Attentional cueing in numerical cognition

On the embodied nature of knowledge: From neurons to numbers

Interdisciplinary investigations of the human mind through the cognitive sciences have identified a key role of the body in representing knowledge. After characterizing knowledge at grounded, embodied, and situated levels, number knowledge is analyzed from this hierarchical perspective. Lateralized cortical processing of coarse versus fine detail is identified as a grounding substrate for the population stereotype few/left, many/right, which then contributes to number-related sensory and motor biases at the embodied and situated levels. Implications of this perspective for education and rehabilitation are discussed.

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

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

Spatial Attention Shifts in Addition and Subtraction Arithmetic: Evidence of Eye Movement

Recently, it has been proposed that solving addition and subtraction problems can evoke horizontal shifts of spatial attention. However, prior to this study, it remained unclear whether orienting shifts of spatial attention relied on actual arithmetic processes (i.e., the activated magnitude) or the semantic spatial association of the operator. In this study, spatial–arithmetic associations were explored through three experiments using an eye tracker, which attempted to investigate the mechanism of those associations. Experiment 1 replicated spatial–arithmetic associations in addition and subtraction problems. Experiments 2 and 3 selected zero as the operand to investigate whether these arithmetic problems could induce shifts of spatial attention. Experiment 2 indicated that addition and subtraction problems (zero as the second operand, i.e., 2 + 0) do not induce shifts of spatial attention. Experiment 3 showed that addition and subtraction arithmetic (zero as the first operand, i.e., 0 + 2) do facilitate rightward and leftward eye movement, respectively. This indicates that the operator alone does not induce horizontal eye movement. However, our findings support the idea that solving addition and subtraction problems is associated with horizontal shifts of spatial attention.

Spatial-numerical associations: Shared symbolic and non-symbolic numerical representations

During the last decades, there have been a large number of studies into the number-related abilities of humans. As a result, we know that humans and non-human animals have a system known as the approximate number system that allows them to distinguish between collections based on their number of items, separately from any counting procedures. Dehaene and others have argued for a model on which this system uses representations for numbers that are spatial in nature and are shared by our symbolic and non-symbolic processing of numbers. However, there is a conflicting theoretical perspective in which there are no representations of numbers underlying the approximate number system, but only quantity-related representations. This perspective would then suggest that there are no shared representations between symbolic and non-symbolic processing. We review the evidence on spatial biases resulting from the activation of numerical representations, for both non-symbolic and symbolic tests. These biases may help decide between the theoretical differences; shared representations are expected to lead to similar biases regardless of the format, whereas different representations more naturally explain differences in biases, and thus behaviour. The evidence is not yet decisive, as the behavioural evidence is split: we expect bisection tasks to eventually favour shared representations, whereas studies on the spatial-numerical association of response codes (SNARC) effect currently favour different representations. We discuss how this impasse may be resolved, in particular, by combining these behavioural studies with relevant neuroimaging data. If this approach is carried forward, then it may help decide which of these two theoretical perspectives on number representations is correct.

Development of number-space associations: SNARC effects and spatial attention in 7- to 11-year-olds

The spatial numerical association of response codes effect, referred to as the SNARC effect, reveals that small numbers elicit faster left than right responses, and conversely, large numbers elicit faster right responses. Here, we explored the development of this number-space association by assessing how 7-, 9-, 11-year-olds, and adults differed in spatial orienting of attention on Posner' paradigm. Compared with the previous research, we examined how the cues would affect the level and strength of the SNARC effect in children under the different attentional conditions. Subjects made parity decisions for endogenous attention (Experiment 1) and exogenous attention (Experiment 2). The results showed that adults displayed the SNARC effect in both experiments, relatively speaking, 7- to 11-year-old Chinese children's ability of spatial numerical association progressed gradually. With endogenous attention, the SNARC effect appeared in all age groups except for 7-year-olds for invalid cues. Compared with the endogenous attention condition, the SNARC effect was more significantly affected by cues in the exogenous attention condition. This result might be owing to the fact that the SNARC effect was not demonstrated in 7-year-olds with either valid or invalid cues. Our results suggest that the differences in the spatial orienting of attention are based on the cognitive load associated with processing number information and that this process can be affected by cues. Further, there may be cross-cultural influences on the SNARC effect, as early family training may explain the results seen in this sample of Chinese 7-year-olds. Thus, reaction times decreased with increasing age in the parity judgment task, and reaction times for valid cues were faster than for invalid cues regardless of the age group in both experiments. The SNARC effect was only present for 7-year-olds for valid cues, for endogenous attention.

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.

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'.

