Perceiving numbers does not cause automatic shifts of spatial attention

The time course of the spatial representation of 'past' and 'future' concepts: New evidence from the STEARC effect

Humans use space to think of and communicate the flow of time. This spatial representation of time is influenced by cultural habits so that in left-to-right reading cultures, short durations and past events are mentally positioned to the left of long durations and future events. The STEARC effect (Space Temporal Association of Response Codes) shows a faster classification of short durations/past events with responses on the left side of space and of long durations/future events with responses on the right side. We have recently showed that during the classification of time durations, space is a late heuristic of time because in this case, the STEARC appears only when manual responses are slow, not when they are fast. Here, we wished to extend this observation to the semantic classification of words as referring to the 'past' or the 'future'. We hypothesised that the semantic processing of 'past' and 'future' concepts would have engaged slower decision processes than the classification of short versus long time durations. According to dual-route models of conflict tasks, if the task-dependent classification/decision process were to proceed relatively slowly, then the effects of direct activation of culturally preferred links between stimulus and response (S-R), i.e., past/left and future/right in the case of the present task, should attain higher amplitudes before the instruction-dependent correct response is selected. This would imply that, at variance with the faster classification of time durations, during the slower semantic classification of time concepts, in incongruent trials, the direct activation of culturally preferred S-R links should introduce significant reaction time (RT) costs and a corresponding STEARC at the fastest manual responses in the experiment too. The study's results confirmed this hypothesis and showed that in the classification of temporal words, the STEARC also increased as a function of the length of RTs. Taken together, the results from sensory duration and semantic classification STEARC tasks show that the occurrence, strength and time course of the STEARC varies significantly as a function of the speed and level of cognitive processing required in the task.

No conclusive evidence for number-induced attentional shifts in a temporal order judgement task

The Spatial Numerical Association of Response Codes (SNARC) effect refers to the observation that relatively small (e.g., 1) and large numbers (e.g., 9) elicit faster left- and right-sided manual responses, respectively. In a variation known as the attentional SNARC effect, merely looking at numbers caused a left- or right-ward shift in covert spatial attention, depending on the number's magnitude. In our study, we probed the notion that numbers induce shifts of spatial attention in accordance with their position on a mental number line (MNL). Critically, we removed any putative spatial response code that may contaminate the responses. We used a square and a tilted square as targets, thereby situating the decisive response dimension in the ventral, non-spatial processing stream. In two experiments where numbers were used as non-informative cues preceding a temporal order judgement (TOJ) task, we did not observe a deflection of the locus of spatial attention as a function of the numerical magnitude of the cue. In a third experiment, finding a significant modulation of TOJ performance as a function of the pointing direction of arrow cues allowed us to rule out the possibility that the absence of any significant modulation in Experiments 1 and 2 was due to a lack of sensitivity of our task set-up. We conclude from the current findings that the spatial codes that the perception and naming of numbers potentially elicit are not in and by themselves sufficient to elicit deflections of spatial attention.

Stronger spatial bias induced more by numbers in mind than numbers in eye: Evidence from event-related potentials

The relationship between number and space is an important issue in numerical cognition. The spatial-numerical association of response codes (SNARC) effect is a classic example of the association between numbers and spaces. It refers to the phenomenon whereby left-handed responses occur faster to small number and right-handed responses occur faster to large number. The current study explored the shared and distinct neural correlates of the SNARC effect considering numbers in eye and numbers in mind, by using event-related potentials (ERPs) technology. In each trial of the task, participants were asked to press freely one of two keys as a response to a number presented visually (numbers in eye) or via imagination (numbers in mind). The behavioral results indicated that the free-choice key presses were affected by the magnitudes of the numbers either in eye or in mind. Electrophysiological results observed that the SNARC effect appeared only in the 110 - 140 ms time window for numbers in eye. In contrast, for numbers in mind, the SNARC effect appeared during a longer time window (110 - 330 ms). These results suggest that both, numbers in eye and numbers in mind, can induce spatial bias at the early stimulus-representation stage, but the time duration of the spatial bias is longer for numbers in mind than numbers in eye. This may reflect a closer connection between numbers in mind and mental number line.

