When Is More Better?

Abstract— We examine three determinants of the relationship between the magnitude of a stimulus and a person's subjective "value" of that stimulus: the process by which value is assessed (either by feeling or by calculation), the evaluability of the relevant magnitude variable (whether the desirability of a given level of that variable can be evaluated independently), and the mode of evaluation (whether stimuli are encountered and evaluated jointly or separately). Reliance on feeling, lack of evaluability, and separate evaluation lead to insensitivity to magnitude. An analysis invoking these factors provides a novel account for why people typically become less sensitive to changes in the magnitude of a stimulus as magnitude increases.     

Individual differences in children's mathematical competence are related to the intentional but not automatic processing of Arabic numerals

In recent years, there has been an increasing focus on the role played by basic numerical magnitude processing in the typical and atypical development of mathematical skills. In this context, tasks measuring both the intentional and automatic processing of numerical magnitude have been employed to characterize how children’s representation and processing of numerical magnitude changes over developmental time. To date, however, there has been little effort to differentiate between different measures of ‘number sense’. The aim of the present study was to examine the relationship between automatic and intentional measures of magnitude processing as well as their relationships to individual differences in children’s mathematical achievement. A group of 119 children in 1st and 2nd grade were tested on the physical size congruity paradigm (automatic processing) as well as the number comparison paradigm to measure the ratio effect (intentional processing). The results reveal that measures of intentional and automatic processing are uncorrelated with one another, suggesting that these tasks tap into different levels of numerical magnitude processing in children. Furthermore, while children’s performance on the number comparison paradigm was found to correlate with their mathematical achievement scores, no such correlations could be obtained for any of the measures typically derived from the physical size congruity task. These findings therefore suggest that different tasks measuring ‘number sense’ tap into different levels of numerical magnitude representation that may be unrelated to one another and have differential predictive power for individual differences in mathematical achievement.     

空间量化的心理表征

空间量化(spacial quantification)是空间知觉的基础, 是对特定空间性质的表达。离散量(discrete magnitude)与连续量(continuous quantity)分别反映了空间分立和连续的性质, 二者有着相似的行为效应, 在神经表达上也有部分重叠, 这些证据暗示了二者可能有共同的表征机制——模拟表征(analog magnitude representation)。数量空间映射(number–space mappings)提供了数量与空间关系的直接证据。但空间量化的研究中还有许多未解之谜, 如：空间量化的动态表征、量化机制的普遍性、参照点问题、复杂和多维空间的量化等。在具身认知(embodied cognition)的框架下, 空间量化的心理表征研究将对空间的性质做出更深刻的回答。     

A colorful walk, but is it on the mental number line? Reply to Cohen Kadosh, Tzelgov, and Henik

Cohen Kadosh, Tzelgov, and Henik [Cohen Kadosh, R., Tzelgov, J., and Henik, A. (2008). A synesthetic walk on the number line: The size effect. Cognition, 106, 548-557] present a new paradigm to probe properties of the mental number line. They describe two experiments which they argue to be inconsistent with the exact small number model proposed by Verguts, Fias, and Stevens [Verguts, T., Fias, W., Stevens, M. (2005). A model of exact small-number representation. Psychonomic Bulletin and Review, 12, 66-80]. We discuss the data, assumptions, and conclusions of Cohen Kadosh et al.'s paper in relation to existing models of numerical cognition.     

Judgments of discrete and continuous quantity: an illusory Stroop effect

Evidence from human cognitive neuroscience, animal neurophysiology, and behavioral research demonstrates that human adults, infants, and children share a common nonverbal quantity processing system with nonhuman animals. This system appears to represent both discrete and continuous quantity, but the proper characterization of the relationship between judgments of discrete and continuous quantity remains controversial. Some researchers have suggested that both continuous and discrete quantity may be automatically extracted from a scene and represented internally, and that competition between these representations leads to Stroop interference. Here, four experiments provide evidence for a different explanation of adults’ performance on the types of tasks that have been said to demonstrate Stroop interference between representations of discrete and continuous quantity. Our well-established tendency to underestimate individual two-dimensional areas can provide an alternative explanation (introduced here as the “illusory-Stroop” hypothesis). Though these experiments were constructed like Stroop tasks, and they produce patterns of performance that initially appear consistent with Stroop interference, Stroop interference effects are not involved. Implications for models of the construction of cumulative area representations and for theories of discrete and continuous quantity processing in large sets are discussed.     

