Larger, smaller, odd or even? Task-specific effects of optokinetic stimulation on the mental number space

A functional role for oculomotor preparation in mental arithmetic evidenced by the abducted eye paradigm

Solving subtraction and addition problems is accompanied by spontaneous leftward and rightward gaze shifts, respectively. These shifts have been related to attentional processes involved in mental arithmetic, but whether these processes induce overt attentional shifts mediated by the activation of the motor programs underlying lateral eye movements or covert shifts only is still unknown. Here, we used the abducted eye paradigm to selectively disrupt activation of the oculomotor system and prevent oculomotor preparation, which affects overt but not covert attentional shifts. Participants had to mentally solve addition and subtraction problems while fixating a screen positioned either in front of them or laterally to their left or right such that they were physically unable to programme and execute saccades further into their temporal field while they still could do so in their nasal field. In comparison to the frontal condition, rightward eye abduction impaired additions (with carrying), and leftward eye abduction impaired subtractions (with borrowing) showing that at least some arithmetic problems rely on processes dedicated to overt attentional shifts. We propose that when solving arithmetic problems requires procedures such as carrying and borrowing, oculomotor mechanisms operating on a mental space transiently built in working memory are recruited to represent one numerical magnitude in relation to another (e.g. the first operand and the result).

Influences of hand action on the processing of symbolic numbers: A special role of pointing?

Embodied and grounded cognition theories state that cognitive processing is built upon sensorimotor systems. In the context of numerical cognition, support to this framework comes from the interactions between numerical processing and the hand actions of reaching and grasping documented in skilled adults. Accordingly, mechanisms for the processing of object size and location during reach and grasp actions might scaffold the development of mental representations of numerical magnitude. The present study exploited motor adaptation to test the hypothesis of a functional overlap between neurocognitive mechanisms of hand action and numerical processing. Participants performed repetitive grasping of an object, repetitive pointing, repetitive tapping, or passive viewing. Subsequently, they performed a symbolic number comparison task. Importantly, hand action and number comparison were functionally and temporally dissociated, thereby minimizing context-based effects. Results showed that executing the action of pointing slowed down the responses in number comparison. Moreover, the typical distance effect (faster responses for numbers far from the reference as compared to close ones) was not observed for small numbers after pointing, while it was enhanced by grasping. These findings confirm the functional link between hand action and numerical processing, and suggest new hypotheses on the role of pointing as a meaningful gesture in the development and embodiment of numerical skills.

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.

The perception of subset quantity and items in an environment with distractors in a population with mathematical learning difficulties

OBJECTIVE

People often perceive a quantity of specific objects that appear as part of an overall group of items (a subset). This study investigates this type of perception among a population with mathematical leaning difficulties (MLD).

METHOD

Sixty-two participants (mean age: 26.82) reported the general and subset quantity of items using a subset quantity detection task or a conjunction visual search task.

RESULTS

MLD had difficulties perceiving both the general quantity presented and the subset quantity of items. They also had difficulties preforming a conjunction visual search task, even when the task did not involve numerical processing.

CONCLUSIONS

MLD has spatial difficulties in the form of visual search and subset quantity detection. The current study suggests that MLD might experience greater difficulties in daily tasks, which might be related to those tasks (e.g., detecting the amount of forks among other items of silverware on the table).

When adding is right: Temporal order judgements reveal spatial attention shifts during two-digit mental arithmetic

