The SNARC effect in two dimensions: Evidence for a frontoparallel mental number plane

The combination of the horizontal and vertical dimensions in mental time representation: the existence of a spatial mental map of time

Horizontal and vertical representations of time (past-left or down and future-right or top) have been demonstrated. However, only a few studies have investigated the existence of a spatial map of time, considering it as the interaction of different spatial dimensions in space. The aim of this study was to investigate the existence of a mental time representation along the diagonal axes, intended as the combination of the horizontal and the vertical dimensions. Seventy-nine Italian participants (85% females; mean age = 25.11 ± 4.86 years; 77 right-handed) performed an online temporal judgment task using 20 Italian temporal expressions presented either always in the center (Experiment 1) or in the four corners of the screen (Experiment 2) and two pairs of response keys ("C" and "U" for the positive diagonal; "R" and "N" for the negative diagonal). Results showed spatial-temporal associations in positive (i.e., time was represented from left-bottom to right-top) and negative (i.e., time was represented from left-top to right-bottom) diagonals, although in Experiment 2 these associations were weak for the negative diagonal. These spatial-temporal associations along both diagonals were confirmed even when participants were free to place different temporal stimuli along a diagonally drawn line, in a Time-to-Position task, indicating that the temporal expressions could be ordered linearly along the diagonal spaces. Finally, these data indicated that the horizontal information was mainly used for determining the spatial-temporal associations along both diagonals, whereas the vertical information was flexible with a bottom-to-top (for positive diagonal) and top-to-bottom (for negative diagonal) temporal representation.

The role of perceptual and action effector strength of graphs and bases of mathematical metaphors in the metaphorical processing of mathematical concepts

Metaphors that describe an abstract concept in terms of a motion concept are widely used to enhance our understanding of abstract concepts. These metaphors are used not only in our daily language but also in learning mathematics. As an example, in the process of understanding the abstract representation of a mathematical concept, a graphical representation may play the role of a mediatory domain. This graphical representation could have a high degree of perceptual and action effector strength. This is particularly the case when a gestures (as a motion) is used to depict the graphical representation. After looking at this example, we discuss perceptual and action effector strength of the base domains of several mathematical metaphors that describe mathematical concepts in terms of spatial and motion concepts. Then, based on the data in the Lancaster Sensorimotor Norms, it is suggested that high degrees of perceptual and action effector strength of the base domains of these metaphors play an important role in the grounding of abstract mathematical concepts in the physical environment.

How to not induce SNAs: The insufficiency of directional force

People respond faster to smaller numbers in their left space and to larger numbers in their right space. Here we argue that movements in space contribute to the formation of spatial-numerical associations (SNAs). We studied the impact of continuous isometric forces along the horizontal or vertical cardinal axes on SNAs while participants performed random number production and arithmetic verification tasks. Our results suggest that such isometric directional force do not suffice to induce SNAs.

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.

Vertical versus horizontal Spatial-Numerical Associations (SNA): A processing advantage for the vertical dimension

Humans have associations between numbers and physical space on both horizontal and vertical dimensions, called Spatial-Numerical Associations (SNAs). Several studies have considered the hypothesis of there being a dominant orientation by examining on which dimension people are more accurate and efficient at responding during various directional SNA tasks. However, these studies have difficulty differentiating between a person’s efficiency at accessing mental representations of numbers in space, and the efficiency at which they exercise motor control functions, particularly bilateral ones, when manifesting a response during an explicit directional SNA task. In this study we use a conflict test employing combined explicit magnitude and spatial directional processing in which pairs of numbers are placed along the diagonal axes and response accuracy/efficiency are considered across the horizontal and vertical dimensions simultaneously. Participants indicated which number in each pair was largest using a joystick that only required unilateral input. The experiment was run in English using Arabic numerals. Results showed that directional SNAs have a vertical rather than horizontal dominance. A moderating factor was also found during post-hoc analysis, where response efficiency, but not accuracy, is conditional on a person’s native language being oriented the same as the language of the experiment, left to right. The dominance of the vertical orientation suggests adopting more vertical display formats for numbers may provide situational advantages, particularly for explicit magnitude comparisons, with some domains like flight controls and the stock market already using these in some cases.

The role of cognitive control in the SNARC effect: A review

The spatial-numerical association of response codes (SNARC) effect, in which people respond to small numbers faster with the left hand and to large numbers faster with the right hand, is a popular topic in cognitive psychology. Some well-known theoretical accounts explaining this effect include the mental number line model, polarity correspondence principle, dual-route model, and working memory account. However, these fail to explain the finding that the size of the SNARC effect is modulated by cognitive control. Here, we propose a new account-a cognitive control-based view of the SNARC effect. This view argues that the SNARC effect is fundamentally determined by cognitive control in resolving conflicts during stimulus-response mapping. Several subcomponents of cognitive control, such as working memory, mental or task set shifting, inhibition control, and conflict adaptation, can easily modulate the SNARC effect. The cognitive control-based view can account for the flexible SNARC effect observed in diverse task situations while providing new insight into its mechanism.

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.

