The development of proportional reasoning: Equivalence matching with continuous vs. discrete quantity

The present study uses a proportional matching task to examine children's ability to make judgments about equivalence among proportions. The matching task starts by presenting a card with a target proportion, which is followed by asking the child to choose an equivalent proportion among three choice cards. In Experiment 1, problems involve discrete quantities. Children fail to make proportional judgments when the problems are presented in a discrete set until the age of 10. 4 and 6-year old children's failures are directly related to their use of an absolute number strategy, counting only the number of focal entities without considering the relation between the number of focal and non-focal entities. In contrast, in Experiment 2, where problems are presented in continuous sets, even 4-year-old children succeed in matching a proportional equivalence without being misled by the absolute quantity of focal entity. The results of this present study provide clear evidence of the effect of the different types of quantity on children's proportional judgments. Our results also show that earlier success in judging proportions in continuous than in discrete quantities is likely attributable to children's use of different strategies in the presence of countable vs. non-countable entities. In the continuous condition, where it is not possible to count the number of items, children use an amount strategy, comparing focal and non-focal region in terms of continuous amount. Children are quite successful in using the amount strategy quite early. In contrast, in the discrete conditions, where countable entities are present, children use the number strategy which they are not successful at using until quite late. In addition, this present study examines the effect of the “half-boundary” on children's making judgments about proportions. Our findings suggest that the half-boundary plays a general and important role in children's proportional judgments for problems involving both continuous and discrete sets.