Spatial frequency equalization does not prevent spatial-numerical associations

There is an intense debate surrounding the origin of spatial-numerical associations (SNAs), according to which small numbers are mapped onto the left side of the space and large numbers onto the right. Despite evidence suggesting that SNAs would emerge as an innate predisposition to map numerical information onto a left-to-right spatially oriented mental representation, alternative accounts have challenged these proposals, maintaining that such a mapping would be the result of a mere spatial frequency (SF) coding of any visual image. That is, any smaller or larger array of objects would naturally contain more low or high SF information and, accordingly, each hemisphere would be preferentially tuned only for one SF range (e.g., right hemisphere tuned for low SF and left hemisphere tuned for high SF). This would determine the typical SNA (e.g., faster RTs for small numerical arrays with the left hand and for large numerical arrays with the right hand). To directly probe the role of SF coding in SNAs, we tested participants in a typical dot-arrays comparison task with two numerical sets: one in which SFs were confounded with numerosity (Experiment 1) and one in which the full SF power spectrum was equalized across all stimuli, keeping this cue uninformative about numerosity (Experiment 2). We found that SNAs emerged in both experiments, independently of whether SF was confounded or not with numerosity. Taken together, these findings suggest that SNAs cannot simply originate from SF power spectrum alone, and, thus, they rule out the brain's asymmetric SF tuning as a primary cause of such an effect.

Pupil size variations reveal covert shifts of attention induced by numbers

The pupil light response is more than a pure reflexive mechanism that reacts to the amount of light entering the eye. The pupil size may also react to the luminance of objects lying in the visual periphery, revealing the locus of covert attention. In the present study, we took advantage of this response to study the spatial coding of abstract concepts with no physical counterpart: numbers. The participants' gaze was maintained fixed in the middle of a screen whose left and right parts were dark or bright, and variations in pupil size were recorded during an auditory number comparison task. The results showed that small numbers accentuated pupil dilation when the darker part of the screen was on the left, while large numbers accentuated pupil dilation when the darker part of the screen was on the right. This finding provides direct evidence for covert attention shifts on a left-to-right oriented mental spatial representation of numbers. From a more general perspective, it shows that the pupillary response to light is subject to modulation from spatial attention mechanisms operating on mental contents.

Effects of attentional shifts along the vertical axis on number processing: An eye-tracking study with optokinetic stimulation

Previous studies suggest that associations between numbers and space are mediated by shifts of visuospatial attention along the horizontal axis. In this study, we investigated the effect of vertical shifts of overt attention, induced by optokinetic stimulation (OKS) and monitored through eye-tracking, in two tasks requiring explicit (number comparison) or implicit (parity judgment) processing of number magnitude. Participants were exposed to black-and-white stripes (OKS) that moved vertically (upward or downward) or remained static (control condition). During the OKS, participants were asked to verbally classify auditory one-digit numbers as larger/smaller than 5 (comparison task; Exp. 1) or as odd/even (parity task; Exp. 2). OKS modulated response times in both experiments. In Exp.1, upward attentional displacement decreased the Magnitude effect (slower responses for large numbers) and increased the Distance effect (slower responses for numbers close to the reference). In Exp.2, we observed a complex interaction between parity, magnitude, and OKS, indicating that downward attentional displacement slowed down responses for large odd numbers. Moreover, eye tracking analyses revealed an influence of number processing on eye movements both in Exp. 1, with eye gaze shifting downwards during the processing of small numbers as compared to large ones; and in Exp. 2, with leftward shifts after large even numbers (6,8) and rightward shifts after large odd numbers (7,9). These results provide evidence of bidirectional links between number and space and extend them to the vertical dimension. Moreover, they document the influence of visuo-spatial attention on processing of numerical magnitude, numerical distance, and parity. Together, our findings are in line with grounded and embodied accounts of numerical cognition.

Number to me, space to you: Joint representation of spatial-numerical associations

Recent work has shown that number concepts activate both spatial and magnitude representations. According to the social co-representation literature which has shown that participants typically represent task components assigned to others together with their own, we asked whether explicit magnitude meaning and explicit spatial coding must be present in a single mind, or can be distributed across two minds, to generate a spatial-numerical congruency effect. In a shared go/no-go task that eliminated peripheral spatial codes, we assigned explicit magnitude processing to participants and spatial processing to either human or non-human co-agents. The spatial-numerical congruency effect emerged only with human co-agents. We demonstrate an inter-personal level of conceptual congruency between space and number that arises from a shared conceptual representation not contaminated by peripheral spatial codes. Theoretical implications of this finding for numerical cognition are discussed.

