Small on the left,large on the right: numbers orient visual attention onto space in preverbal infants

Numbers are represented as ordered magnitudes along a spatially oriented number line. While culture and formal education modulate the direction of this number–space mapping, it is a matter of debate whether its emergence is entirely driven by cultural experience. By registering 8–9‐month‐old infants’ eye movements, this study shows that numerical cues are critical in orienting infants’ visual attention towards a peripheral region of space that is congruent with the number's relative position on a left‐to‐right oriented representational continuum. This finding provides the first direct evidence that, in humans, the association between numbers and oriented spatial codes occurs before the acquisition of symbols or exposure to formal education, suggesting that the number line is not merely a product of human invention.     

Images of numbers, or "When 98 is upper left and 6 sky blue".

Some people declare that they possess a personal visual representation of numbers: some automatically "see" the numbers they are confronted with in a precise location in a structured mental space, others "associate" specific colours with given numbers. Such visuo-spatial representations of numbers were first described by Galton in 1880 but have since received little attention from psychologists. It is the aim of this article to describe these mental representations and discuss their role in number processing. The authors first review Galton's observations, and then present their own. Finally, they discuss the relevance of these visuo-spatial representations in relation to contemporary debates on number representation and calculation.     

人类数能力的演化基础——数能力比较研究的启示

人类的数能力可以发展到很高的抽象水平,但大量研究表明,人类婴儿和非人灵长类也具有基本的数表征和数推理能力。文章从表征内容和表征形式两个方面系统地比较了人类婴儿和非人灵长类的数能力,并对比了成人和非人灵长类数表征的生理基础。人类和非人灵长类在这三方面存在的相似性揭示了他们可能享有相同的数表征系统。在今后的研究中,进一步探讨这两者相似的数表征核心系统,可以加深我们对人类数能力起源和本质的理解     

The Relational SNARC: Spatial Representation of Nonsymbolic Ratios

Recent research in numerical cognition has begun to systematically detail the ability of humans and nonhuman animals to perceive the magnitudes of nonsymbolic ratios. These relationally defined analogs to rational numbers offer new potential insights into the nature of human numerical processing. However, research into their similarities with and connections to symbolic numbers remains in its infancy. The current research aims to further explore these similarities by investigating whether the magnitudes of nonsymbolic ratios are associated with space just as symbolic numbers are. In two experiments, we found that responses were faster on the left for smaller nonsymbolic ratio magnitudes and faster on the right for larger nonsymbolic ratio magnitudes. These results further elucidate the nature of nonsymbolic ratio processing, extending the literature of spatial–numerical associations to nonsymbolic relative magnitudes. We discuss potential implications of these findings for theories of human magnitude processing in general and how this general processing relates to numerical processing.     

A mental number line in human newborns

Humans represent numbers on a mental number line with smaller numbers on the left and larger numbers on the right side. A left‐to‐right oriented spatial–numerical association, (SNA), has been demonstrated in animals and infants. However, the possibility that SNA is learnt by early exposure to caregivers’ directional biases is still open. We conducted two experiments: in Experiment 1, we tested whether SNA is present at birth and in Experiment 2, we studied whether it depends on the relative rather than the absolute magnitude of numerousness. Fifty‐five‐hour‐old newborns, once habituated to a number (12), spontaneously associated a smaller number (4) with the left and a larger number (36) with the right side (Experiment 1). SNA in neonates is not absolute but relative. The same number (12) was associated with the left side rather than the right side whenever the previously experienced number was larger (36) rather than smaller (4) (Experiment 2). Control on continuous physical variables showed that the effect is specific of discrete magnitudes. These results constitute strong evidence that in our species SNA originates from pre‐linguistic and biological precursors in the brain.     

Moving attention along the mental number line in preschool age: Study of the operational momentum in 3‐ to 5‐year‐old children's non‐symbolic arithmetic

