How Does Working Memory Enable Number-Induced Spatial Biases?

The effect of spatial distance on numerical distance processing

The close relationship between numerical and spatial representation has been widely studied. However, little is known regarding the influence of spatial distance on the processing of numerical distance. The purpose of this study was to examine this relationship by employing a modified numerical Stroop task, in which the spatial distance was either congruent or incongruent with the numerical distance. That is, numerical and spatial distances were either compatible with each other or incompatible. Experiment 1 demonstrated that when participants were directly requested to assess the numerical distance, spatial distance influenced task performance, thereby revealing a novel effect-the spatial-numerical distance congruency effect. Experiment 2 demonstrated that these relations are asymmetrical and revealed that numerical distance did not influence spatial distance when the numerical distance was task-irrelevant. Experiment 3 revealed that the spatial-numerical distance congruency effect can also be obtained automatically by employing a numerical comparison task, which is considered a marker for indirect distance processing. In addition, also tested across the three experiments was whether spatial alignment on the screen (i.e., left, centre, and right) can influence the spatial-numerical distance congruency effect. Results revealed that when numbers were presented more naturally (on the left and centre of the screen), a larger effect was obtained compared with when stimuli were presented on the right side. Together, these findings shed new light regarding the relationship between numerical distance and spatial distance and whether and how these aspects influence each other.

Beyond fixed sets: boundary conditions for obtaining SNARC-like effects with continuous semantic magnitudes

Previous research has demonstrated the presence of an effect (i.e., the spatial-numerical association of response codes or SNARC) in both numerical parity and magnitude judgment tasks in which smaller numerical magnitudes are manually responded to faster on the left side and larger numerical magnitudes on the right side. Such a result has typically been attributed to a spatially based representation of numerical magnitude in long-term memory, the format of which has recently been postulated to be positional in line with learning of a canonically ordered number sequence. As a test of this view, in the current research, participants made classification judgments involving either the size (N = 88) or the living-nonliving status (N = 114) corresponding to the names of animals/objects etc. to which no learned canonical ordering of size exists. Names were taken from a very large set of 400 animals/objects etc. and each name was presented only once in an experimental session. Responses were made using left and right manual keypresses. In this work, the relation between response time and the relative sizes of the animals/objects did not differ across the left-right side of response indicating that SNARC-like effects did not occur. As such, the results suggest that space is not an inherent aspect of the long-term representation of magnitude in the brain and that some form of positional coding of magnitude is necessary for SNARC-like effects to occur.

Separating spatial representations from polarity encoding in the processing of number and sequence stimuli in a four-way classification task

Although the SNARC effect in the processing of most magnitude stimuli and sequence stimuli has been reported for the past 30 years, it remains unclear whether this effect is caused by the spatial representation or polarity encoding of stimuli. In the present study, we designed five experiments using a four-way classification task to evaluate the ability of spatial representation theory and polarity encoding theory to explain the SNARC effect in the processing of number and sequence stimuli. In all five experiments in the present study, stimuli (Experiments 1 and 4: four different Arabic numbers, Experiment 2: sequence stimuli, Experiment 3: ordinal sequences relevant to working memory, Experiment 5: Chinese characters without any implicit spatial information) were centrally presented. Participants were asked to respond to specific number or sequence stimuli by pressing the A, S, K, and L keys in consistent trials (or the L, K, S, and A keys in inconsistent trials). The results showed that (1) the SNARC effect occurred in the processing of number and sequence stimuli both when only one specific number was mapped to one specific key (Experiments 1, 2 and 3) and when two numbers were mapped to one specific key (Experiment 4). (2) There was not a SNARC effect when the numbers were replaced with Chinese characters without any implicit spatial information (Experiment 5). The results of these five experiments imply that the SNARC effect in the processing of magnitude stimuli, including numbers and sequences, originates from the spatial representation of stimuli, supporting spatial representation theory.

The reciprocity of spatial-numerical associations of vocal response codes depends on stimulus mode

Individuals make faster left responses to small/er numbers and faster right responses to large/r numbers than vice versa. This "spatial-numerical association of response codes" (SNARC) effect represents evidence for an overlap between the cognitive representations of number and space. Theories of the SNARC effect differ in whether they predict bidirectional S-R associations between number and space or not. We investigated the reciprocity of S-R priming effects between number and location in three experiments with vocal responses. In Experiments 1 and 2, participants completed a number-location task, with digits as stimuli and location words as responses, and a location-number task, with physical locations as stimuli and number words as responses. In addition, we varied the S-R mapping in each task. Results revealed a strong SNARC effect in the number-location task, but no reciprocal SNARC effect in the location-number task. In Experiment 3, we replaced physical location stimuli with location words and digit stimuli with number words. Results revealed a regular and a reciprocal SNARC effect of similar size. Reciprocal SNARC effects thus seem to emerge with verbal location stimuli and vocal responses, but not with physical location stimuli and vocal responses. The S-R associations underlying the SNARC effect with vocal responses thus appear bidirectional and symmetrical for some combinations of stimulus and response sets, but not for others. This has implications for theoretical accounts of the SNARC effect which need to explain how stimulus mode affects the emergence of reciprocal but not regular SNARC effects.

