The mental representation of ordinal sequences is spatially organized

In the domain of numbers the existence of spatial components in the representation of numerical magnitude has been convincingly demonstrated by an association between number magnitude and response preference with faster left- than right-hand responses for small numbers and faster right- than left-hand responses for large numbers (Dehaene, S., Bossini, S., & Giraux, P. (1993) The mental representation of parity and number magnitude. Journal of Experimental Psychology: General, 122, 371-396). Because numbers convey not only real or integer meaning but also ordinal meaning, the question of whether non-numerical ordinal information is spatially coded naturally follows. While previous research failed to show an association between ordinal position and spatial response preference, we present two experiments involving months (Experiment 1) and letters (Experiment 2) in which spatial coding is demonstrated. Furthermore, the response-side effect was obtained with two different stimulus-response mappings. The association occurred both when ordinal information was relevant and when it was irrelevant to the task, showing that the spatial component of the ordinal representation can be automatically activated.     

Moving the eyes along the mental number line: comparing SNARC effects with saccadic and manual responses

Bimanual parityjudgments about numerically small (large) digits are faster with the left (right) hand, even though parity is unrelated to numerical magnitude per se (the SNARC effect; Dehaene, Bossini, & Giraux, 1993). According to one model, this effect reflects a space-related representation of numerical magnitudes (mental number line) with a genuine left-to-right orientation. Alternatively, it may simply reflect an overlearned motor association between numbers and manual responses--as, for example, on typewriters or computer keyboards--in which case it should be weaker or absent with effectors whose horizontal response component is less systematically associated with individual numbers. Two experiments involving comparisons of saccadic and manual parity judgment tasks clearly support the first view; they also establish a vertical SNARC effect, suggesting that our magnitude representation resembles a number map, rather than a number line.     

负数的空间表征引起的空间注意转移

本研究采用数字线索提示的刺激探测任务, 通过三个实验探讨负数的低水平加工能否,以及怎样引起空间注意的转移。实验一探讨只有负数单独呈现作为线索时能否引起空间注意的转移。结果表明：对负数绝对值大小的加工能引起空间注意的转移。实验二进一步探讨在正数、负数和零混合作为线索时能否引起空间注意的转移。结果表明：对负数数量大小的加工能引起空间注意的转移。实验三再次用正数, 负数和0三种数字混合作为探测刺激前的线索, 但仅对负数和零作为提示线索之后的探测刺激进行反应, 又一次得到了由有效提示线索所引发的对数字数量大小加工引起的空间注意的转移。本研究表明, 对负数的低水平加工可以引起空间注意的转移, 然而, 是对绝对值的加工还是数量大小的加工引起注意转移依赖于共同参与的其它数字加工产生的影响。     

Spatial structure of quantitative representation of numbers: Evidence from the SNARC effect

Dehaene, Bossini, and Giraux (1993) revealed that subjects responded to large numbers faster with the choice on the right than with the choice on the left, whereas the reverse held true for small numbers (SNARC effect). According to Dehaene et al. (1993), the SNARC effect depends on the quantitative representation of number, such as a left-to-right-oriented analog number line. The main goal of the present study was twofold: first, to investigate whether the vertical SNARC effect could be observed, and, second, to verify whether Dehaene et al.'s (1993) explanation of the SNARC effect is correct. Experiments 2A and 2B showed the vertical SNARC effect in a parity judgment task. Subjects responded to large numbers faster with the top choice than with the bottom choice, whereas the reverse held true for small numbers. However, Experiment 3 failed to show the SNARC effect in a number magnitude judgment task, suggesting that the quantitative representation could be dissociated from the spatial code that produces the SNARC effect.     

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。     

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     

The representational space of numerical magnitude: Illusions of length

In recent years, a growing amount of evidence concerning the relationships between numerical and spatial representations has been interpreted, by and large, in favour of the mental number line hypothesis—namely, the analogue continuum where numbers are spatially represented (Dehaene, 1992 Dehaene, S. 1992. Varieties of numerical abilities. Cognition, 44: 1–42. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]; Dehaene, Piazza, Pinel, & Cohen, 2003 Dehaene, S., Piazza, M., Pinel, P. and Cohen, L. 2003. Three parietal circuits of number processing. Cognitive Neuropsychology, 20: 487–506. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). This numerical representation is considered the core of number meaning and, accordingly, needs to be accessed whenever numbers are semantically processed. The present study explored, by means of a length reproduction task, whether besides the activation of lateralized spatial codes, numerical processing modulates the mental representation of a horizontal spatial extension. Mis-estimations of length induced by Arabic numbers are interpreted in terms of a cognitive illusion, according to which the elaboration of magnitude information brings about an expansion or compression of the mental representation of spatial extension. These results support the hypothesis that visuo-spatial resources are involved in the representation of numerical magnitude.     

