Functions, operations, and decalage in the development of transitivity.

Abstract

The hypothesis that some attempts to reduce the performance demands of concrete operational tasks may have allowed children to solve those tasks with preoperational functions was tested by administering two previously used versions of the transitivity task for length and weight to 120 children 6 to 9 years of age. In the standard Piagetian version of the task, comparison objects were presented only two at a time, with no correlation between length or weight and spatial position. In the alternate version, objects were ordered in space by length or weight, but placed too far apart for differences in length to be seen. As predicted: (a) the standard version was solvable only through an operational composition of premise relations, but the alternate version allowed children to recode length or weight as a function of spatial position; (b) the functional solution was easier and developed at an earlier age than the operational solution for length as for weight. In addition, the decalage between length and weight was found to result from children's tendency to infer weight as a function of size rather than from an operational composition of premise relations.

How to detect reasoning-remembering dependence (and how not to)

“…In Popperian terms, the hypothesis has survived a test of falsification. Thus, Chapman and Lindenberger's (1988) finding that increasing numbers of premises affected reasoning in their "standard" condition but not in their "alternate" condition provides evidence for the hypothesis by which those results were predicted (that premise comparisons were represented separately in the one condition but not the other). Against Reyna and Brainerd's analogy between children's use of an external array in the transitivity task and the use of a scratch pad for adding a series of numbers, we would argue that the external array condition is more like providing people with sets of objects corresponding to the numbers to be added so that they can obtain the correct sum without adding the numbers at all, but by counting all the individual objects instead.…”

“…Both Brainerd and Kingma (1984) and Chapman and Lindenberger (1988) agreed that reasoning-remembering dependencies should be found under some conditions. According to Brainerd and Kingma, such dependencies should be found when retention of the explicit background facts is a "logical precondition for correct reasoning" (1984, p.…”

“…No relation between correct judgments and memory for a given set of premises should be found if children can and do infer a correct judgment from an entirely different set of premises. In a comparison between two commonly used versions of the transitive reasoning task, for example, Chapman and Lindenberger (1988) found that one version (the "alternate" version) could be solved through the use of spatial position cues, because the relevant physical dimension was correlated with spatial position (e.g., comparison objects were arranged from right to left in order of increasing length). The other version (the "standard" version) could not be solved in the same way, because the relevant dimension was uncorrelated with position.…”

“…Following this line of reasoning, dependence between adjacent and nonadjacent comparisons would be expected only when the extraneous correlations necessary for functional reasoning have been eliminated. Chapman and Lindenberger (1988) cited three sources of evidence for their conclusion that children used different forms of reasoning depending on whether the relevant dimension in a transitivity task was objectively correlated with right-left spatial orientation (their "alternate" condition) or uncorrelated with spatial orientation (their "standard" condition): (a) Children justified correct answers differently under the two conditions, giving functional justifications (e.g., "object A is longer than object C, because A is on the right") in the "alternate" condition and operational justifications ("A is longer than C, because A is longer than B and B is longer than C") in the "standard" condition; (b) the task was considerably easier and solved by children at younger ages in the "alternate" condition than in the "standard" control condition; (c) the number of CHAPMAN AND LINDENBERGER premise comparisons affected performance only in the "standard," and not in the "alternate" condition. Chapman and Lindenberger argued that these findings provided converging evidence for the hypothesis that children inferred their judgments as a function of spatial orientation in the "alternate" condition and through a composition of premise relations in the "standard" condition-in other words, that they tended to use the same forms of reasoning in inferring their judgments as in constructing their justifications across conditions.…”

“…Significantly, the "alternate" version was modelled on the task used by Kingma (1984, 1985) for testing the dependency between memory and reasoning. According to Chapman and Lindenberger (1988), one would not expect any reasoning-remembering dependency for that version in any case, because it can be solved without recourse to the set of premises ("A is longer than B, " "B is longer than C") for which memory was tested (see also Chapman, 1989).…”