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Comparison-induced distortion theory (Choplin 2007; Choplin and Hummel 2002) describes how comparison words like “better” suggest quantitative differences between compared values. When a comparison word is used to contrast a personal attribute value with some standard (e.g. “Your score is better than average”), the comparison-suggested difference for the word may bias estimates or recall of personal attribute values. Three studies investigated how comparison-suggested differences determine the effect of social comparison on estimates or recall of personal attribute values. The first study demonstrated that estimates of attributes are biased towards (assimilation) or away from (contrast) a comparison standard depending on whether the difference between the compared attribute values exceeds or falls below the comparison-suggested difference. The second study showed that the comparison language selected by participants (through the difference suggested by the language) mediated the effect of standard similarity on attribute estimates following a social comparison. The third study demonstrated concurrent assimilation and contrast effects in recall of attribute values due to the size of the observed difference between the self and the standard for the attribute. Unlike in previous research on social comparison, assimilation and contrast patterns in these studies can be explained through a single process.
Traditional theories of how children learn the positive integers start from infants' abilities in detecting the quantity of physical objects. Our target article examined this view and found no plausible accounts of such development. Most of our commentators appear to agree that no adequate developmental theory is presently available, but they attempt to hold onto a role for early enumeration. Although some defend the traditional theories, others introduce new basic quantitative abilities, new methods of transformation, or new types of end states. A survey of these proposals, however, shows that they do not succeed in bridging the gap to knowledge of the integers. We suggest that a better theory depends on starting with primitives that are inherently structural and mathematical.
Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe that these abilities set the stage for later mathematics: natural numbers and arithmetic. However, the connection between these early and later skills is far from obvious. We evaluate two possible routes to mathematics and argue that neither is sufficient: (1) We first sketch what we think is the most likely model for infant abilities in this domain, and we examine proposals for extrapolating the natural number concept from these beginnings. Proposals for arriving at natural number by (empirical) induction presuppose the mathematical concepts they seek to explain. Moreover, standard experimental tests for children's understanding of number terms do not necessarily tap these concepts. (2) True concepts of number do appear, however, when children are able to understand generalizations over all numbers; for example, the principle of additive commutativity (a+b=b+a). Theories of how children learn such principles usually rely on a process of mapping from physical object groupings. But both experimental results and theoretical considerations imply that direct mapping is insufficient for acquiring these principles. We suggest instead that children may arrive at natural numbers and arithmetic in a more top-down way, by constructing mathematical schemas.
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