Mapping numerical magnitudes onto symbols: the numerical distance effect and individual differences in children's mathematics achievement

Although it is often assumed that abilities that reflect basic numerical understanding, such as numerical comparison, are related to children’s mathematical abilities, this relationship has not been tested rigorously. In addition, the extent to which symbolic and nonsymbolic number processing play differential roles in this relationship is not yet understood. To address these questions, we collected mathematics achievement measures from 6- to 8-year-olds as well as reaction times from a numerical comparison task. Using the reaction times, we calculated the size of the numerical distance effect exhibited by each child. In a correlational analysis, we found that the individual differences in the distance effect were related to mathematics achievement but not to reading achievement. This relationship was found to be specific to symbolic numerical comparison. Implications for the role of basic numerical competency and the role of accessing numerical magnitude information from Arabic numerals for the development of mathematical skills and their impairment are discussed.     

Flexible and unique representations of two-digit decimals

We examined the representation of two-digit decimals through studying distance and compatibility effects in magnitude comparison tasks in four experiments. Using number pairs with different leftmost digits, we found both the second digit distance effect and compatibility effect with two-digit integers but only the second digit distance effect with two-digit pure decimals. This suggests that both integers and pure decimals are processed in a compositional manner. In contrast, neither the second digit distance effect nor the compatibility effect was observed in two-digit mixed decimals, thereby showing no evidence for compositional processing of two-digit mixed decimals. However, when the relevance of the rightmost digit processing was increased by adding some decimals pairs with the same leftmost digits, both pure and mixed decimals produced the compatibility effect. Overall, results suggest that the processing of decimals is flexible and depends on the relevance of unique digit positions. This processing mode is different from integer analysis in that two-digit mixed decimals demonstrate parallel compositional processing only when the rightmost digit is relevant. Findings suggest that people probably do not represent decimals by simply ignoring the decimal point and converting them to natural numbers.     

Dyslexia and dyscalculia: Two learning disorders with different cognitive profiles

This study tests the hypothesis that dyslexia and dyscalculia are associated with two largely independent cognitive deficits, namely a phonological deficit in the case of dyslexia and a deficit in the number module in the case of dyscalculia. In four groups of 8- to 10-year-olds (42 control, 21 dyslexic, 20 dyscalculic, and 26 dyslexic/dyscalculic), phonological awareness, phonological and visual-spatial short-term and working memory, naming speed, and basic number processing skills were assessed. A phonological deficit was found for both dyslexic groups, irrespective of additional arithmetic deficits, but not for the dyscalculia-only group. In contrast, deficits in processing of symbolic and nonsymbolic magnitudes were observed in both groups of dyscalculic children, irrespective of additional reading difficulties, but not in the dyslexia-only group. Cognitive deficits in the comorbid dyslexia/dyscalculia group were additive; that is, they resulted from the combination of two learning disorders. These findings suggest that dyslexia and dyscalculia have separable cognitive profiles, namely a phonological deficit in the case of dyslexia and a deficient number module in the case of dyscalculia.     

Numerical comparison of two-digit numbers: How differences at encoding can involve differences in processing

The study of two-digit numbers processing has recently gathered a growing interest. Here, we examine whether differences at encoding of two-digit oral verbal numerals induce differences in the type of processing involved. Twenty-four participants were submitted to a comparison task to 55. Differences at encoding were introduced by the use of dichotic listening and synchronous (synchronous condition) or asynchronous presentation (tens-first and units-first conditions) of the two-digit numerals' components. Our results showed that differences at the encoding stage of two-digit numerals involve: (1) different comparison processes (tens-first and units-first conditions: parallel comparison; synchronous condition: parallel and holistic comparison); and (2) differences in the weight of the tens- and units-effects. Therefore, attentional mechanisms determining at the encoding stage how much attention is paid to the two-digit numerals' components might account for the different types of processing found with two-digit numbers.     

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.     

