A study of the development of the concept of quantity by scalogram analysis

A developmental analysis was made of the sequential ordering of eleven quantity tasks selected on the basis of Piaget's theory of intelligence. The following set of tasks, in the hypothesized order of difficulty, comprised the test series: (1) subtraction/addition of discontinuous substance; (2) addition/subtraction of discontinuous substance; (3) gross comparisons; (4) intensive comparisons; (5) extensive comparisons; (6) conservation of continuous substance; (7) conservation of weight; (8) transitivity of weight; (9) conservation of occupied volume; (10) conservation of displacement volume; (11) calculation of volume. Green's (1956) method of scalogram analysis was the principal analytic technique for evaluating the data. Scoring criteria adapted from Smedslund (1961) were found to be highly reliable. The sample consisted of 100 children, ages five, six, eight, ten and twelve years; they were of average intelligence, members of the Caucasian race, with an equal number of children of each sex in each age group. The five age groups did not differ significantly in intelligence. In addition to the quantity test series each subject was administered the Peabody Picture Vocabulary Test. The hypothesis that the components of the concept of quantity conform to a hierarchy of increasing difficulty and reflect the properties Guttman (1950) has described for an ordinal scale was not accepted. However, the data are considered adequate to support a claim for the existence of what Guttman describes as a quasiscale. The hypothesis of an invariant, sequential relationship among subtraction-addition, addition/subtraction and conservation of discontinuous substance was also accepted. The prediction that the infra-logical operation of conservation of weight takes genetic precedence over the logical operation of transitivity of weight was also confirmed. The hypothesis that there is an invariant order of increasing difficulty involved in making gross, intensive and extensive comparisons of discontinuous substance was given only partial support with the major discrepancy being a reversal in the predicted order of difficulty for gross and intensive comparisons. Likewise, only partial support was available for the prediction of an invariant, sequential relationship among the component concepts of conservation of substance, weight and volume. Whereas the volume concept emerged as the most advanced of the three components, no relationship was found between the other two. No support was given to the theoretical inference of developmental priority for conservation of discontinuous over continuous substance, nor was the prediction upheld that there is a systematic relationship among the separate components of the volume concept distinguished in Piaget's theory. The fact that conservation of displacement volume was acquired significantly earlier than the other two components of the volume concept is consistent with other validational evidence. The predictions of overall differences in total scale scores for age but not for sex were confirmed. A curvilinear relationship between chronological age and total scale score was obtained, and this finding was interpreted as support for the conclusion that the scale points represent a quasi-developmental scale. Moreover, it was suggested, on the basis of the latter findings, that Piaget's stage of formal operational thought may begin at an earlier age than twelve years. Other areas of theoretical significance were also discussed in relation to the findings. An outline of some of the general implications of the study was made, and suggestions were offered of some possible applications in the field of education.