Is the attentional SNARC effect truly attentional? Using temporal order judgements to differentiate attention from response

The spatial–numerical association of response codes (SNARC) effect reflects the phenomenon that low digits are responded to faster with the left hand and high digits with the right. Recently, a particular variant of the SNARC effect known as the attentional SNARC (which reflects that attention can be shifted in a similar manner) has had notable replicability issues. However, a potentially useful method for measuring it was revealed by Casarotti et al. using a temporal order judgement (TOJ) task. Accordingly, the present study evaluated whether Casarotti et al.’s results were reproducible by presenting a low (1) or high (9) digit prior to a TOJ task where participants had to indicate which of two peripherally presented targets appeared first (Experiment 1) or second (Experiment 2). In Experiment 1, it was revealed that the findings of Casarotti et al.’s were indeed observable upon replication. In Experiment 2, when attention and response dimensions were put in opposition, the SNARC effect corresponded to the side of response rather than attention. Taken together, the present study confirms the robustness of the attentional SNARC in TOJ tasks, but that it is not likely due to shifts in attention.

Perceiving numerosity does not cause automatic shifts of spatial attention

It is debated whether the representation of numbers is endowed with a directional-spatial component so that perceiving small-magnitude numbers triggers leftward shifts of attention and perceiving large-magnitude numbers rightward shifts. Contrary to initial findings, recent investigations have demonstrated that centrally presented small-magnitude and large-magnitude Arabic numbers do not cause leftward and rightward shifts of attention, respectively. Here we verified whether perceiving small or large non-symbolic numerosities (i.e., clouds of dots) drives attention to the left or the right side of space, respectively. In experiment 1, participants were presented with central small (1, 2) vs large-numerosity (8, 9) clouds of dots followed by an imperative target in the left or right side of space. In experiment 2, a central cloud of dots (i.e., five dots) was followed by the simultaneous presentation of two identical dot-clouds, one on the left and one on the right side of space. Lateral clouds were both lower (1, 2) or higher in numerosity (8, 9) than the central cloud. After a variable delay, one of the two lateral clouds turned red and participants had to signal the colour change through a unimanual response. We found that (a) in Experiment 1, the small vs large numerosity of the central cloud of dots did not speed up the detection of left vs right targets, respectively, (b) in Experiment 2, the detection of colour change was not faster in the left side of space when lateral clouds were smaller in numerosity than the central reference and in the right side when clouds were larger in numerosity. These findings show that perceiving non-symbolic numerosity does not cause automatic shifts of spatial attention and suggests no inherent association between the representation of numerosity and that of directional space.

Electrophysiological Evidence of Space-Number Associations in 9-Month-Old Infants

Infant research is providing accumulating evidence that number-space mappings appear early in development. Here, a Posner cueing paradigm was used to investigate the neural mechanisms underpinning the attentional bias induced by nonsymbolic numerical cues in 9-month-old infants (N = 32). Event-related potentials and saccadic reaction time were measured to the onset of a peripheral target flashing right after the offset of a centered small or large numerical cue, with the location of the target being either congruent or incongruent with the number's relative position on a left-to-right oriented representational continuum. Results indicated that the cueing effect induced by numbers on infants' orienting of eye gaze brings about sensory facilitation in processing visual information at the cued location.

Spatial-numerical associations from a novel paradigm support the mental number line account

Multiple tasks have been used to demonstrate the relation between numbers and space. The classic interpretation of these directional spatial-numerical associations (d-SNAs) is that they are the product of a mental number line (MNL), in which numerical magnitude is intrinsically associated with spatial position. The alternative account is that d-SNAs reflect task demands, such as explicit numerical judgements and/or categorical responses. In the novel "Where was The Number?" task, no explicit numerical judgements were made. Participants were simply required to reproduce the location of a numeral within a rectangular space. Using a between-subject design, we found that numbers, but not letters, biased participants' responses along the horizontal dimension, such that larger numbers were placed more rightward than smaller numbers, even when participants completed a concurrent verbal working memory task. These findings are consistent with the MNL account, such that numbers specifically are inherently left-to-right oriented in Western participants.

