两种数量表征系统

数量表征是人类数学能力的基础,数量表征研究中的一个争论焦点在于是否存在两种不同的数量表征系统：对小数的精确表征系统和对大数的近似表征系统。通过综述不同研究领域对数量表征的研究,总结了支持两种表征系统分离的证据：对1~3范围内小数的表征受数量大小的限制,基于指向物体本身的注意,更依赖于物体的知觉特征,对物体及其数量进行精确表征;而对4以上的数量的近似表征系统则受韦伯定律的限制,基于指向数量的模拟幅度的表征,而不依赖单个物体的知觉特征,是对数量的近似的、心理的表征。fMRI、PET和ERP的脑成像研究结果迄今尚无定论,但认知神经科学研究的深入开展将最终阐明数量表征的机制     

The representations of the arithmetic operations include functional relationships

Current theories of mathematical problem solving propose that people select a mathematical operation as the solution to a problem on the basis of a structure mapping between their problem representation and the representation of the mathematical operations. The structure-mapping hypothesis requires that the problem and the mathematical representations contain analogous relations. Past research has demonstrated that the problem representation consists of functional relationships, or principles. The present study tested whether people represent analogous principles for each arithmetic operation (i.e., addition, subtraction, multiplication, and division). For each operation, college (Experiments 1 and 2) and 8th grade (Experiment 2) participants were asked to rate the degree to which a set of completed problems was a good attempt at the operation. The pattern of presented answers either violated one of four principles or did not violate any principles. The distance of the presented answers from the correct answers was independently manipulated. Consistent with the hypothesis that people represent the principles, (1) violations of the principles were rated as poorer attempts at the operation, (2) operations that are learned first (e.g., addition) had more extensive principle representations than did operations learned later (multiplication), and (3) principles that are more frequently in evidence developed more quickly. In Experiment 3, college participants rated the degree to which statements were indicative of each operation. The statements were either consistent or inconsistent with one of two principles. The participants' ratings showed that operations with longer developmental histories had strong principle representations. The implications for a structure-mapping approach to mathematical problem solving are discussed.     

Representation of inflected nouns in the internal lexicon

The lexical representation of Serbo-Croatian nouns was investigated in a lexical decision task. Because Serbo-Croatian nouns are declined, a noun may appear in one of several grammatical cases distinguished by the inflectional morpheme affixed to the base form. The grammatical cases occur with different frequencies, although some are visually and phonetically identical. When the frequencies of identical forms are compounded, the ordering of frequencies is not the same for masculine and feminine genders. These two genders are distinguished further by the fact that the base form for masculine nouns is an actual grammatical case, the nominative singular, whereas the base form for feminine nouns is an abstraction in that it cannot stand alone as an independent word. Exploiting these characteristics of the Serbo-Croatian language, we contrasted three views of how a noun is represented: (1) the independent-entries hypothesis, which assumes an independent representation for each grammatical case, reflecting its frequency of occurrence; (2) the derivational hypothesis, which assumes that only the base morpheme is stored, with the individual cases derived from separately stored inflectional morphemes and rules for combination; and (3) the satellite-entries hypothesis, which assumes that all cases are individually represented, with the nominative singular functioning as the nucleus and the embodiment of the noun’s frequency and around which the other cases cluster uniformly. The evidence strongly favors the satellite-entries hypothesis.     

负数的空间表征机制

本研究采用快速数字大小分类范式,每次试验呈现一个数字,要求被试快速判断即时呈现的数字大于或小于-5（或5）,探讨负数在心理数字线上的表征方向问题。实验一将负数（-1～-9）和正数（1～9）分两组分别呈现;实验二将正负数混合呈现,仅对负数进行反应。结果表明,负数按照其绝对值大小表征在心理数字线上,绝对值小的负数表征在心理数字线的左侧,绝对值大的负数表征在心理数字线的右侧。该结果支持系统进化论假说     

风险寻求中的图形表征效应

在负性风险信息沟通中,采用文本表征与图形表征,结果发现图形表征方式更能引起风险回避行为。基于此,本研究以获益方案作为风险沟通信息,分别采用文本表征与图形表征来探索信息表征方式对风险寻求的影响。结果显示:在文本表征方式中,个体决策主要受备择方案之间风险大小的影响;而图形表征方式中,个体不仅受备择方案之间风险大小的影响,而且更受备择方案之间获益大小的影响。结果说明:相对文本表征,图形表征方式更能引起决策者为了获益而冒更大的风险,从而表现出风险寻求偏好,从而证实"图形效应"在风险决策中存在普遍性,为我们风险决策信息提供依据。     

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.     