The link between mental rotation ability and basic numerical representations

Mental rotation and number representation have both been studied widely, but although mental rotation has been linked to higher-level mathematical skills, to date it has not been shown whether mental rotation ability is linked to the most basic mental representation and processing of numbers. To investigate the possible connection between mental rotation abilities and numerical representation, 43 participants completed four tasks: 1) a standard pen-and-paper mental rotation task; 2) a multi-digit number magnitude comparison task assessing the compatibility effect, which indicates separate processing of decade and unit digits; 3) a number-line mapping task, which measures precision of number magnitude representation; and 4) a random number generation task, which yields measures both of executive control and of spatial number representations. Results show that mental rotation ability correlated significantly with both size of the compatibility effect and with number mapping accuracy, but not with any measures from the random number generation task. Together, these results suggest that higher mental rotation abilities are linked to more developed number representation, and also provide further evidence for the connection between spatial and numerical abilities.     

Dissociation of the distance effect and size effect in one-digit numbers

Magnitude comparison of single digits is robustly characterized by a distance effect (close numbers are more difficult to compare than numbers further apart) and a size effect (for a given distance, comparison difficulty increases with increasing size). The distance effect indicates access to the mental number line (Dehaene, 1997), and the size effect is usually interpreted as indicating that the mental number line represents larger numbers more vaguely than smaller ones. In contrast, we have argued earlier (Verguts, Fias, & Stevens, 2005) that for symbolic numbers (Arabic or verbal notation), the size effect does not originate from the mental number line but, instead, originates from mappings to relevant output components that are specific for magnitude comparison. If the latter is true, it should be possible to dissociate the distance effect from the size effect in tasks other than magnitude comparison. In two experiments, we observed a robust distance effect insame/different judgments, which implies access to the mental number line. Yet the size effect was absent. Consistent with our prediction, this finding establishes a dissociation between the size effect and the distance effect.     

SNARC效应: 现状、理论与建议

SNARC效应是当对数字进行奇偶判断时,即使数的奇偶性与数的大小无关,但右手（左手）对相对大（小）的数的反应快。首先介绍SNARC效应的起源和理论解释,然后总结SNARC效应的特性,论述SNARC效应和Simon效应以及MARC 效应的关系,并对SNARC效应的脑机制进行了概述,最后提出3个有待深入研究的问题：（1）SNARC效应的加工处理机制;（2）SANRC效应的理论探索;（3）SNARC效应的本质。     

Numbers and space: a computational model of the SNARC effect

The SNARC (spatial numerical associations of response codes) effect reflects the tendency to respond faster with the left hand to relatively small numbers and with the right hand to relatively large numbers (S. Dehaene, S. Bossini, & P. Giraux, 1993). Using computational modeling, the present article aims to provide a framework for conceptualizing the SNARC effect. In line with models of spatial stimulus-response congruency, the authors modeled the SNARC effect as the result of parallel activation of preexisting links between magnitude and spatial representation and short-term links created on the basis of task instructions. This basic dual-route model simulated all characteristics associated with the SNARC effect. In addition, 2 experiments tested and confirmed new predictions derived from the model.     

Numerical and nonnumerical estimation in children with and without mathematical learning disabilities