Rational numbers: Componential versus holistic representation of fractions in a magnitude comparison task

This study investigated whether the mental representation of the fraction magnitude was componential and/or holistic in a numerical comparison task performed by adults. In Experiment 1, the comparison of fractions with common numerators (x/a_x/b) and of fractions with common denominators (a/x_b/x) primed the comparison of natural numbers. In Experiment 2, fillers (i.e., fractions without common components) were added to reduce the regularity of the stimuli. In both experiments, distance effects indicated that participants compared the numerators for a/x_b/x fractions, but that the magnitudes of the whole fractions were accessed and compared for x/a_x/b fractions. The priming effect of x/a_x/b fractions on natural numbers suggested that the interference of the denominator magnitude was controlled during the comparison of these fractions. These results suggested a hybrid representation of their magnitude (i.e., componential and holistic). In conclusion, the magnitude of the whole fraction can be accessed, probably by estimating the ratio between the magnitude of the denominator and the magnitude of the numerator. However, adults might prefer to rely on the magnitudes of the components and compare the magnitudes of the whole fractions only when the use of a componential strategy is made difficult.     

Exploring the developmental changes in automatic two-digit number processing

Even when two-digit numbers are irrelevant to the task at hand, adults process them. Do children process numbers automatically, and if so, what kind of information is activated? In a novel dot-number Stroop task, children (Grades 1-5) and adults were shown two different two-digit numbers made up of dots. Participants were asked to select the number that contained the larger dots. If numbers are processed automatically, reaction time for dot size judgment should be affected by numerical characteristics. The results suggest that, like adults, children process two-digit numbers automatically. Based on the current findings, we propose a developmental trend for automatic two-digit number processing that goes from decomposed sequential (activation of decade digit followed by that of unit digit) to decomposed parallel processing (simultaneous activation of decade and unit digits).     

Reaching towards an end: Numerical end and distance effects in motor movements

Numerical comparisons are affected by the distance between the numbers and by the presence of an end stimulus. In line with embodied cognition approaches, past studies found evidence for the distance effect in continuous motor movements. The present study is the first to provide evidence for the end effect (i.e., faster comparisons for pairs that include an end stimulus of a set) using continuous motor movements. Two digits were presented horizontally on a screen and participants reached towards the larger one using a computer mouse cursor. Response trajectories were straighter (1) when the number pair included the end stimulus of 1, and (2) when the numerical distance between the numbers was large. Importantly, the end effect appeared earlier in the motor trajectory than the distance effect. The implications of this pattern for the cognitive processes underlying the end and the distance effects are discussed.     

Cross-notational semantic priming between symbolic and nonsymbolic numerosity

Symbolic and nonsymbolic numerosities produce similar behavioural effects and activate the same brain areas. These results have usually been interpreted in terms of a common, notation-independent magnitude representation. However, semantic priming between symbolic and nonsymbolic inputs has been somehow elusive (e.g., Koechlin, Naccache, Block, & Dehaene, 1999 Koechlin, E., Naccache, L., Block, E. and Dehaene, S. 1999. Primed numbers: Exploring the modularity of numerical representations with masked and unmasked semantic priming. Journal of Experimental Psychology: Human Perception and Performance, 25: 1882–1905. [Crossref], [Web of Science ®] , [Google Scholar]). In Experiment 1, we looked at whether cross-notational semantic priming depends on exact numerical meaning. Dice faces and digits were mixed as prime and target. Semantic priming occurred when prime and target were in the same notation as much as when they were in different notation. In Experiment 2, we found cross-notation semantic priming even when the nonsymbolic numerosity was presented as a set of random dots. Priming, however, occurred only from sets of dots to digit, not vice versa. These data support the computational model recently proposed by Verguts and Fias (2004 Verguts, T. and Fias, W. 2004. Representation of number in animals and humans: A neural model. Journal of Cognitive Neuroscience, 16: 1493–1504. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]; Verguts, Fias, & Stevens, 2005 Verguts, T., Fias, W. and Stevens, M. 2005. A model of exact small-number representation. Psychonomic Bulletin & Review, 112: 66–80.  [Google Scholar]).     

发展性计算障碍儿童的数量转换缺陷

本研究拟考察发展性计算障碍儿童的认知缺陷成因。实验1要求被试在三种形式(点/点,数/数,点/数)下进行数量比较,实验2仅将点集替换为汉字数字词。结果表明障碍组和正常组在数/数、点/数和汉字/汉字比较任务上的成绩存在显著差异,而在点/点和汉字/汉字比较上没有差异。据此推论,计算障碍儿童符号加工能力受到损伤,符号与非符号数量转换能力存在缺陷,但非符号加工能力和不同符号间数量转换没有缺陷,支持语义提取缺陷假设。     

Holistic or compositional representation of two-digit numbers? Evidence from the distance, magnitude, and SNARC effects in a number-matching task