Number space is made by response space: Evidence from left spatial neglect

Whether the semantic representation of numbers is endowed with an intrinsic spatial component, so that smaller numbers are inherently represented to the left of larger ones on a Mental Number Line (MNL), is a central matter of debate in numerical cognition. To gain an insight into this issue, we investigated the performance of right brain damaged patients with left spatial neglect (N+) in a bimanual Magnitude Comparison SNARC task and in a uni-manual Magnitude Comparison Go/No-Go task (i.e. "is the number smaller or larger than 5?"). While the first task requires the use of contrasting left/right spatial codes for response selection, the second task does not require the use of these codes. In line with previous evidence, in the SNARC task N+ patients displayed a significant asymmetry in Reaction Times (RTs), with slower RTs to number "4", that was immediately precedent to the numerical reference "5", with respect to the number "6", that immediately followed the same reference. This RTs asymmetry was correlated with lesion of white matter tracts, i.e. Fronto-Occipital-Fasciculus, that allows prefrontal Ba 8 and 46 to regulate the distribution of attention on sensory and memory traces in posterior occipital, temporal and parietal areas. In contrast, no similar RTs asymmetry was found in the Go/No-Go task. These findings suggest that while in the SNARC task numbers get mentally organised from left-to-right as a function of their increasing magnitude, so that N+ patients display a delay in the processing of number-magnitudes that are immediately smaller than a given numerical reference, in the Go/No-Go task no left-to-right organization is activated. These results support the idea that it is the use of contrasting left/right spatial codes, whether motor or conceptual, that triggers the generation of a spatially left-to-right organised MNL and that the representation of number magnitude is not endowed with an inherent spatial component.

How to trigger and keep stable directional Space-Number Associations (SNAs)

Humans are prone to mentally organise the ascending series of integers according to reading habits so that in western cultures small numbers are positioned to the left of larger ones on a mental number line. Despite 140 years since seminal observations by Sir Francis Galton (Galton, 1880a, b), the functional mechanisms that give rise to directional Space-Number Associations (SNAs) remain elusive. Here, we contrasted three different experimental conditions, each including a different version of a Go/No-Go task with intermixed numerical and arrow-targets (Shaki and Fischer, 2018; Pinto et al., 2019a). We show that directional SNAs are not "all or none" phenomena. We demonstrate that SNAs get progressively less noisy and more stable the more contrasting small/large magnitude-codes and contrasting left/right spatial-codes are explicitly and fully combined in the task set. The analyses of the time-course of space-number congruency effects showed that both the absence and presence of the SNA were independent of the speed of reaction times. In agreement with our original proposal (Aiello et al., 2012), these findings show that conceptualising the ascending series of integers in spatial terms depends on the use of spatial codes in the numerical task at hand rather than on the presence of an inherent spatial dimension in the semantic representation of numbers. This evidence suggests that directional SNAs, like the SNARC effect, are secondary to the primary transfer of spatial response codes to number stimuli, rather than deriving from a primary congruency or incongruence between independent spatial-response and spatial-number codes.

Shifting attention does not influence numerical processing

Many theories of numerical cognition assume that numbers and space share a common representation at the response level. For example, observers are faster to respond to small numbers with their left hand and large numbers with their right hand (the SNARC effect). There is also evidence that viewing numbers can produce spatial shifts of attention, suggesting that attention may play a role in the spatial representation of numbers. In the present study, we assessed whether shifts of attention can influence numerical processing. Participants viewed a leftward or rightward peripheral cue followed by a centrally presented number, then judged whether the number was odd or even. Participants responded faster and made fewer errors when the number magnitude and response side were compatible, revealing a response-based SNARC effect. Participants also responded faster when the cue direction and response side were compatible, revealing a Simon effect. However, participants did not respond faster when the cue direction and number magnitude were compatible. Similar findings were observed when the association between numbers and space was relatively explicit. Moreover, although we failed to observe a response-based SNARC effect when number magnitude was directly relevant to observers' task, we observed a large Simon effect. Together, these findings suggest that although numbers and space share a common representation at the response level, attention does not play a substantial role in the spatial representation of numbers.

Registered Replication Report on Fischer, Castel, Dodd, and Pratt (2003)

The attentional spatial-numerical association of response codes (Att-SNARC) effect (Fischer, Castel, Dodd, & Pratt, 2003)—the finding that participants are quicker to detect left-side targets when the targets are preceded by small numbers and quicker to detect right-side targets when they are preceded by large numbers—has been used as evidence for embodied number representations and to support strong claims about the link between number and space (e.g., a mental number line). We attempted to replicate Experiment 2 of Fischer et al. by collecting data from 1,105 participants at 17 labs. Across all 1,105 participants and four interstimulus-interval conditions, the proportion of times the effect we observed was positive (i.e., directionally consistent with the original effect) was .50. Further, the effects we observed both within and across labs were minuscule and incompatible with those observed by Fischer et al. Given this, we conclude that we failed to replicate the effect reported by Fischer et al. In addition, our analysis of several participant-level moderators (finger-counting habits, reading and writing direction, handedness, and mathematics fluency and mathematics anxiety) revealed no substantial moderating effects. Our results indicate that the Att-SNARC effect cannot be used as evidence to support strong claims about the link between number and space.

