Increased proficiency when solving arithmetic problems may have two origins. During learning, problems may be associated with their respective outcome and retrieved from memory when probed. Alternatively, acceleration of procedural solving strategies may lead to the observed improvements. The study examined the effects of a four-day drill training on complex addition and subtraction problems at both behavioral and neural levels. Significant improvement in performance was observed, reflected by a decrease in response times and increase in accuracy for specific groups of problems. At the neural level, we observed increased univariate activation in the parieto-occipital regions for problem groups showing the most pronounced training effect. These areas were previously linked to eye-movements and could hence accommodate more proficient visuospatial transformations, leading to an increase in procedural efficiency. Multivariate decoding showed changes in activity patterns in all ROIs between sessions, suggesting automatization of arithmetic strategies. Right hippocampus activation patterns indicated training-induced changes linked to memory retrieval. Representational similarity analysis, however, revealed that training-specific memory effects are not related to increased procedural efficiency alone. On the univariate level, the current data seem to support the acceleration of the procedural solving hypothesis. However, the inconclusiveness of the multivariate analyses and the involvement of the hippocampus indicate parallel use of arithmetic fact retrieval. The study highlights the importance of complementary analysis techniques and demonstrates the context-sensitivity of arithmetic learning by using complex addition and subtraction problems (two-digit addition and subtraction problems).

Inhibition has been suggested to contribute to symbolic and non-symbolic quantity processing, but conclusions remain inconsistent. Using Structural Equation Modelling, the structure of inhibition and its contributions to symbolic and non-symbolic arithmetic are explored in predominantly White French 5- and 7-year-olds (N= 331, 169 females). Results suggest a lack of support for a unitary or binary (Response Inhibition and Distractor Suppression) inhibition construct in light of poor correlations between inhibition measures and highlight a link in both age groups between the Stop-Signal Task and symbolic arithmetic, and a link between the Flanker task and non-symbolic arithmetic only in 7-year-olds. The current study allows to dissect the differential contributions of inhibition facets to arithmetic development in a critical time window.

The temporal momentum effect (TME) refers to a systematic distortion in temporal arithmetic whereby addition tends to produce overestimation, and subtraction tends to produce little underestimation or no bias. To date, studies investigating the TME have relied exclusively on reproduction paradigms, leaving open whether the bias emerges during perceptual combination or motor reproduction. Across two experiments we examined the functional locus of the TME. In Experiment 1, we isolated temporal arithmetic from motor output using a psychophysical comparison staircase paradigm. Contrary to the classic TME observed in reproduction paradigms, participants underestimated addition outcomes and overestimated subtraction outcomes, yielding a reversed TME. In Experiment 2, we tested whether this reversal could be explained by anchoring effects arising from differences in operand ranges across operations. To this end, we varied the position of the longer duration within addition problems (first vs. second operand). No effect of operand order emerged, indicating that anchoring on the first operand cannot account for the reverse TME. Taken together, these findings suggest that motoric reproduction contributes substantially to the classical TME, and that removing motor demands reveals additional influences, such as attentional allocation or methodological factors, that shape temporal arithmetic in perceptual tasks. We discuss how pacemaker–counter models, generalized counting models, and attentional gate mechanisms may each account for different aspects of the findings. Overall, the results indicate that the TME reflects an interaction between motoric, attentional, and representational processes rather than a single underlying mechanism.

The Temporal Momentum Effect (TME) describes the systematic overestimation of reproduced estimates when adding two temporal intervals and underestimation when subtracting them. The TME has been attributed to a spatial component in the representation of time. However, previous studies on the TME used continuous white noise to indicate the operands, which confounds duration with stimulus energy. In Experiment 1, we separated sensory energy from temporal duration by presenting operands defined as the silent interval between a start and a stop signal. We replicated the TME both with continuous white noise and empty intervals, implying that it reflects temporal cognition. Experiment 2 tested the hypothesis that the TME merely reflects basic perceptual principles such as central tendency and range effects that govern behavior in various psychophysical domains. Under this hypothesis, the TME should have been mainly reflecting the impact of stimulus range. While Experiment 2 revealed a significant impact of stimulus range, the overall pattern of results did not entirely support this mechanism as the main driver of the TME. These results establish the TME as a cognitive bias that allows studying the principles that govern the mental manipulation of quantitative information such as time, space, and number.