Recent research suggests that addition and subtraction induce horizontal shifts of attention. Previous studies used single-digit (1d) problems or verification paradigms that lend themselves to alternative solution strategies beyond mental arithmetic. To measure spatial attention during the active production of solutions to complex two-digit arithmetic problems (2d) without manual motor involvement, we used a temporal order judgement (TOJ) paradigm in which two lateralised targets were sequentially presented on screen with a varying stimulus onset asynchrony (SOA). Participants verbally indicated which target appeared first. By varying the delay between the arithmetic problem presentation and the TOJ task, we investigated how arithmetically induced attention shifts develop over time (Experiment 1, n = 31 and Experiment 2, n = 58). In Experiment 2, we additionally varied the carry property of the arithmetic task to examine how task difficulty modulates the effects. In the arithmetic task, participants were first presented with the arithmetic problem via headphones and performed the TOJ task after the delay before responding to the arithmetic task. To account for spontaneous attentional biases, a baseline TOJ was run without arithmetic processing. Both experiments revealed that addition induces shifts of spatial attention to the right suggesting that visuospatial attention mechanisms are recruited during complex arithmetic. We observed no difference in spatial attention between the carry and noncarry condition (Experiment 2). No shifts were observed for subtraction problems. No common and conclusive influence of delay was observed across experiments. Qualitative differences between addition and subtraction and the role of task difficulty are discussed.

Spatial grounding of symbolic arithmetic: an investigation with optokinetic stimulation

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

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.

Time course of overt attentional shifts in mental arithmetic: Evidence from gaze metrics

Processing numbers induces shifts of spatial attention in probe detection tasks, with small numbers orienting attention to the left and large numbers to the right side of space. This has been interpreted as supporting the concept of a mental number line with number magnitudes ranging from left to right, from small to large numbers. Recently, the investigation of this spatial-numerical link has been extended to mental arithmetic with the hypothesis that solving addition or subtraction problems might induce attentional displacements, rightward or leftward, respectively. At the neurofunctional level, the activations elicited by the solving of additions have been shown to resemble those induced by rightward eye movements. However, the possible behavioural counterpart of these activations has not yet been observed. Here, we investigated overt attentional shifts with a target detection task primed by addition and subtraction problems (2-digit ± 1-digit operands) in participants whose gaze orientation was recorded during the presentation of the problems and while calculating. No evidence of early overt attentional shifts was observed while participants were hearing the first operand, the operator or the second operand, but they shifted their gaze towards the right during the solving step of addition problems. These results show that gaze shifts related to arithmetic problem solving are elicited during the solving procedure and suggest that their functional role is to access, from the first operand, the representation of the result.

Order Information in Verbal Working Memory Shifts the Subjective Midpoint in Both the Line Bisection and the Landmark Tasks

A largely substantiated view in the domain of working memory is that the maintenance of serial order is achieved by generating associations of each item with an independent representation of its position, so-called position markers. Recent studies reported that the ordinal position of an item in verbal working memory interacts with spatial processing. This suggests that position markers might be spatial in nature. However, these interactions were so far observed in tasks implying a clear binary categorization of space (i.e., with left and right responses or targets). Such binary categorizations leave room for alternative interpretations, such as congruency between non-spatial categorical codes for ordinal position (e.g., begin and end) and spatial categorical codes for response (e.g., left and right). Here we discard this interpretation by providing evidence that this interaction can also be observed in a task that draws upon a continuous processing of space, the line bisection task. Specifically, bisections are modulated by ordinal position in verbal working memory, with lines bisected more towards the right after retrieving items from the end compared to the beginning of the memorized sequence. This supports the idea that position markers are intrinsically spatial in nature.

Contribution of visuospatial attention, short-term memory and executive functions to performance in number interval bisection

Number interval bisection consists of estimating the mid-number within a pair (1-9=>5). Healthy adults and right-brain damage patients can show biased performance in this task, underestimating and overestimating the mid-number, respectively. The role of visuospatial attention during this task, and its interplay with other cognitive abilities (e.g., working memory) is still object of debate. In this study we explored the relation between visuospatial attention and individual differences in working memory and executive functions during number interval bisection. To manipulate the deployment of visuospatial attention, healthy participants tracked a dot moving to the left or moving to the right while bisecting numerical intervals. We also collected information concerning verbal and visuospatial short-term memory span, and concerning verbal and visuospatial fluency scores. Beside replicating what is typically observed in this task (e.g., underestimation bias), a correlation was observed between verbal short-term memory and bisection bias, and an interesting relation between performance in the number interval bisection, verbal short-term memory, and visuospatial attention. Specifically, performance of those participants with low verbal span was affected by the direction of the moving dot, underestimating at a larger extent when the dot moved leftward than rightward. Finally, it was also observed that participants' verbal fluency ability contributed in the generation of biases in the numerical task. The finding of the involvement of abilities belonging to the verbal domain contributes to unveil the multi-componential nature of number interval bisection. Considering the debate on the nature of number interval bisection and its use in the clinical assessment of deficits following brain damage, this finding may be interesting also from a clinical perspective.