Does Number Perception Cause Automatic Shifts of Spatial Attention? A Study of the Att-SNARC Effect in Numbers and Chinese Months

The Attentional Spatial Numerical Association of Response Codes (Att-SNARC) effect has shown that number perception induces shifts in spatial attention (Fischer et al., 2003; Dodd et al., 2008). However, many replications were attempted and they often failed. In the present study, we investigated whether the Att-SNARC effect can be found for numbers in different notations: months in Arabic form, Simplified Chinese form, Traditional Chinese form (includes numerical ordinal information) and in Chinese non-numerical form (an ordinal sequence). By varying the cognitive task, we also examined whether the effect is a consequence of automatic perceptual processing. In Experiment 1, an Att-SNARC effect was observed for numbers regardless of notation. In Experiment 2 (order-irrelevant task) and Experiment 3 (order-relevant task), the effect was also found consistently for months in Arabic form, Simplified Chinese form, and Traditional Chinese form. This effect was not observed for months in Chinese non-numerical form in Experiment 3. These results show that number and numerical sequence perception automatically causes a spatial shift of attention. Our study provides positive evidence for the Att-SNARC effect and indicates that the effect can generalize to other numerical ordinal sequences that contain numeric information.

Spatial biases in mental arithmetic are independent of reading/writing habits: Evidence from French and Arabic speakers

The representation of numbers in human adults is linked to space. In Western cultures, small and large numbers are associated respectively with the left and right sides of space. An influential framework attributes the emergence of these spatial-numerical associations (SNAs) to cultural factors such as the direction of reading and writing, because SNAs were found to be reduced or inverted in right-to-left readers/writers (e.g., Arabic, Farsi, or Hebrew speakers). However, recent cross-cultural and animal studies cast doubt on the determining role of reading and writing directions on SNAs. In this study, we assessed this role in mental arithmetic, which requires explicit number manipulations and has revealed robust leftward or rightward biases in Western participants. We used a temporal order judgement task in French and Arabic speakers, two languages that have opposite reading/writing directions. Participants had to solve subtraction and addition problems presented auditorily while at the same time determining which of a left or right visual target appeared first on a screen. The results showed that the right target was favoured more often when solving additions than when solving subtractions both in the French- (n = 31) and Arabic-speaking (n = 25) groups. This was true even in Arabic-speaking participants whose preference for ordering of various series of numerical and non-numerical stimuli went from right to left (n = 10). These results indicate that SNAs in mental arithmetic cannot be explained by the direction of reading/writing habits and call for a reconsideration of current models to acknowledge the pervasive role of biological factors in SNAs in adults.

The spatial-numerical association of response codes effect and math skills: why related?

Evidence from multiple studies conducted in the past few decades converges on the conclusion that numerical properties can be associated with specific directions in space. Such spatial-numerical associations (SNAs), as a signature of elementary number processing, seem to be a likely correlate of math skills. Nevertheless, almost three decades of research on the spatial-numerical association of response codes (SNARC) effect, the hallmark of SNAs, has not provided conclusive results on whether there is a relation with math skills. Here, going beyond reviewing the existing literature on the topic, we try to answer a more fundamental question about why the SNARC effect should (and should not) be related to math skills. We propose a multiroute model framework for a SNARC-math skills relationship. We conclude that the relationship is not straightforward and that several other factors should be considered, which under certain circumstances or in certain groups can cause effects of opposite directions. The model can account for conflicting results, and thus may be helpful for deriving predictions in future studies.

Can Implicit or Explicit Time Processing Impact Numerical Representation? Evidence From a Dual Task Paradigm

Whether the human brain processes various types of magnitude, such as numbers and time, through a shared representation or whether there are different representations for each type of magnitude is still debated. Here, we investigated two aspects of number-time interaction: the effects of implicit and explicit processing of time on numbers and the bi-directional interaction between time and number processing. Thirty-two participants were randomly assigned into two experimental groups that performed, respectively, a Single task (number comparison, with implicit time processing) and a Dual task (number comparison as a primary task, with explicit time processing as a secondary task). Results showed that participants, only in the Dual task, were faster and more accurate when processing large numbers paired with long rather than short durations, whereas the opposite pattern was not evident for small numbers. Moreover, participants were more accurate when judging long durations after having processed large rather than small numbers, whereas the opposite pattern emerged for short durations. We propose that number processing influences time processing more than vice versa, suggesting that numbers and time might be at least partially independently represented. This finding can pave the way for investigating the hierarchical representation of space, numbers, and time.