元认知在画图表征策略和小学生数学问题解决能力中的中介作用

该研究采用自然实验和问卷调查法,对268名小学五年级学生进行画图表征策略训练,在此基础上根据策略学习情况筛选出掌握了该策略的218名学生,以这些学生为被试,探讨元认知在画图表征策略和小学生数学问题解决能力中的作用。结果表明:(1)策略学习后,小学生的画图表征策略水平和数学问题解决成绩均显著高于策略学习前。(2)画图表征策略通过元认知这一中介变量对小学生数学问题解决能力有显著的促进作用。     

Comparison of the distortion of probability information in decision under risk and an equivalent visual task

Decision makers typically overweight small probabilities and underweight large probabilities. However, there are recent reports that when probability is presented in the form of relative frequencies, this typical pattern reverses. We tested this hypothesis by comparing decision making in two tasks: In one task, probability was stated numerically, and in the other task, it was conveyed through a visual representation. In the visual task, participants chose whether a "stochastic bullet" should be fired at either a large target for a small reward or a small target for a large reward. Participants' knowledge of probability in the visual task was the result of extensive practice firing bullets at targets. In the classical numerical task, participants chose between pairs of lotteries with probabilities and rewards matched to the probabilities and rewards in the visual task. We found that participants' probability-weighting functions were significantly different in the two tasks, but the pattern for the visual task was the typical, not the reversed, pattern.     

Food and token quantity discrimination in capuchin monkeys (<Emphasis Type="Italic">Cebus apella</Emphasis>)

Quantity discrimination is adaptive in a variety of ecological contexts and different taxa discriminate stimuli differing in numerousness, both in the wild and in laboratory settings. Quantity discrimination between object arrays has been suggested to be more demanding than between food arrays but, to our knowledge, the same paradigm has never been used to directly compare them. We investigated to what extent capuchin monkeys’ relative numerousness judgments (RNJs) with food and token are alike. Tokens are inherently non-valuable objects that acquire an associative value upon exchange with the experimenter. Our aims were (1) to assess capuchins’ RNJs with food (Experiment 1) and with tokens (Experiment 2) by presenting all the possible pair-wise choices between one to five items, and (2) to evaluate on which of the two proposed non-verbal mechanisms underlying quantity discrimination (analogue magnitude and object file system) capuchins relied upon. In both conditions capuchins reliably selected the larger amount of items, although their performance was higher with food than with tokens. The influence of the ratio between arrays on performance indicates that capuchins relied on the same system for numerical representation, namely analogue magnitude, regardless of the type of stimuli (food or tokens) and across both the small and large number ranges.     

Semantic numerical representation in blind subjects: The role of vision in the spatial format of the mental number line

Does vision play a role in the elaboration of the semantic representation of small and large numerosities, notably in its spatial format? To investigate this issue, we decided to compare in the auditory modality the performance of congenitally and early blind people with that of a sighted control group, in two number comparison tasks (to 5 and to 55) and in one parity judgement task. Blind and sighted participants presented exactly the same distance and SNARC (Spatial Numerical Association of Response Codes) effects, indicating that they share the same semantic numerical representation. In consequence, our results suggest that the spatial dimension of the numerical representation is not necessarily attributable to the visual modality and that the absence of vision does not preclude the elaboration of this representation for 1-digit (Experiment 1) and 2-digit numerosities (Experiment 2). Moreover, as classical semantic numerical effects were observed in the auditory modality, the postulate of the amodal nature of the mental number line for both small and large magnitudes was reinforced.     

Semantic numerical representation in blind subjects: the role of vision in the spatial format of the mental number line

Does vision play a role in the elaboration of the semantic representation of small and large numerosities, notably in its spatial format? To investigate this issue, we decided to compare in the auditory modality the performance of congenitally and early blind people with that of a sighted control group, in two number comparison tasks (to 5 and to 55) and in one parity judgement task. Blind and sighted participants presented exactly the same distance and SNARC (Spatial Numerical Association of Response Codes) effects, indicating that they share the same semantic numerical representation. In consequence, our results suggest that the spatial dimension of the numerical representation is not necessarily attributable to the visual modality and that the absence of vision does not preclude the elaboration of this representation for 1-digit (Experiment 1) and 2-digit numerosities (Experiment 2). Moreover, as classical semantic numerical effects were observed in the auditory modality, the postulate of the amodal nature of the mental number line for both small and large magnitudes was reinforced.     