Children's comparison of different-length numbers: Managing different attributes in multidigit number processing

In everyday life the comparison of numbers usually occurs between numbers with different numbers of digits. However, experimental research here is scarce. Recent research has shown that adults respond faster to congruent pairs (the initial digit in the number with more digits is larger, e.g., 2384 vs. 107) than to incongruent pairs (the initial digit is larger in the number with fewer digits, e.g., 2675 vs. 398). This has been interpreted as support for the processing of multiple attributes in parallel and against serial accounts. The current research asked whether there is a change in the relevance of these attributes as school grades increase. School-age children from the second to sixth grades (N = 206) were presented with pairs of numbers that had either the same number of digits (3 vs. 3 or 4 vs. 4) or a different number of digits (3 vs. 4). In this latter condition, the stimuli, matched by distance, could be either length/digit congruent (e.g., 2384 vs. 107) or length/digit incongruent (e.g., 2675 vs. 398). Linear mixed models showed a length/digit congruity effect from second graders. Interestingly, in the response time measure, congruity interacted with school grade and the side in which the larger number of the pair was presented. Whereas these results support a model that considers number comparison as a process that weighs different attributes in parallel, it is also argued that developmental changes are associated with differences in the level of automatization of the componential skills involved in the comparison.

Regional specificity of cathodal transcranial direct current stimulation effects on spatial-numerical associations: Comparison of four stimulation sites

Neuromodulation with transcranial direct current stimulation (tDCS) is an increasingly popular research tool to experimentally manipulate cortical areas and probe their causal involvements in behavior, but its replicability and regional specificity are not clear. This registered report investigated cathodal tDCS effects on spatial-numerical associations (i.e., the SNARC effect), the numerical distance effect (NDE), and inhibitory control (i.e., stop-signal reaction time; SSRT). Healthy adults (N = 160) were randomly assigned to one of five groups to receive sham tDCS or 1 mA cathodal tDCS to one of four stimulation sites (left/right prefrontal cortex [PFC], left/right posterior parietal cortex) with extracephalic return. We replicated that cathodal tDCS over the left PFC reduced the SNARC effect compared to sham tDCS and to tDCS over the left parietal cortex. However, neither NDE nor SSRT were modulated in the main analyses. Post hoc contrasts and exploratory analyses showed that cathodal tDCS over the right PFC had a time-dependent effect by delayed practice-related improvements in SSRT. Math anxiety moderated changes in the NDE in the groups receiving tDCS to the right parietal cortex. With few exceptions, the replicability and regional specificity of tDCS effects on behavior were weak and partially moderated by individual differences. Future research needs to characterize the parameter settings for effective neuromodulation.

Effect of egocentric and allocentric reference frames on spatial-numerical associations

From an embodied view of cognition, sensorimotor mechanisms are strongly involved in abstract processing, such as Arabic number meanings. For example, spatial cognition can influence number processing. These spatial-numerical associations (SNAs) have been deeply explored since the seminal SNAs of response code (SNARC) effect (i.e., faster left/right sided responses to small/large magnitude numbers, respectively). Although these SNAs along the transverse plane (left-to-right axis) have been extensively studied in cognitive sciences, no systematic assessment of other planes of the tridimensional space has been afforded. Moreover, there is no evidence of how SNAs organise themselves throughout the changes in spatial body-reference frames (egocentric and allocentric). Hence, this study aimed to explore how SNAs organise themselves along the transverse and sagittal planes when egocentric and allocentric changes are processed during body displacements in the environment. In the first experiment, the results revealed that, when the participants used an egocentric reference, SNAs were observed only along the sagittal plane. In a second experiment that used an allocentric reference, the reversed pattern of results was observed: SNAs were present only along the transverse plane of the body. Overall, these findings suggest that, depending on the spatial reference frames of the body, SNAs are strongly flexible.

A sensorimotor perspective on numerical cognition

Numbers are present in every part of modern society and the human capacity to use numbers is unparalleled in other species. Understanding the mental and neural representations supporting this capacity is of central interest to cognitive psychology, neuroscience, and education. Embodied numerical cognition theory suggests that beyond the seemingly abstract symbols used to refer to numbers, their underlying meaning is deeply grounded in sensorimotor experiences, and that our specific understanding of numerical information is shaped by actions related to our fingers, egocentric space, and experiences with magnitudes in everyday life. We propose a sensorimotor perspective on numerical cognition in which number comprehension and numerical proficiency emerge from grounding three distinct numerical core concepts: magnitude, ordinality, and cardinality.

