Single-digit Arabic numbers do not automatically activate magnitude representations in adults or in children: Evidence from the symbolic same–different task

We investigated whether the mere presentation of single-digit Arabic numbers activates their magnitude representations using a visually-presented symbolic same–different task for 20 adults and 15 children. Participants saw two single-digit Arabic numbers on a screen and judged whether the numbers were the same or different. We examined whether reaction time in this task was primarily driven by (objective or subjective) perceptual similarity, or by the numerical difference between the two digits. We reasoned that, if Arabic numbers automatically activate magnitude representations, a numerical function would best predict reaction time; but if Arabic numbers do not automatically activate magnitude representations, a perceptual function would best predict reaction time. Linear regressions revealed that a perceptual function, specifically, subjective visual similarity, was the best and only significant predictor of reaction time in adults and in children. These data strongly suggest that, in this task, single-digit Arabic numbers do not necessarily automatically activate magnitude representations in adults or in children. As the first study to date to explicitly study the developmental importance of perceptual factors in the symbolic same–different task, we found no significant differences between adults and children in their reliance on perceptual information in this task. Based on our findings, we propose that visual properties may play a key role in symbolic number judgements.     

Response-retrieval and negative priming

The existence of across-notation automatic numerical processing of two-digit (2D) numbers was explored using size comparisons tasks. Participants were Arabic speakers, who use two sets of numerical symbols—Arabic and Indian. They were presented with pairs of 2D numbers in the same or in mixed notations. Responses for a numerical comparison task were affected by decade difference and unit-decade compatibility and global distance in both conditions, extending previous findings with Arabic digits (Nuerk, Weger, & Willmes, 2001). Responses for a physical comparison task were affected by congruency with the numerical size, as indicated by the size congruency effect (SiCE). The SiCE was affected by unit-decade compatibility but not by global distance, thus suggesting that the units and decades digits of the 2D numbers, but not the whole number value were automatically translated into a common representation of magnitude. The presence of similar results for same- and mixed-notation pairs supports the idea of an abstract representation of magnitude.     

Integers do not automatically activate their quantity representation

Researchers have generally come to the conclusion that integers automatically activate the quantity they symbolize and that this quantity dominates responding. I conducted a strong test of this hypothesis with two numerical same/different experiments. On each trial, I presented the participant an integer between 1 and 9 and asked him or her to identify whether that symbol was a 5. If quantity information dominates responding, participants’ reaction times (RTs) should be a function of the numerical distance between the target and the distractor. If quantity information is not activated, the integer is merely a shape, and participants’ RTs should be a function of the physical similarity of the target and the distractor. The data from Experiments 1 and 2 demonstrate that quantity information exerts no control and that physical similarity is the primary controlling factor. These findings demonstrate that integers maintain a level of independence from their quantity representations.     

Semantic processing of Arabic,Kanji, and Kana numbers: Evidence from interference in physical and numerical size judgments

Two experiments were conducted with the following objectives (1) to replicate the finding of similar semantic representation of Arabic and written-word (Kanji and Kana) numbers with a direct numerical task, (2) to investigate the automatic semantic processing of Arabic and written-word numbers, and (3) to verify whether the assumption of a common semantic representation is valid in an indirect numerical task. Subjects were asked to judge which of two numbers (e.g, 6-8) was larger either in its numerical size (Experiment 1) or in its physical size (Experiment 2) using the three notations. Effects of two factors were analyzed: the congruity between numerical and physical size and the numerical distance. The effects of these factors were very similar across the three notations in Experiment 1, but were drastically different in Experiment 2. The results of Experiment 2 demonstrated the nonsemantic processing of Kana numbers, and suggest that there may be separate semantic representations for Arabic and Kanji numbers.     

Across-notation automatic numerical processing

In this article, the authors explored the existence of across-notation automatic numerical processing using size comparison and same-different paradigms. Participants were Arabic speakers, who used 2 sets of numerical symbols -- Arabic and Indian. They were presented with number pairs in the same notation (Arabic or Indian) or in different ones (Arabic and Indian). In the size comparison paradigm, 2 digits differing both numerically and physically were compared on the physical dimension. Nevertheless, there was evidence that participants automatically processed the irrelevant numerical dimension in different notation pairs. In the same-different paradigm, 2 digits were presented either in the same or in different notations. Participants had to indicate whether the 2 digits were physically the same. The results again showed evidence for the automatic processing of numerical magnitude for pairs in different notations. Findings of both experiments suggest that numbers in different notations are automatically translated into a common representation of magnitude, in line with M. McCloskey's (1992) abstract representation model.     