The relation between spatial skill and early number knowledge: The role of the linear number line

[Correction Notice: An Erratum for this article was reported in Vol 48(5) of Developmental Psychology (see record 2012-11771-001). The grey boxes around the faces in Figure 2 are missing. The correct version is presented in the erratum.] Spatial skill is highly related to success in math and science (e.g., Casey, Nuttall, Pezaris, & Benbow, 1995). However, little work has investigated the cognitive pathways by which the relation between spatial skill and math achievement emerges. We hypothesized that spatial skill plays a crucial role in the development of numerical reasoning by helping children to create a spatially meaningful, powerful numerical representation-the linear number line. In turn, a strong linear number representation improves other aspects of numerical knowledge such as arithmetic estimation. We tested this hypothesis using 2 longitudinal data sets. First, we found that children's spatial skill (i.e., mental transformation ability) at the beginning of 1st and 2nd grades predicted improvement in linear number line knowledge over the course of the school year. Second, we found that children's spatial skill at age 5 years predicted their performance on an approximate symbolic calculation task at age 8 and that this relation was mediated by children's linear number line knowledge at age 6. The results are consistent with the hypothesis that spatial skill can improve children's development of numerical knowledge by helping them to acquire a linear spatial representation of numbers. (PsycINFO Database Record (c) 2012 APA, all rights reserved).     

Cognitive skills in mathematical problem solving in Grade 3

Background. Research on the relationship between cognitive skills and mathematical problem solving is usually conducted on adults or on participants with acquired deficits associated with brain injury (e.g. Cipolotti, 1995 ; Cohen, Dehaene, & Verstichel, 1994 ; McCloskey, 1992 ). Aims. In these studies we wanted to make a contribution to the field of children's mathematical problem solving. The first aim of this study was to investigate whether mathematical problem solving in children is merely determined by semantic elaboration, as hypothesized in some of the models of adult processing (semantic hypothesis). In addition, we aimed to investigate whether there is a continuum from very good to very poor mathematical problem solving among children with mathematical learning disabilities showing immature cognitive skills (maturational lag hypothesis). Sample. The participants were 376 third graders and 107 second graders. Method. The internal structure of the data was analysed with a principal components analysis. In addition, two MANOVA were conducted to compare children with learning disabilities or problems with age‐matched and performance‐matched subjects. Results. Two components, a semantic and a non‐semantic one, were needed to account for an adequate fit of the dataset. In addition, children with mathematical learning disabilities had less‐developed cognitive skills compared with peers without learning disabilities, but they did not differ from younger children on seven of the nine cognitive skills. Conclusions. This study highlighted that children's mathematical problem solving is not determined by one general component. The picture is more complex, since two mathematics components were found. In addition, although our findings point in the direction of the maturational lag hypothesis it may be important to assess the different cognitive skills and especially assess the number system knowledge, since it seems below average in children with mathematical learning disabilities, compared with the knowledge of younger children with comparable skills in mathematics.     

The perception of back vowels: Centre of gravity hypothesis

According to the formant centre of gravity (FCOG) hypothesis, two vowel formants in close proximity are merged during perceptual analysis, and their contribution to vowel quality depends on the centre of gravity of the formant cluster. Findings consistent with this hypothesis are that two formants can be replaced by a single formant of intermediate centre frequency, provided their separation is less than 3-3.5 Bark; and that changes in their relative amplitudes produce systematic shifts in vowel quality. In Experiment 1, listeners adjusted the frequencies of F1 and F2 in a synthesized 6-formant vowel (with the F1-F2 separation fixed at 250 Hz, i.e. less than 3 Bark) to find the best phonetic match to a reference vowel with modified formant amplitudes. Contrary to FCOG predictions, F2 attenuation did not produce lower frequency matches. Raising the amplitude of F2 led to predicted upward shifts in formant frequencies of the matched vowel, but with increased variability of matches for some stimuli. In Experiment 2, listeners identified synthesized vowels with a range of separations of F1 and F2. Formant amplitude manipulations had no effect on listeners' judgements when the fundamental frequency was low (125 Hz). Small shifts in vowel quality appeared for stimuli with a high fundamental (250 Hz), but the shifts were significantly larger for F1-F2 separations greater than 3.5 Bark. These effects of formant amplitude are qualitatively different from those observed with single-formant vowels and are generally incompatible with a formant-averaging mechanism.     