There are currently multiple explanations for mathematical learning disabilities (MLD). The present study focused on those assuming that MLD are due to a basic numerical deficit affecting the ability to represent and to manipulate number magnitude ( Butterworth, 1999 , 2005 ; A. J. Wilson & Dehaene, 2007 ) and/or to access that number magnitude representation from numerical symbols ( Rousselle & No?l, 2007 ). The present study provides an original contribution to this issue by testing MLD children (carefully selected on the basis of preserved abilities in other domains) on numerical estimation tasks with contrasting symbolic (Arabic numerals) and nonsymbolic (collection of dots) numbers used as input or output. MLD children performed consistently less accurately than control children on all the estimation tasks. However, MLD children were even weaker when the task involved the mapping between symbolic and nonsymbolic numbers than when the task required a mapping between two nonsymbolic numerical formats. Moreover, in the estimation of nonsymbolic numerosities, MLD children relied more than control children on perceptual cues such as the cumulative area of the dots. Finally, the task requiring a mapping from a nonsymbolic format to a symbolic format was the best predictor of MLD. In order to explain these present results, as well as those reported in the literature, we propose that the impoverished number magnitude representation of MLD children may arise from an initial mapping deficit between number symbols and that magnitude representation.     

The universal SNARC effect: the association between number magnitude and space is amodal

It is thought that number magnitude is represented in an abstract and amodal way on a left-to-right oriented mental number line. Major evidence for this idea has been provided by the SNARC effect (Dehaene, Bossini, & Giraux, 1993): responses to relatively larger numbers are faster for the right hand, those to smaller numbers for the left hand, even when number magnitude is irrelevant. The SNARC effect has been used to index automatic access to a central semantic and amodal magnitude representation. However, this assumption of modality independence has never been tested and it remains uncertain if the SNARC effect exists in other modalities in a similar way as in the visual modality. We have examined this question by systematically varying modality/notation (auditory number word, visual Arabic numeral, visual number word, visual dice pattern) in a within-participant design. The SNARC effect was found consistently for all modality/notation conditions, including auditory presentation. The size of the SNARC effect in the auditory condition did not differ from the SNARC effect in any visual condition. We conclude that the SNARC effect is indeed a general index of a central semantic and amodal number magnitude representation.     

Automatic response activation of implicit spatial information: Evidence from the SNARC effect

In the present paper, we focus on how irrelevant implicit spatial information is processed. By irrelevant we mean information that is not required to fulfill the task and by implicit we mean information that is not directly available in the external stimulus. A good example of a task in which such information exists is the SNARC task [Dehaene, S., Bossini, S., & Giraux, P. (1993). The mental representation of parity and number magnitude. Journal of Experimental Psychology: General, 122, 371-396]. The SNARC effect shows that the magnitude of a number, although irrelevant to the task, activates spatial codes that may interfere with the task-related response. These spatial associations exist both for the horizontal and the vertical direction. In Experiment 1, response keys were discriminating in the vertical or the horizontal direction. It is shown that the impact of the numerical spatial codes on overt behavior, although automatic, depends on the response discrimination of the horizontal or the vertical dimension. In Experiment 2, response keys were assigned such that both the horizontal and the vertical direction of the response were discriminating. In this case, the horizontal and the vertical dimension of the irrelevant numerical spatial codes were shown to interact. In general, the results are in line with the response-discrimination account [Ansorge, U., & Wühr, P. (2004). A response-discrimination account of the Simon effect. Journal of Experimental Psychology: Human Perception and Performance, 30, 365-377].     

数形联觉——数字空间表征研究“新”取向

数字空间表征是人类对数字进行表征的重要方式。数形联觉（number-form synesthesia）是一种数字可以有意识地引起空间知觉的独特现象,与此类似的是非联觉者中广泛存在的无意识的心理数字线（mental number line）现象。两者在行为和脑机制上存在着很多重叠,也存在着值得思考的差异。数形联觉的研究能够提供实质性的行为和脑机制数据,用以解决数字空间表征研究中出现的问题,加强对于数字空间表征的理解;也为更加全面深入地开展进一步研究提供了新的启示,成为数字空间表征研究中值得推崇的新取向。     

Mental number line disruption in a right-neglect patient after a left-hemisphere stroke

A right-neglect patient with focal left-hemisphere damage to the posterior superior parietal lobe was assessed for numerical knowledge and tested on the bisection of numerical intervals and visual lines. The semantic and verbal knowledge of numbers was preserved, whereas the performance in numerical tasks that strongly emphasize the visuo-spatial layout of numbers (e.g. number bisection) was impaired. The behavioral pattern of error in the two bisection tasks mirrored the one previously described in left-neglect patients. In other words, our patient misplaced the subjective midpoint (numerical or visual) to the left as function of the interval size. These data, paired with the patient's lesion site are strictly consistent with the tripartite organization of number-related processes in the parietal lobes as proposed by Dehaene and colleagues. According to these authors, the posterior superior parietal lobe on both hemispheres underpins the attentional orientation on the putative mental number line, the horizontal segment of the intraparietal sulcus is bilaterally related to the semantic of the numerical domain, whereas the left angular gyrus subserves the verbal knowledge of numbers. In summary, our results suggest that the processes involved in the navigation along the mental number line, which are related to the parietal mechanisms for spatial attention, and the processes involved in the semantic and verbal knowledge of numbers, are dissociable.     

