Dynamics Versus Development in Numerosity Estimation: A Computational Model Accurately Predicts a Developmental Reversal

Perceptual judgments result from a dynamic process, but little is known about the dynamics of number-line estimation. A recent study proposed a computational model that combined a model of trial-to-trial changes with a model for the internal scaling of discrete numbers. Here, we tested a surprising prediction of the model—a situation in which children's estimates of numerosity would be better than those of adults. Consistent with the model simulations, task contexts led to a clear developmental reversal: children made more adult-like, linear estimates when to-be-estimated numbers were descending over trials (i.e., backward condition), whereas adults became more like children with logarithmic estimates when numbers were ascending (i.e., forward condition). In addition, adults’ estimates were subject to inter-trial differences regardless of stimulus order. In contrast, children were not able to use the trial-to-trial dynamics unless stimuli varied systematically, indicating the limited cognitive capacity for dynamic updates. Together, the model adequately predicts both developmental and trial-to-trial changes in number-line tasks.     

非符号数量表征和符号分数表征的关系

个体学习符号分数的一个关键是能对其数值形成准确表征。现有研究假设符号分数表征的认知基础是人类自婴幼儿期就具有的非符号数量表征(如表征两个集合各自的数量, 或两个数量的比例)。其证据包括表征非符号数量(尤其是非符号数量比例关系)和表征符号分数在行为和大脑神经活动层面上都表现出相关性。然而要说明非符号数量表征是符号分数表征的认知基础, 还需更多研究表明两者在数量概念上的独特相关和因果联系, 并阐明符号分数表征形成的认知机制。     

Preference structures and threshold models

Preferences are often represented in terms of a function, in the deterministic case as well as in the probabilistic case. In the present paper we develop a new numerical representation of preference structures for which the strict preference relation (P) is without circuit but not necessarily transitive. Moreover, we investigate the consequences of the representation for the usual preference structures. In particular, we propose new formulations for the numerical representation of the interval order structure.     

The effects of reinforcement magnitude on functional analysis outcomes

The duration or magnitude of reinforcement has varied and often appears to have been selected arbitrarily in functional analysis research. Few studies have evaluated the effects of reinforcement magnitude on problem behavior, even though basic findings indicate that this parameter may affect response rates during functional analyses. In the current study, 6 children with autism or developmental disabilities who engaged in severe problem behavior were exposed to three separate functional analyses, each of which varied in reinforcement magnitude. Results of these functional analyses were compared to determine if a particular reinforcement magnitude was associated with the most conclusive outcomes. In most cases, the same conclusion about the functions of problem behavior was drawn regardless of the reinforcement magnitude.     

On the effects of signaling reinforcer probability and magnitude in delayed matching to sample

Two experiments examined whether postsample signals of reinforcer probability or magnitude affected the accuracy of delayed matching to sample in pigeons. On each trial, red or green choice responses that matched red or green stimuli seen shortly before a variable retention interval were reinforced with wheat access. In Experiment 1, the reinforcer probability was either 0.2 or 1.0 for both red and green responses. Reinforcer probability was signaled by line or cross symbols that appeared after the sample had been presented. In Experiment 2, all correct responses were reinforced, and the signaled reinforcer durations were 1.0 s and 4.5 s. Matching was more accurate when larger or more probable reinforcers were signaled, independently of retention interval duration. Because signals were presented postsample, the effects were not the result of differential attention to the sample.     

Further analysis of variables that affect self-control with aversive events

The purpose of this study was to examine variables that affect self-control in the context of academic task completion by elementary school children with autism. In the baseline assessment of Study 1, mathematics problem completion was shown to be an aversive event, and sensitivity to task magnitude, task difficulty, and delay to task completion were measured. The effects of manipulating values of those parameters on self-control then were assessed. For all participants, self-control increased as a function of one or more changes in task parameter values. In Study 2, the effects of a commitment response on self-control was assessed. Results indicated that for all participants, levels of self-control were higher when the opportunity to commit to the immediate aversive event was available.     