Apparent reversal frequencies in squares and trapezia: the effect of invariant and variant information

Three studies of apparent reversals (ARs) in rotating rectangular and trapezoid arrays are reported, together with a mathematical analysis. Study 1 explores the effect of different degrees of outline suppression on the frequency and distribution of ARs in squares and trapezia. Study 2 examines how modifications in both types of array reduce AR frequency, demonstrates a manipulation whereby ARs in rotating squares are eliminated, and suggests that ARs in squares and trapezia are of different origins. In the mathematical analysis, an invariant characteristic of the transformations of rotating rectangles is reported, and from this a numerical index is developed that expresses the extent to which a rotating trapezium may appear to pause just before each point of oscillation. In study 3 it is shown that this measure is a better predictor of AR frequencies than the heuristic proposed by Braunstein.     

How numbers mean: Comparing random walk models of numerical cognition varying both encoding processes and underlying quantity representations

How do people derive meaning from numbers? Here, we instantiate the primary theories of numerical representation in computational models and compare simulated performance to human data. Specifically, we fit simulated data to the distributions for correct and incorrect responses, as well as the pattern of errors made, in a traditional “relative quantity” task. The results reveal that no current theory of numerical representation can adequately account for the data without additional assumptions. However, when we introduce repeated, error-prone sampling of the stimulus (e.g., Cohen, 2009) superior fits are achieved when the underlying representation of integers reflects linear spacing with constant variance. These results provide new insights into (i) the detailed nature of mental numerical representation, and, (ii) general perceptual processes implemented by the human visual system.     

从数量表征到数表征：具身认知视角下的人类数能力获得

数能力是数学认知的基本成分。与动物所具有的基本数能力不同,人类不仅具备数量表征能力,更重要的是还拥有对数概念进行表征的数表征能力。虽然具身认知与离身认知都对数概念的表征问题进行了解释,但二者却存在明显理论分歧。具身认知观点主要从具身数量表征和数能力发展的具身认知机制两方面为人类独特数能力的获得提供了理论支撑及实证证据。这启示人们需要重视具身学习在数能力形成实践中的关键作用,重视具身数量表征在数学教学中的作用,仍需进一步揭示其内在的心理和神经基础。     

Numerical-spatial representation affects spatial coding: binding errors across the numerical distance effect

Does numerical-spatial representation affect feature binding? Studies of visual attention show that poor spatial coding leads to illusory conjunctions (ICs). In numerical cognition, it has been shown that numbers and space are not totally dissociated. This association underlies the numerical distance effect (DE): faster responses as the distance between the compared digits becomes larger (2 7 vs. 2 4). We used the DE to test whether numerical-spatial representation is available to visual processes that rely on spatial coding, such as feature binding. Participants reported the larger of two colored numbers. Both numerical distance (distances 2 and 5) and number–space congruity (e.g., congruent pair, 1 3; incongruent pair, 3 1) were analyzed. Results showed a higher proportion of ICs for distance 2 than for distance 5, providing strong evidence that numerical-spatial representation (1) entails a strong location code and (2) is available to visual processes that rely on location information.     

Differences in Fear of Isolation as an explanation of Cultural Differences: Evidence from memory and reasoning

Previous research suggests that members of East Asian cultures show a greater preference for dialectical thinking and sensitivity to context information than do Westerners. We suggest this difference is rooted in a greater chronic Fear of Isolation (FOI) in East Asians than in Westerners. To support this hypothesis, we manipulated FOI in a group of Westerners and assessed their relative preference for dialectical proverbs and sensitivity to context. For cross-cultural validation of our hypothesis, we assessed the relationship between chronic levels of FOI and dialectical reasoning in Koreans. Consistent with our proposal, both experimentally primed FOI (Experiment 1A and 2) and chronic levels of FOI (Experiment 1B) were positively related to relative preference for dialectical proverbs. This effect was independent of participants’ level of negative mood (Experiment 2). A third experiment showed that sensitivity to context was affected by FOI in a manner consistent with previous studies of cultural differences (Experiment 3).     

Numerical Subtraction in the Pigeon: Evidence for a Linear Subjective Number Scale

When humans and animals compare two numbers, responding is faster and more accurate with increasing numerical disparity and decreasing numerical size. Researchers explaining these distance and size effects often assume that the subjective number continuum is logarithmically compressed. An alternative hypothesis is that the subjective number continuum is linear, but positions farther along it are proportionately fuzzier, that is, less precisely located. These two hypotheses have been treated as functionally equivalent because of their similar empirical predictions. The current experiment sought to resolve this issue with a paradigm originally developed to address the subjective representation of time (time left). In our adaptation, pigeons were required to compare a constant number with the number remaining after a numerical subtraction. Our results indicate that subjective number is linearly, not logarithmically, related to objective number.