The Effect of Verbal Task Instruction on Spatial-Numerical Associations of Response Codes Effect Coding of Spatial-Numerical Associations: Evidence From Event-Related Potential

The spatial-numerical associations of response codes (SNARC) effect reveals that individuals can represent numbers spatially. In this study, event-related potential (ERP) technology was used to probe the effect of verbal-spatial task instructions on spatial-numerical association coding by using digit parity and magnitude judgment tasks, with the numbers 1–9 (except 5) and Chinese word labels (“left” and “right”) as experimental materials. The behavioral results of Experiment 1 showed that the SNARC effect was mainly based on verbal-spatial coding and appeared when the stimulus onset asynchrony (SOA) between the presentation of the verbal labels and the target digit was 0 ms. ERP results did not reveal any significant SNARC-related effects in either the N1 or P3 components. The behavioral results of Experiment 2 again showed that the SNARC effect was dominated by verbal-spatial coding. ERP results showed that significant effects related to verbal-spatial coding were found in both the early positive deflection of the stimulus-locked lateralized readiness potential (S-LRP) and the latency of the response-locked LRP (R-LRP). Hence, in this study, the nature of the spatial coding of the digit magnitudes was influenced by the processing of the word labels and affected both the response selection and response preparation stages.

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.

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.

An implicit task reveals space-time associations along vertical and diagonal axes

In previous studies investigating the space-time compatibility effect, the experimental task always invites explicit spatial or temporal processing or both. In this study, we kept space and time irrelevant to the task. In a go/no-go task, participants (N = 50) were asked to either press a single button when they found the target or refrain from responding when there was no target in a search array. We manipulated the duration of the target-alone presentation that preceded a 7 × 7 search array consisting of either target plus distractors or distractors alone. The results revealed faster responses to shorter durations when the target appeared in the upper relative to the lower space. A similar effect also appeared along the diagonal axis with faster responses to shorter durations in upper-left relative to lower-right space. In contrast, no such difference was found along the horizontal axis. We hypothesize that vertical and diagonal space-time associations arise from the grounding of mental representation of time in physical experiences.

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.

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.

Is 'heavy' up or down? Testing the vertical spatial representation of weight

Smaller numbers are typically responded to faster with a bottom than a top key, whereas the opposite occurs for larger numbers (a vertical spatial-numerical association of response codes: i.e. the vertical SNARC effect). Here, in four experiments, we explored whether a vertical spatial-magnitude association can emerge for lighter vs. heavier items. Participants were presented with a central target stimulus that could be a word describing a material (e.g. 'paper', 'iron': Experiment 1), a numerical quantity of weight (e.g. '1 g', '1 kg': Experiment 2) or a picture associated with a real object that participants weighed before the experiment (Experiments 3a/3b). Participants were asked to respond either to the weight (Experiments 1-3a) or to the size (i.e. weight was task-irrelevant; Experiment 3b) of the stimuli by pressing vertically placed keys. In Experiments 1 and 2, faster responses emerged for the lighter-bottom/heavier-top mapping-in line with a standard SNARC-like effect-whereas in Experiment 3a the opposite mapping emerged (lighter-top/heavier-bottom). No evidence of an implicit weight-space association emerged in Experiment 3b. Overall, these results provide evidence indicating a possible context-dependent vertical spatial representation of weight.

Mental Number Representations in 2D Space

There is evidence both for mental number representations along a horizontal mental number line with larger numbers to the right of smaller numbers (for Western cultures) and a physically grounded, vertical representation where "more is up." Few studies have compared effects in the horizontal and vertical dimension and none so far have combined both dimensions within a single paradigm where numerical magnitude was task-irrelevant and none of the dimensions was primed by a response dimension. We now investigated number representations over both dimensions, building on findings that mental representations of numbers and space co-activate each other. In a Go/No-go experiment, participants were auditorily primed with a relatively small or large number and then visually presented with quasi-randomly distributed distractor symbols and one Arabic target number (in Go trials only). Participants pressed a central button whenever they detected the target number and elsewise refrained from responding. Responses were not more efficient when small numbers were presented to the left and large numbers to the right. However, results indicated that large numbers were associated with upper space more strongly than small numbers. This suggests that in two-dimensional space when no response dimension is given, numbers are conceptually associated with vertical, but not horizontal space.

The spatial representation of number, time, and serial order following sensory deprivation: A systematic review

The spatial representation of numerical and temporal information is thought to be rooted in our multisensory experiences. Accordingly, we may expect visual or auditory deprivation to affect the way we represent numerical magnitude and time spatially. Here, we systematically review recent findings on how blind and deaf individuals represent abstract concepts such as magnitude and time (e.g., past/future, serial order of events) in a spatial format. Interestingly, available evidence suggests that sensory deprivation does not prevent the spatial "re-mapping" of abstract information, but differences compared to normally sighted and hearing individuals may emerge depending on the specific dimension considered (i.e., numerical magnitude, time as past/future, serial order). Herein we discuss how the study of sensory deprived populations may shed light on the specific, and possibly distinct, mechanisms subserving the spatial representation of these concepts. Furthermore, we pinpoint unresolved issues that need to be addressed by future studies to grasp a full understanding of the spatial representation of abstract information associated with visual and auditory deprivation.