4-5岁幼儿数数行为的规则性与策略化应用特点的研究

本研究的主要目的是考察4-5岁儿童数数行为的规则性和策略化应用特点。研究选取了104名幼儿园中班儿童(年龄范围4-5岁),男女各半,采用现场实验法进行了逐一测查,测查任务为标准计数任务和数字复制任务。结果发现:4-5岁儿童的数数行为已经遵循一些数数规则,如固定顺序原则、一一对应原则和基数原则;但该年龄段的儿童还不会自发地把数数作为问题解决的策略应用于比较两组物体数目的问题上。     

Nature and culture of finger counting: diversity and representational effects of an embodied cognitive tool

Studies like the one conducted by Domahs et al. (2010, in Cognition) corroborate that finger counting habits affect how numbers are processed, and legitimize the assumption that this effect is culturally modulated. The degree of cultural diversity in finger counting, however, has been grossly underestimated in the field at large, which, in turn, has restricted research questions and designs. In this paper, we demonstrate that fingers as a tool for counting are not only naturally available, but are also-and crucially so-culturally encoded. To substantiate this, we outline the variability in finger counting and illustrate each of its types with instances from the literature. We argue that the different types of finger counting all constitute distinct representational systems, and we use their properties-dimensionality, dimensional representation, base and sub-base values, extendibility and extent, sign count, and regularity-to devise a typology of such systems. This allows us to explore representational effects, that is, the cognitive implications these properties may have, for instance, for the efficiency of information encoding and representation, ease of learning and mastering the system, or memory retrieval and cognitive load. We then highlight the ambivalent consequences arising from structural inconsistencies between finger counting and other modes of number representation like verbal or notational systems, and we discuss how this informs questions on the evolution and development of counting systems. Based on these analyses, we suggest some directions for future research in the field of embodied cognition that would profit substantially from taking into account the cultural diversity in finger counting.     

Knowledge of counting principles: How relevant is order irrelevance?

Most children who are older than 6 years of age apply essential counting principles when they enumerate a set of objects. Essential principles include (a) one-to-one correspondence between items and count words, (b) stable order of the count words, and (c) cardinality—that the last number refers to numerosity. We found that the acquisition of a fourth principle, that the order in which items are counted is irrelevant, follows a different trajectory. The majority of 5- to 11-year-olds indicated that the order in which objects were counted was relevant, favoring a left-to-right, top-to-bottom order of counting. Only some 10- and 11-year-olds applied the principle of order irrelevance, and this knowledge was unrelated to their numeration skill. We conclude that the order irrelevance principle might not play an important role in the development of children’s conceptual knowledge of counting.     

Numerical competence in a chimpanzee (Pan troglodytes)

A chimpanzee (Pan troglodytes), trained to count foods and objects by using Arabic numbers, demonstrated the ability to sum arrays of 0-4 food items placed in 2 of 3 possible sites. To address representational use of numbers, we next baited sites with Arabic numbers as stimuli. In both cases performance was significantly above chance from the first sessions, which suggests that without explicit training in combining arrays, the animal was able to select the correct arithmetic sum for arrays of foods or Arabic numbers under novel test conditions. These findings demonstrate that counting strategies and the representational use of numbers lie within the cognitive domain of the chimpanzee and compare favorably with the spontaneous use of addition algorithms demonstrated in preschool children.     

Qualitative differences in the representation of abstract versus concrete words: Evidence from the visual-world paradigm

In the present visual-world experiment, participants were presented with visual displays that included a target item that was a semantic associate of an abstract or a concrete word. This manipulation allowed us to test a basic prediction derived from the qualitatively different representational framework that supports the view of different organizational principles for concrete and abstract words in semantic memory. Our results confirm the assumption of a primary organizational principle based on association for abstract words, different from the semantic similarity principle proposed for concrete words, and provide the first piece of evidence in support of this view obtained from healthy participants. The results shed light on the representational structure of abstract and concrete concepts.     

Children's understanding of counting

This study examines the abstractness of children's mental representation of counting, and their understanding that the last number word used in a count tells how many items there are (the cardinal word principle). In the first experiment, twenty-four 2- and 3-year-olds counted objects, actions, and sounds. Children counted objects best, but most showed some ability to generalize their counting to actions and sounds, suggesting that at a very young age, children begin to develop an abstract, generalizable mental representation of the counting routine. However, when asked "how many" following counting, only older children (mean age 3.6) gave the last number word used in the count a majority of the time, suggesting that the younger children did not understand the cardinal word principle. In the second experiment (the "give-a-number" task), the same children were asked to give a puppet one, two, three, five, and six items from a pile. The older children counted the items, showing a clear understanding of the cardinal word principle. The younger children succeeded only at giving one and sometimes two items, and never used counting to solve the task. A comparison of individual children's performance across the "how-many" and "give-a-number" tasks shows strong within-child consistency, indicating that children learn the cardinal word principle at roughly 3 1/2 years of age. In the third experiment, 18 2- and 3-year-olds were asked several times for one, two, three, five, and six items, to determine the largest numerosity at which each child could succeed consistently. Results indicate that children learn the meanings of smaller number words before larger ones within their counting range, up to the number three or four. They then learn the cardinal word principle at roughly 3 1/2 years of age, and perform a general induction over this knowledge to acquire the meanings of all the number words within their counting range.     

