The understanding of die as an object that has numerical functions. A longitudinal study using two children from the ages of 24 to 36 months interacting with an adult / La comprensión del dado como objeto con funciones numéricas. Un estudio longitudinal con dos niños desde los 24 a los 36 meses en interacción con un adulto

Abstract

We investigated the process through which children understand the conventional use of numbers in interaction with an adult. We analysed previous semiotic systems that support, transform and adjust to this new semiotic system — the numerical — and we investigated the communicative-educative basis that makes it possible. We conducted an exploratory and longitudinal study with one boy and one girl at 24, 27, 30, 33 and 36 months of age in a triadic interaction (adult-child-object) with their mothers at home. We adopt the pragmatic-semiotic perspective of the object, which considers that triadic interactions are the unit of analysis of early development. Our results show a great variety of semiotic mediators used by children and adults and both children’s progressive comprehension of the meaning and use of the die as a numerical object.     

Both non-symbolic and symbolic quantity processing are important for arithmetical computation but not for mathematical reasoning

This study investigated whether numerical processing was important for two types of mathematical competence: arithmetical computation and mathematical reasoning. Thousand eight hundred and fifty-seven Chinese primary school children in third through sixth grades took eight computerised tasks: numerical processing (numerosity comparison, digit comparison), arithmetical computation, number series completion, non-verbal matrix reasoning, mental rotation, choice reaction time, and word rhyming. Hierarchical regressions showed that both non-symbolic numerical processing (numerosity comparison) and symbolic numerical processing (digit comparison) were independent predictors of arithmetical computation but neither was a predictor of mathematical reasoning (assessed by number series completion). These findings suggest that the cognitive basis of mathematical performance varies depending on the type of mathematical competence measured.     

Symbolic representation of the number three: a study with three-year-old children from contrasting socioeconomic environments

The goal of the present study was to compare a range of aspects in children’s symbolic knowledge about the number three among two groups of three-year-olds from contrasting socioeconomic backgrounds. Every child was presented with five tasks that focused on the number three and that had cognitive demands of different complexity: expressing their age, reciting the conventional number series up to three, quantifying a collection of three, and two tasks requiring the use of visually presented quantitative information.The results showed the same order of difficulty of the tasks in both socioeconomic groups and a clear performance difference depending on socioeconomic background. These findings show that symbolic knowledge about the number three does not come in an all or none fashion. Rather, different aspects of this symbolic competence become apparent in response to different tasks, and seem to depend largely on the socioeconomic environment in which children develop.     

How the Abstract Becomes Concrete: Irrational Numbers Are Understood Relative to Natural Numbers and Perfect Squares

Mathematical cognition research has largely emphasized concepts that can be directly perceived or grounded in visuospatial referents. These include concrete number systems like natural numbers, integers, and rational numbers. Here, we investigate how a more abstract number system, the irrationals denoted by radical expressions like , is understood across three tasks. Performance on a magnitude comparison task suggests that people interpret irrational numbers (specifically, the radicands of radical expressions) as natural numbers. Strategy self‐reports during a number line estimation task reveal that the spatial locations of irrationals are determined by referencing neighboring perfect squares. Finally, perfect squares facilitate the evaluation of arithmetic expressions. These converging results align with a constellation of related phenomena spanning tasks and number systems of varying complexity. Accordingly, we propose that the task‐specific recruitment of more concrete representations to make sense of more abstract concepts (referential processing) is an important mechanism for teaching and learning mathematics.     

数字空间表征的在线建构：来自干扰情境中数字SNARC效应的证据

尽管已有研究发现数字以空间方式表征在人类记忆系统, 但是人脑如何完成数字的空间表征尚存争议。本研究两个实验在不同比例的数字字母(实验1)和不同比例的数字汉字(实验2)混合情境中考察了数字空间表征特点及其机制, 对上述争议进行了深入研究。结果发现：(1)当数字字母比例为“1 : 1”时, 数字加工中不出现SNARC效应。当数字字母比例为“1 : 6”和“6 : 1”时, 数字加工中均出现SNARC效应。即数字字母比例与数字SNARC效应之间呈倒“U”型关系。(2)数字汉字混合情境中数字汉字比例与数字SNARC效应之间同样呈倒“U”型关系。结果说明：(1)干扰刺激与数字混合呈现会影响数字SNARC效应。(2)干扰刺激加工对数字SNARC效应的影响受到数字与干扰刺激比例的调节, 且具有跨干扰材料的稳定性。研究结果意味着数字的空间表征是人类通过统计学习在线建构的, 支持了工作记忆理论。     

The left digit effect in a complex judgment task: Evaluating hypothetical college applicants

