The development of a proportionality concept: Judging relative fullness

Abstract

Six experiments were performed to investigate children's understanding of a proportionality concept, the concept of fullness. Experiments 1 and 2 were intended to rigorously test Bruner and Kenney's (1966) hypothesized developmental progression for the fullness of a water jar task. The results of the experiments supported Bruner and Kenney's contention that 6-year-olds base their spontaneous judgments of fullness on the relative heights of liquid columns, that 10-year-olds base their judgments on the relative volumes, and that neither age group relies on the proportion of filled to empty space. Experiments 3 through 6 were devoted to testing a variety of possible explanations for why children did not use the Proportionality Rule. These experiments failed to support the hypotheses that nonuse of the Proportionality Rule was due to: (a) misunder-standing of instructions; (b) failure to attend to both filled and empty space within each glass; (c) inability to apply appropriate labels to proportions; or (d) inability to compare the magnitudes of proportions. Instead, it appeared that children of both ages could execute all of these component operations, but did not know how to put them together; simple instruction in how to combine the components led to 90% of the children of each age mastering the Proportionality Rule. The results pointed to the importance of distinguishing between the processes of learning and invention.