Children's representation of symbolic and nonsymbolic magnitude examined with the priming paradigm

How people process and represent magnitude has often been studied using number comparison tasks. From the results of these tasks, a comparison distance effect (CDE) is generated, showing that it is easier to discriminate two numbers that are numerically further apart (e.g., 2 and 8) compared with numerically closer numbers (e.g., 6 and 8). However, it has been suggested that the CDE reflects decisional processes rather than magnitude representation. In this study, therefore, we investigated the development of symbolic and nonsymbolic number processes in kindergartners and first, second, and sixth graders using the priming paradigm. This task has been shown to measure magnitude and not decisional processes. Our findings revealed that a priming distance effect (PDE) is already present in kindergartners and that it remains stable across development. This suggests that formal schooling does not affect magnitude representation. No differences were found between the symbolic and nonsymbolic PDE, indicating that both notations are processed with comparable precision. Finally, a poorer performance on a standardized mathematics test seemed to be associated with a smaller PDE for both notations, possibly suggesting that children with lower mathematics scores have a less precise coding of magnitude. This supports the defective number module hypothesis, which assumes an impairment of number sense.     

Transitional and steady-state choice behavior under an adjusting-delay schedule

Twelve rats made repeated choices on an adjusting-delay schedule between a smaller reinforcer (A) that was delivered immediately after a response and a larger reinforcer (B) that was delivered after a delay which increased or decreased by 20% depending on the subject's choices in successive blocks of trials. In two phases of the experiment (100 sessions and 40 sessions), reinforcer sizes were selected which enabled theoretical parameters expressing the rate of delay discounting and sensitivity to reinforcer size to be estimated from the ratio of the indifference delays obtained in the two phases. Indifference delays, calculated from adjusting delays in the last 10 sessions of each phase, were shorter when the sizes of A and B were 14 and 25 μl of a 0.6 M sucrose solution than when they were 25 and 100 μl of the same solution. The ratio of the indifference delays was significantly smaller than that predicted on the basis of an assumed linear relation between reinforcer size and instantaneous reinforcer value, consistent with a previous proposal that this relation may be hyperbolic in form. Estimates of the rate of delay discounting based on the ratio of the two indifference delays (mean, 0.08 s(-1)) were similar to values obtained previously using different intertemporal choice protocols. Estimates of the size-sensitivity parameter (mean 113 μl) were similar to estimates recently derived from performance on progressive-ratio schedules. In both phases of the experiment, adjusting delays in successive blocks of trials were analyzed using the Fourier transform. The power spectrum obtained from individual rats had a dominant frequency that corresponded to a period of oscillation of the adjusting delay between 30 and 100 trial blocks (mean, 78). Power in the dominant frequency band was highest in the early sessions of the first phase and declined with extended training. It is suggested that this experimental protocol may have utility in neurobehavioral studies of intertemporal choice.     

Are historic years understood as numbers or events? An fMRI study of numbers with semantic associations

While numbers generally cue processing of quantity or order, they can also contain semantic information, as in the case of historic years (e.g., “1492” calls forth associations of Columbus sailing the ocean blue). Whether these dates are processed as quantities or events may depend on the context in which they occur. We examined such “ambiguous number” processing in two different contexts using a paired-comparison task, recording both behavioral responses and brain activity. Participants were either asked to think of all items as numbers and to choose the larger number, or were told to treat the comparators as events and to choose the later event. Behaviorally, all events showed a normal distance effect, establishing that they may be understood and compared in an ordinal sequence. Functional magnetic resonance imaging (fMRI) demonstrated significant differences between years when treated as numbers versus as events. Dates in both contexts shared activity in parietal lobe regions previously implicated in number processing. Dates as numbers showed no extra-numeric activity, while dates thought of as events evoked activity in temporal semantic processing and frontal semantic retrieval areas. These differences suggest that extra-numeric information may be easily accessed and incorporated during processing when supported by even a weak context. This work supports previous studies showing a dissociation between quantity and meaning, and illustrates the brain areas involved in these different aspects.