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• W2767510062 abstract "Number Representations and their Development: A Connectionist Model of Number Comparison Mark Rose Lewis (lewis505@umn.edu) Department of Educational Psychology, 56 East River Road Minneapolis, MN 55455 Sashank Varma (sashank@umn.edu) Department of Educational Psychology, 56 East River Road Minneapolis, MN 55455 Abstract Building on prior work, the current study evaluated whether connectionist models can account for the distance and size effects in adults and the development of the distance effect in children. A family of models was constructed by orthogonally varying training environment (naturalistic versus non-naturalistic) and number representation (one-to-one versus magnitude). The ability of the models to account for the adult distance and size effects depended critically on a naturalistic training environment but was relatively independent of number representation. With respect to the developmental data, the naturalistic/one-to-one model provided a good account of response times and errors. The relation between the current models and prior models and avenues for future exploration are discussed. Keywords: number comparison; distance effect; size effect; connectionism; models; development Introduction The nature of number representations is an enduring question in cognitive science. One clue to this representation is the distance effect: the time it takes to judge the greater (or lesser) of two numbers decreases with the distance between the numbers (Moyer & Landauer, 1967). For example, 1 vs. 9 is judged faster than 1 vs. 3. Another clue is the size effect: the time to judge the greater (or lesser) of two numbers that are a fixed distance apart increases with the absolute magnitude of the numbers (Parkman, 1971). For example, 7 vs. 9 is judged more slowly than 1 vs. 3. The distance and size effects conform to psychophysical laws (i.e., ) and are therefore commonly interpreted as evidence that numbers are represented as analog representations, perhaps localized to the intra-parietal sulcus (Dehaene, Piazza, Pinel, & Cohen, 2003). Researchers have proposed various implementations of these analog representations. The classic ones are as points on a compressed mental number line (Dehaene & Mehler, 1992; Rule, 1969) and as points on a linear mental number line associated with increasing variability (e.g., Gallistel & Gelman, 2000). More recently, two connectionist models of number representation have appeared. Zorzi and Butterworth (1999) assumed magnitude representations whereby numbers are represented by banks of overlapping units. This model was able to account for the adult distance effect. By contrast, Verguts, Fias, and Steven (2005) assumed a coarse-coded representation, with each number corresponding primarily to one unit, but with graded activation of adjacent units. This model was able to account for the adult distance and size effects. The purpose of the current study was to evaluate the ability of connectionist models to (1) account for the adult distance and size effects as a function of training environment and number representation and to (2) account for the development of the distance effect. In these regards, the reported simulations are the first of their kind. With respect to training environment, some connectionist models (Zorzi & Butterworth, 1999) have employed a non- naturalistic training environment (i.e., every one-digit number appears with equal likelihood). However, corpus studies indicate that the frequency of a number falls off as a power function of its magnitude (Dehaene & Mehler, 1992), implying that one-digit numbers are non-uniformly distributed in a naturalistic environment. Some connectionist models have employed a naturalistic training environment (Verguts et al., 2005). We sampled comparisons (i.e., pairs of one-digit numbers) from these contrasting training environments to evaluate whether the distance and size effects were contingent upon naturalistic input. With respect to number representation, we considered the magnitude representation implemented by the Zorzi and Butterworth (1999) model and a one-to-one variant of the coarse-coded representation implemented by the Verguts et al. (2005) model 1 . Finally, in the first study to model the development of the distance effect, we evaluated whether improvements in model performance throughout training parallel improvements in children’s response times and error rates throughout development. Method We developed four connectionist models by orthogonally varying training environment (naturalistic versus non- naturalistic) and number representation (magnitude versus one-to-one). The models were implemented within a common connectionist architecture patterned after Verguts et al. (2005). Both of these codings represent exact numbers. We use the label “magnitude” to reflect the fact that the number of representation nodes activated in this coding corresponds to the number being compared." @default.
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• W2767510062 title "Number Representations and their Development: A Connectionist Model of Number Comparison" @default.
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