Pointing to numbers and grasping magnitudes

Effect of spatial training on space-number mapping: a situated cognition account

From an embodied perspective of cognition, number processing influences the spatial organization of motor responses showing faster left/right responses to small/large numbers. Recent evidence suggests that such spatial-numerical associations (SNAs) along the transverse and sagittal planes are mutually exclusive with respect to the spatial reference frames used by the participant. Specifically, in egocentric and allocentric frames, SNAs appear along the sagittal and transverse plane, respectively. The first aim of this study was to replicate previous findings. The second aim was to explore the role of switching spatial reference frames in SNAs occurrence according to the processed plane. Consequently, during a referential frame switching (RFS) training, participants were required to identify targets based on an embodied avatar's perspective. Using a random number generation (RNG) task after observing an avatar's displacement, we investigated the effect of RFS training on SNAs organization across the different planes (Experiment 1 & 2 for the egocentric and allocentric perspectives, respectively). Both experiments replicated previous results, but more importantly, RFS training enables the development of new situated cognition strategies from egocentric perspectives and the generalization of transverse SNAs to other planes from allocentric perspectives.

A cross-cultural comparison of finger-based and symbolic number representations

The current study examined predictions from embodied cognition for effects of finger counting on number processing. Although finger counting is spontaneous and nearly universal, counting habits reflect learning and culture. European cultures use a sub-base-five system, requiring a full hand plus additional fingers to express numbers exceeding 5. Chinese culture requires only one hand to express such numbers. We investigated the differential impact of early-acquired finger-based number representations on adult symbolic number processing. In total, 53 European and 56 Chinese adults performed two versions of the magnitude classification task, where numbers were presented either as Arabic symbols or as finger configurations consistent with respective cultural finger-counting habits. Participants classified numbers as smaller/larger than 5 with horizontally aligned buttons. Finger-based size and distance effects were larger in Chinese compared with Europeans. These differences did not, however, induce reliably different symbol processing signatures. This dissociation challenges the idea that sensory and motor habits shape our conceptual representations and implies notation-specific processing patterns.

Conceptual Metaphor and Graphical Convention Influence the Interpretation of Line Graphs

Many metaphors in language reflect conceptual metaphors that structure thought. In line with metaphorical expressions such as 'high number', experiments show that people associate larger numbers with upward space. Consistent with this metaphor, high numbers are conventionally depicted in high positions on the y-axis of line graphs. People also associate good and bad (emotional valence) with upward and downward locations, in line with metaphorical expressions such as 'uplifting' and 'down in the dumps'. Graphs depicting good quantities (e.g., vacation days) are consistent with graphical convention and the valence metaphor, because 'more' of the good quantity is represented by higher y-axis positions. In contrast, graphs depicting bad quantities (e.g., murders) are consistent with graphical convention, but not the valence metaphor, because more of the bad quantity is represented by higher (rather than lower) y-axis positions. We conducted two experiments (N = 300 per experiment) where participants answered questions about line graphs depicting good and bad quantities. For some graphs, we inverted the conventional axis ordering of numbers. Line graphs that aligned (versus misaligned) with valence metaphors (up = good) were easier to interpret, but this beneficial effect did not outweigh the adverse effect of inverting the axis numbering. Line graphs depicting good (versus bad) quantities were easier to interpret, as were graphs that depicted quantity using the x-axis (versus y-axis). Our results suggest that conceptual metaphors matter for the interpretation of line graphs. However, designers of line graphs are warned against subverting graphical convention to align with conceptual metaphors.

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.

Children's spatial-numerical associations on horizontal, vertical, and sagittal axes

There is substantial evidence linking numerical magnitude to the physical properties of space. The most influential support for this connection comes from the SNARC effect (spatial-numerical association of response codes), in which responses to small/large numbers are faster on the left/right side of space, respectively. The SNARC effect has been extensively replicated, and is understood as horizontal mapping of numerical magnitude. However, much less is known about how numbers are represented on the vertical and sagittal axes, and whether spatial-numerical associations on different axes emerge during childhood. To that end, we tested two groups of children, aged 5-7 years and 8 and 9 years, on a single-digit magnitude comparison task with response buttons positioned either upper/lower (vertical), left/right (horizontal) or near/far (sagittal). Our results provide evidence of spatial-numerical mapping on all three axes for both age groups that are similar in strength. This indicates that, even at an early stage of formal education, children can flexibly assign numerical magnitude to any spatial dimension. To examine the contribution of extracorporeal space and spatio-anatomical mapping to the SNARC effect across axes, these sources were pitted against each other by swapping the position of the response hands in Experiment 1b. Switching hand position did not reveal convincing evidence for SNARC effects on any axis. Results are discussed with respect to the utility of three-dimensional mental number lines, and potential avenues for future research are outlined.

