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Previous studies have found that children have difficulty solving proportional reasoning problems involving discrete units until 10-to 12-years of age, but can solve parallel problems involving continuous quantities by 6-years of age. The present studies examine where children go wrong in processing proportions that involve discrete quantities. A computerized proportional equivalence choice task was administered to kindergartners through fourth-graders in Study 1, and to first-and third-graders in Study 2. Both studies involved four between-subjects conditions that were formed by pairing continuous and discrete target proportions with continuous and discrete choice alternatives. In Study 1, target and choice alternatives were presented simultaneously and in Study 2 target and choice alternatives were presented sequentially. In both studies, children performed significantly worse when both the target and choice alternatives were represented with discrete quantities than when either or both of the proportions involved continuous quantities. Taken together, these findings indicate that children go astray on proportional reasoning problems involving discrete units only when a numerical match is possible, suggesting that their difficulty is due to an overextension of numerical equivalence concepts to proportional equivalence problems. KeywordsProportional Reasoning; Continuous and Discrete Quantities; Mathematical Development; Intuitive and Explicit Processing Proportional reasoning involves understanding the multiplicative relationships between rational quantities (a/b = c/d), and is a form of reasoning that characterizes important structural relationships in mathematics and science, as well as in every day life (Cramer & Post, 1993;Lesh, Post, & Behr, 1988). As Ahl, Moore, and Dixon (1992) emphasized, "Proportional reasoning is a pervasive activity that transcends topical barriers in adult life." Proportional information is crucial in dealing with such diverse topics as economic values, relational spatial contrasts, temperatures, densities, concentrations, velocities, chemical compositions, demographic information, and recipe formulation (Karplus, Pulos, & Stage, 1983;Moore, Dixon, & Haines, 1991;Siegler & Vago, 1978;Sophian & Wood, 1997;Spinillo & Bryant, 1999). For example, when baking, one needs to think proportionally about the relative measures of each ingredient (e.g., 2-and-½ cups flour, 1/3 cup sugar, and ¼ cup butter), and must maintain these proportions whenever deviating from the recipe (e.g., whenever doubling or halving the intended amount). In chemistry, proportionality is central to balancing chemical equations. NIH-PA Author ManuscriptNIH-PA Author Manuscript NIH-PA Author Manuscript geographical locations based on the proportion of the population represented by specific demographic target groups. Understanding of proportionality is also central to mathematics; it is the basis of rational number operations, unit partitioning, and basic algebra and geometry problem solving (Empson, 1999;Fuso...

We present a consensus-based checklist to improve and document the transparency of research reports in social and behavioural research. An accompanying online application allows users to complete the form and generate a report that they can submit with their manuscript or post to a public repository.

In 3 studies (N = 188) we tested the hypothesis that children use a perceptual access approach to reason about mental states before they understand beliefs. The perceptual access hypothesis predicts a U-shaped developmental pattern of performance in true belief tasks, in which 3-year-olds who reason about reality should succeed, 4- to 5-year-olds who use perceptual access reasoning should fail, and older children who use belief reasoning should succeed. The results of Study 1 revealed the predicted pattern in 2 different true belief tasks. The results of Study 2 disconfirmed several alternate explanations based on possible pragmatic and inhibitory demands of the true belief tasks. In Study 3, we compared 2 methods of classifying individuals according to which 1 of the 3 reasoning strategies (reality reasoning, perceptual access reasoning, belief reasoning) they used. The 2 methods gave converging results. Both methods indicated that the majority of children used the same approach across tasks and that it was not until after 6 years of age that most children reasoned about beliefs. We conclude that because most prior studies have failed to detect young children's use of perceptual access reasoning, they have overestimated their understanding of false beliefs. We outline several theoretical implications that follow from the perceptual access hypothesis.

In recent years research on automatic imitation has received considerable attention because it represents an experimental platform for investigating a number of inter-related theories suggesting that the perception of action automatically activates corresponding motor programs.A key debate within this research centers on whether automatic imitation is any different than other long-term S-R associations, such as spatial stimulus-response compatibility. One approach to resolving this issue is to examine whether automatic imitation shows similar response characteristics as other classes of stimulus-response compatibility. This hypothesis was tested by comparing imitative and spatial compatibility effects with a two alternative forced-choice stimulus-response compatibility paradigm and two tasks: one that involved selecting a response to the stimulus (S-R) and one that involved selecting a response to the opposite stimulus (OS-R), i.e., the one not presented. The stimulus for both tasks was a left or right hand with either the index or middle finger tapping down. Speeded responses were performed with the index or middle finger of the right hand in response to the finger identity or the left-right spatial position of the fingers. Based on previous research and a connectionist model, we predicted standard compatibility effects for both spatial and imitative compatibility in the S-R task, and a reverse compatibility effect for spatial compatibility but not for imitative compatibility in the OS-R task.The results from the mean response times, mean percentage of errors, and response time distributions all converged to support these predictions. A second noteworthy result was that the recoding of the finger identity in the OS-R task required significantly more time than the recoding of the left-right spatial position, but the encoding time for the two stimuli in the S-R task was equivalent. In sum, this evidence suggests that the processing of spatial and imitative Automatic Imitation & Spatial Compatibility 3 compatibility is dissociable with regard to two different processes in dual processing models of stimulus-response compatibility.