The role of gesture in children's learning to count.

The role of spontaneous gesture was examined in children's counting and in their assessment of counting accuracy. Eighty-five 2-, 3-, and 4-year-olds counted 6 sets of 2-, 4-, and 6-object arrays. In addition, children assessed the counting accuracy of a puppet whose gestures varied as he counted (i.e., gesture matched the number words, gesture mismatched the number words, no gesture at all). Results showed that the correspondence of children's speech and gesture varied systematically across the age groups and that children adhered to the one-to-one correspondence principle in gesture prior to speech. Moreover, children's correspondence of speech and gesture, adherence to the one-to-one principle in gesture, and assessment of the puppet's counting accuracy were related to children's counting accuracy. Findings are discussed in terms of the role that gesture may play in children's understanding of counting.     

Does time fly when you're counting down? The effect of counting direction on subjective time judgment

We show that counting downward while performing a task shortens the perceived duration of the task compared to counting upward. People perceive that less time has elapsed when they were counting downward versus upward while using a product (Studies 1 and 3) or watching geometrical shapes (Study 2). The counting direction effect is obtained using both prospective and retrospective time judgments (Study 3), but only when the count range begins with the number “1” (Study 2). Furthermore, the counting direction affects peoples' attitude toward the product, their likelihood of using it again, and their purchase intentions. We test several plausible accounts for the counting direction effect, including task difficulty, numerical anchoring, and arousal. We find preliminary evidence that downward counting feels shorter because it is more arousing than upward counting.     

Find the picture of eight turtles: a link between children's counting and their knowledge of number word semantics

An essential part of understanding number words (e.g., eight) is understanding that all number words refer to the dimension of experience we call numerosity. Knowledge of this general principle may be separable from knowledge of individual number word meanings. That is, children may learn the meanings of at least a few individual number words before realizing that all number words refer to numerosity. Alternatively, knowledge of this general principle may form relatively early and proceed to guide and constrain the acquisition of individual number word meanings. The current article describes two experiments in which 116 children (2½- to 4-year-olds) were given a Word Extension task as well as a standard Give-N task. Results show that only children who understood the cardinality principle of counting successfully extended number words from one set to another based on numerosity—with evidence that a developing understanding of this concept emerges as children approach the cardinality principle induction. These findings support the view that children do not use a broad understanding of number words to initially connect number words to numerosity but rather make this connection around the time that they figure out the cardinality principle of counting.     

Teleological reasoning in infancy: the nai;ve theory of rational action

Converging evidence demonstrates that one-year-olds interpret and draw inferences about other's goal-directed actions. We contrast alternative theories about how this early competence relates to our ability to attribute mental states to others. We propose that one-year-olds apply a non-mentalistic interpretational system, the 'teleological stance' to represent actions by relating relevant aspects of reality (action, goal-state and situational constraints) through the principle of rational action, which assumes that actions function to realize goal-states by the most efficient means available. We argue that this early inferential principle is identical to the rationality principle of the mentalistic stance - a representational system that develops later to guide inferences about mental states.     

Distributed representation as a principle for the analysis of cockpit information displays

This article examines the representational properties of cockpit information displays from the perspective of distributed representations (Zhang & Norman, 1994). The basic idea is that the information needed for many tasks in a cockpit is distributed across the external information displays in the cockpit and the internal minds of the pilots. It is proposed that the relative distribution of internal and external information is the major factor of a display's representational efficiency. Several functionally equivalent but representationally different navigation displays are selected to illustrate how the principle of distributed representations is applied to the analysis of the representational efficiencies of cockpit information displays.     

The effect of external representations on numeric tasks

This article explores the effect of external representations on numeric tasks. Through several minor modifications on the previously reported two-digit number comparison task, we obtained different results. Rather than holistic comparison, we found parallel comparison. We argue that this difference was a reflection of different representational forms: The comparison was based on internal representations in previous studies but on external representations in our present study. This representational effect is discussed under a framework of distributed number representations. We propose that in numerical tasks involving external representations, numbers should be considered as distributed representations, and the behaviour in these tasks should be considered as the interactive processing of internal and external information through the interplay of perceptual and cognitive processes. We suggest that theories of number representations and process models of numerical cognition should consider external representations as an essential component.     

儿童数字线表征的发展——心理长度的影响

探讨儿童在数字线估计任务中心理长度的发展及其对数量表征模式的影响.选取7~9岁儿童共109名进行数字线估计任务测试,设置了10cm和20cm两种长度条件,要求儿童完成根据位置判断数字任务(position to number,PN任务).结果表明儿童在数字线PN任务中存在心理长度,且7岁儿童心理长度的范围超过10,随着儿童年龄的增长,他们的心理长度范围不断缩小;心理长度范围影响儿童的表征模式,随着心理长度范围的缩小,儿童的数字线表征出现从指数模式到线性模式的变化趋势;与表征模式的发展趋势一致,儿童估计的精确性随年龄增长逐渐提高.     

