The present study is concerned with the mathematical concept of ratio, and more specifically with the development of the child's understanding of fullness as a ratio. The idea of ratio might be said to involve a quantitative relation between two or more features that remains invariant regardless of the nature of the component features. Such a quantitative relation is typified by the idea of a physical proportion, and it is this which will be investigated: with proportion as embodied in the question, "Which of two containers is fuller (or emptier)?" "Fullness" is a very interesting concept to investigate, for it involves in its very definition a ratio or proportion between the volume of a container and the volume of the substance contained. Thus in the case of proportion, looked at formally, there is an especially great difficulty for the child: in order to make a correct judgment of fullness or emptiness, the child must resist the temptation to follow perceptual appearances, and instead must use a symbolic operation somewhat like computing a ratio. He must estimate the volume of the container that is filled and then relate this volume to the total volume of the container. This is quite_a difficult task for the child to master, but it is plain that in some primitive form the child can deal with 'full' and 'empty' long before he understands ratios. The child's first use of the terms is limited and he does not grasp what is meant by the comparative forms 'fuller' and 'emptier'. However, at some stage the child goes beyond this partial and restricted idea to a more complete understanding of fullness as a ratio. At the Harvard Centre for Cognitive Studies, Jerome S. Bruner (1964, 1966) has recently been concerned with this very aspect of the child's cognitive development. He investigated the concept of fullness by requiring subjects aged five, six, seven, nine and eleven to judge which of two glasses was fuller and which emptier and presented eleven pairs of glasses in all differing with respect to the height of the glasses, the diameter of the glasses, the height of the water, the volume of the water, the volume of unfilled space in each glass, and finally the proportion full and proportion empty. On the basis of his results he claims that it is not until age eleven that the child can resist perceptual cues and judge fullness correctly. There were, however, several shortcomings in both Bruner's design and his procedure which cast doubt upon these findings. It was therefore decided to replicate this experiment modifying and elaborating where considered necessary. (Hanlon, 1967) The results of this experiment showed that the eleven year old child, contrary to Bruner's conclusions, did not fully understand the concept of ratio through the idea of fullness so the age range studied was subsequently extended to 15 years (Moore and McWhirter, in press). However, it soon became clear that even the older children were not responding correctly, and it seemed that the concept was causing difficulty through misunderstanding rather than inability to perform the necessary operations. The children judged according to their interpretation of the concept and in so doing gave sporadic glimpses that if the problem were communicated to them more precisely they could respond correctly. A further extension was therefore undertaken in which an attempt was made to communicate the problem more exactly to a representative sample of children in the same age range and a comparison was also made between the performance of adult subjects in the original task and in the new experiment where aid was given. The original subjects served - 3 - as a control group. The results were considered both within the context of the development of the child's understanding of fullness as a ratio and also in relation to the findings of Bruner and Kenney.