Impact of optokinetic stimulation on mental arithmetic

Spatial coding of arithmetic operations in early learning: an eye tracking study in first-grade elementary school children

A growing body of evidence indicates that mental calculation in adults is accompanied by horizontal attention shifts along a mental continuum representing the range of plausible answers. The fast deployment of spatial attention suggests a predictive role in guiding the search for the answer. The link between arithmetic and spatial functions is theoretically justified by the need to alleviate the cognitive load of mental calculation, but the question of how this link establishes during development gives rise to opposing views emphasizing either biological or cultural factors. The role of education, in particular, remains debated in the absence of data covering the period when children learn arithmetic. In this study, we measured gaze movements, as a proxy for attentional shifts, while first-grade elementary school children solved single-digit additions and subtractions. The investigation was scheduled only a few weeks after the formal teaching of symbolic subtraction to assess the role of spatial attention in early learning. Gaze patterns revealed horizontal- but not vertical- attentional shifts, with addition shifting the gaze more rightward than subtraction. The shift was observed as soon as the first operand and the operator were presented, corroborating the view that attention is used to predictively identify the portion of the numerical continuum where the answer is likely to be located, as adult studies suggested. The finding of a similar gaze pattern in adults and six-year-old children who have just learned how to subtract single digits challenges the idea that arithmetic problem solving requires intensive practice to be linked to spatial attention.

Anchoring bias in mental arithmetic

Mental arithmetic is widely studied, both with symbolic digits and with non-symbolic dot patterns that require operand estimation. Several studies reported surprising biases in adults' performance with both formats while their direction (over/underestimation in addition/subtraction) remains controversial (operational momentum effect or OM; Prado & Knops, Prado and Knops, Psychonomic Bulletin & Review, in Press., 2024). Theoretical accounts of OM make opposing predictions, thus enabling a decisive test: Using symbolic stimuli and responses, we enabled accurate operand encoding and result reporting, thus leaving mental calculation as only source of bias. Importantly, we manipulated operand order through calculation instructions (e.g., "29 + 19" vs. "add 19 to 29") to assess the crucial role of first operand size as cognitive anchor. With both auditory (Experiment 1, N = 30) and visual presentation (Experiment 2, N = 30), we observed reverse OM, i.e., overestimations in subtraction and underestimations in addition. Importantly, this instance of operation-based anchoring was independent of a second anchoring effect related to operand order: A large operand is a stronger anchor when mentioned first. Our discovery of both operation-based and order-based anchoring extends the well-known anchoring effect into mental arithmetic and eliminates several competing theories about the origin of OM.

Spatial attention in mental arithmetic: A literature review and meta-analysis

We review the evidence for the conceptual association between arithmetic and space and quantify the effect size in meta-analyses. We focus on three effects: (a) the operational momentum effect (OME), which has been defined as participants' tendency to overestimate results of addition problems and underestimate results of subtraction problems; (b) the arithmetic cueing effect, in which arithmetic problems serve as spatial cues in target detection or temporal order judgment tasks; and (c) the associations between arithmetic and space observed with eye- and hand-tracking studies. The OME was consistently found in paradigms that provided the participants with numerical response alternatives. The OME shows a large effect size, driven by an underestimation during subtraction while addition was unbiased. In contrast, paradigms in which participants indicated their estimate by transcoding their final estimate to a spatial reference frame revealed no consistent OME. Arithmetic cueing studies show a reliable small to medium effect size, driven by a rightward bias for addition. Finally, eye- and hand-tracking studies point to replicable associations between arithmetic and eye or hand movements. To account for the complexity of the observed pattern, we introduce the Adaptive Pathways in Mental Arithmetic (APiMA) framework. The model accommodates central notions of numerical and arithmetic processing and helps identifying which pathway a given paradigm operates on. It proposes that the divergence between OME and arithmetic cueing studies comes from the predominant use of non-symbolic versus symbolic stimuli, respectively. Overall, our review and findings clearly support an association between arithmetic and spatial processing.

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).

Spatial biases in mental arithmetic are independent of reading/writing habits: Evidence from French and Arabic speakers

The representation of numbers in human adults is linked to space. In Western cultures, small and large numbers are associated respectively with the left and right sides of space. An influential framework attributes the emergence of these spatial-numerical associations (SNAs) to cultural factors such as the direction of reading and writing, because SNAs were found to be reduced or inverted in right-to-left readers/writers (e.g., Arabic, Farsi, or Hebrew speakers). However, recent cross-cultural and animal studies cast doubt on the determining role of reading and writing directions on SNAs. In this study, we assessed this role in mental arithmetic, which requires explicit number manipulations and has revealed robust leftward or rightward biases in Western participants. We used a temporal order judgement task in French and Arabic speakers, two languages that have opposite reading/writing directions. Participants had to solve subtraction and addition problems presented auditorily while at the same time determining which of a left or right visual target appeared first on a screen. The results showed that the right target was favoured more often when solving additions than when solving subtractions both in the French- (n = 31) and Arabic-speaking (n = 25) groups. This was true even in Arabic-speaking participants whose preference for ordering of various series of numerical and non-numerical stimuli went from right to left (n = 10). These results indicate that SNAs in mental arithmetic cannot be explained by the direction of reading/writing habits and call for a reconsideration of current models to acknowledge the pervasive role of biological factors in SNAs in adults.

Semantic associations between arithmetic and space: Evidence from temporal order judgements

Spatial biases associated with subtraction or addition problem solving are generally considered as reflecting leftward or rightward attention shifts along a mental numerical continuum, but an alternative hypothesis not implying spatial attention proposes that the operator (plus or minus sign) may favour a response to one side of space (left or right) because of semantic associations. We tested these two accounts in a series of temporal order judgement experiments that consisted in the auditory presentation of addition or subtraction problems followed 200 ms (Experiments 1-2) or 800 ms (Experiment 3) later by the display of two lateralized targets in close temporal succession. To dissociate the side where the operation first brought their attention from the side they had to respond to, we asked participants to report which of the left or right target appeared first or last on screen. Under the attention-orienting account, addition should elicit more rightward responses than subtraction when participants have to focus on the first target, but more leftward responses when they have to focus on the last target, because the latter is opposite to the side where the operation first brought their attention. Under the semantic account, addition should elicit more rightward responses than subtraction, no matter the focus is on the first or last target, because participants should systematically favour the side conceptually linked to the operator. The results of the three experiments converge to indicate that, in lateralized target detection tasks, the spatial biases induced by arithmetic operations stem from semantic associations.

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.

Spatial Bias Induced by Simple Addition and Subtraction: From Eye Movement Evidence

The associations between number and space have been intensively investigated. Recent studies indicated that this association could extend to more complex tasks, such as mental arithmetic. However, the mechanism of arithmetic-space associations in mental arithmetic was still a topic of debate. Thus, in the current study, we adopted an eye-tracking technology to investigate whether spatial bias induced by mental arithmetic was related with spatial attention shifts on the mental number line or with semantic link between the operator and space. In Experiment 1, participants moved their eyes to the corresponding response area according to the cues after solving addition and subtraction problems. The results showed that the participants moved their eyes faster to the leftward space after solving subtraction problems and faster to the right after solving addition problems. However, there was no spatial bias observed when the second operand was zero in the same time window, which indicated that the emergence of spatial bias may be associated with spatial attention shifts on the mental number line. In Experiment 2, participants responded to the operator (operation plus and operation minus) with their eyes. The results showed that mere presentation of operator did not cause spatial bias. Therefore, the arithmetic–space associations might be related with the movement along the mental number line.