Representations in Distributed Cognitive Tasks

In this article we propose a theoretical framework of distributed representations and a methodology of representational analysis for the study of distributed cognitive tasks—tasks that require the processing of information distributed across the internal mind and the external environment. The basic principle of distributed representations Is that the representational system of a distributed cognitive task is a set of internal and external representations, which together represent the abstract structure of the task. The basic strategy of representational analysis is to decompose the representation of a hierarchical task into its component levels so that the representational properties at each level can be independently examined. The theoretical framework and the methodology are used to analyze the hierarchical structure of the Tower of Hanoi problem. Based on this analysis, four experiments are designed to examine the representational properties of the Tower of Hanoi. Finally, the nature of external representations is discussed.     

The Roles of Emotional Comprehension and Representational Drawing Skill in Children's Expressive Drawing

The purpose of the present study was to investigate the roles of emotional comprehension and representational drawing skill in children's expressive drawing. Fifty 7‐ to 10‐year‐olds were asked to produce two (happy and sad) expressive drawings, two representational drawings (drawing of a man running and drawing of a house) and to answer the Test of Emotion Comprehension (Pons & Harris, 2000). The expressive drawings were assessed on the number of expressive subject matter themes (‘content expression’) and the overall quality of expression on a 5‐point scale. Each of the representational drawings was measured on a scale assessing detail and visual realism criteria, and contributed to a single representational drawing skill score. In line with our predictions, we found that both emotional comprehension and representational drawing skill accounted for a significant variance in children's expressive drawings. We explain that children's developing emotional comprehension may allow them to consider more detailed and poignant expressive ideas for their drawings and that their developing representational drawing skill facilitates the graphic execution of these emotional ideas. Copyright © 2014 John Wiley & Sons, Ltd.     

Representational change and magnitude estimation: why young children can make more accurate salary comparisons than adults

Development of estimation has been ascribed to two sources: (1) a change from logarithmic to linear representations of number and (2) development of general mathematical skills. To test the representational change hypothesis, we gave children and adults a task in which an automatic, linear representation is less adaptive than the logarithmic representation: estimating the value of salaries given in fractional notation. The representational change hypothesis generated the surprising (and accurate) prediction that when estimating the magnitude of salaries given in fractional notation, young children would outperform adults, whereas when estimating the magnitude of the same salaries given in decimal notation, adults would outperform children.     

Stochastic approaches to understanding dissociations in inflectional morphology

Computer modelling research has undermined the view that double dissociations in behaviour are sufficient to infer separability in the cognitive mechanisms underlying those behaviours. However, all these models employ multi-modal representational schemes, where functional specialisation of processing emerges from the training process. Targeted lesioning of different regions of functional specialisation leads to varied but predictable deficits in model performance. We argue that multi-modal representational schemes are not a necessary condition for the observation of double dissociations in an information processing system that shares resources across multiple tasks. Using a uni-modal representational system, we demonstrate that double dissociations may also result from stochastic processes. Lesioning experiments on a single-route, uni-modal connectionist model of regular and irregular noun and verb morphology confirm and extend earlier work demonstrating that selective impairment across tasks can result from damage to a distributed information processing system. A systematic investigation of the degree to which performance deteriorates across different inflectional classes reveals that simple and double dissociations can occur in this single-route, uni-modal model. An important prediction of the model is that double dissociations between regular and irregular inflection, resulting from stochastic processes should be extremely rare. However, they are particularly likely to occur when the researcher uses test batteries consisting of a small number of items. Given that cognitive neuropsychologists rarely provide details about the distribution of performance in a disordered population, it is concluded that a stochastic interpretation of double dissociations may have wider applicability than is normally supposed.     

Representation for reciprocal agent–environment interaction

Declarative modelling approaches in principle assume a notion of representation or representational content for the modelling concepts. The notion of representational content as discussed in literature in cognitive science and philosophy of mind shows complications as soon as agent and environment have an intense reciprocal interaction. In such cases an internal agent state is affected by the way in which internal and external aspects are interwoven during (ongoing) interaction. In this paper it is shown that the classical correlational approach to representational content is not applicable, but the temporal-interactivist approach is. As this approach involves more complex temporal relationships, formalisation was used to define specifications of the representational content more precisely. These specifications have been validated by automatically checking them on traces generated by a simulation model. Moreover, by mathematical proof it was shown how these specifications are entailed by the basic local properties.     

Representational content and computation in the human visual system

Summary Information-processing systems can be characterized by their ability to transform systematically certain internal representational states (symbols) into one another. The presence of such an information-processing capacity calls for an explanation. How could such an explanation in principle be formulated? How is it possible to specify internal representational states and to ascribe to them certain representational contents? What has to be demonstrated by such explanations is how an information-processing capacity is actually instantiated in a system. In this paper, the outlines of an explanation by instantiation are sketched for a specific human visual capacity. In addition, some fundamental problems facing the development of this explanation will be discussed.