Contrasting left/right codes for response selection must not be necessarily associated with contrasting numerical features to get the SNARC

The SNARC effect consists of faster reaction times to small numerical magnitudes when manual responses are delivered in the left-side of space and to large magnitudes when responses are delivered in the right-side. This spatial compatibility effect points at the interaction between the representations of space and that of numbers. Several studies have highlighted that an important determinant for the production of the SNARC is the use of contrasting left/right spatial codes in the selection of motor responses. In these studies, one spatial code for response selection, e.g. "left", is usually associated with one number feature, e.g. "lower than 5", while the contrasting spatial code, e.g. "right", is associated with the contrasting number feature, e.g. "higher than 5". Using a task with intermixed number and letter targets, here we show that significant and reliable SNARC effects are also produced when: a) one spatial response is associated with the intra-categorical discrimination of a number feature (i.e. magnitude or parity) and the contrasting response with the simple detection of letter targets; b) or when one spatial response is associated with the intra-categorical discrimination of the position of a letter in the alphabet (i.e. before or after "m") and the contrasting spatial response with the simple detection of numerical targets (Experiments 1 and 2). In contrast, no reliable SNARC is found when no intra-categorical number or letter discrimination is required and contrasting left/right spatial response codes are simply associated with the discrimination between numbers and letters, e.g. "push left if the target is a number/push right if it is a letter" (Experiment 3). In a final control test (Experiment 4), we found no SNARC when the magnitude or parity classification of Arabic digits presented at central fixation was made unimanually with a left side or a right side-key and no left/right contrast was present in response selection. These results show that the use of contrasting left/right spatial response codes elicits reliable SNARC effects independently of their assignment to contrasting number features and confirm the important role played by the use of spatial codes in the genesis of the number-space interaction.

The SNARC and MARC effects measured online: Large-scale assessment methods in flexible cognitive effects

The Spatial-Numerical Association of Response Codes (SNARC) effect (i.e., faster reactions to small/large numbers on the left-/right-hand side) is usually observed along with the linguistic Markedness of Response Codes (MARC) effect-that is, faster left-/right-hand responses to odd/even numbers. The SNARC effect is one of the most thoroughly investigated phenomena in numerical cognition. However, almost all SNARC and MARC studies to date were conducted with sample sizes smaller than 100. Here we report on a study with 1,156 participants from various linguistic and cultural backgrounds performing a typical parity judgment task. We investigated whether (1) the SNARC and MARC effects can be observed in an online setup, (2) the properties of these effects observed online are similar to those observed in laboratory setups, (3) the effects are reliable, and (4) they are valid. We found robust SNARC and MARC effects. Their magnitude and reliabilities were comparable to values previously reported in in-lab studies. Furthermore, we reproduced commonly observed validity correlations of the SNARC and MARC effects. Namely, SNARC and MARC correlated with mean reaction times and intraindividual variability in reaction times. Additionally, we found interindividual differences in the SNARC and MARC effects (e.g., finger-counting routines for the SNARC and handedness for the MARC). Large-scale testing via web-based data acquisition not only produces SNARC and MARC effects and validity correlations similar to those from small, in-lab studies, but also reveals substantial insights with regard to interindividual differences that usually cannot be revealed in the offline laboratory, due to power considerations.

Multiple left-to-right spatial representations of number magnitudes? Evidence from left spatial neglect

The SNARC effect reflects the observation that when healthy observers with left-to-right reading habits are asked to compare the magnitude or to judge the parity of numbers, they provide faster reaction times (RT) to small numbers with left-sided responses and faster RTs to large numbers with right-sided responses. In magnitude comparison (MC), right brain damaged patients with left-sided neglect typically show a pathologically enlarged SNARC for large numbers and selective slowing to numbers that are immediately lower than the numerical reference (e.g. 4 for reference 5). This asymmetry has been taken as evidence that small numbers are mentally positioned to the left of the reference and, therefore, are processed less efficiently by patients neglecting the left side of space. In parity judgement (PJ), on the other hand, the size of the SNARC effect is unaffected by neglect. This dissociation is typically attributed to the disturbed explicit processing of number magnitude in MC and preserved implicit processing of magnitude in PJ. Before accepting this interpretation, however, it remains to be investigated whether neglect patients show the same RT pattern that characterizes the performance of healthy participants (i.e. left-side RTs that increase linearly as a function of number magnitude and right-side RTs that decrease linearly as a function of magnitude). Clarifying this point is crucial, because an equally sized SNARC can originate from different RT patterns. Here we demonstrate that the RT pattern of neglect patients during PJ is entirely comparable to those of patients without neglect and healthy controls, while the same neglect patients show selective slowing to numbers that are immediately lower than the numerical reference in MC. These findings suggest the existence of multiple left-to-right spatial representations of number magnitude and provides an explanation of the functional dissociation between MC and PJ tasks.