The approximate number system (ANS) is thought to mediate symbolic and non-symbolic numerical magnitude comparison. Challenging this view, the dual system model stipulates that non-symbolic comparisons rely on the ANS while symbolic comparisons rely on a discrete semantic system (DSS). In three experiments, the current study tests whether symbolic and non-symbolic magnitude comparisons rely on a common ANS or a DSS by examining the correlation between the size and distance effects in numerical magnitude comparison. We replicated previous studies, which used one-digit numbers 1 to 9, but also aimed to increase variance by using less familiar number ranges. Experiment 1 used a fixed-reference paradigm (reference = 55) with two-digit integers (11-99). Experiments 2 and 3 extended the design to decimals (0.01-0.98) with variable (Experiment 2) or fixed reference (Experiment 3). All experiments additionally included non-symbolic dot comparison in which the expected negative correlation between size and distance effect emerged. Across experiments, size and distance effects in less familiar number ranges were uncorrelated when presented in symbolic format, corroborating the idea that symbolic number comparison relies on a DSS. These findings were moderated by the observation of a significant correlation between size and distance effects in a subsample of participants who showed significant size and distance effects at the individual level. Interpretation of the current results must take into account limitations concerning specificities of multi-digit number processing, the reliability of the effects, and the possible role of unmeasured external factors in shaping the observed correlations.

French number words provide a unique window into the relationship between numerical cognition and language, because numbers above 60 follow a vigesimal (base-20) word structure (e.g., 72 = "60-12"). In a two-digit magnitude comparison task with sixty French native speakers, we replicated the classic unit-decade compatibility effect (UDCE; slower responses when unit and decade comparisons conflict) and within-decade effect (faster responses when decades are identical), reflecting the place-value structure of Arabic numerals. Given the French vigesimal system, we expected not only the classic UDCE and within-decade effect but also their vigesimal counterparts driven by magnitudes of number words: a unit-vigintade compatibility effect (UVCE) and a within-vigintade effect, in which pairs sharing the same decade word (e.g., "soixante" for the 60s and 70s) are processed faster than other between-decade pairs. Linear mixed models revealed both a UDCE for numbers larger than 60 and a UVCE, indicating that number words were accessed during processing. Participants also responded faster to within-vigintade items (86 vs. 95) than to between-vigintade items (76 vs. 85) and as fast as to within-decade items (82 vs. 85), indicating a verbal equivalent of the within-decade effect. This effect is unaffected by decade distance and can only be explained by access to number words so that the decades were identical ("80-6" vs. "80-15"). Overall, our data indicate that verbal representations can shape basic numerical judgments and that number processing may be more closely tied to language than previously assumed.

The temporal momentum effect (TME) refers to a systematic distortion in temporal arithmetic whereby addition tends to produce overestimation and subtraction tends to produce little underestimation or no bias. Previous investigations have relied predominantly on reproduction paradigms, leaving open whether the effect reflects motor processes or more general aspects of temporal cognition. Across two experiments, we examined the TME using a psychophysical comparison staircase paradigm, which minimizes motor demands. In Experiment 1, participants underestimated addition outcomes and overestimated subtraction outcomes. Although this pattern appears reversed relative to reproduction tasks, it corresponds, when expressed in terms of subjective duration, to the classical TME. In Experiment 2, we tested whether the TME could be explained by anchoring effects arising from differences in operand ranges across operations. To this end, we varied whether the longer duration appeared first or second in addition problems. No effect of operand order emerged, indicating that anchoring on the first operand cannot account for the TME under these conditions. Taken together, these results suggest that TME-like biases can also emerge in comparison paradigms and are therefore not restricted to reproduction procedures alone. More broadly, the findings are compatible with the idea that the TME reflects biases arising during the mental manipulation of temporal intervals, although the mechanisms underlying these distortions remain to be established.

In non-symbolic numerical comparison tasks (Approximate Number System or ANS tasks) where participants determine which of two dot arrays is more numerous, judgments can be biased by irrelevant non-numerical dimensions of magnitude, such as dot size or the total occupied area. While inhibition may help overcome these conflicts, the mechanisms guiding attention toward relevant numerical information remain unclear. We hypothesized that modes of attentional processing - focusing on individual elements (local processing) versus the overall configuration (global processing) - may influence how participants resolve numerical conflicts. Fifty-four adults (Mage = 32.07 years, SD = 11.38) completed ANS trials preceded by prime items designed to activate either a local or global processing mode, or no prime in a control condition. Global priming did not significantly affect performance, while local priming modulated numerosity/magnitude conflicts across magnitude dimensions, notably reducing the impact of convex hull. These findings shed light on how perceptual modes impact adults’ ability to extract numerical information in the face of visual conflict.

