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).     

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]).     

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.     

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.     

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。     

Typical and atypical development of basic numerical skills in elementary school

Deficits in basic numerical processing have been identified as a central and potentially causal problem in developmental dyscalculia; however, so far not much is known about the typical and atypical development of such skills. This study assessed basic number skills cross-sectionally in 262 typically developing and 51 dyscalculic children in Grades 2, 3, and 4. Findings indicate that the efficiency of number processing improves over time and that dyscalculic children are generally less efficient than children with typical development. For children with typical arithmetic development, robust effects of numerical distance, size congruity, and compatibility of ten and unit digits in two-digit numbers could be identified as early as the end of Grade 2. Only the distance effect for comparing symbolic representations of numerosities changed developmentally. Dyscalculic children did not show a size congruity effect but showed a more marked compatibility effect for two-digit numbers. We did not find strong evidence that dyscalculic children process numbers qualitatively differently from children with typical arithmetic development.     

The SNARC effect in the processing of second-language number words: Further evidence for strong lexico-semantic connections

We present new evidence that word translation involves semantic mediation. It has been shown that participants react faster to small numbers with their left hand and to large numbers with their right hand. This SNARC (spatial-numerical association of response codes) effect is due to the fact that in Western cultures the semantic number line is oriented from left (small) to right (large). We obtained a SNARC effect when participants had to indicate the parity of second-language (L2) number words, but not when they had to indicate whether L2 number words contained a particular sound. Crucially, the SNARC effect was also obtained in a translation verification task, indicating that this task involved the activation of number magnitude.     

Cognitive deficits in developmental disorders

The existence of specific developmental disorders such as dyslexia and autism raises interesting issues about the structure of the normally developing mind. In these disorders distinct cognitive deficits can explain a range of behavioural impairments and have the potential to be linked to specific brain abnormalities. One possibility is that there are specific mechanisms dedicated to particular types of information processing. These mechanisms may function independently of more general information processing systems and may have a distinct anatomical basis in the brain.     

符号数量和非符号数量的空间表征:5岁儿童的SNARC效应和距离效应

对物理刺激的数量信息表征是符号数字表征的前提和基础,据此假设在儿童的SNARC效应发生的时序问题上,非符号数量(如面积)的空间表征早于符号数量(如阿拉伯数字)的空间表征。本研究邀请5岁幼儿完成数字比较和面积比较两类任务,结果发现在数字比较任务中没有出现SNARC效应,但却存在距离效应;在面积比较任务中出现了SNARC效应和距离效应。可以推断,在阿拉伯数字的空间表征出现之前,儿童已经能够对非符号数量信息进行空间表征。     

The effect of psychological distance on perceptual level of construal

Three studies examined the effect of primed psychological distance on level of perceptual construal, using Navon's paradigm of composite letters (global letters that are made of local letters). Relative to a control group, thinking of the more distant future (Study 1), about more distant spatial locations (Study 2), and about more distant social relations (Study 3) facilitated perception of global letters relative to local letters. Proximal times, spatial locations, and social relations had the opposite effect. The results are discussed within the framework of Construal Level Theory of psychological distance ( Liberman & Trope, 2008 ; Trope & Liberman, 2003 ).