People tend to underestimate subtraction and overestimate addition outcomes and to associate subtraction with the left side and addition with the right side. These two phenomena are collectively labeled 'operational momentum' (OM) and thought to have their origins in the same mechanism of 'moving attention along the mental number line'. OM in arithmetic has never been tested in children at the preschool age, which is critical for numerical development. In this study, 3–5 years old were tested with non‐symbolic addition and subtraction tasks. Their level of understanding of counting principles (CP) was assessed using the give‐a‐number task. When the second operand's cardinality was 5 or 6 (Experiment 1), the child's reaction time was shorter in addition/subtraction tasks after cuing attention appropriately to the right/left. Adding/subtracting one element (Experiment 2) revealed a more complex developmental pattern. Before acquiring CP, the children showed generalized overestimation bias. Underestimation in addition and overestimation in subtraction emerged only after mastering CP. No clear spatial‐directional OM pattern was found, however, the response time to rightward/leftward cues in addition/subtraction again depended on stage of mastering CP. Although the results support the hypothesis about engagement of spatial attention in early numerical processing, they point to at least partial independence of the spatial‐directional and magnitude OM. This undermines the canonical version of the number line‐based hypothesis. Mapping numerical magnitudes to space may be a complex process that undergoes reorganization during the period of acquisition of symbolic representations of numbers. Some hypotheses concerning the role of spatial‐numerical associations in numerical development are proposed.     

Duration and numerical estimation in right brain‐damaged patients with and without neglect: Lack of support for a mental time line

Previous studies have shown that left neglect patients are impaired when they have to orient their attention leftward relative to a standard in numerical comparison tasks. This finding has been accounted for by the idea that numerical magnitudes are represented along a spatial continuum oriented from left to right with small magnitudes on the left and large magnitudes on the right. Similarly, it has been proposed that duration could be represented along a mental time line that shares the properties of the number continuum. By comparing directly duration and numerosity processing, this study investigates whether or not the performance of neglect patients supports the hypothesis of a mental time line. Twenty‐two right brain‐damaged patients (11 with and 11 without left neglect), as well as 11 age‐matched healthy controls, had to judge whether a single dot presented visually lasted shorter or longer than 500 ms and whether a sequence of flashed dots was smaller or larger than 5. Digit spans were also assessed to measure verbal working memory capacities. In duration comparison, no spatial‐duration bias was found in neglect patients. Moreover, a significant correlation between verbal working memory and duration performance was observed in right brain‐damaged patients, irrespective of the presence or absence of neglect. In numerical comparison, only neglect patients showed an enhanced distance effect for numerical magnitude smaller than the standard. These results do not support the hypothesis of the existence of a mental continuum oriented from left to right for duration. We discuss an alternative account to explain the duration impairment observed in right brain‐damaged patients.     

Core systems of number

What representations underlie the ability to think and reason about number? Whereas certain numerical concepts, such as the real numbers, are only ever represented by a subset of human adults, other numerical abilities are widespread and can be observed in adults, infants and other animal species. We review recent behavioral and neuropsychological evidence that these ontogenetically and phylogenetically shared abilities rest on two core systems for representing number. Performance signatures common across development and across species implicate one system for representing large, approximate numerical magnitudes, and a second system for the precise representation of small numbers of individual objects. These systems account for our basic numerical intuitions, and serve as the foundation for the more sophisticated numerical concepts that are uniquely human.     

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.     

Elementary school children’s attentional biases in physical and numerical space

Numbers are conceptualized spatially along a horizontal mental line. This view is supported by mounting evidence from healthy adults and patients with unilateral spatial neglect. Little is known about children's representation of numbers with respect to space. This study investigated elementary school children's directional biases in physical and numerical space to better understand the relation between space and number. We also examined the nature of spatial organization in numerical space. In two separate tasks, children (n = 57) were asked to bisect a physical line and verbally estimate the midpoint of number pairs. In general, results indicated leftward biases in both tasks, but the degree of deviation did not correlate between the tasks. In the number bisection task, leftward bias (underestimating the midpoint) increased as a function of numerical magnitude and interval between number pairs. In contrast, a rightward deviation was found for smaller number pairs. These findings suggest that different underlying spatial attentional mechanisms might be directed in physical and numerical space in young school children, which would be integrated in adulthood.     

Remembering the past and thinking about the future: Is it really about time?

In this paper we discuss some literature relating to episodic memory, future episodic thinking and mental time travel in humans and non-human animals. We discuss the concept of mental time travel and argue that the concept relies on subjective phenomena such as consciousness and on this basis is not useful when studying episodic memory and future episodic thinking, particularly in non-human animals. We discuss recent work which emphasizes views of both episodic memory and future thinking which do not rely on such mental time travel and, more importantly, give less prominence to the concept of time. The implications of such a view for research into future thinking in non-human animals are considered.     

Varieties of numerical abilities.