Spatio-Numerical Mapping in 3D

The close link between number and space is illustrated by the Spatial Numerical Association of Response Codes (SNARC) effect. The current research focuses on the flexibility of the SNARC across three dimensions. Shaki and Fischer (2018) pointed out that spatial attributes of stimuli and response effectors can favor an ad hoc spatial representation. In this paper, we aimed to broaden this perspective using two Go/NoGo experiments with digits being presented at two spatial locations while a central response was required. In Experiment 1, stimuli appeared either to the left or right (horizontal) and below or above fixation (vertical). In Experiment 2, as the monitor was laying down flat on the desk, stimuli appeared either to the left or right (horizontal) and either close or far from the observer (midsagittal). The results of Experiment 1 show significant effects for the two dimensions (horizontal, vertical), while in Experiment 2, we observe only a barely significant effect for the sagittal axis. We interpret these findings as showing (1) the importance of motor response spatialization in eliciting the SNAs and (2) the dominance of the vertical axis over the horizontal when the spatial component of the motor response is removed.

Thinking about order: a review of common processing of magnitude and learned orders in animals

Rich behavioral and neurobiological evidence suggests cognitive and neural overlap in how quantitatively comparable dimensions such as quantity, time, and space are processed in humans and animals. While magnitude domains such as physical magnitude, time, and space represent information that can be quantitatively compared (4 "is half of" 8), they also represent information that can be organized ordinally (1→2→3→4). Recent evidence suggests that the common representations seen across physical magnitude, time, and space domains in humans may be due to their common ordinal features rather than their common quantitative features, as these common representations appear to extend beyond magnitude domains to include learned orders. In this review, we bring together separate lines of research on multiple ordinal domains including magnitude-based and learned orders in animals to explore the extent to which there is support for a common cognitive process underlying ordinal processing. Animals show similarities in performance patterns across natural quantitatively comparable ordered domains (physical magnitude, time, space, dominance) and learned orders (acquired through transitive inference or simultaneous chaining). Additionally, they show transfer and interference across tasks within and between ordinal domains that support the theory of a common ordinal representation across domains. This review provides some support for the development of a unified theory of ordinality and suggests areas for future research to better characterize the extent to which there are commonalities in cognitive processing of ordinal information generally.

The SNARC effect: a preregistered study on the interaction of horizontal, vertical, and sagittal spatial-numerical associations

Small numbers are processed faster through left-sided than right-sided responses, whereas large numbers are processed faster through right-sided than left-sided responses [i.e., the Spatial-Numerical Association of Response Codes (SNARC) effect]. This effect suggests that small numbers are mentally represented on the left side of space, whereas large numbers are mentally represented on the right side of space, along a mental number line. The SNARC effect has been widely investigated along the horizontal Cartesian axis (i.e., left-right). Aleotti et al. (Cognition 195:104111, 2020), however, have shown that the SNARC effect could also be observed along the vertical (i.e., small numbers-down side vs. large numbers-up side) and the sagittal axis (i.e., small numbers-near side vs. large numbers-far side). Here, we investigated whether the three Cartesian axes could interact to elicit the SNARC effect. Participants were asked to decide whether a centrally presented Arabic digit was odd or even. Responses were collected through an ad hoc-made response box on which the SNARC effect could be compatible for one, two, or three Cartesian axes. The results showed that the higher the number of SNARC-compatible Cartesian axes, the stronger the SNARC effect. We suggest that numbers are represented in a three-dimensional number space defined by interacting Cartesian axes.

Resourceful Event-Predictive Inference: The Nature of Cognitive Effort

Pursuing a precise, focused train of thought requires cognitive effort. Even more effort is necessary when more alternatives need to be considered or when the imagined situation becomes more complex. Cognitive resources available to us limit the cognitive effort we can spend. In line with previous work, an information-theoretic, Bayesian brain approach to cognitive effort is pursued: to solve tasks in our environment, our brain needs to invest information, that is, negative entropy, to impose structure, or focus, away from a uniform structure or other task-incompatible, latent structures. To get a more complete formalization of cognitive effort, a resourceful event-predictive inference model (REPI) is introduced, which offers computational and algorithmic explanations about the latent structure of our generative models, the active inference dynamics that unfold within, and the cognitive effort required to steer the dynamics-to, for example, purposefully process sensory signals, decide on responses, and invoke their execution. REPI suggests that we invest cognitive resources to infer preparatory priors, activate responses, and anticipate action consequences. Due to our limited resources, though, the inference dynamics are prone to task-irrelevant distractions. For example, the task-irrelevant side of the imperative stimulus causes the Simon effect and, due to similar reasons, we fail to optimally switch between tasks. An actual model implementation simulates such task interactions and offers first estimates of the involved cognitive effort. The approach may be further studied and promises to offer deeper explanations about why we get quickly exhausted from multitasking, how we are influenced by irrelevant stimulus modalities, why we exhibit magnitude interference, and, during social interactions, why we often fail to take the perspective of others into account.

