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 SNARC effect does not imply a mental number line

In this study, we directly contrast two approaches that have been proposed to explain the SNARC effect. The traditional direct mapping account suggests that a direct association exists between the position of a number on the mental number line and the location of the response. On the other hand, accounts are considered that propose an intermediate step in which numbers are categorized as either small or large between the number magnitude and the response representations. In a magnitude comparison task, we departed from the usual bimanual left/right response dimension and instead introduced the unimanual close/far dimension. A spatial-numerical association was observed: small numbers were associated with a close response, while large numbers were associated with a far response, regardless of the movement direction (left/right). We discuss why these results cannot be explained by assuming a direct mapping from the representation of numbers on a mental number line to response locations and discuss how the results can be explained by the alternative accounts.     

Bad–good constraints on a polarity correspondence account for the spatial–numerical association of response codes (SNARC) and markedness association of response codes (MARC) effects

Performance in numerical classification tasks involving either parity or magnitude judgements is quicker when small numbers are mapped onto a left-sided response and large numbers onto a right-sided response than for the opposite mapping (i.e., the spatial–numerical association of response codes or SNARC effect). Recent research by Gevers et al. [Gevers, W., Santens, S., Dhooge, E., Chen, Q., Van den Bossche, L., Fias, W., & Verguts, T. (2010). Verbal-spatial and visuospatial coding of number–space interactions. Journal of Experimental Psychology: General, 139, 180–190] suggests that this effect also arises for vocal “left” and “right” responding, indicating that verbal–spatial coding has a role to play in determining it. Another presumably verbal-based, spatial–numerical mapping phenomenon is the linguistic markedness association of response codes (MARC) effect whereby responding in parity tasks is quicker when odd numbers are mapped onto left-sided responses and even numbers onto right-sided responses. A recent account of both the SNARC and MARC effects is based on the polarity correspondence principle [Proctor, R. W., & Cho, Y. S. (2006). Polarity correspondence: A general principle for performance of speeded binary classification tasks. Psychological Bulletin, 132, 416–442]. This account assumes that stimulus and response alternatives are coded along any number of dimensions in terms of – and + polarities with quicker responding when the polarity codes for the stimulus and the response correspond. In the present study, even–odd parity judgements were made using either “left” and “right” or “bad” and “good” vocal responses. Results indicated that a SNARC effect was indeed present for the former type of vocal responding, providing further evidence for the sufficiency of the verbal–spatial coding account for this effect. However, the decided lack of an analogous SNARC-like effect in the results for the latter type of vocal responding provides an important constraint on the presumed generality of the polarity correspondence account. On the other hand, the presence of robust MARC effects for “bad” and “good” but not “left” and “right” vocal responses is consistent with the view that such effects are due to conceptual associations between semantic codes for odd–even and bad–good (but not necessarily left–right).     

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.     

Bisecting the mental number line in near and far space

Much evidence suggests that common posterior parietal mechanisms underlie the orientation of attention in physical space and along the mental number line. For example, the small leftward bias (pseudoneglect) found in paper-and-pencil line bisection is also found when participants “bisect” number pairs, estimating (without calculating) the number midway between two others. For bisection of physical lines, pseudoneglect has been found to shift rightward as lines are moved from near space (immediately surrounding the body) to far space. We investigated whether the presentation of stimuli in near or far space also modulated spatial attention for the mental number line. Participants bisected physical lines or number pairs presented at four distances (60, 120, 180, 240 cm). Clear rightward shifts in bias were observed for both tasks. Furthermore, the rate at which this shift occurred in the two tasks, as measured by least-squares regression slopes, was significantly correlated across participants, suggesting that the transition from near to far distances induced a common modulation of lateral attention in physical and numerical space. These results demonstrate a tight coupling between number and physical space, and show that even such prototypically abstract concepts as number are modulated by our on-line interactions with the world.     

When the mental number line involves a delay: the writing of numbers by children of different arithmetical abilities

Our study focused on number transcoding in children. It investigated how 9-year-olds with and without arithmetical disabilities wrote Arabic digits after they had heard them as number words. Planning time before writing each digit was registered. Analyses revealed that the two groups differed not only in arithmetical abilities but also in verbal and reading abilities. Children with arithmetical disabilities were overall slower in planning Arabic digits than were control children with normal arithmetical abilities. In addition, they showed a number size effect for numbers smaller than 10, suggesting a semantically mediated route in number processing. Control children did not need more planning time for large numbers (e.g., 8) than for small numbers (e.g., 3), suggesting a direct nonsemantic route. For both two- and three-digit numbers, both groups of children showed a number size effect, although the effect was smaller each time for control children. The presence of the stronger number size effect for children with arithmetical disabilities was seen as a delay in the development of quick and direct transcoding. The relation between transcoding problems and arithmetical disabilities is discussed. A defect in the linking of numerical symbols to analog numerical representations is proposed as an explanation for the transcoding problems found in some children.     