非语言情境中时间加工与空间距离加工的关系

时间加工与空间距离加工的关系主要有两种, 一种是在空距离–空时距以及动物的研究中, 时间加工影响空间距离加工, 空间距离加工影响时间加工, 两者相互影响, 存在对称的干扰, 这种关系与注意和记忆有关。另一种是在实距离–实时距中, 空间距离加工影响时间加工, 而时间加工不受空间距离加工的影响, 时间加工和空间距离加工之间具有不对称的干扰。时空关系的不对称与隐喻理论、人们的感知运动经验和时空信息的显著性有关。神经机制上, 右顶叶皮质、侧顶内区、左顶叶皮质、小脑、前额等参与时间加工和空间距离加工, 但两种加工涉及到的神经机制也有部分差异。未来可以从时间加工的分段性、不同条件下时间加工和空间距离加工的心理机制以及脑机制进行研究。     

The effects of reinforcer magnitude on timing in rats

The relation between reinforcer magnitude and timing behavior was studied using a peak procedure. Four rats received multiple consecutive sessions with both low and high levels of brain stimulation reward (BSR). Rats paused longer and had later start times during sessions when their responses were reinforced with low-magnitude BSR. When estimated by a symmetric Gaussian function, peak times also were earlier; when estimated by a better-fitting asymmetric Gaussian function or by analyzing individual trials, however, these peak-time changes were determined to reflect a mixture of large effects of BSR on start times and no effect on stop times. These results pose a significant dilemma for three major theories of timing (SET, MTS, and BeT), which all predict no effects for chronic manipulations of reinforcer magnitude. We conclude that increased reinforcer magnitude influences timing in two ways: through larger immediate after-effects that delay responding and through anticipatory effects that elicit earlier responding.     

4～5岁儿童一一对应任务中的比例效应

一一对应和数量比较是幼儿数概念发展的两个重要方面。在本研究中．40名4岁和39名5岁的儿童分别完成了不同比例数量的一一对应任务和数量比较任务。结果表明：（1）两种实验任务下均出现比例效应．说明儿童在解决一一对应问题时,仍然用数量比较的方法来进行判断;（2）任务类型的主效应显著,一一对应任务下的正确率要显著低于数量比较任务下的正确率;（3）知觉线索更多地影响数量比较任务。     

An electro-physiological temporal principal component analysis of processing stages of number comparison in developmental dyscalculia

Developmental dyscalculia (DD) still lacks a generally accepted definition. A major problem is that the cognitive component processes contributing to arithmetic performance are still poorly defined. By a reanalysis of our previous event-related brain potential (ERP) data (Soltész et al., 2007) here our objective was to identify and compare cognitive processes in adolescents with DD and in matched control participants in one-digit number comparison. To this end we used temporal principal component analysis (PCA) on ERP data. First, PCA has identified four major components explaining the 85.8% of the variance in number comparison. Second, the ERP correlate of the most frequently used marker of the so-called magnitude representation, the numerical distance effect, was intact in DD during all processing stages identified by PCA. Third, hemispheric differences in the first temporal component and group differences in the second temporal component suggest executive control differences between DD and controls.     

Children's mappings of large number words to numerosities

Previous studies have suggested that children's learning of the relation between number words and approximate numerosities depends on their verbal counting ability, and that children exhibit no knowledge of mappings between number words and approximate numerical magnitudes for number words outside their productive verbal counting range. In the present study we used a numerical estimation task to explore children's knowledge of these mappings. We classified children as Level 1 counters (those unable to produce a verbal count list up to 35), Level 2 counters (those who were able to count to 35 but not 60) and Level 3 counters (those who counted to 60 or above) and asked children to estimate the number of items on a card. Although the accuracy of children's estimates depended on counting ability, children at all counting skill levels produced estimates that increased linearly in proportion to the target number, for numerosities both within and beyond their counting range. This result was obtained at the group level (Experiment 1) and at the level of individual children (Experiment 2). These findings provide evidence that even the least skilled counters do exhibit some knowledge of the form of the mapping between large number words and approximate numerosities.