The Developmental Trajectory of the Operational Momentum Effect

Mental calculation is thought to be tightly related to visuospatial abilities. One of the strongest evidence for this link is the widely replicated operational momentum (OM) effect: the tendency to overestimate the result of additions and to underestimate the result of subtractions. Although the OM effect has been found in both infants and adults, no study has directly investigated its developmental trajectory until now. However, to fully understand the cognitive mechanisms lying at the core of the OM effect it is important to investigate its developmental dynamics. In the present study, we investigated the development of the OM effect in a group of 162 children from 8 to 12 years old. Participants had to select among five response alternatives the correct result of approximate addition and subtraction problems. Response alternatives were simultaneously presented on the screen at different locations. While no effect was observed for the youngest age group, children aged 9 and older showed a clear OM effect. Interestingly, the OM effect monotonically increased with age. The increase of the OM effect was accompanied by an increase in overall accuracy. That is, while younger children made more and non-systematic errors, older children made less but systematic errors. This monotonous increase of the OM effect with age is not predicted by the compression account (i.e., linear calculation performed on a compressed code). The attentional shift account, however, provides a possible explanation of these results based on the functional relationship between visuospatial attention and mental calculation and on the influence of formal schooling. We propose that the acquisition of arithmetical skills could reinforce the systematic reliance on the spatial mental number line and attentional mechanisms that control the displacement along this metric. Our results provide a step in the understanding of the mechanisms underlying approximate calculation and an important empirical constraint for current accounts on the origin of the OM effect.

Probing the Neural Correlates of Number Processing

The cognitive and neural mechanisms that enable humans to encode and manipulate numerical information have been subject to an increasing number of experimental studies over the past 25 years or so. Here, I highlight recent findings about how numerical information is neurally coded, focusing on the theoretical implications derived from the most influential theoretical framework in numerical cognition—the Triple Code Model. At the core of this model is the assumption that bilateral parietal cortex hosts an approximate number system that codes for the cardinal value of perceived numerals. I will review studies that ask whether or not the numerical coding within this system is invariant to varying input notation, format, or modality, and whether or not the observed parietal activity is number-specific over and above the parietal involvement in response-related processes. Extant computational models of numerosity (the number of objects in a set) perception are summarized and related to empirical data from human neuroimaging and monkey neurophysiology.

Ocular drift along the mental number line

We examined the spontaneous association between numbers and space by documenting attention deployment and the time course of associated spatial-numerical mapping with and without overt oculomotor responses. In Experiment 1, participants maintained central fixation while listening to number names. In Experiment 2, they made horizontal target-direct saccades following auditory number presentation. In both experiments, we continuously measured spontaneous ocular drift in horizontal space during and after number presentation. Experiment 2 also measured visual-probe-directed saccades following number presentation. Reliable ocular drift congruent with a horizontal mental number line emerged during and after number presentation in both experiments. Our results provide new evidence for the implicit and automatic nature of the oculomotor resonance effect associated with the horizontal spatial-numerical mapping mechanism.

Inhibition of return and attentional facilitation: Numbers can be counted in, letters tell a different story

Prior research has provided strong evidence for spatial-numerical associations. Single digits can for instance act as attentional cues, orienting visuo-spatial attention to the left or right hemifield depending on the digit's magnitude, thus facilitating target detection in the cued hemifield (left/right hemifield after small/large digits, respectively). Studies using other types of behaviourally or biologically relevant central cues known to elicit automated symbolic attention orienting effects such as arrows or gaze have shown that the initial facilitation of cued target detection can turn into inhibition at longer stimulus onset asynchronies (SOAs). However, no studies so far investigated whether inhibition of return (IOR) is also observed using digits as uninformative central cues. To address this issue we designed an attentional cueing paradigm using SOAs ranging from 500 ms to 1650 ms. As expected, the results showed a facilitation effect at the relatively short 650 ms SOA, replicating previous findings. At the long 1650 ms SOA, however, participants were faster to detect targets in the uncued hemifield compared to the cued hemifield, showing an IOR effect. A control experiment with letters showed no such congruency effects at any SOA. These findings provide the first evidence that digits not only produce facilitation effects at shorter intervals, but also induce inhibitory effects at longer intervals, confirming that Arabic digits engage automated symbolic orienting of attention.

Spatial biases during mental arithmetic: evidence from eye movements on a blank screen

While the influence of spatial-numerical associations in number categorization tasks has been well established, their role in mental arithmetic is less clear. It has been hypothesized that mental addition leads to rightward and upward shifts of spatial attention (along the “mental number line”), whereas subtraction leads to leftward and downward shifts. We addressed this hypothesis by analyzing spontaneous eye movements during mental arithmetic. Participants solved verbally presented arithmetic problems (e.g., 2 + 7, 8–3) aloud while looking at a blank screen. We found that eye movements reflected spatial biases in the ongoing mental operation: Gaze position shifted more upward when participants solved addition compared to subtraction problems, and the horizontal gaze position was partly determined by the magnitude of the operands. Interestingly, the difference between addition and subtraction trials was driven by the operator (plus vs. minus) but was not influenced by the computational process. Thus, our results do not support the idea of a mental movement toward the solution during arithmetic but indicate a semantic association between operation and space.