Developing number–space associations: SNARC effects using a color discrimination task in 5-year-olds

Human adults’ numerical representation is spatially oriented; consequently, participants are faster to respond to small/large numerals with their left/right hand, respectively, when doing a binary classification judgment on numbers, known as the SNARC (spatial–numerical association of response codes) effect. Studies on the emergence and development of the SNARC effect remain scarce. The current study introduces an innovative new paradigm based on a simple color judgment of Arabic digits. Using this task, we found a SNARC effect in children as young as 5.5 years. In contrast, when preschool children needed to perform a magnitude judgment task necessitating exact number knowledge, the SNARC effect started to emerge only at 5.8 years. Moreover, the emergence of a magnitude SNARC but not a color SNARC was linked to proficiency with Arabic digits. Our results suggest that access to a spatially oriented approximate magnitude representation from symbolic digits emerges early in ontogenetic development. Exact magnitude judgments, on the other hand, rely on experience with Arabic digits and, thus, necessitate formal or informal schooling to give access to a spatially oriented numerical representation.     

亮度对空间-数字反应编码联合效应的影响

本研究探讨亮度对空间-数字反应编码联合效应（Spatial-Numerical Association of Response Codes,简称SNARC效应）的影响及其机制。通过三个实验设计不同的亮度对比水平,要求被试对阿拉伯数字1~9（5除外）进行奇偶判断。实验一将数字亮度设为最高值255时,结果出现了数字的SNARC效应。实验二将数字的亮度值分别设为255和213时,结果仍存在SNARC效应。实验三将亮度值分别设置为213和42时,数字的SNARC效应却消失了。这些结果表明亮度会激活或抑制数字的空间表征,可能与亮度对比值的高低及所消耗认知资源的多少有关。     

Implicit response-irrelevant number information triggers the SNARC effect: Evidence using a neural overlap paradigm

There is evidence from the SNARC (spatial–numerical association of response codes) effect and NDE (numerical distance effect) that number activates spatial representations. Most of this evidence comes from tasks with explicit reference to number, whether through presentation of Arabic digits (SNARC) or through magnitude decisions to nonsymbolic representations (NDE). Here, we report four studies that use the neural overlap paradigm developed by Fias, Lauwereyns, and Lammertyn (2001) to examine whether the presentation of implicit and task-irrelevant numerosity information (nonsymbolic arrays and auditory numbers) is enough to activate a spatial representation of number. Participants were presented with either numerosity arrays (1–9 circles or triangles) to which they made colour (Experiment 1) or orientation (Experiment 2) judgements, or auditory numbers coupled with an on-screen stimulus to which they made a colour (Experiment 3) or orientation (Experiment 4) judgement. SNARC effects were observed only for the orientation tasks. Following the logic of Fias et al., we argue that this SNARC effect occurs as a result of overlap in parietal processing for number and orientation judgements irrespective of modality. Furthermore, we found stronger SNARC effects in the small number range (1–4) than in the larger number range (6–9) for both nonsymbolic displays and auditory numbers. These results suggest that quantity is extracted (and interferes with responses in the orientation task) but this is not exact for the entire number range. We discuss a number of alternative models and mechanisms of numerical processing that may account for such effects.     

Task instructions determine the visuospatial and verbal–spatial nature of number–space associations

Evidence for number–space associations comes from the spatial–numerical association of response codes (SNARC) effect, consisting in faster reaction times to small/large digits with the left/right hand, respectively. Two different proposals are commonly discussed concerning the cognitive origin of the SNARC effect: the visuospatial account and the verbal–spatial account. Recent studies have provided evidence for the relative dominance of verbal–spatial over visuospatial coding mechanisms, when both mechanisms were directly contrasted in a magnitude comparison task. However, in these studies, participants were potentially biased towards verbal–spatial number processing by task instructions based on verbal–spatial labels. To overcome this confound and to investigate whether verbal–spatial coding mechanisms are predominantly activated irrespective of task instructions, we completed the previously used paradigm by adding a spatial instruction condition. In line with earlier findings, we could confirm the predominance of verbal–spatial number coding under verbal task instructions. However, in the spatial instruction condition, both verbal–spatial and visuospatial mechanisms were activated to an equal extent. Hence, these findings clearly indicate that the cognitive origin of number–space associations does not always predominantly rely on verbal–spatial processing mechanisms, but that the spatial code associated with numbers is context dependent.     

