The Acquisition of Inverse Proportionality: A Training Experiment

School children often experience difficulty in mastering the concept of proportionality (Hart, 1981; Inhelder & Piaget, 1970; Noelting, 1980a, 1980b, 1983; Piaget, 1946; Vergnaud, 1980, 1983). Its development results from the quantification of a two-fold qualitative relation (Inhelder & Piaget, 1970; Sinclair, 1968). Vergnaud (1980) and Karplus, Pulos, and Stage (1983a, 1983b) have studied the distinction between two types of relationships (within and between) among variables, and Vergnaud and Ricco (1977) have observed the between comparison to be more "natural." No such thorough investigations have been devoted, however, to the case of inverse proportionality. The study conducted by Linn and Pulos (1983) gave more information on sex differences than on developmental acquisition. Siegler (1976, 1981) has analyzed the qualitative development of inverse proportionality without attending in any detail to the various quantificational stages involved. In this paper, a model is proposed in which the quantification of inverse proportionality is developed beyond the initial qualitative relation. This general model was based on a task analysis and was tested in a training experiment. The effects of training were assessed by means of a balance task (Inhelder & Piaget, 1970).