Is the future near or far depending on the verb tense markers used? An experimental investigation into the effects of the grammaticalization of the future

Psycholinguistic approaches that study the effects of language on mental representations have ignored a potential role of the grammaticalization of the future (i.e., how the future manifests linguistically). We argue that the grammaticalization of the future may be an important aspect, as thinking about the future is omnipresent in our everyday life. The aim of this study was to experimentally manipulate the degree of future time references (i.e., present and future verb tense and temporal adverbials) to address their impact on the perceived location of future events. Across four experiments, two in French and two in German, no effect was found, irrespective of our verb and adverbial manipulations, and contrary to our hypotheses. Bayes factors confirmed that our null effects were not due to a lack of power. We present one of the first empirical accounts investigating the role of the grammaticalization of the future on effects of mental representations. We discuss possible reasons for these null results and illustrate further avenues for future research.

Early is left and up: Saccadic responses reveal horizontal and vertical spatial associations of serial order in working memory

Maintaining serial order in working memory is crucial for cognition. Recent theories propose that serial information is achieved by positional coding of items on a spatial frame of reference. In line with this, an early-left and late-right spatial-positional association of response code (SPoARC) effect has been established. Various theoretical accounts have been put forward to explain the SPoARC effect (the mental whiteboard hypothesis, conceptual metaphor theory, polarity correspondence, or the indirect spatial-numerical association effect). Crucially, while all these accounts predict a left-to-right orientation of the SPoARC effect, they make different predictions regarding the direction of a possible vertical SPoARC effect. In this study, we therefore investigated SPoARC effects along the horizontal and vertical spatial dimension by means of saccadic responses. We replicated the left-to-right horizontal SPoARC effect and established for the first time an up-to-down vertical SPoARC effect. The direction of the vertical SPoARC effect was in contrast to that predicted by metaphor theory, polarity correspondence, or by the indirect spatial-numerical association effect. Rather, our results support the mental whiteboard-hypothesis, according to which positions can be flexibly coded on an internal space depending on the task demands. We also found that the strengths of the horizontal and vertical SPoARC effects were correlated, showing that some people are more prone than others to use spatial references for position coding. Our results therefore suggest that context templates used for position marking are not necessarily spatial in nature but depend on individual strategy preferences.

Reading direction and spatial effects in parity and arithmetic tasks

This study investigated the relationship between numerical and spatial processing and reading direction, conducting conceptual replications of the Shaki et al. (Psychonomic Bulletin & Review 16(2): 328-331, 2009) parity task and the Mathieu et al. (Cognition 146: 229-239, 2016, Experiment 1) simple addition (e.g., 3 + 2) and subtraction (e.g., 3 - 2) task. Twenty-four left-to-right readers (LTR) and 24 right-to-left readers (RTL) were tested. The response time (RT) analysis of the parity task presented a robust spatial-numerical association of response codes (SNARC) effect (left-side response advantage for smaller numbers and right-side advantage for larger numbers) for LTR but not RTL readers. In the arithmetic task, the three problem elements (e.g., 3 + 4) were presented sequentially with the second operand displaced slightly to the left or right of fixation. RTL but not LTR readers presented a RT advantage for subtraction relative to addition with a right-shifted second operand compared to it being left-shifted. This is consistent with a spatial bias linked to native reading direction. For both reading-direction groups, effects of the left vs. right side manipulation in the arithmetic or parity task did not correspond to parallel effects in the other task. The results imply that the parity-based SNARC effects and side-related effects in cognitive arithmetic are not equivalent measures of space-related processes in cognitive number processing and likely reflect distinct mechanisms.

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.

Interaction mechanism between location and sequence in letter cognition

A previous study used days as a sequence symbol to investigate the interaction mechanism between location and sequence in sequence symbol cognition; the study findings suggested that the spatial stimulus-response compatibility effect and the Simon effect could not co-exist with the SNARC-like effect when processing sequence symbols. The previous study did not include the influence of the difficulty of identifying sequence symbols on the Simon effect in the investigation, so it is unclear whether the conclusion about processing sequence symbols with considerable identification difficulty can be extended to the processing of sequence symbols with less identification difficulty. Therefore, the present study explored letters that have a low level of identification difficulty to investigate the interaction mechanism between location and sequence in sequence symbol cognition. Participants were asked to classify a probe letter, which was randomly displayed on the left or right side of the screen, according to its location (Experiment 1), its sequence (Experiment 2) or its colour (Experiment 3). The results indicated that (1) only the spatial stimulus-response compatibility effect and Simon effect were present in the letter location classification task and letter colour classification task, respectively. (2) The Simon effect co-existed with the SNARC-like effect, and these two effects interacted with each other in the letter sequence classification task. From these results, it can be concluded that the task determines whether the Simon effect and the SNARC-like effect can co-exist, with differences presented across sequence symbols.

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.

Measuring spontaneous and automatic processing of magnitude and parity information of Arabic digits by frequency-tagging EEG

Arabic digits (1–9) are everywhere in our daily lives. These symbols convey various semantic information, and numerate adults can easily extract from them several numerical features such as magnitude and parity. Nonetheless, since most studies used active processing tasks to assess these properties, it remains unclear whether and to what degree the access to magnitude and especially to parity is automatic. Here we investigated with EEG whether spontaneous processing of magnitude or parity can be recorded in a frequency-tagging approach, in which participants are passively stimulated by fast visual sequences of Arabic digits. We assessed automatic magnitude processing by presenting a stream of frequent small digit numbers mixed with deviant large digits (and the reverse) with a sinusoidal contrast modulation at the frequency of 10 Hz. We used the same paradigm to investigate numerical parity processing, contrasting odd digits to even digits. We found significant brain responses at the frequency of the fluctuating change and its harmonics, recorded on electrodes encompassing right occipitoparietal regions, in both conditions. Our findings indicate that both magnitude and parity are spontaneously and unintentionally extracted from Arabic digits, which supports that they are salient semantic features deeply associated to digit symbols in long-term memory.

