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. (shrink)

We explored the relationship between qualities of victims in hypothetical scenarios and the appearance of framing effects. In past studies, participants’ feelings about the victims have been demonstrated to affect whether framing effects appear, but this relationship has not been directly examined. In the present study, we examined the relationship between caring about the people at risk, the perceived interdependence of the people at risk, and frame. Scenarios were presented that differed in the degree to which participants could be expected (...) to care about the group and the extent to which the group could be construed as interdependent. A framing effect was found only for the scenario describing the victims as the participants’ friends who did not know each other (high caring/low interdependence), and this went in the opposite direction from typical framing effects. Finally, perceived interdependence and caring affected choice both within and across scenarios, with more risky choices made by participants with high interdependence ratings and high caring ratings. (shrink)

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. (shrink)