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The Child's Understanding of Correspondence Relations


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A number of quantitative comparison tasks were designed to tap knowledge of injective and surjective correspondences, one-directional compositions (greater + greater yields greater), countervailing compositions (greater + lesser yields ?), and length-density relations in 4- to 7-year-olds. The results indicated that performance on the comparison tasks was related to performance on a number conservation test as well as to age. Nonconservers performed at better than chance levels on tasks that tapped an elementary knowledge of injective and surjective correspondences; concrete-operational children, however, tended to perform better on all tasks. Uncorrected and disattenuated correlation coefficients revealed considerable consistency across measures. Factor analyses, with and without age included, yielded a unitary factor. An explanation of the results based on perceptual salience was ruled out. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
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... As anticipated in hypotheses three and four, the correlational and factor-analytical findings reveal consistency across measures and suggest that a unitary, developmental factor underlies performance differences. Consistent with Schonfeld (1990), the findings suggest that children's performance on comparison tasks improves in two ways: at advanced levels (1) the understanding of more complex functional relations emerges and (2) thinking becomes increasingly exact, facilitating improvement in cognitive performances in which the more advanced preoperational (i.e. OL-2) children demonstrate some preliminary competence (i.e. ...
... TPO items). The findings pertaining to the specific cognitive accomplishments of pre-operational children were consistent with results on cognate discrete-quantity tasks (Schonfeld, 1986Schonfeld, , 1990). The correlations among the liquid scales tended to be lower than the correlations among the discrete-quantity scales (median r=.52, median corrected r=.7O in Schonfeld, 1990; r= .68 for Scales 1 and 2, corrected r= .84 in Schonfeld, 1986). ...
... The most reasonable explanation for the difference is that the liquid scales had considerably less variance than the discrete-quantity scales. Each liquid scale comprised only three items and the discrete quantity scales included as many as 14 items (Schonfeld, 1990). The average size of the correlation coefficients among the liquid scales, however, was consistent with findings obtained by de Ribaupierre, Rieben & Lautrey (1985), who employed a different set of Piagetian tasks and a more conservative computational method. ...
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Four liquid comparison tasks were designed to assess children's knowledge of functional relations, one-directional compositions of functional relations (greater + greater yield greater) and countervailing compositions (greater + lesser yield ?) in 4- to 7-year-olds. On one task, in which height indexed quantity, children of every age group performed well. Success on the other comparison tasks was related to operative level, as indexed by conservation performance, and age. More advanced pre-operational children evidenced a degree of success on the one-directional composition task. Consistent with Schonfeld (1990), the results suggested that at more advanced operative levels: (1) the understanding of increasingly complex functional relations emerges and (2) thinking becomes increasingly exact. Correlational results revealed consistency across measures and factor-analytical findings suggested that a unitary developmental factor underlies performance differences. An attentional explanation of the findings was ruled out. The findings highlighted the multifaceted nature of children's progress toward integrating information from different dimensions of the comparison tasks.
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Highlights the similarities and differences of the Genevan and Cattell-Horn theories of intelligence and reports an investigation of the relation of operative level and set of performance on tasks indexing children's knowledge of correspondence relations. 105 children (aged 4 yrs to 7 yrs 11 mo) completed counting, instructional set, conservation of number, and static numerical comparison tasks. Findings indicate that performance on quantitative comparision task reflecting Ss' understanding of correspondence relations was highly related to operative level and that Ss' capacity to implement solution aids in making quantitative comparison was moderated by their level of operative development. While findings lend support to the Genevan theory, this theory tends to neglect issues pertaining to localized functioning such as the question of relative efficacy of rival solution approaches. In contrast, the Cattell-Horn theory emphasizes that well-learned knowledge-producing skills constitute potential solution aids and that children differ in what solution aids they learn to implement. (44 ref)
Years ago, prompted by Grize, Apostel and Papert, we undertook the study of functions, but until now we did not properly understand the relations between functions and operations, and their increasing interactions at the level of 'constituted functions'. By contrast, certain recent studies on 'constitutive functions', or preoperatory functional schemes, have convinced us of the existence of a sort of logic of functions (springing from the schemes of actions) which is prior to the logic of operations (drawn from the general and reversible coordinations between actions). This preoperatory 'logic' accounts for the very general, and until now unexplained, primacy of order relations between 4 and 7 years of age, which is natural since functions are ordered dependences and result from oriented 'applications'. And while this 'logic' ends up in a positive manner in formalizable structures, it has gaps or limitations. Psychologically, we are interested in understanding the system atic errors due to this primacy of order, such .as the undifferentiation of 'longer' and 'farther', or the non-conservations caused by ordinal estimations (of levels, etc. ), as opposed to extensive or metric evaluations. In a sense which is psychologically very real, this preoperatory logic of constitutive functions represents only the first half of operatory logic, if this can be said, and it is reversibility which allows the construction of the other half by completing the initial one-way structures."
The role of stimulus factors and logical judgments in conservation of discontinuous quantity was explored using 2-choice, nonverbal discrimination tasks. 40 preschoolers estimated the amount of candy in 2 containers under conditions varying the relative diameters of the 2 containers (identical or different) and the type of judgment required (direct comparison, identity, and equivalence estimates). The tasks formed a Guttman scale. The easiest items involved identical containers. There was a significant interaction between stimulus setting and judgments. When identical containers were used, identity judgments were least accurate, but when the 2 containers differed, identity was easiest followed by equivalence and comparison judgments.
An analysis of the quantitative processes underlying conservation of quantity is presented. Models of quantitative operators (subitizing, counting, estimation) are derived from adult performance in quantification tasks, and some features of the operators are described. The emergence of conservation is described in terms of the development of the operators and a set of rules which evoke them and coordinate their results. Empirical data related to the developmental argument is discussed.