The influence of locative expressions on context-dependency of endpoint control in aiming

It has been claimed that increased reliance on context, or allocentric information, develops when aiming movements are more consciously monitored and/or controlled. Since verbalizing target features requires strong conscious monitoring, we expected an increased reliance on allocentric information when verbalizing a target label (i.e. target number) during movement execution. We examined swiping actions towards a global array of targets embedded in different local array configurations on a tablet under no-verbalization and verbalization conditions. The global and local array configurations allowed separation of contextual-effects from any possible numerical magnitude biases triggered from calling out specific target numbers.The patterns of constant errors in the target directionwere used to assess differences between conditions. Variation in the target context configuration systematically biased movement endpoints in both the no-verbalization and verbalization conditions. Ultimately, our results do not support the assertion that calling out target numbers during movement execution increases the context-dependency of targeted actions.

Spatialization in working memory: can individuals reverse the cultural direction of their thoughts?

A recent study based on the SPoARC effect (spatial position association response codes) showed that culture heavily shapes cognition and more specifically the way thought is organized; when Western adults are asked to keep in mind a sequence of colors, they mentally organize them from left to right, whereas right-to-left reading/writing adults spatialize them in the opposite direction. Here, we investigate if the spontaneous direction of spatialization in Westerners can be reversed. Lists of five consonants were presented auditorily at a rate of 3 s per item, participants were asked to mentally organize the memoranda from right to left. Each list was followed by a probe. Participants had to indicate whether the probe was part of the sequence by pressing a "yes" key or a "no" key with the left or right index finger. Left/right-hand key assignment was switched after half of the trials were completed. The results showed a reverse SPoARC effect that was comparable in magnitude to the spontaneous left-to-right SPoARC effect found in a previous study. Overall, our results suggest that individuals can reverse the cultural direction of their thoughts.

SNARC effect modulated by central executive control: revealed in a cue-based trisection task

People respond to small numbers faster with the left hand and respond to large numbers faster with the right hand, a phenomenon known as the Spatial-Numerical Association of Response Codes (SNARC) effect. Whether the SNARC effect originates from culturally determined long-term experience or the task-set-influenced temporary associations among spaces, locations, and numerical magnitudes in working memory (WM) is still controversial. In the present study, we used a trisection paradigm in which numbers were divided into three categories (small: 1, 2; middle: 4, 5, 6; and large: 8, 9) to explore whether the central executive control can modulate the SNARC effect. Participants were serially presented with a cue and a target number. The cue denoted a task rule, which informed participants to compare the target number with either 3 or 7. The cue was either switched or repeated across trials. We found that the SNARC effects were observed in the cue-switching condition. In the cue-repeat condition, the SNARC effect disappeared. These findings suggest that the SNARC effect is modulated by set-shifting-related central executive control in WM, supporting the view that the SNARC effect is WM-dependent.

The influence of children's mathematical competence on performance in mental number line, time knowledge and time perception

A growing body of research suggests that space, time and number are represented within a common system. Other studies have shown this relationship is related to the mathematical competency. Here we examined the influence of the mathematical capacities of 8-12 years old children, grouped into high (n = 63) and low (n = 58) on performance in mental number line, time knowledge and time perception. The results revealed that mathematical competency influences mental number line and time knowledge, but with regard to time perception the effects were only observed in time production task. In addition, the results of correlation analysis revealed interaction between time knowledge, time production (but not reproduction) and mental number line. Finally, the findings are discussed within the framework of the recent theories regarding representation of space, time and number.

The spatial coding mechanism of ordinal symbols: a study based on the ordinal position effect

The ordinal position effect refers to a phenomenon in which items positioned early in an ordinal sequence receive a faster response with the left key than with the right key, and the opposite response pattern occurs when items are positioned later in an ordinal sequence. Previous studies have suggested that ordinal symbols are spatially represented from left to right, thus leading to the ordinal position effect; however, the spatial coding mechanism of ordinal symbols remains unclear. Therefore, the present study explored the ordinal position effect as an index to judge the spatial coding of ordinal symbols, and three experiments were performed to investigate the spatial coding mechanism of ordinal symbols. In particular, a novel transitory ordinal sequence was induced by presenting successive dots of different colors centrally (Experiment 1), from left to right or from right to left (Experiments 2 and 3), and participants were asked to memorize the successive dots in the correct order. Then, the participants were asked to press a key to provide a response corresponding to a probe dot's ordinal position (Experiments 1 and 2) or its spatial location (Experiment 3). The following results were identified: (1) The ordinal position effect occurred when responses were based on the ordinal position regardless of the presentation direction, and (2) the ordinal position effect was overridden when responses were based on the spatial locations of the ordinal symbols. From these results, we concluded that the spatial coding of ordinal symbols is flexible and that ordinal symbols are encoded depending on the specific experimental context.