Attention allows the SNARC effect to operate on multiple number lines

To investigate whether participants can activate only one spatially oriented number line at a time or multiple number lines simultaneously, they were asked to solve a unit magnitude comparison task (unit smaller/larger than 5) and a parity judgment task (even/odd) on two-digit numbers. In both these primary tasks, decades were irrelevant. After some of the primary task trials (randomly), participants were asked to additionally solve a secondary task based on the previously presented number. In Experiment 1, they had to decide whether the two-digit number presented for the primary task was larger or smaller than 50. Thus, for the secondary task decades were relevant. In contrast, in Experiment 2, the secondary task was a color judgment task, which means decades were irrelevant. In Experiment 1, decades' and units' magnitudes influenced the spatial association of numbers separately. In contrast, in Experiment 2, only the units were spatially associated with magnitude. It was concluded that multiple number lines (one for units and one for decades) can be activated if attention is focused on multiple, separate magnitude attributes.

Non-symbolic magnitudes are represented spatially: Evidence from a non-symbolic SNARC task

A core proposition in numerical cognition is numbers are represented spatially. Evidence for this proposition comes from the "spatial numerical association of response codes" effect (SNARC) in which faster responses are made by the left/right hand judging whether one of a pair of Arabic digits is smaller/larger than the other. Less is known if a similar SNARC effect exists for non-symbolic magnitudes; and research that has been conducted used stimuli which could be translated into symbolic terms. To overcome this limitation, we employed a referent-to-target judgment paradigm in which a referent dot array (n = 30 dots) was follow by a second array of dots (e.g., n = 45 or 15 dots)-participants judged if the second array contained fewer or more dots than the referent array. Dot arrays with fewer dots were judged more quickly with the left hand compared to the right hand (i.e., a SNARC effect). Not all participants demonstrated a SNARC effect, however. Neither visuospatial working memory nor math ability was associated with the presence/absence of a non-symbolic SNARC effect. Implications of the non-symbolic SNARC effect for accounts of numerical cognition are discussed.

Spatial grounding of symbolic arithmetic: an investigation with optokinetic stimulation

Growing evidence suggests that mental calculation might involve movements of attention along a spatial representation of numerical magnitude. Addition and subtraction on nonsymbolic numbers (numerosities) seem to induce a “momentum” effect, and have been linked to distinct patterns of neural activity in cortical regions subserving attention and eye movements. We investigated whether mental arithmetic on symbolic numbers, a cornerstone of abstract mathematical reasoning, can be affected by the manipulation of overt spatial attention induced by optokinetic stimulation (OKS). Participants performed additions or subtractions of auditory two-digit numbers during horizontal (experiment 1) or vertical OKS (experiment 2), and eye movements were concurrently recorded. In both experiments, the results of addition problems were underestimated, whereas results of subtractions were overestimated (a pattern that is opposite to the classic Operational Momentum effect). While this tendency was unaffected by OKS, vertical OKS modulated the occurrence of decade errors during subtractions (i.e., fewer during downward OKS and more frequent during upward OKS). Eye movements, on top of the classic effect induced by OKS, were affected by the type of operation during the calculation phase, with subtraction consistently leading to a downward shift of gaze position and addition leading to an upward shift. These results highlight the pervasive nature of spatial processing in mental arithmetic. Furthermore, the preeminent effect of vertical OKS is in line with the hypothesis that the vertical dimension of space–number associations is grounded in universal (physical) constraints and, thereby, more robust than situated and culture-dependent associations with the horizontal dimension.

On the genesis of spatial-numerical associations: Evolutionary and cultural factors co-construct the mental number line

Mapping numbers onto space is a common cognitive representation that has been explored in both behavioral and neuroimaging contexts. Empirical work probing the diverse nature of these spatial-numerical associations (SNAs) has led researchers to question 1) how the human brain links numbers with space, and 2) whether this link is biologically vs. culturally determined. We review the existing literature on the development of SNAs and situate that empirical work within cognitive and neuroscientific theoretical frameworks. We propose that an evolutionarily-ancient frontal-parietal circuit broadly tuned to multiple magnitude dimensions provides the phylogenetic substrate for SNAs, while enculturation and sensorimotor experience shape their specific profiles. We then use this perspective to discuss educational implications and highlight promising avenues for future research.