Implicit response-irrelevant number information triggers the SNARC effect: Evidence using a neural overlap paradigm

There is evidence from the SNARC (spatial-numerical association of response codes) effect and NDE (numerical distance effect) that number activates spatial representations. Most of this evidence comes from tasks with explicit reference to number, whether through presentation of Arabic digits (SNARC) or through magnitude decisions to nonsymbolic representations (NDE). Here, we report four studies that use the neural overlap paradigm developed by Fias, Lauwereyns, and Lammertyn (2001) to examine whether the presentation of implicit and task-irrelevant numerosity information (nonsymbolic arrays and auditory numbers) is enough to activate a spatial representation of number. Participants were presented with either numerosity arrays (1-9 circles or triangles) to which they made colour (Experiment 1) or orientation (Experiment 2) judgements, or auditory numbers coupled with an on-screen stimulus to which they made a colour (Experiment 3) or orientation (Experiment 4) judgement. SNARC effects were observed only for the orientation tasks. Following the logic of Fias et al., we argue that this SNARC effect occurs as a result of overlap in parietal processing for number and orientation judgements irrespective of modality. Furthermore, we found stronger SNARC effects in the small number range (1-4) than in the larger number range (6-9) for both nonsymbolic displays and auditory numbers. These results suggest that quantity is extracted (and interferes with responses in the orientation task) but this is not exact for the entire number range. We discuss a number of alternative models and mechanisms of numerical processing that may account for such effects.     

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.     

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。     

数字空间表征的在线建构：来自干扰情境中数字SNARC效应的证据

尽管已有研究发现数字以空间方式表征在人类记忆系统, 但是人脑如何完成数字的空间表征尚存争议。本研究两个实验在不同比例的数字字母(实验1)和不同比例的数字汉字(实验2)混合情境中考察了数字空间表征特点及其机制, 对上述争议进行了深入研究。结果发现：(1)当数字字母比例为“1 : 1”时, 数字加工中不出现SNARC效应。当数字字母比例为“1 : 6”和“6 : 1”时, 数字加工中均出现SNARC效应。即数字字母比例与数字SNARC效应之间呈倒“U”型关系。(2)数字汉字混合情境中数字汉字比例与数字SNARC效应之间同样呈倒“U”型关系。结果说明：(1)干扰刺激与数字混合呈现会影响数字SNARC效应。(2)干扰刺激加工对数字SNARC效应的影响受到数字与干扰刺激比例的调节, 且具有跨干扰材料的稳定性。研究结果意味着数字的空间表征是人类通过统计学习在线建构的, 支持了工作记忆理论。     

Coding strategies in number space: memory requirements influence spatial-numerical associations

The tendency to respond faster with the left hand to relatively small numbers and faster with the right hand to relatively large numbers (spatial numerical association of response codes, SNARC effect) has been interpreted as an automatic association of spatial and numerical information. We investigated in two experiments the impact of task-irrelevant memory representations on this effect. Participants memorized three Arabic digits describing a left-to-right ascending number sequence (e.g., 3-4-5), a descending sequence (e.g., 5-4-3), or a disordered sequence (e.g., 5-3-4) and indicated afterwards the parity status of a centrally presented digit (i.e., 1, 2, 8, or 9) with a left/right keypress response. As indicated by the reaction times, the SNARC effect in the parity task was mediated by the coding requirements of the memory tasks. That is, a SNARC effect was only present after memorizing ascending or disordered number sequences but disappeared after processing descending sequences. Interestingly, the effects of the second task were only present if all sequences within one experimental block had the same type of order. Taken together, our findings are inconsistent with the idea that spatial-numerical associations are the result of an automatic and obligatory cognitive process but do suggest that coding strategies might be responsible for the cognitive link between numbers and space.     

Two routes for the processing of verbal numbers: evidence from the SNARC effect

The functional locus of the semantic system is an important issue in number processing. In the present article, the necessity of addressing a central semantic magnitude system in the processing of printed verbal number words is evaluated by looking at the presence of a spatial-numerical association of response codes or SNARC effect. This effect consists of an association of number magnitude and response-preference (preferred responses to small numbers with the left hand and to large numbers with the right hand) and reflects semantic access. Two experiments were run. In Experiment 1, participants performed a parity judgment task which requires access to number semantics. A SNARC effect was observed. In Experiment 2 a phoneme monitoring task was used, which can, in principle, be performed through direct asemantic transcoding. No SNARC effect occurred. Apparently, written number words access the semantic system only if this is necessary for correct task completion. Hence, a semantic and an asemantic route can be postulated for the processing of word numerals. These observations contrast with the processing of Arabic numerals for which semantic effects are omnipresent. Implications of this explicit demonstration of a dissimilarity between the processing of digits and of number words are discussed. Received: 25 October 2000 / Accepted: 21 May 2001     

