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This paper examines Piantadosi, Tenenbaum, and Goodman’s (2012) model for how children learn the relation between number words (“one” through “ten”) and cardinalities (sizes of sets with one through ten elements). This model shows how statistical learning can induce this relation, reorganizing its procedures as it does so in roughly the way children do. We question, however, Piantadosi et al.’s claim that the model performs “Quinian bootstrapping,” in the sense of Carey (2009) . Unlike bootstrapping, the concept it learns is not discontinuous with the concepts it starts with. Instead, the model learns by recombining its primitives into hypotheses and confirming them statistically. As such, it accords better with earlier claims ( Fodor, 1975 , Fodor, 1981 ) that learning does not increase expressive power. We also question the relevance of the simulation for children’s learning. The model starts with a preselected set of 15 primitives, and the procedure it learns differs from children’s method. Finally, the partial knowledge of the positive integers that the model attains is consistent with an infinite number of nonstandard meanings—for example, that the integers stop after ten or loop from ten back to one.

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.

Research on framing effects has revealed cases where the type of outcome at risk (e.g., human lives vs. animal lives) affects the magnitude of the framing effect. Some authors have appealed to the shape of the value function as predicting when framing effects will occur: The more valuable the outcome type, the more nonlinear its value function, and the larger the resulting framing effect (Levin & Chapman, 1990). However, having a more or less nonlinear value function cannot explain situations in which participants strongly prefer the same option in both frames. Another factor that may be at work in these types of outcome effects is an aspiration level (AL; Lopes, 1987; Schneider, 1992), which determines how acceptable the options are and combines (or competes) with the risk attitude encouraged by frame. The results described here indicate that differences in the shape of the value function between outcome types are evident but are inconsistent between framed losses and gains, though nonlinearity in the value function can be increased with a manipulation that also encourages framing effects. The results also demonstrate that an AL can lead to the same predominant risk preference in the positive and negative frame. These findings indicate that the shape of the value function and the AL each play a role in outcome type interactions with frame, and in some cases, a combination of the two factors may be at work.

According to one theory about how children learn the meaning of the words for the positive integers, they first learn that "one," "two," and "three" stand for appropriately sized sets. They then conclude by inductive inference that the next numeral in the count sequence denotes the size of sets containing one more object than the size denoted by the preceding numeral. We have previously argued, however, that the conclusion of this Induction does not distinguish the standard meaning of the integers from nonstandard meanings in which, for example, "ten" could mean set sizes of 10, 20, 30,... elements. Margolis and Laurence [Margolis, E., & Laurence, S. (2008). How to learn the natural numbers: Inductive inference and the acquisition of number concepts. Cognition, 106, 924-939] believe that our argument depends on attributing to children "radically indeterminate" concepts. We show, first, that our conclusion is compatible with perfectly determinate meanings for "one" through "three." Second, although the inductive inference is indeed indeterminate - which is why it is consistent with nonstandard meanings - making it determinate presupposes the constraints that the inference is supposed to produce.

According to one theory about how children learn the concept of natural numbers, they first determine that "one", "two", and "three" denote the size of sets containing the relevant number of items. They then make the following inductive inference (the Bootstrap): The next number word in the counting series denotes the size of the sets you get by adding one more object to the sets denoted by the previous number word. For example, if "three" refers to the size of sets containing three items, then "four" (the next word after "three") must refer to the size of sets containing three plus one items. We argue, however, that the Bootstrap cannot pick out the natural number sequence from other nonequivalent sequences and thus cannot convey to children the concept of the natural numbers. This is not just a result of the usual difficulties with induction but is specific to the Bootstrap. In order to work properly, the Bootstrap must somehow restrict the concept of "next number" in a way that conforms to the structure of the natural numbers. But with these restrictions, the Bootstrap is unnecessary.

We explored the relationship between qualities of victims in hypo- thetical scenarios and the appearance of framing effects. In past studies, participants' feelings about the victims have been demonstrated to affect whe- ther 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.

Past research provides conflicting evidence for the role of value in the appearance of framing effects. In this study, the effects of frame and group size were examined using scenarios about less valuable and more valuable groups (animal vs. human). In addition, two picture manipulations, intended to increase the value of the group, were presented. Choice patterns differed for the human and animal groups, with participants exhibiting greater risk seeking overall for the human scenario and showing a framing effect for humans but not animals when no pictures were presented. A small group size increased the proportion of risky choices for both the animal and human scenarios. Presenting pictures with names did lead to framing effects for animals, but providing pictures or pictures and names eliminated framing effects for the human scenario. These findings suggest that the relationship between value and framing effects is a matter of degree.

To establish what is currently known about factors that affect foreign language listening comprehension, with a focus on characteristics of the listener, passage, and testing conditions. Research on second language (L2) listening comprehension strongly supports the importance of a number of factors, for example, a listener s working memory capacity and the number of ideas in a passage. Much of the research, however, reports weak or inconclusive results, leaving many factors and complex interactions among factors unresolved and in need of further investigation. Identifying the factors that affect L2 listening comprehension will help Defense Language Institute Proficiency Test (DLPT) designers anticipate how qualities of selected authentic materials will impact listening comprehension.

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.