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.

Regional specificity of cathodal transcranial direct current stimulation (tDCS) effects on spatial-numerical associations: Comparison of four stimulation sites

Based on a theory of impulsive and reflective human behavior, we test the effects of transcranial direct current stimulation (tDCS) targeting either prefrontal or parietal cortex in either hemisphere. In a confirmatory registered report, cathodal tDCS is administered to conceptually reproduce tDCS modulations of implicit spatial-numerical associations, numerical distance effects, and response inhibition. Those cognitive operations are hypothesized to draw on left prefrontal, parietal, and right prefrontal activations, respectively, thereby susceptible to inhibitory, cathodal tDCS across those regions. Vice versa, the mutual regional and behavioral specificity of tDCS effects on these behavioral indices is examined and expected to produce double dissociations. In a mixed within-subjects (baseline, during tDCS, post-tDCS) and between-subjects (target electrode: left/right prefrontal cortex/posterior parietal cortex, or sham tDCS) design, we collect (a) confirmatory data on the robustness of cathodal tDCS effects on three behavioral effects and (b) differential data on the specificity of regional targets in male and female human participants. Results will provide crucial tests of theories of cortical organization implied by implicit associations and explicit regulation, which can direct future brain stimulation studies.

Symbol grounding of number words in the subitization range

In three experiments, we explored whether number words are grounded in a nonsymbolic representation of numerosity. We used a sentence-picture verification task, where participants are required to check whether the concept given in a sentence corresponds to the subsequently presented object. We concurrently manipulated numerical congruency by orthogonally varying the number word attached to the concept and the quantity of objects. The number words and numerosities varied from one to four in Experiment 1 and from six to nine in Experiment 2. In Experiment 3, we employed number words six and eight with the constraint that, in the incongruent condition, a constant number-to-numerosity ratio of 2:1 was used. In Experiment 1, we found that participants were faster and more efficient when concept-object matches were accompanied by numerical congruency relative to incongruency. On the other hand, no such difference was observed in Experiments 2 and 3 for numbers falling outside of the subitization range. The results are consistent with the hypothesis that number words from one to four are grounded in a nonsymbolic representation of the size of small sets.

The effects of risk magnitude training on mapping risks on space

Risk perception has recently been shown to reveal a mental spatial representation, with people responding faster to low-risk items on the left side, and high-risk items on the right side. Subjective risk perception has a stronger spatial representation than objective risk perception; however, both reveal small effect sizes. With risk magnitude being a new domain within spatial mapping literature, we sought to explore its nuances. Following discussion surrounding the relationship between spatial mapping and level of expertise, this study investigated the effect of training an objective risk magnitude sequence on mental spatial representations. Participants (n = 34) used their left and right hands to indicate whether eight risk stimuli were lower or higher risk than a referent activity, both before and after training. Training involved repetitively learning the objectively correct order of the same eight risk stimuli for approximately 15 min. Pre-training results demonstrated the expected spatial representations. Contrary to our predictions, the spatial representation did not get stronger post-training, but instead disappeared. Previous research has demonstrated a loss of spatial-numerical mappings with increased task load. An increase in post-training reaction times could reflect an increase in task load due to a lack of adequate knowledge of risk stimulus order; thus revealing no mental spatial representation. However, failure to find training effects highlights the flexibility of weaker spatial representations, and supports research demonstrating spatial representation flexibility.

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 influence of semantic processing and response latency on the SNARC effect

The Spatial-Numerical Association of Response Code (SNARC) effect refers to the finding that small or large numbers elicit faster leftward or rightward responses, respectively. Traditionally, this effect has been thought to reflect the intrinsic spatial orientation of the mental number line (MNL account) and thus to be modulated by the amount of semantic processing required by the task. This study aimed to test this hypothesis. Participants performed two tasks requiring semantic processing (magnitude classification and parity judgement) and two tasks requiring the processing of non-semantic features of the number (phoneme detection and color judgement). Contrary to the MNL account, the SNARC effect in the four tasks was not modulated by the amount of semantic processing, but rather by response latency. These results provide evidence against the MNL account and in favor of alternative accounts (dual-route model, working memory account).

Scanning of speechless comics changes spatial biases in mental model construction

The mental representation of both time and number shows lateral spatial biases, which can be affected by habitual reading and writing direction. However, this effect is in place before children begin to read. One potential early cause is the experiences of looking at picture books together with a carer, as those images also follow the directionality of the script. What is the underlying mechanism for this effect? In the present study, we test the possibility that such experiences induce spatial biases in mental model construction, a mechanism which is a good candidate to induce the biases observed with numbers and times. We presented a speechless comic in either standard (left-to-right) or mirror-reversed (right-to-left) form to adult Spanish participants. We then asked them to draw the scene depicted by sentences like 'the square is between the cross and the circle'. The position of the lateral objects in these drawings reveals the spatial biases at work when building mental models in working memory. Under conditions of highly consistent directionality, the mirror comic changed pre-existing lateral biases. Processes of mental model construction in working memory stand as a potential mechanism for the generation of spatial biases for time and number.This article is part of the theme issue 'Varieties of abstract concepts: development, use and representation in the brain'.

