Finger Tracking Reveals the Covert Stages of Mental Arithmetic

Negation mitigates rather than inverts the neural representations of adjectives

Combinatoric linguistic operations underpin human language processes, but how meaning is composed and refined in the mind of the reader is not well understood. We address this puzzle by exploiting the ubiquitous function of negation. We track the online effects of negation ("not") and intensifiers ("really") on the representation of scalar adjectives (e.g., "good") in parametrically designed behavioral and neurophysiological (MEG) experiments. The behavioral data show that participants first interpret negated adjectives as affirmative and later modify their interpretation towards, but never exactly as, the opposite meaning. Decoding analyses of neural activity further reveal significant above chance decoding accuracy for negated adjectives within 600 ms from adjective onset, suggesting that negation does not invert the representation of adjectives (i.e., "not bad" represented as "good"); furthermore, decoding accuracy for negated adjectives is found to be significantly lower than that for affirmative adjectives. Overall, these results suggest that negation mitigates rather than inverts the neural representations of adjectives. This putative suppression mechanism of negation is supported by increased synchronization of beta-band neural activity in sensorimotor areas. The analysis of negation provides a steppingstone to understand how the human brain represents changes of meaning over time.

Spatiotemporal Dynamics of Successive Activations across the Human Brain during Simple Arithmetic Processing

Previous neuroimaging studies have offered unique insights about the spatial organization of activations and deactivations across the brain; however, these were not powered to explore the exact timing of events at the subsecond scale combined with a precise anatomical source of information at the level of individual brains. As a result, we know little about the order of engagement across different brain regions during a given cognitive task. Using experimental arithmetic tasks as a prototype for human-unique symbolic processing, we recorded directly across 10,076 brain sites in 85 human subjects (52% female) using the intracranial electroencephalography. Our data revealed a remarkably distributed change of activity in almost half of the sampled sites. In each activated brain region, we found juxtaposed neuronal populations preferentially responsive to either the target or control conditions, arranged in an anatomically orderly manner. Notably, an orderly successive activation of a set of brain regions-anatomically consistent across subjects-was observed in individual brains. The temporal order of activations across these sites was replicable across subjects and trials. Moreover, the degree of functional connectivity between the sites decreased as a function of temporal distance between regions, suggesting that the information is partially leaked or transformed along the processing chain. Our study complements prior imaging studies by providing hitherto unknown information about the timing of events in the brain during arithmetic processing. Such findings can be a basis for developing mechanistic computational models of human-specific cognitive symbolic systems.

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.

Spatiotemporal dynamics of successive activations across the human brain during simple arithmetic processing

Previous neuroimaging studies have offered unique insights about the spatial organization of activations and deactivations across the brain, however these were not powered to explore the exact timing of events at the subsecond scale combined with precise anatomical source information at the level of individual brains. As a result, we know little about the order of engagement across different brain regions during a given cognitive task. Using experimental arithmetic tasks as a prototype for human-unique symbolic processing, we recorded directly across 10,076 brain sites in 85 human subjects (52% female) using intracranial electroencephalography (iEEG). Our data revealed a remarkably distributed change of activity in almost half of the sampled sites. Notably, an orderly successive activation of a set of brain regions - anatomically consistent across subjects-was observed in individual brains. Furthermore, the temporal order of activations across these sites was replicable across subjects and trials. Moreover, the degree of functional connectivity between the sites decreased as a function of temporal distance between regions, suggesting that information is partially leaked or transformed along the processing chain. Furthermore, in each activated region, distinct neuronal populations with opposite activity patterns during target and control conditions were juxtaposed in an anatomically orderly manner. Our study complements the prior imaging studies by providing hitherto unknown information about the timing of events in the brain during arithmetic processing. Such findings can be a basis for developing mechanistic computational models of human-specific cognitive symbolic systems.

Significance statement

Our study elucidates the spatiotemporal dynamics and anatomical specificity of brain activations across >10,000 sites during arithmetic tasks, as captured by intracranial EEG. We discovered an orderly, successive activation of brain regions, consistent across individuals, and a decrease in functional connectivity as a function of temporal distance between regions. Our findings provide unprecedented insights into the sequence of cognitive processing and regional interactions, offering a novel perspective for enhancing computational models of cognitive symbolic systems.