Whether two-digit numbers are represented holistically (each digit pair processed as one number) or compositionally (each digit pair processed separately as a decade digit and a unit digit) remains unresolved. Two experiments were conducted to examine the distance, magnitude, and SNARC effects in a number-matching task involving two-digit numbers. Forty undergraduates were asked to judge whether two two-digit numbers (presented serially in Experiment 1 and simultaneously in Experiment 2) were the same or not. Results showed that, when numbers were presented serially, unit digits did not make unique contributions to the magnitude and distance effects, supporting the holistic model. When numbers were presented simultaneously, unit digits made unique contributions, supporting the compositional model. The SNARC (Spatial-Numerical Association of Response Codes) effect was evident for the whole numbers and the decade digits, but not for the unit digits in both experiments, which indicates that two-digit numbers are represented on one mental number line. Taken together, these results suggested that the representation of two-digit numbers is on a single mental number line, but it depends on the stage of processing whether they are processed holistically or compositionally.     

Language effects in magnitude comparison: small, but not irrelevant

It is assumed that number magnitude comparison is performed by assessing magnitude representation on a single analog mental number line. However, we have observed a unit-decade-compatibility effect in German which is inconsistent with this assumption (Nuerk, Weger, & Willmes, 2001). Incompatible magnitude comparisons in which decade and unit comparisons lead to different responses (e.g., 37_52 for which 3<5, but 7>2) are slower and less accurately responded to than compatible trials in which decade and unit comparisons lead to the same response (e.g., 42_57, for which 4<5 and 2<7). As overall distance was held constant, a single holistic magnitude representation could not account for this compatibility effect. However, because of the inversion property of the corresponding German two-digit number words ("einundzwanzig" ), the language-generality of the effect is questionable. We have therefore examined the compatibility effect with native English speakers. We were able to replicate the compatibility effect using Arabic notation. Thus, the compatibility effect is not language-specific. However, in cross-linguistic analyses language-specific modulations were observed not only for number words but also for Arabic numbers. The constraints imposed on current models by the verbal mediation of Arabic number comparison are discussed.     

In touch with numbers: Embodied and situated effects in number magnitude comparison

The idea of embodied numerosity denotes that seemingly abstract number concepts (e.g., magnitude) are rooted in bodily experiences and situated action. In the present study we evaluated whether there is an embodied representation of the place–value structure of the Arabic number system and if so whether this representation is influenced by situated aspects. In a two-digit number magnitude comparison task participants had to directly touch the larger of two numbers. Importantly, pointing responses were systematically biased toward the decade digit of the target number. Additionally, this leftward bias towards the tens digit was reduced in unit–decade incompatible number pairs. Thereby, we demonstrated an influence of place–value processing on manual pointing movement. Our results therefore corroborate the notion of an embodied representation of the place–value structure of Arabic numbers which is modulated by situated aspects.     

Numerical and non-numerical ordinality processing in children with and without developmental dyscalculia: Evidence from fMRI

Ordinality is – beyond numerical magnitude (i.e., quantity) – an important characteristic of the number system. There is converging empirical evidence that (intra)parietal brain regions mediate number magnitude processing. Furthermore, recent findings suggest that the human intraparietal sulcus (IPS) supports magnitude and ordinality in a domain-general way. However, the latter findings are derived from adult studies and with respect to children (i.e., developing brain systems) both the neural correlates of ordinality processing and the precise role of the IPS (domain-general vs. domain-specific) in ordinality processing are thus far unknown. The present study aims at filling this gap by employing functional magnetic resonance imaging (fMRI) to investigate numerical and non-numerical ordinality knowledge in children with and without developmental dyscalculia. In children (without DD) processing of numerical and non-numerical ordinality alike is supported by (intra)parietal cortex, thus extending the notion of a domain-general (intra)parietal cortex to developing brain systems. Moreover, activation extents in response to numerical ordinality processing differ significantly between children with and without dyscalculia in inferior parietal regions (supramarginal gyrus and IPS).     