Individual Differences in Implicit and Explicit Spatial Processing of Fractions

Recent studies have explored the foundations of mathematical skills by linking basic numerical processes to formal tests of mathematics achievement. Of particular interest is the relationship between spatial-numerical associations-specifically, the Spatial Numerical Association of Response Codes (SNARC) effect-and various measures of math ability. Thus far, studies investigating this relationship have yielded inconsistent results. Here, we investigate how individual implicit and explicit spatial representations of fractions relate to fraction knowledge and other formal measures of math achievement. Adult participants (n = 105) compared the magnitude of single digit, irreducible fractions to ½, a task that has previously produced a reliable SNARC effect. We observed a significant group-level SNARC effect based on overall fraction magnitude, with notable individual variability. While individual SNARC effects were correlated with performance on a fraction number-line estimation (NLE) task, only NLE significantly predicted scores on a fractions test and basic standardized math test, even after controlling for IQ, mean accuracy, and mean reaction time. This suggests that-for fractions-working with an explicit number line is a stronger predictor of math ability than implicit number line processing. Neither individual SNARC effects nor NLE performance were significant predictors of algebra scores; thus, the mental number line may not be as readily recruited during higher-order mathematical concepts, but rather may be a foundation for thinking about simpler problems involving rational magnitudes. These results not only characterize the variability in adults' mental representations of fractions, but also detail the relative contributions of implicit (SNARC) and explicit (NLE) spatial representations of fractions to formal math skills.

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.

Mechanism of the SNARC Effect in Numerical Magnitude, Time Sequence, and Spatial Sequence Tasks: Involvement of LTM and WM

The spatial-numerical association of response codes (SNARC) effect refers to the phenomenon that responses involving small numbers are faster with the left hand and responses involving large numbers are faster with the right hand. Previous studies have investigated the mechanism of the SNARC effect only when the time sequence is induced by centrally presented successive numbers. No study has investigated the mechanism of the SNARC effect when the spatial sequence is induced. Given that the induction of a spatial sequence together with a time sequence provides a new temporary reference for the serial order to be coded in working memory (WM), it would be interesting to examine the SNARC effect when both the time sequence and spatial sequence are induced. Therefore, a novel priming paradigm that simultaneously induced the time sequence and spatial sequence was employed in the present study to investigate the mechanism of the SNARC effect. Specifically, the time sequence and spatial sequence were induced by presenting a series of self-paced successive numbers, centrally or in a left-to-right or right-to-left direction, on the screen. Following the presentation of successive numbers, the probe number was centrally presented on the screen and university students were asked to distinguish to which time sequence or spatial sequence the probe number belonged by pressing a specified key of a qwerty keyboard. The results indicated that (1) the SNARC effect simultaneously appeared in the processing of the number magnitude and time sequence when only the time sequence was induced. (2) The SNARC effect disappeared in the processing of the time sequence; however, the SNARC effect coexisted in the processing of the numerical magnitude and spatial sequence when the spatial sequence was induced and participants performed a time sequence relevant task. (3) The SNARC effect coexisted in the processing of the numerical magnitude, time sequence, and spatial sequence when the spatial sequence was induced and participants performed a spatial sequence relevant task. Based on these results, we conclude that whether the SNARC effect coexists in the processing of the numerical magnitude, the time sequence and spatial sequence were influenced by the spatial sequence and relevant task. The results further support the mental whiteboard hypothesis and extended the WM account. Implications for theories on the SNARC effect were discussed.

The SNARC effect is associated with worse mathematical intelligence and poorer time estimation

Interactions between the ways we process space, numbers and time may arise from shared and innate generic magnitude representations. Alternatively or concurrently, such interactions could be due to the use of physical magnitudes, like spatial extent, as metaphors for more abstract ones, like number and duration. That numbers might be spatially represented along a mental number line is suggested by the SNARC effect: faster left-side responses to small single digits, like 1 or 2, and faster right-side responses to large ones, like 8 or 9. Previously, we found that time estimation predicts mathematical intelligence and speculated that it may predict spatial ability too. Here, addressing this issue, we test-on a relatively large sample of adults and entirely within subjects-the relationships between (a) time: proficiency at producing and evaluating durations shorter than one second, (b) space: the ability to mentally rotate objects, (c) numbers: mathematical reasoning skills, and (d) space-number associations: the SNARC effect. Better time estimation was linked to greater mathematical intelligence and better spatial skills. Strikingly, however, stronger associations between space and numbers predicted worse mathematical intelligence and poorer time estimation.