A left digit effect has been broadly observed across judgment and decision-making contexts ranging from product evaluation to medical treatment decisions to number line estimation. For example, $3.00 is judged to be a much greater cost than $2.99, and “801” is estimated strikingly too far to the right of “798” on a number line. Although the consequences of the effects for judgment and decision behavior have been documented, the sources of the effects are not well established. The goal of the current work is to extend investigations of the left digit effect to a new complex judgment activity and to assess whether the magnitude of the effect at the individual level can be predicted from performance on a simpler number skills task on which the left digit effect has also recently been observed. In three experiments (N = 434), adults completed a judgment task in which they rated the strength of hypothetical applicants for college admission and a self-paced number line estimation task. In all experiments, a small or medium left digit effect was found in the college admissions task, and a large effect was found in number line estimation. Individual-level variation was observed, but there was no relationship between the magnitudes of the effects in the two tasks. These findings provide evidence of a left digit effect in a novel multiattribute judgment task but offer no evidence that such performance can be predicted from a simple number skills task such as number line estimation.     

听觉通道下负数的低水平加工引起注意的SNARC效应

采用刺激探测任务,探讨了听觉通道下负数的低水平加工能否引起注意的SNARC效应。结果表明:(1)负数的低水平加工能够产生注意的SNARC效应,即线索为绝对值小的负数时,被试对左耳目标反应更快;线索为绝对值大的负数时,被试对右耳目标反应更快。(2)负数的加工所引起听觉空间注意的转移并不依赖于负数所在的情境,而仅通过其绝对值大小与空间表征发生关联。这一结论与视觉条件下所得到的结论不一致,并支持了系统发生论假说。     

Mirror neurons and their function in cognitively understood empathy

The current renewal of interest in empathy is closely connected to the recent neurobiological discovery of mirror neurons. Although the concept of empathy has been widely deployed, we shall focus upon one main psychological function it serves: enabling us to understand other peoples’ intentions. In this essay we will draw on neuroscientific, psychological, and philosophical literature in order to investigate the relationships between mirror neurons and empathy as to intention understanding. Firstly, it will be explored whether mirror neurons are the neural basis of our empathic capacities: a vast array of empirical results appears to confirm this hypothesis. Secondly, the higher level capacity of reenactive empathy will be examined and the question will be addressed whether philosophical analysis alone is able to provide a foundation for this more abstract level of empathy. The conclusion will be drawn that both empirical evidence and philosophical analysis can jointly contribute to the clarification of the concept of empathy.     

Evaluating the Sampling Performance of Exploratory and Cross-Validated DETECT Procedure with Imperfect Models

Among the methods proposed for identifying the number of latent traits in multidimensional IRT models, DETECT has attracted the attention of both methodologists and applied researchers as a nonparametric counterpart to other procedures. The current study investigated the overall performance of the DETECT procedure and its outcomes using a real-data sampling design recommended by MacCallum (2003 MacCallum, R.C. (2003). Working with imperfect models. Multivariate Behavioral Research, 38 (1), 113–139. http://dx.doi.org/10.1207/S15327906MBR3801_5[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and compared the results from a purely simulated data set that was generated with a well-specified “perfect” model. The comparison revealed that the sampling behavior of the maximized DETECT value and R-ratio statistics was quite robust to minor factors and other model misspecifications that potentially exist in the real data set, as there were negligible differences between the results of the real and simulated data sets. Item classification accuracy was also nearly identical for the real and simulated data sets. The accuracy of the identified number of dimensions reported by DETECT was the only outcome with an obvious difference between the purely simulated data set and the real data set. While the difference was small for smaller sample sizes, the identified number of dimensions was more accurate for larger sample sizes when the population data set was purely simulated. In many instances, exploratory DETECT analysis outperformed the cross-validated DETECT analysis in terms of overall accuracy.     

Ultralarge lotteries: Analyzing the Lottery Paradox using non-standard analysis

A popular way to relate probabilistic information to binary rational beliefs is the Lockean Thesis, which is usually formalized in terms of thresholds. This approach seems far from satisfactory: the value of the thresholds is not well-specified and the Lottery Paradox shows that the model violates the Conjunction Principle. We argue that the Lottery Paradox is a symptom of a more fundamental and general problem, shared by all threshold-models that attempt to put an exact border on something that is intrinsically vague. We propose application of the language of relative analysis—a type of non-standard analysis—to formulate a new model for rational belief, called Stratified Belief. This contextualist model seems well-suited to deal with a concept of beliefs based on probabilities ‘sufficiently close to unity’ and satisfies a moderately weakened form of the Conjunction Principle. We also propose an adaptation of the model that is able to deal with beliefs that are less firm than ‘almost certainty’. The adapted version is also of interest for the epistemicist account of vagueness.