The representation of pitch in auditory imagery: Evidence from S-R compatibility and distance effects

In three experiments we explored the nature of representations constructed during the perception and imagination of pitch. We employed a same–different task to eliminate the influence of nonauditory information and to minimise use of cognitive strategies on auditory imagery. A reference tone of frequency 1000, 1500, or 2000 Hz, or an imagined tone of a pitch indicated by a visual cue, was followed by a comparison tone (1000, 1500, or 2000 Hz) to which either a speeded same or different response was required. In separate experiments, same–different judgements were mapped to vertically (Experiments 1 and 2) and horizontally arranged responses (Experiment 3). Judgements of tones closer in pitch yielded longer reaction times and higher error rates than more distant tones, indicating a pitch distance effect for perceptual and imagery tasks alike. In addition, in the imagery task, same–different responses were faster when low-pitched tones demanded a bottom or left key response and high-pitched tones a top or right response than vice versa, suggesting that pitch is coded spatially. Together, these behavioural effects support the assumption that both perceived and imagined pitch are translated into an analogical representation in the spatial domain.     

Whorf reloaded: language effects on nonverbal number processing in first grade--a trilingual study

The unit-decade compatibility effect is interpreted to reflect processes of place value integration in two-digit number magnitude comparisons. The current study aimed at elucidating the influence of language properties on the compatibility effect of Arabic two-digit numbers in Austrian, Italian, and Czech first graders. The number word systems of the three countries differ with respect to their correspondence between name and place value systems; the German language is characterized by its inversion of the order of tens and units in number words as compared with digital notations, whereas Italian number words are generally not inverted and there are both forms for Czech number words. Interestingly, the German-speaking children showed the most pronounced compatibility effect with respect to both accuracy and speed. We interpret our results as evidence for a detrimental influence of an intransparent number word system place value processing. The data corroborate a weak Whorfian hypothesis in children, with even nonverbal Arabic number processing seeming to be influenced by linguistic properties in children.     

Number processing of Arabic and Hebrew bilinguals: Evidence supporting the distance effect

In the current study, a direct assessment of the effect of language lexical‐syntactic structure and magnitude semantic access on numerical processing was made by contrasting the performance of Arabic/Hebrew bilinguals in a digital (Hindi‐digits/Arabic‐digits) and verbal numerical comparison task (Arabic, an inverted language: Units‐Decades, Hebrew, a non‐inverted language: Decades‐Units). Our data revealed in the digital presentation format (Experiment 1) a regular distance effect in Arabic language‐Hindi digits and Hebrew language‐Arabic digits, characterized by an inverse relation between reaction times and numerical distance with no difference in the mean reaction times of participants in Arabic‐L1 and Hebrew‐L2. This indicates that both lexical digits of two‐digit numbers in L1 and L2 are similarly processed and semantically accessed. However in the verbal presentation format (Experiment 2) a similar pattern of distance effect was found, but the mean reaction times in Arabic were lower than in Hebrew in each numerical distance. This indicates that the processing of two‐digit number words in L1 and L2 is semantically accessed and determined by the syntactic structure of each language.     

发展性计算障碍的最新研究进展

发展性计算障碍是一种影响算术技能获得的特定的学习障碍。截至目前,有关发展性计算障碍的认知与神经机制的理论尚存分歧,有关诊断与鉴别标准也未统一。近年来,对于发展性计算障碍的理论假设有从一般认知因素取向到数学特定因素取向发展的趋势。而随着脑成像技术的不断发展,对于发展性计算障碍神经机制的研究也从针对单个脑区的特异性功能发展到从功能连接网络角度进行研究。并且研究者们开始尝试开发基于数学认知基本理论的干预方法,并采用了利用生物技术手段的新方法。基因-脑-行为的整合研究将有助于全面揭示发展性计算障碍的发生机制,而建立在系统理论基础上经过科学评估的干预手段将可能有效促进障碍者的计算能力。     

发展性计算障碍儿童的数认知

发展性计算障碍作为一种特异性学习障碍,影响儿童的数学成就与数学学习观念,因而引起了多学科研究的关注和重视。文章分析了发展性计算障碍领域在数认知方面的已有研究,揭示了发展性计算障碍的数认知缺陷及内部差异,从一般认知加工障碍与数字加工模块障碍两个角度阐述了发展性计算障碍的可能成因,最后对当前研究进行了小结与展望,认为数认知缺陷的复杂关系、亚类型特征、成因之间的相互作用与机制还需进一步研究     

一～四年级小学生发展性计算障碍的亚类型研究

本研究以1~4年级小学生为被试,探讨发展性计算障碍的亚类型.对筛选出的54名发展性计算障碍学生进行聚类分析,结果表明:在小学1~4年级学生中,存在着4种发展性计算障碍的亚类型,分别是听觉型、视觉型、类比型和听觉——类比型,这几种亚类型在数字加工和计算能力不同功能模块上的缺陷模式存在着差异.