This paper provides a tutorial introduction to numerical cognition, with a review of essential findings and current points of debate. A tacit hypothesis in cognitive arithmetic is that numerical abilities derive from human linguistic competence. One aim of this special issue is to confront this hypothesis with current knowledge of number representations in animals, infants, normal and gifted adults, and brain-lesioned patients. First, the historical evolution of number notations is presented, together with the mental processes for calculating and transcoding from one notation to another. While these domains are well described by formal symbol-processing models, this paper argues that such is not the case for two other domains of numerical competence: quantification and approximation. The evidence for counting, subitizing and numerosity estimation in infants, children, adults and animals is critically examined. Data are also presented which suggest a specialization for processing approximate numerical quantities in animals and humans. A synthesis of these findings is proposed in the form of a triple-code model, which assumes that numbers are mentally manipulated in an arabic, verbal or analogical magnitude code depending on the requested mental operation. Only the analogical magnitude representation seems available to animals and preverbal infants.     

Orienting numbers in mental space: Horizontal organization trumps vertical

While research on the spatial representation of number has provided substantial evidence for a horizontally oriented mental number line, recent studies suggest vertical organization as well. Directly comparing the relative strength of horizontal and vertical organization, however, we found no evidence of spontaneous vertical orientation (upward or downward), and horizontal trumped vertical when pitted against each other (Experiment 1). Only when numbers were conceptualized as magnitudes (as opposed to nonmagnitude ordinal sequences) did reliable vertical organization emerge, with upward orientation preferred (Experiment 2). Altogether, these findings suggest that horizontal representations predominate, and that vertical representations, when elicited, may be relatively inflexible. Implications for spatial organization beyond number, and its ontogenetic basis, are discussed.     

Orienting numbers in mental space: horizontal organization trumps vertical

While research on the spatial representation of number has provided substantial evidence for a horizontally oriented mental number line, recent studies suggest vertical organization as well. Directly comparing the relative strength of horizontal and vertical organization, however, we found no evidence of spontaneous vertical orientation (upward or downward), and horizontal trumped vertical when pitted against each other (Experiment 1). Only when numbers were conceptualized as magnitudes (as opposed to nonmagnitude ordinal sequences) did reliable vertical organization emerge, with upward orientation preferred (Experiment 2). Altogether, these findings suggest that horizontal representations predominate, and that vertical representations, when elicited, may be relatively inflexible. Implications for spatial organization beyond number, and its ontogenetic basis, are discussed.     

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, Piazza, Pinel, & Cohen, 2003). 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.     

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。     

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 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.     

Semantic numerical representation in blind subjects: The role of vision in the spatial format of the mental number line

Does vision play a role in the elaboration of the semantic representation of small and large numerosities, notably in its spatial format? To investigate this issue, we decided to compare in the auditory modality the performance of congenitally and early blind people with that of a sighted control group, in two number comparison tasks (to 5 and to 55) and in one parity judgement task. Blind and sighted participants presented exactly the same distance and SNARC (Spatial Numerical Association of Response Codes) effects, indicating that they share the same semantic numerical representation. In consequence, our results suggest that the spatial dimension of the numerical representation is not necessarily attributable to the visual modality and that the absence of vision does not preclude the elaboration of this representation for 1-digit (Experiment 1) and 2-digit numerosities (Experiment 2). Moreover, as classical semantic numerical effects were observed in the auditory modality, the postulate of the amodal nature of the mental number line for both small and large magnitudes was reinforced.     

Semantic numerical representation in blind subjects: the role of vision in the spatial format of the mental number line

Does vision play a role in the elaboration of the semantic representation of small and large numerosities, notably in its spatial format? To investigate this issue, we decided to compare in the auditory modality the performance of congenitally and early blind people with that of a sighted control group, in two number comparison tasks (to 5 and to 55) and in one parity judgement task. Blind and sighted participants presented exactly the same distance and SNARC (Spatial Numerical Association of Response Codes) effects, indicating that they share the same semantic numerical representation. In consequence, our results suggest that the spatial dimension of the numerical representation is not necessarily attributable to the visual modality and that the absence of vision does not preclude the elaboration of this representation for 1-digit (Experiment 1) and 2-digit numerosities (Experiment 2). Moreover, as classical semantic numerical effects were observed in the auditory modality, the postulate of the amodal nature of the mental number line for both small and large magnitudes was reinforced.