The numerical distance effect is task dependent

Number comparison tasks produce a distance effect e.g., Moyer & Landauer (Nature 215: 1519-1520, 1967). It has been suggested that this effect supports the existence of semantic mental representations of numbers. In a matching task, a distance effect also appears, which suggests that the effect has an automatic semantic component. Recently, Cohen (Psychonomic Bulletin & Review 16: 332-336, 2009) suggested that in both automatic and intentional tasks, the distance effect might reflect not a semantic number representation, but a physical similarity between digits. The present article (1) compares the distance effect in the automatic matching task with that in the intentional number comparison task and suggests that, in the latter, the distance effect does include an additional semantic component; and (2) indicates that the distance effect in the standard automatic matching task is questionable and that its appearance in previous matching tasks was based on the specific analysis and design that were applied.     

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.     

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.     

Differences between literates and illiterates on symbolic but not nonsymbolic numerical magnitude processing

The study of numerical magnitude processing provides a unique opportunity to examine interactions between phylogenetically ancient systems of semantic representations and those that are the product of enculturation. While nonsymbolic representations of numerical magnitude are processed similarly by humans and nonhuman animals, symbolic representations of numerical magnitude (e.g., Hindu–Arabic numerals) are culturally invented symbols that are uniquely human. Here, we report a comparison of symbolic and nonsymbolic numerical magnitude processing in two groups of participants who differ substantially in their level of literacy. In this study, level of literacy is used as an index of level of school-based numeracy skill. The data from these groups demonstrate that while the processing of nonsymbolic numerical magnitude (numerical distance effect) is unaffected by an individual’s level of literacy, the processing of Hindu–Arabic numerals differs between literate and illiterate individuals who live in a literature culture and have limited symbolic recognition skills. These findings reveal that nonsymbolic numerical magnitude processing is unaffected by enculturation, while the processing of numerical symbols is modulated by literacy.     

Representing quantity beyond whole numbers: some, none, and part.

Previous research has demonstrated how children develop the ability to use notational representations to indicate simple quantities. These studies have shown a developmental shift from the use of idiosyncratic, to analogical, to conventional, numerical notations. The present paper extends these findings by reporting the results from a study in which children from 3 to 7 years old were asked to write a representation to indicate a quantity presented in a game-like scenario. Three kinds of quantities were included: whole numbers, zeros, and fractions. The children's notations were shown to them shortly after they were produced and then again two weeks later to see if children could interpret them. The results showed the familiar developmental pattern towards increased use of conventional notations for all quantities. The ability to read the notations was greatest for conventional numbers where performance was at ceiling, lower for analogue representations, and very poor for idiosyncratic global recordings. Children's choice of a notational format was influenced almost entirely by their age and not by the quantity being represented. Children were able to solve the zero problems almost as well as they could the whole numbers, but their understanding and use of representations for fractions was very limited.     

Are Arabic and verbal numbers processed in different ways?

Four experiments were conducted in order to examine effects of notation--Arabic and verbal numbers--on relevant and irrelevant numerical processing. In Experiment 1, notation interacted with the numerical distance effect, and irrelevant physical size affected numerical processing (i.e., size congruity effect) for both notations but to a lesser degree for verbal numbers. In contrast, size congruity had no effect when verbal numbers were the irrelevant dimension. In Experiments 2 and 3, different parameters that could possibly affect the results, such as discriminability and variability (Experiment 2) and the block design (Experiment 3), were controlled. The results replicated the effects obtained in Experiment 1. In Experiment 4, in which physical size was made more difficult to process, size congruity for irrelevant verbal numbers was observed. The present results imply that notation affects numerical processing and that Arabic and verbal numbers are represented separately, and thus it is suggested that current models of numerical processing should have separate comparison mechanisms for verbal and Arabic numbers.     

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.     