Cross-Representational Interactions: Interface and Overlap Mechanisms

A crucial question facing cognitive science concerns the nature of conceptual representations as well as the constraints on the interactions between them. One specific question we address in this paper is what makes cross-representational interplay possible? We offer two distinct theoretical scenarios: according to the first scenario, co-activated knowledge representations interact with the help of an interface established between them via congruent activation in a mediating third-party general cognitive mechanism, e.g., attention. According to the second scenario, co-activated knowledge representations interact due to an overlap between their features, for example when they share a magnitude component. First, we make a case for cross-representational interplay based on grounded and situated theories of cognition. Second, we discuss interface-based interactions between distinct (i.e., non-overlapping) knowledge representations. Third, we discuss how co-activated representations may share their architecture via partial overlap. Finally, we outline constraints regarding the flexibility of these proposed mechanisms.

Hemispatial Neglect Shows That "Before" Is "Left"

Recent research has led to the hypothesis that events which unfold in time might be spatially represented in a left-to-right fashion, resembling writing direction. Here we studied fourteen right-hemisphere damaged patients, with or without neglect, a disorder of spatial awareness affecting contralesional (here left) space processing and representation. We reasoned that if the processing of time-ordered events is spatial in nature, it should be impaired in the presence of neglect and spared in its absence. Patients categorized events of a story as occurring before or after a central event, which acted as a temporal reference. An asymmetric distance effect emerged in neglect patients, with slower responses to events that took place before the temporal reference. The event occurring immediately before the reference elicited particularly slow responses, closely mirroring the pattern found in neglect patients performing numerical comparison tasks. Moreover, the first item elicited significantly slower responses than the last one, suggesting a preference for a left-to-right scanning/representation of events in time. Patients without neglect showed a regular and symmetric distance effect. These findings further suggest that the representation of events order is spatial in nature and provide compelling evidence that ordinality is similarly represented within temporal and numerical domains.

Composite body movements modulate numerical cognition: evidence from the motion-numerical compatibility effect

A recent hierarchical model of numerical processing, initiated by Fischer and Brugger (2011) and Fischer (2012), suggested that situated factors, such as different body postures and body movements, can influence the magnitude representation and bias numerical processing. Indeed, Loetscher et al. (2008) found that participants’ behavior in a random number generation task was biased by head rotations. More small numbers were reported after leftward than rightward head turns, i.e., a motion-numerical compatibility effect. Here, by carrying out two experiments, we explored whether similar motion-numerical compatibility effects exist for movements of other important body components, e.g., arms, and for composite body movements as well, which are basis for complex human activities in many ecologically meaningful situations. In Experiment 1, a motion-numerical compatibility effect was observed for lateral rotations of two body components, i.e., the head and arms. Relatively large numbers were reported after making rightward compared to leftward movements for both lateral head and arm turns. The motion-numerical compatibility effect was observed again in Experiment 2 when participants were asked to perform composite body movements of congruent movement directions, e.g., simultaneous head left turns and arm left turns. However, it disappeared when the movement directions were incongruent, e.g., simultaneous head left turns and arm right turns. Taken together, our results extended Loetscher et al.’s (2008) finding by demonstrating that their effect is effector-general and exists for arm movements. Moreover, our study reveals for the first time that the impact of spatial information on numerical processing induced by each of the two sensorimotor-based situated factors, e.g., a lateral head turn and a lateral arm turn, can cancel each other out.