The impact of inhibition capacities and age on number–space associations

Numerical and spatial representations are tightly linked, i.e., when doing a binary classification judgment on Arabic digits, participants are faster to respond with their left/right hand to small/large numbers, respectively (Spatial-Numerical Association of Response Codes, SNARC effect, Dehaene et al. in J Exp Psychol Gen 122:371–396, 1993). To understand the underlying mechanisms of the well-established SNARC effect, it seems essential to explore the considerable inter-individual variability characterizing it. The present study assesses the respective roles of inhibition, age, working memory (WM) and response speed. Whereas these non-numerical factors have been proposed as potentially important factors to explain individual differences in SNARC effects, none (except response speed) has so far been explored directly. Confirming our hypotheses, the results show that the SNARC effect was stronger in participants that had weaker inhibition abilities (as assessed by the Stroop task), were relatively older and had longer response times. Interestingly, whereas a significant part of the age influence was mediated by cognitive inhibition, age also directly impacted the SNARC effect. Similarly, cognitive inhibition abilities explained inter-individual variability in number–space associations over and above the factors age, WM capacity and response speed. Taken together our results provide new insights into the nature of number–space associations by describing how these are influenced by the non-numerical factors age and inhibition.     

Loudness counts: Interactions between loudness,number magnitude,and space

ATOM (a theory of magnitude) suggests that magnitude information of different formats (numbers, space, and time) is processed within a generalized magnitude network. In this study we investigated whether loudness, as a possible indicator of intensity and magnitude, interacts with the processing of numbers. Small and large numbers, spoken in a quiet and a loud voice, were simultaneously presented to the left and right ear (Experiments 1a and 1b). Participants judged whether the number presented to the left or right ear was louder or larger. Responses were faster when the smaller number was spoken in a quiet voice, and the larger number in a loud voice. Thus, task-irrelevant numerical information influenced the processing of loudness and vice versa. This bi-directional link was also confirmed by classical SNARC paradigms (spatial–numerical association of response codes; Experiments 2a–2c) when participants again judged the magnitude or loudness of separately presented stimuli. In contrast, no loudness–number association was found in a parity judgment task. Regular SNARC effects were found in the magnitude and parity judgment task, but not in the loudness judgment task. Instead, in the latter task, response side was associated with loudness. Possible explanations for these results are discussed.     

Language statistics explain the spatial–numerical association of response codes

The spatial–numerical association of response codes (SNARC) has shown that parity judgments with participants’ left hands yield faster response times (RTs) for smaller numbers than for larger numbers, with the opposite result for right-hand responses. These findings have been explained by participants perceptually simulating magnitude on a mental number line. In three RT experiments, we showed that the SNARC effect can also be explained by language statistics. Participants made parity judgments of number words (Exp. 1) and Arabic numerals (Exp. 2). Linguistic frequencies of the number words and numbers mirrored the SNARC effect, explaining aspects of processing that a perceptual simulation account could not. In Experiment 3, we investigated whether high- and low-frequency nonnumerical words would also elicit a SNARC-like effect. Again, RTs were faster for high-frequency words for left-hand responses, with the opposite result for right-hand responses. These results demonstrate that what has only been attributed to perceptual simulation should also be attributed to language statistics.     

书写习惯对数字空间表征SNARC效应的影响

本文设计两个实验,采用大小判断任务,选取读写习惯相反的维、汉大学生为研究对象,探讨两类被试对阿拉伯数字和母语数字空间表征的一致性与差异性。结果发现：（1）维、汉被试均表现出对阿拉伯小数反应左手快于右手,对大数反应右手快于左手,即出现正向SNARC效应;（2）维吾尔族被试表现出对母语小数反应右手快于左手,对大数反应左手快于右手,即对母语数字出现反向SNARC效应;汉族被试表现出对母语小数反应左手快于右手,对大数反应右手快于左手,即对母语数字出现正向SNARC效应。结论：（1）维、汉被试对于阿拉伯数字的SNARC效应存在一致性;（2）维、汉被试母语数字的SNARC效应存在差异性;（3）数字空间对应性与书写习惯是导致结果的重要因素。     