Time dependency of the SNARC effect for different number formats: evidence from saccadic responses

In line with the suggestion that the strength of the spatial numerical association of response codes (SNARC) effect was time dependent, the aim of the present study was to assess whether the association strength depends on the processing time of numerical quantity and/or of the time to initiate responses. More specifically, we examined whether and how the SNARC effect could be modulated by number format and effector type. Experiment 1 compared the effect induced by Arabic numbers and number words on the basis of saccadic responses in a parity judgment task. Indeed, previous studies have shown that Arabic numbers lead to faster processing than number words. The results replicated the SNARC effect with Arabic numbers, but not with number words. Experiment 2 was similar to Experiment 1, but this time manual responses (i.e., responses far slower than saccadic ones) were recorded. A strong SNARC effect was observed for both number formats. Further analyses revealed a correlation between mean individual response times and the strength of the SNARC effect. We proposed that the initiation times for saccadic responses may be too short for the SNARC effect to appear, in particular with the written number format for which activation of magnitude takes time. We conclude in terms of time variations resulting from processing specificities related with number format, effector type and also individual reaction and processing speed.

Which way is which? Examining symbolic control of attention with compound arrow cues

Spatial symbols can generate attentional biases toward peripheral locations compatible with the symbol's meaning. An important question concerns how one symbol is selected when competing symbols are present. Studies examining this issue for spatially distinct symbols have suggested that selection depends on the task goals. In the present study, we examined whether the influence of competing symbolic stimuli (arrows) at different levels of structure on attentional control also depends on the task goals. Participants made simple detection responses to a peripheral target preceded by a spatially uninformative compound arrow (global arrow composed of local arrows). In addition, participants were required to perform a secondary task in which they matched the orientation of the global arrow (global task) or the location of a uniquely colored local arrow (local task) to a test display presented immediately following a detection response. When the global and local arrows pointed at opposite locations, a local cueing effect emerged in the local task, and a global cueing effect in the global task, indicating that the task goals influenced the selection of the level of structure. However, when the local level was spatially neutral (global arrow, local rectangles), a cueing effect was observed independent of task, and when the global level was spatially neutral (global rectangle, local arrows), a cueing effect was observed in the local task only, suggesting that global processing was obligatory and local processing optional. These findings suggest that attentional effects triggered by the global level are more strongly reflexive than those triggered by the local level.

Biased numerical cognition impairs economic decision-making in Parkinson's disease

Objective

Previous findings suggest a context‐dependent bihemispheric allocation of numerical magnitude. Accordingly, we predicted that lateralized motor symptoms in Parkinson's disease (PD), which reflect hemispheric asymmetries, would induce systematic lateralized biases in numerical cognition and have a subsequent influence on decision‐making.

Methods

In 20 PD patients and matched healthy controls we assessed numerical cognition using a number‐pair bisection and random number generation task. Decision‐making was assessed using both the dictator game and a validated questionnaire.

Results

PD patients with predominant right‐sided motor symptoms exhibited pathological biases toward smaller numerical magnitudes and formulated less favorable prosocial choices during a neuroeconomics task (i.e., dictator game). Conversely, patients with left‐sided motor symptoms exhibited pathological biases toward larger numerical magnitudes and formulated more generous prosocial choices. Our account of context‐dependent hemispheric allocation of numerical magnitude in PD was corroborated by applying our data to a pre‐existing computational model and observing significant concordance. Notably, both numerical biasing and impaired decision‐making were correlated with motor asymmetry.

Interpretation

Accordingly, motor asymmetry and functional impairment of cognitive processes in PD can be functionally intertwined. To conclude, our findings demonstrate context‐dependent hemispheric allocation and encoding of numerical magnitude in PD and how biases in numerical magnitude allocation in Parkinsonian patients can correspondingly impair economic decision‐making.

How Does Working Memory Enable Number-Induced Spatial Biases?

Number-space associations are a robust observation, but their underlying mechanisms remain debated. Two major accounts have been identified. First, spatial codes may constitute an intrinsic part of number representations stored in the brain – a perspective most commonly referred to as the Mental Number Line account. Second, spatial codes may be generated at the level of working memory when number (or other) representations are coordinated in function of a specific task. The aim of the current paper is twofold. First, whereas a pure Mental Number Line account cannot capture the complexity of observations reported in the literature, we here explore if and how a pure working memory account can suffice. Second, we make explicit (more than in our earlier work) the potential building blocks of such a working memory account, thereby providing clear and concrete foci for empirical efforts to test the feasibility of the account.