The Developing Mental Number Line: Does Its Directionality Relate to 5- to 7-Year-Old Children's Mathematical Abilities?

Spatial representations of number, such as a left-to-right oriented mental number line, are well documented in Western culture. Yet, the functional significance of such a representation remains unclear. To test the prominent hypothesis that a mental number line may support mathematical development, we examined the relation between spatial-numerical associations (SNAs) and math proficiency in 5- to 7-year-old children. We found evidence of SNAs with two tasks: a non-symbolic magnitude comparison task, and a symbolic "Where was the number?" (WTN) task. Further, we found a significant correlation between these two tasks, demonstrating convergent validity of the directional mental number line across numerical format. Although there were no significant correlations between children's SNAs on the WTN task and math ability, children's SNAs on the magnitude comparison task were negatively correlated with their performance on a measure of cross-modal arithmetic, suggesting that children with a stronger left-to-right oriented mental number line were less competent at cross-modal arithmetic, an effect that held when controlling for age and a set of general cognitive abilities. Despite some evidence for a negative relation between SNAs and math ability in adulthood, we argue that the effect here may reflect task demands specific to the magnitude comparison task, not necessarily an impediment of the mental number line to math performance. We conclude with a discussion of the different properties that characterize a mental number line and how these different properties may relate to mathematical ability.

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.

Spatial-Numerical Associations Enhance the Short-Term Memorization of Digit Locations

Little is known about how spatial-numerical associations (SNAs) affect the way individuals process their environment, especially in terms of learning and memory. In this study, we investigated the potential effects of SNAs in a digit memory task in order to determine whether spatially organized mental representations of numbers can influence the short-term encoding of digits positioned on an external display. To this aim, we designed a memory game in which participants had to match pairs of identical digits in a 9 × 2 matrix of cards. The nine cards of the first row had to be turned face up and then face down, one by one, to reveal a digit from 1 to 9. When a card was turned face up in the second row, the position of the matching digit in the first row had to be recalled. Our results showed that performance was better when small numbers were placed on the left side of the row and large numbers on the right side (i.e., congruent) as compared to the inverse (i.e., incongruent) or a random configuration. Our findings suggests that SNAs can enhance the memorization of digit positions and therefore that spatial mental representations of numbers can play an important role on the way humans process and encode the information around them. To our knowledge, this study is the first that reaches this conclusion in a context where digits did not have to be processed as numerical values.

Systematic spatial patterns of the sense of familiarity: Hierarchical modelling based on eye-tracking experiments

Using different types of stimuli, such as pictures, horizontally written Japanese words, and vertically written Japanese words, this study investigated the spatial patterns of the sense of familiarity within the visual field. The perceptual asymmetry theory predicted that stimuli in the lower visual field would be processed more fluently and would therefore be perceived as more familiar. The working memory theory, originally proposed in space-number research, envisaged type-specific spatial patterns for different stimuli. Participants made old/new recognition memory judgements for stimuli, presented at random positions, while their eye movements were recorded. The observed spatial patterns changed according to the stimulus type (e.g., "more left = older" for horizontally written words and "upper = older" for vertically written words), and this flexibility is encapsulated by the working memory theory as follows: (a) stimulus-type-specific spatial configurations are encoded in long-term memory on the basis of one's experience (e.g., vertically written words are empirically associated with the "upper = older" spatial configuration), (b) the presentation of a stimulus automatically cues the temporal activation of the associated spatial configuration in working memory, and (c) the referential process between the stimulus and configuration unconsciously affects the viewer's sense of familiarity.

Implicit and Explicit Number-Space Associations Differentially Relate to Interference Control in Young Adults With ADHD

Behavioral evidence for the link between numerical and spatial representations comes from the spatial-numerical association of response codes (SNARC) effect, consisting in faster reaction times to small/large numbers with the left/right hand respectively. The SNARC effect is, however, characterized by considerable intra- and inter-individual variability. It depends not only on the explicit or implicit nature of the numerical task, but also relates to interference control. To determine whether the prevalence of the latter relation in the elderly could be ascribed to younger individuals' ceiling performances on executive control tasks, we determined whether the SNARC effect related to Stroop and/or Flanker effects in 26 young adults with ADHD. We observed a divergent pattern of correlation depending on the type of numerical task used to assess the SNARC effect and the type of interference control measure involved in number-space associations. Namely, stronger number-space associations during parity judgments involving implicit magnitude processing related to weaker interference control in the Stroop but not Flanker task. Conversely, stronger number-space associations during explicit magnitude classifications tended to be associated with better interference control in the Flanker but not Stroop paradigm. The association of stronger parity and magnitude SNARC effects with weaker and better interference control respectively indicates that different mechanisms underlie these relations. Activation of the magnitude-associated spatial code is irrelevant and potentially interferes with parity judgments, but in contrast assists explicit magnitude classifications. Altogether, the present study confirms the contribution of interference control to number-space associations also in young adults. It suggests that magnitude-associated spatial codes in implicit and explicit tasks are monitored by different interference control mechanisms, thereby explaining task-related intra-individual differences in number-space associations.