Differences between digit naming and number word reading in a flanker task

Numbers can be represented as number words or as digits, but are the two notations processed differently? Two experiments in which a flanker paradigm with a naming task was used were conducted, with digits and number words as targets and flankers. Reaction times were shortest when the flanker denoted the same numerical value as the target. The numerical distance between the target and a numerically different flanker modulated reaction times in all conditions, except for number word targets with digit flankers. The direction of this effect—targets were named faster when the flanker was numerically close than when it was far—indicates that the numerical magnitude representations of numbers are associatively connected. When the target and the flanker were presented in the same format, no difference was observed in the distance effects for the two formats. This indicates that number words activate the abstract representation of their numerical value in a way that is very similar to that for digits.     

不同注意条件下大数与小数的加工差异

考察在注意（注视点）与非注意（非注视点）条件下数字加工的距离效应和符号效应。采用小数（1－4）和大数（6－9）的中文与阿拉伯数字为材料,以判断数字是否大于5为任务。实验结果表明：⑴ 在注意条件下,大小数都出现了距离效应;而在非注意条件下,只有小数出现距离效应;⑵ 在注意条件下,大小数都没有出现符号效应;而在非注意条件下,只有小数出现符号效应,中文数字绩效显著好于阿拉伯数字。     

不同手指表征方式对数量表征能力的影响

采用距离启动范式,考察中国文化背景下不同手指表征方式对数量表征能力的影响。实验首先验证单手表征中不同手指数量表征方式对小数字（1～5）认知表征的影响;实验2则进一步采用中国人特有的单手手指表征,考察其对大数字（5～9）认知表征的影响。结果表明,小数字中出现了标准手指表征方式语义层面的位置编码、非标准手指表征方式知觉层面总和编码的激活;但大数字中两种手指表征方式均出现了语义层面位置编码的激活。此结果与计算模型理论一致,说明当手指数量从少到多变化时,标准手指表征方式为语义性的符号数量表征;而非标准手指表征方式由知觉性的非符号向语义性符号数量表征过渡。     

Toward a scalable holographic word-form representation

Phenomena in a variety of verbal tasks—for example, masked priming, lexical decision, and word naming—are typically explained in terms of similarity between word-forms. Despite the apparent commonalities between these sets of phenomena, the representations and similarity measures used to account for them are not often related. To show how this gap might be bridged, we build on the work of Hannagan, Dupoux, and Christophe, Cognitive Science 35:79-118, (2011) to explore several methods of representing visual word-forms using holographic reduced representations and to evaluate them on their ability to account for a wide range of effects in masked form priming, as well as data from lexical decision and word naming. A representation that assumes that word-internal letter groups are encoded relative to word-terminal letter groups is found to predict qualitative patterns in masked priming, as well as lexical decision and naming latencies. We then show how this representation can be integrated with the BEAGLE model of lexical semantics (Jones & Mewhort, Psychological Review 114:1–37, 2007) to enable the model to encompass a wider range of verbal tasks.     

听障和听力正常人群空间主导性和空间参照框架的交互作用

通过要求被试分别在近处空间和远处空间完成空间参照框架的判断任务, 考察了听障和听力正常人群空间主导性和空间参照框架的交互作用。结果表明：(1)相对于听力正常人群, 听障人群完成自我参照框架判断任务的反应时更长, 而在完成环境参照框架判断任务无显著差异; (2)听障人群和听力正常人群空间主导性和空间参照框架交互作用呈现出相反模式。研究表明, 听障人群在听力功能受损后, 其空间主导性和空间参照框架的交互作用也产生了变化。     

Decade breaks in the mental number line? Putting the tens and units back in different bins.

Most models of number recognition agree that among other number representations there is a central semantic magnitude representation which may be conceptualized as a logarithmically compressed mental number line. Whether or not this number line is decomposed into different representations for tens and units is, however, controversial. We investigated this issue in German participants in a magnitude comparison (selection) task in which the larger of two visually presented Arabic two-digit numbers had to be selected. Most importantly, we varied unit-decade-compatibility: a number pair was defined as compatible if the decade magnitude comparison and the unit magnitude comparison of the two numbers would lead to the same response (e.g. 52 and 67) and as incompatible if this was not the case (e.g. 47 and 62). While controlling for overall numerical distance, size and other variables, we consistently found compatibility effects. A control experiment showed that this compatibility effect was not due to perceptual presentation characteristics. We conclude that the idea of one single number line representation that does not additionally assume different magnitude representations for tens and units is not sufficient to account for the data. Finally, we discuss why decade effects were not found in other experimental settings.     

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.