Infants as young as 9 months demonstrate both inhibitory control, as revealed by the Spatial Negative Priming (SNP) effect, and numerical biases, such as the Operational Momentum (OM) effect - an overestimation in addition and underestimation in subtraction. Despite their early and parallel emergence, no studies have investigated whether these two cognitive abilities are related in infancy. In the present study, we tested 9- and 12-month-old infants in Paris on both a SNP task, measuring inhibitory skills, and an OM task, assessing sensitivity to directional biases in ordinal numerical sequences. Our results revealed a significant negative correlation between SNP and OM at 12 months, and between SNP at 9 months and OM at 12 months: infants with stronger inhibitory control showed reduced sensitivity to OM. These findings suggest that early inhibitory mechanisms may play a role in the development of OM bias during infancy.

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.

Abstract In this chapter, the authors first present the latest findings regarding the developing and the learning brain, and in particular that such development is nonlinear and dynamic and more affected by the environment than originally conceived. The authors then provide a comprehensive review of the latest findings on the domain-specific and domain-general neurocognitive processes involved in learning to read and learning mathematics in typically developing children and children with learning disabilities. In the fourth part of the chapter, the authors focus on the role of executive functions and inhibitory control more specifically, in school learning, such as arithmetic word problem and deductive reasoning. In addition, the authors present evidence that pedagogical interventions based on teaching to inhibit a given misleading strategy can be effective in helping students overcome systematic errors in different school learning. The authors conclude this chapter by providing a note of caution and a framework to bridge the gap between educational neuroscience in the lab and pedagogical practice in the classroom. More specifically, the authors argue that translating findings from cognitive neuroscience into intervention is a complex and arduous process that takes time but is worth a try given the importance of reading, mathematical, and reasoning skills in industrialized societies and the struggles that many students experience in learning these cultural tools.

A broad variety of domain-specific (e.g., non-symbolic magnitude comparison and arithmetic) and domain-general (e.g., spatial skills and inhibition) skills have been identified as precursors to mathematics achievement. However, due to the increasing complexity of mathematics education with age and the associated increase in cognitive demands, it can be assumed that a divergent set of skills is predictive of mathematics achievement at different ages. This cross-sectional study in children aged 3, 5, and 7 years aims at identifying the differential contribution of domain-specific and domain-general contributors to mathematics and delineating their developmental dynamics. Our results reveal a consistent role for non-symbolic magnitude comparison across all age groups, non-symbolic arithmetic starting from the age of 5 years and visuospatial memory only in 5-year-olds. These findings support the notion that mathematics cannot be conceived as a unitary cognitive skill and provide a fine-grained analysis of the cognitive requirements of mathematical skills at different ages. On a more general note, they are in line with the idea that the ANS provides a critical scaffold for the development of mathematical skills but challenge the view that all ANS measures tap into the same underlying process.

The Spatial Numerical Association of Response Codes (SNARC) effect refers to the observation that relatively small (e.g., 1) and large numbers (e.g., 9) elicit faster left- and right-sided manual responses, respectively. In a variation known as the attentional SNARC effect, merely looking at numbers caused a left- or right-ward shift in covert spatial attention, depending on the number's magnitude. In our study, we probed the notion that numbers induce shifts of spatial attention in accordance with their position on a mental number line (MNL). Critically, we removed any putative spatial response code that may contaminate the responses. We used a square and a tilted square as targets, thereby situating the decisive response dimension in the ventral, non-spatial processing stream. In two experiments where numbers were used as non-informative cues preceding a temporal order judgement (TOJ) task, we did not observe a deflection of the locus of spatial attention as a function of the numerical magnitude of the cue. In a third experiment, finding a significant modulation of TOJ performance as a function of the pointing direction of arrow cues allowed us to rule out the possibility that the absence of any significant modulation in Experiments 1 and 2 was due to a lack of sensitivity of our task set-up. We conclude from the current findings that the spatial codes that the perception and naming of numbers potentially elicit are not in and by themselves sufficient to elicit deflections of spatial attention.