The revelation of the negative side of a mental number line depends on list context: Evidence from a sign priming paradigm

Two parity judgement experiments examined how the activation of spatial-numerical associations of a single, centrally presented digit, reflected by the Spatial-Numerical Association Response Codes (SNARC) effect, is modulated by a preceding + (plus) or ? (minus) prime. The centrally presented prime prior to a digit presentation presumably triggers its positive or negative attributes. When the plus- and minus-primed trials were blocked, the left-small/right-large SNARC effects occurred regardless of prime type. When the plus- and minus-primed trials were randomly intermixed, this left-small/right-large SNARC effect occurred for plus-primed digits, but was reversed for minus-primed digits. The implications of this finding for context-dependent SNARC effects are discussed.     

A colorful walk on the mental number line: Striving for the right direction

Verguts and Van Opstal [Verguts, T., & Van Opstal, F. (2008). A colorful walk, but is it on the mental number line? Reply to Cohen Kadosh, Tzelgov, and Henik, Cognition, 106, 558-563] cleverly explained the results of Cohen Kadosh, Tzelgov, and Henik [Cohen Kadosh, R., Tzelgov, J., & Henik, A. (2008). A synesthetic walk on the mental number line: The size effect, Cognition, 106, 548-557] as a result of different association strength between the size of a number and its color in synesthesia. Here we present three challenges to their alternative explanation, and support our original suggestion.     

Eye movements during mental time travel follow a diagonal line

Recent research showed that past events are associated with the back and left side, whereas future events are associated with the front and right side of space. These spatial–temporal associations have an impact on our sensorimotor system: thinking about one’s past and future leads to subtle body sways in the sagittal dimension of space (Miles, Nind, & Macrae, 2010). In this study we investigated whether mental time travel leads to sensorimotor correlates in the horizontal dimension of space. Participants were asked to mentally displace themselves into the past or future while measuring their spontaneous eye movements on a blank screen. Eye gaze was directed more rightward and upward when thinking about the future than when thinking about the past. Our results provide further insight into the spatial nature of temporal thoughts, and show that not only body, but also eye movements follow a (diagonal) “time line” during mental time travel.     

A synesthetic walk on the mental number line: the size effect

Are small and large numbers represented similarly or differently on the mental number line? The size effect was used to argue that numbers are represented differently. However, recently it has been argued that the size effect is due to the comparison task and is not derived from the mental number line per se. Namely, it is due to the way that the mental number line is mapped onto the task-relevant output component. Here synesthesia was used to disentangle these two alternatives. In two naming experiments a digit-color synesthete showed that the congruity effect was modulated by number size. These results support the existence of a mental number line with a vaguer numerical representation as numbers increase in size. In addition, the results show that in digit-color synesthesia, colors can evoke numerical representation automatically.     

The mental time line: An analogue of the mental number line in the mapping of life events

A crucial aspect of the human mind is the ability to project the self along the time line to past and future. It has been argued that such self-projection is essential to re-experience past experiences and predict future events. In-depth analysis of a novel paradigm investigating mental time shows that the speed of this “self-projection” in time depends logarithmically on the temporal-distance between an imagined “location” on the time line that participants were asked to imagine and the location of another imagined event from the time line. This logarithmic pattern suggests that events in human cognition are spatially mapped along an imagery mental time line. We argue that the present time-line data are comparable to the spatial mapping of numbers along the mental number line and that such spatial maps are a fundamental basis for cognition.     

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.     

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.     

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     

The mental representation of integers: an abstract-to-concrete shift in the understanding of mathematical concepts

Mathematics has a level of structure that transcends untutored intuition. What is the cognitive representation of abstract mathematical concepts that makes them meaningful? We consider this question in the context of the integers, which extend the natural numbers with zero and negative numbers. Participants made greater and lesser judgments of pairs of integers. Experiment 1 demonstrated an inverse distance effect: When comparing numbers across the zero boundary, people are faster when the numbers are near together (e.g., −1 vs. 2) than when they are far apart (e.g., −1 vs. 7). This result conflicts with a straightforward symbolic or analog magnitude representation of integers. We therefore propose an analog-x hypothesis: Mastering a new symbol system restructures the existing magnitude representation to encode its unique properties. We instantiate analog-x in a reflection model: The mental negative number line is a reflection of the positive number line. Experiment 2 replicated the inverse distance effect and corroborated the model. Experiment 3 confirmed a developmental prediction: Children, who have yet to restructure their magnitude representation to include negative magnitudes, use rules to compare negative numbers. Taken together, the experiments suggest an abstract-to-concrete shift: Symbolic manipulation can transform an existing magnitude representation so that it incorporates additional perceptual-motor structure, in this case symmetry about a boundary. We conclude with a second symbolic-magnitude model that instantiates analog-x using a feature-based representation, and that begins to explain the restructuring process.