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.

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.

Placing Abstract Concepts in Space: Quantity, Time and Emotional Valence

Research has shown that abstract concepts are often conceptualized along horizontal and vertical axes. However, there are mixed results concerning which axis is preferred for which type of conceptual domain. For instance, it has been suggested that the vertical axis may be preferred for quantity in tasks using linguistic stimuli (e.g., ‘more,’ ‘less’), whereas numerals (e.g., ‘1,’ ‘2,’ ‘3’) may be more prone to horizontal conceptualization. In this study, we used a task with free response options to see where participants would place quantity words (‘most,’ ‘more,’ ‘less,’ ‘least’), numerals (‘2,’ ‘4,’ ‘7,’ ‘9’), time words (‘past,’ ‘future,’ ‘earliest,’ ‘earlier,’ ‘later,’ ‘latest’) and emotional valence words (‘best,’ ‘better,’ ‘worse,’ ‘worst’). We find that for quantity words, the vertical axis was preferred; whereas for numerals, participants preferred the horizontal axis. For time concepts, participants preferred the horizontal axis; and for emotional valence, they preferred the vertical axis. Across all tasks, participants tended to use specific axes (horizontal, vertical), rather than combining these two axes in diagonal responses. These results shed light on the spatial nature of abstract thought.

Number concepts: abstract and embodied

Numerical knowledge, including number concepts and arithmetic procedures, seems to be a clear-cut case for abstract symbol manipulation. Yet, evidence from perceptual and motor behaviour reveals that natural number knowledge and simple arithmetic also remain closely associated with modal experiences. Following a review of behavioural, animal and neuroscience studies of number processing, we propose a revised understanding of psychological number concepts as grounded in physical constraints, embodied in experience and situated through task-specific intentions. The idea that number concepts occupy a range of positions on the continuum between abstract and modal conceptual knowledge also accounts for systematic heuristics and biases in mental arithmetic, thus inviting psycho-logical approaches to the study of the mathematical mind.This article is part of the theme issue 'Varieties of abstract concepts: development, use and representation in the brain'.

On the linear representation of numbers: evidence from a new two-numbers-to-two positions task

In the number-to-position methodology, a number is presented on each trial and the observer places it on a straight line in a position that corresponds to its felt subjective magnitude. In the novel modification introduced in this study, the two-numbers-to-two-positions method, a pair of numbers rather than a single number is presented on each trial and the observer places them in appropriate positions on the same line. Responses in this method indicate not only the subjective magnitude of each single number but, simultaneously, provide a direct estimation of their subjective numerical distance. The results of four experiments provide strong evidence for a linear representation of numbers and, commensurately, for the linear representation of numerical distances. We attribute earlier results that indicate a logarithmic representation to the ordered nature of numbers and to the task used and not to a truly non-linear underlying representation.

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.

Creating semantics in tool use

This article presents the first evidence for a functional link between tool use and the processing of abstract symbols like Arabic numbers. Participants were required to perform a tool-use task after the processing of an Arabic number. These numbers represented either a small (2 or 3) or a large magnitude (8 or 9). The tool-use task consisted in using inverse pliers for gripping either a small or a large object. The inverse pliers enable to dissociate the hand action from the tool action in relation to the object (i.e., closing the hand led to an opening of the tool and vice versa). The number/tool hypothesis predicts that the quantity representation associated with Arabic numbers will interact with the action of the tool toward the object. Conversely, the number/hand hypothesis predicts that the quantity associated with numbers will interact with the action of the hand toward the tool. Results confirmed the first hypothesis and rejected the second. Indeed, large numbers interacted with the action of the tool, such that participants were longer to perform an "opening-hand/closing-tool" action after the processing of large numbers. Moreover, no effect was detected for small numbers, confirming previous studies which used only finger movements. Altogether, our finding suggests that the well-known finger/number interaction can be reversed with tool use.