Developmental dyscalculia is a specific learning disorder that persists over lifetime and can have an enormous impact on personal, health-related, and professional aspects of life. Despite its central importance, the origin both at the cognitive and neural level is not yet well understood. Several classification schemas of dyscalculia have been proposed, sometimes together with an associated deficit at the neural level. However, these explanations are (a) not providing an exhaustive framework that is at levels with the observed complexity of developmental dyscalculia at the behavioral level and (b) are largely mono-causal approaches focusing on gray matter deficits. We suggest that number processing is instead the result of context-dependent interaction of two anatomically largely separate, distributed but overlapping networks that function/cooperate in a closely integrated fashion. The proposed two-network framework (TNF) is the result of a series of studies in adults on the neural correlates underlying magnitude processing and arithmetic fact retrieval, which comprised neurofunctional imaging of various numerical tasks, the application of probabilistic fiber tracking to obtain well-defined connections, and the validation and modification of these results using disconnectome mapping in acute stroke patients. Emerged from data in adults, it represents the endpoint of the acquisition and use of mathematical competencies in adults. Yet, we argue that its main characteristics should already emerge earlier during development. Based on this TNF, we develop a classification schema of phenomenological subtypes and their underlying neural origin that we evaluate against existing propositions and the available empirical data.

The Triple-Code Model stipulates that numerical information from different formats and modalities converges on a common magnitude representation in the Intraparietal Sulcus (IPS). To what extent the representations of all numerosity forms overlap remains unsolved. It has been postulated that the representation of symbolic numerosities (for example, Arabic digits) is sparser and grounded in an existing representation that codes for non-symbolic numerosity information (i.e., sets of objects). Other theories argue that numerical symbols represent a separate number category that emerges only during education. Here, we tested a unique group of sighted tactile Braille readers with numerosities 2, 4, 6 and 8 in three number notations: Arabic digits, sets of dots, tactile Braille numbers. Using univariate methods, we showed a consistent overlap in activations evoked by these three number notations. This result shows that all three used notations are represented in the IPS, which may suggest at least a partial overlap between the representations of the three notations used in this experiment. Using MVPA, we found that only non-automatized number information (Braille and sets of dots) allowed successful number classification. However, the numerosity of one notation could not be predicted above chance from the brain activation patterns evoked by another notation (no cross-classification). These results show that the IPS may host independent number codes in overlapping cortical circuits. In addition, they suggest that the level of training in encoding a given type of number information is an important factor that determines the amount of exploitable information and needs to be controlled for in order to identify the neural code underlying numerical information per se.

The notion that mental arithmetic is associated with shifts of spatial attention along a spatially organised mental number representation has received empirical support from three lines of research. First, participants tend to overestimate results of addition and underestimate those of subtraction problems in both exact and approximate formats. This has been termed the operational momentum (OM) effect. Second, participants are faster in detecting right-sided targets presented in the course of addition problems and left-sided targets in subtraction problems (attentional bias). Third, participants are biased toward choosing right-sided response alternatives to indicate the results of addition problems and left-sided response alternatives for subtraction problems (Spatial Association Of Responses [SOAR] effect). These effects potentially have their origin in operation-specific shifts of attention along a spatially organised mental number representation: rightward for addition and leftward for subtraction. Using a lateralised target detection task during the calculation phase of non-symbolic additions and subtractions, the current study measured the attentional focus, the OM and SOAR effects. In two experiments, we replicated the OM and SOAR effects but did not observe operation-specific biases in the lateralised target-detection task. We describe two new characteristics of the OM effect: First, a time-resolved, block-wise analysis of both experiments revealed sequential dependency effects in that the OM effect builds up over the course of the experiment, driven by the increasing underestimation of subtraction over time. Second, the OM effect was enhanced after arithmetic operation repetition compared to trials where arithmetic operation switched from one trial to the next. These results call into question the operation-specific attentional biases as the sole generator of the observed effects and point to the involvement of additional, potentially decisional processes that operate across trials.

The dual-route model explains the SNARC (Spatial-Numerical Association of Response Codes) effect assuming two routes of parallel information processing: the unconditional route (automatic activation of pre-existing links) and the conditional route (activation of task-specific links). To test predictions derived from this model, we evaluated whether response latency in superficial number processing modulates the SNARC effect in a color task (participants judged the color of a number). In Experiment 1, participants performed a parity task, an easy color task (short RTs), and a difficult color task (RTs similar to those of the parity task). A SNARC effect emerged only in the parity task. In Experiment 2, participants performed a color task and a secondary task under four conditions chosen to orthogonally manipulate response latency (short vs. long) and processing depth (semantic vs. perceptual). Only the long-latency perceptual-processing condition elicited a SNARC effect. To explain these results, we suggest that the cognitive resources required by a secondary task might dilute the SNARC effect. Our results indicate that the dual-route model should be modified to take into account additional factors (e.g., working memory load) that influence the level of activation of the unconditional route.