The mental number line electrified: brain potentials in a numerical flanker task

It has been suggested that the mental representation of numbers is spatial in nature such that numbers are ordered on a mental number line. In the present investigation we use a variant of the Eriksen flanker task requiring a magnitude decision (smaller or larger than 5) for a central target number by pressing a response button with the right or left hand. The target number is flanked by irrelevant distracters that are either identical to the target, different from the target but biasing the same response, or different from the target and biasing a different response. Response latencies and event-related brain potentials were obtained in a group of healthy adults. Besides the typical response congruency effects on response latency and the N2 component of the ERP, we observed several other effects. First, numerical distance of the target to the standard 5 influenced decision latencies and amplitude and latency of the P3 component with smaller distances leading to longer decision latencies, longer P3 latencies and smaller P3 amplitudes. Second, smaller numerical distance between target and distracters led to faster decisions for response congruent and to slower decisions for response-incongruent trials. For response-incongruent trials P3 amplitude was small/large and P3 latency was long/short for small/large distances. These findings underscore the spatial character of number representation and further show that the relation between targets and distracters, although task irrelevant, is assessed automatically with facilitatory and inhibitory effects driven by spatial distance on the mental number line.     

On the mental representation of negative numbers: Context-dependent SNARC effects with comparative judgments

In one condition, positive and negative number pairs were compared in separate blocks of trials. In another condition, the positive and the negative number pairs were intermixed. In the intermixed condition, comparisons involving negative numbers were faster with the left hand than with the right, and comparisons were faster with the right hand than with the left hand with the positive numbers; that is, a spatial numerical association of response codes (SNARC) effect was obtained, in which the mental number line was extended leftward with the negative numbers. On the other hand, in the blocked condition, a reverse SNARC effect was obtained with the negative numbers; that is, negative number pairs have the same underlying spatial representation as the positive numbers in this context. Nongraded semantic congruity effects, obtained in both the blocked and the intermixed conditions, are consistent with the idea that magnitude information is extracted prior to the generation of discrete semantic codes.     

"1-2-3": is there a temporal number line? Evidence from a serial comparison task

Evidence suggests that numbers are intimately related to space (Dehaene, Bossini, & Giraux, 1993; Hubbard, Piazza, Pinel, & Dehaene, 2005). Recently, Walsh (2003) suggested that numbers might also be closely related to time. To investigate this hypothesis we asked participants to compare two digits that were presented in a serial manner, i.e., one after another. Temporally ascending digit pairs (such as 2-3) were responded to faster than temporally descending pairs (3-2). This effect was, in turn, qualified by a local SNARC (spatial numerical association of response codes) effect and a local semantic congruity effect (SCE). Moreover, we observed a global numerical SCE only for temporally descending digit pairs. However, we did not observe a global SNARC effect, i.e., an interaction of numerical magnitude and the right/left response hand. We discuss our results in terms of overlearned forward-associations ("1-2-3") as formed by our ubiquitous cognitive routines to count off objects or events.     

Sensori-motor spatial training of number magnitude representation

An adequately developed spatial representation of number magnitude is associated with children's general arithmetic achievement. Therefore, a new spatial-numerical training program for kindergarten children was developed in which presentation and response were associated with a congruent spatial numerical representation. In particular, children responded by a full-body spatial movement on a digital dance mat in a magnitude comparison task. This spatial-numerical training was more effective than a non-spatial control training in enhancing children's performance on a number line estimation task and a subtest of a standardized mathematical achievement battery (TEDI-MATH). A mediation analysis suggested that these improvements were driven by an improvement of children's mental number line representation and not only by unspecific factors such as attention or motivation. These results suggest a benefit of spatial numerical associations. Rather than being a merely associated covariate, they work as an independently manipulated variable which is functional for numerical development.