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

The existence of an association between numbers and space is known for a long time. The most prominent demonstration of this relationship is the spatial numerical association of response codes (SNARC) effect, describing the fact that participants' reaction times are shorter with the left hand for small numbers and with the right hand for large numbers, when being asked to judge the parity of a number (Dehaene et al., J. Exp. Psychol., 122, 371-396, 1993). The SNARC effect is commonly seen as support for the concept of a mental number line, i.e. a mentally conceived line where small numbers are represented more on the left and large numbers are represented more on the right. The SNARC effect has been demonstrated for all three cardinal axes and recently a transverse SNARC plane has been reported (Chen et al., Exp. Brain Res., 233(5), 1519-1528, 2015). Here, by employing saccadic responses induced by auditory or visual stimuli, we measured the SNARC effect within the same subjects along the horizontal (HM) and vertical meridian (VM) and along the two interspersed diagonals. We found a SNARC effect along HM and VM, which allowed predicting the occurrence of a SNARC effect along the two diagonals by means of linear regression. Importantly, significant differences in SNARC strength were found between modalities. Our results suggest the existence of a frontoparallel mental number plane, where small numbers are represented left and down, while large numbers are represented right and up. Together with the recently described transverse mental number plane our findings provide further evidence for the existence of a three-dimensional mental number space.

Numbers in Action

Humans show a remarkable tendency to describe and think of numbers as being placed on a mental number line (MNL), with smaller numbers located on the left and larger ones on the right. Faster responses to small numbers are indeed performed on the left side of space, while responses to large numbers are facilitated on the right side of space (spatial-numerical association of response codes, SNARC effect). This phenomenon is considered the experimental demonstration of the MNL and has been extensively replicated throughout a variety of paradigms. Nevertheless, the majority of previous literature has mainly investigated this effect by means of response times and accuracy, whereas studies considering more subtle and automatic measures such as kinematic parameters are rare (e.g., in a reaching-to-grasp movement, the grip aperture is enlarged in responding to larger numbers than in responding to small numbers). In this brief review we suggest that numerical magnitude can also affect the what and how of action execution (i.e., temporal and spatial components of movement). This evidence could have large implications in the strongly debated issue concerning the effect of experience and culture on the orientation of MNL.

Fast or slow? Compressions (or not) in number-to-line mappings

We investigated, in a university student population, spontaneous (non-speeded) fast and slow number-to-line mapping responses using non-symbolic (dots) and symbolic (words) stimuli. Seeking for less conventionalized responses, we used anchors 0–130, rather than the standard 0–100. Slow responses to both types of stimuli only produced linear mappings with no evidence of non-linear compression. In contrast, fast responses revealed distinct patterns of non-linear compression for dots and words. A predicted logarithmic compression was observed in fast responses to dots in the 0–130 range, but not in the reduced 0–100 range, indicating compression in proximity of the upper anchor 130, not the standard 100. Moreover, fast responses to words revealed an unexpected significant negative compression in the reduced 0–100 range, but not in the 0–130 range, indicating compression in proximity to the lower anchor 0. Results show that fast responses help revealing the fundamentally distinct nature of symbolic and non-symbolic quantity representation. Whole number words, being intrinsically mediated by cultural phenomena such as language and education, emphasize the invariance of magnitude between them—essential for linear mappings, and therefore, unlike non-symbolic (psychophysical) stimuli, yield spatial mappings that don’t seem to be influenced by the Weber-Fechner law of psychophysics. However, high levels of education (when combined with an absence of standard upper anchors) may lead fast responses to overestimate magnitude invariance on the lower end of word numerals.

Pushing forward in embodied cognition: may we mouse the mathematical mind?

Freely available software has popularized “mousetracking” to study cognitive processing; this involves the on-line recording of cursor positions while participants move a computer mouse to indicate their choice. Movement trajectories of the cursor can then be reconstructed off-line to assess the efficiency of responding in time and across space. Here we focus on the process of selecting among alternative numerical responses. Several studies have recently measured the mathematical mind with cursor movements while people decided about number magnitude or parity, computed sums or differences, or simply located numbers on a number line. After some general methodological considerations about mouse tracking we discuss several conceptual concerns that become particularly evident when “mousing” the mathematical mind.

Spatial Associations in Numerical Cognition—From Single Digits to Arithmetic

The literature on spatial associations during number processing is dominated by the SNARC (spatial–numerical association of response codes) effect. We describe spatial biases found for single digits and pairs of numbers, first in the “original” speeded parity task and then extending the scope to encompass different tasks, a range of measures, and various populations. Then we review theoretical accounts before surveying the emerging evidence for similar spatial associations during mental arithmetic. We conclude that the mental number line hypothesis and an embodied approach are useful frameworks for further studies.