不同加工深度非符号数量信息的SNARC效应：眼动证据

通过两项眼动实验考察了个体在水平和垂直方向上对点阵数量进行较浅(浏览)和较深(数字比较)程度加工时的数量空间表征联系。结果显示：点阵在水平方向上出现了SNARC效应, 且SNARC效应的大小不受加工深浅的影响, 而在垂直方向上没有出现SNARC效应。表明(1)点阵数量SNARC效应的稳定性不如阿拉伯数字; (2)方向对点阵SNARC效应的影响更大, 加工程度的影响则不明显。结合加工程度、空间方向和点阵数量的自身特征等对研究发现进行了讨论, 认为数字的空间表征在多个因素的共同影响下可能呈现出多样性。     

SNARC effects with numerical and non-numerical symbolic comparative judgments: instructional and cultural dependencies

With English-language readers in an experiment requiring pairwise comparative judgments of the sizes of animals, the nature of the association between the magnitudes of the animal pairs and the left or right sides of response (i.e., the SNARC effect) was reversed depending on whether the participants had to choose either the smaller or the larger member of the pair. In contrast, such a dependence of the direction of the SNARC effect on the form of the comparative instructions was not evident for pairwise comparisons of numerical magnitude made by a similar group of participants. Furthermore, exactly the same configuration of findings was obtained for a single group of Israeli-Palestinian right-to-left reading and writing participants, except that the spatial direction of the SNARC effects for both the animal-size and number comparisons were completely reversed. In a final experiment with English readers, SNARC effects paralleling those for the animal-size comparisons were obtained for pairwise comparative judgments involving the just-learned height relations between 6 imaginary individuals. As will be discussed, such results serve to extend the generality of the SNARC effect far beyond the current modal view that it simply reflects culturally influenced, long-term learned associations between numerical magnitudes and the locations on a fixed mental number line. The implications that these results have for both the Proctor and Cho (2006) polarity correspondence view and the Gevers, Verguts, Reynvoet, Caessens, and Fias (2006) computational model of the SNARC effect will also discussed.     

数字的空间特性

当要求被试对数字进行奇偶判断时,左手对小数的反应较快,而右手对大数的反应较快,该现象被称为空间数字反应编码联合效应（SNARC效应）。大量研究证实了SNARC效应的存在,该效应表明人类对数字的加工受空间表征和空间注意的影响。该文系统地回顾了SNARC效应存在及其发生阶段的证据,对比了Simon效应与SNARC效应,最后尝试着从空间注意的角度挖掘数字和空间的本质联     

Coding strategies in number space: Memory requirements influence spatial–numerical associations

The tendency to respond faster with the left hand to relatively small numbers and faster with the right hand to relatively large numbers (spatial numerical association of response codes, SNARC effect) has been interpreted as an automatic association of spatial and numerical information. We investigated in two experiments the impact of task-irrelevant memory representations on this effect. Participants memorized three Arabic digits describing a left-to-right ascending number sequence (e.g., 3–4–5), a descending sequence (e.g., 5–4–3), or a disordered sequence (e.g., 5–3–4) and indicated afterwards the parity status of a centrally presented digit (i.e., 1, 2, 8, or 9) with a left/right keypress response. As indicated by the reaction times, the SNARC effect in the parity task was mediated by the coding requirements of the memory tasks. That is, a SNARC effect was only present after memorizing ascending or disordered number sequences but disappeared after processing descending sequences. Interestingly, the effects of the second task were only present if all sequences within one experimental block had the same type of order. Taken together, our findings are inconsistent with the idea that spatial–numerical associations are the result of an automatic and obligatory cognitive process but do suggest that coding strategies might be responsible for the cognitive link between numbers and space.     

The mental representation of numerical fractions: real or integer?

Numerical fractions are commonly used to express ratios and proportions (i.e., real numbers), but little is known about how they are mentally represented and processed by skilled adults. Four experiments employed comparison tasks to investigate the distance effect and the effect of the spatial numerical association of response codes (SNARC) for fractions. Results showed that fractions were processed componentially and that the real numerical value of the fraction was not accessed, indicating that processing the fraction's magnitude is not automatic. In contrast, responses were influenced by the numerical magnitude of the components and reflected the simple comparison between numerators, denominators, and reference, depending on the strategy adopted. Strategies were used even by highly skilled participants and were flexibly adapted to the specific experimental context. In line with results on the whole number bias in children, these findings suggest that the understanding of fractions is rooted in the ability to represent discrete numerosities (i.e., integers) rather than real numbers and that the well-known difficulties of children in mastering fractions are circumvented by skilled adults through a flexible use of strategies based on the integer components.