Specialization in the human brain: the case of numbers

How numerical representation is encoded in the adult human brain is important for a basic understanding of human brain organization, its typical and atypical development, its evolutionary precursors, cognitive architectures, education, and rehabilitation. Previous studies have shown that numerical processing activates the same intraparietal regions irrespective of the presentation format (e.g., symbolic digits or non-symbolic dot arrays). This has led to claims that there is a single format-independent, numerical representation. In the current study we used a functional magnetic resonance adaptation paradigm, and effective connectivity analysis to re-examine whether numerical processing in the intraparietal sulci is dependent or independent on the format of the stimuli. We obtained two novel results. First, the whole brain analysis revealed that format change (e.g., from dots to digits), in the absence of a change in magnitude, activated the same intraparietal regions as magnitude change, but to a greater degree. Second, using dynamic causal modeling as a tool to disentangle neuronal specialization across regions that are commonly activated, we found that the connectivity between the left and right intraparietal sulci is format-dependent. Together, this line of results supports the idea that numerical representation is subserved by multiple mechanisms within the same parietal regions.

The Cultural Number Line: A Review of Cultural and Linguistic Influences on the Development of Number Processing

Approximate processing of numerosities is a universal and preverbal skill, while exact number processing above 4 involves the use of culturally acquired number words and symbols. The authors first review core concepts of numerical cognition, including number representation in the brain and the influential view that numbers are associated with space along a “mental number line.” Then, they discuss how cultural influences, such as reading direction, finger counting, and the transparency of the number word system, can influence the representation and processing of numbers. Spatial mapping of numbers emerges as a universal cognitive strategy. The authors trace the impact of cultural factors on the development of number skills and conclude that a cross-cultural perspective can reveal important constraints on numerical cognition.

Using Mental Representations of Space When Words Are Unavailable: Studies of Enumeration and Arithmetic in Indigenous Australia

Here the authors describe the nature and use of spatial strategies in a standard nonverbal addition task in two groups of children who speak only languages in which counting words are not available, as compared with children who were raised speaking English. They tested speakers of Warlpiri and Anindilyakwa aged between 4 and 7 years at two remote sites in the Northern Territory of Australia. These children used spatial strategies extensively and were significantly more accurate when they did so. English-speaking children used spatial strategies very infrequently but relied on an enumeration strategy supported by counting words to do the addition task. The main spatial strategy exploited the known visual memory strengths of Aboriginals and involved matching the spatial patterns of the augend and addend sets. These findings suggest that counting words, far from being necessary for exact arithmetic, offer one strategy among others. They also suggest that spatial models for numbers do not need to be one-dimensional vectors, as in a mental number line (MNL), but can be at least two-dimensional.

Finger Counting Habits in Middle Eastern and Western Individuals: An Online Survey

The current study documents the presence of cultural differences in the development of finger counting strategies. About 900 Middle Eastern (i.e., Iranian) and Western (i.e., European and American) individuals reported in an online survey how they map numbers onto their fingers when counting from 1 to 10. The analysis of these bimanual counting patterns revealed clear cross-cultural differences in the hand and finger starting preferences: While most Western individuals started counting with the left hand and associated the number 1 with their thumb, most Middle Eastern respondents preferred to start counting with the right hand and preferred to map the number 1 onto their little finger. The transition between the two hands during counting showed equal proportions of symmetry-based and spatial continuity-based patterns in the two cultures. Implications of these findings for numerical cognition and for the origin of the well-known association between numbers and space are discussed.

Extending the Mental Number Line

Multi-digit number processing is ubiquitous in our everyday life – even in school, multi-digit numbers are computed from the first year onward. Yet, many problems children and adults have are about the relation of different digits (for instance with fractions, decimals, or carry effects in multi-digit addition). Cognitive research has mainly focused on single-digit processing, and there is no comprehensive review of the different multi-digit number processing types and effects. The current review aims to fill this gap. First, we argue that effects observed in single-digit tasks cannot simply be transferred to multi-digit processing. Next, we list 16 effect types and processes which are specific for multi-digit number processing. We then discuss the development of multi-digit number processing, its neurocognitive correlates, its cultural or language-related modulation, and finally some models for multi-digit number processing. We finish with conclusions and perspectives about where multi-digit number processing research may or should be heading in following years.

Non-abstractness as mental simulation in the representation of number

Abstraction is instrumental for our understanding of how numbers are cognitively represented. We propose that the notion of abstraction becomes testable from within the framework of simulated cognition. We describe mental simulation as embodied, grounded, and situated cognition, and report evidence for number representation at each of these levels of abstraction.