The development of simple addition problem solving in children: Reliance on automatized counting or memory retrieval depends on both expertise and problem size

In an experiment, 98 children aged 8 to 9, 10 to 12, and 13 to 15 years solved addition problems with a sum up to 10. In another experiment, the same children solved the same calculations within a sign priming paradigm where half the additions were displayed with the "+" sign 150 ms before the addends. Therefore, size effects and priming effects could be considered conjointly within the same populations. Our analyses revealed that small problems, constructed with addends from 1 to 4, presented a linear increase of solution times as a function of problem sums (i.e., size effect) in all age groups. However, an operator priming effect (i.e., facilitation of the solving process with the anticipated presentation of the "+" sign) was observed only in the group of oldest children. These results support the idea that children use a counting procedure that becomes automatized (as revealed by the priming effect) around 13 years of age. For larger problems and whatever the age group, no size or priming effects were observed, suggesting that the answers to these problems were already retrieved from memory at 8 to 9 years of age. For this specific category of large problems, negative slopes in solution times demonstrate that retrieval starts from the largest problems during development. These results are discussed in light of a horse race model in which procedures can win over retrieval.

Continuous manipulation of mental representations is compromised in cerebellar degeneration

We introduce a novel perspective on how the cerebellum might contribute to cognition, hypothesizing that this structure supports dynamic transformations of mental representations. In support of this hypothesis, we report a series of neuropsychological experiments comparing the performance of individuals with degenerative cerebellar disorders on tasks that either entail continuous, movement-like mental operations or more discrete mental operations. In the domain of visual cognition, the cerebellar disorders group exhibited an impaired rate of mental rotation, an operation hypothesized to require the continuous manipulation of a visual representation. In contrast, the cerebellar disorders group showed a normal processing rate when scanning items in visual working memory, an operation hypothesized to require the maintenance and retrieval of remembered items. In the domain of mathematical cognition, the cerebellar disorders group was impaired at single-digit addition, an operation hypothesized to primarily require iterative manipulations along a mental number-line; this group was not impaired on arithmetic tasks linked to memory retrieval (e.g. single-digit multiplication). These results, obtained in tasks from two disparate domains, point to a potential constraint on the contribution of the cerebellum to cognitive tasks. Paralleling its role in motor control, the cerebellum may be essential for coordinating dynamic, movement-like transformations in a mental workspace.

Elementary math in elementary school: the effect of interference on learning the multiplication table

Memorizing the multiplication table is a major challenge for elementary school students: there are many facts to memorize, and they are often similar to each other, which creates interference in memory. Here, we examined whether learning would improve if the degree of interference is reduced, and which memory processes are responsible for this improvement. In a series of 16 short training sessions over 4 weeks, first-grade children learned 16 multiplication facts-4 facts per week. In 2 weeks the facts were dissimilar from each other (low interference), and in 2 control weeks the facts were similar (high interference). Learning in the low-similarity, low-interference weeks was better than in the high-similarity weeks. Critically, this similarity effect originated in the specific learning context, i.e., the grouping of facts to weeks, and could not be explained as an intrinsic advantage of certain facts over others. Moreover, the interference arose from the similarity between facts in a given week, not from the similarity to previously learned facts. Similarity affected long-term memory-its effect persisted 7 weeks after training has ended; and it operated on long-term memory directly, not via the mediation of working memory. Pedagogically, the effectiveness of the low-interference training method, which is dramatically different from currently used pedagogical methods, may pave the way to enhancing how we teach the multiplication table in school.

Preschool language and visuospatial skills respectively predict multiplication and addition/subtraction skills in middle school children

A converging body of evidence from neuroimaging, behavioral and neuropsychology studies suggests that different arithmetic operations rely on distinct neuro-cognitive processes: while addition and subtraction may rely more on visuospatial reasoning, multiplication would depend more on verbal abilities. In this paper, we tested this hypothesis in a longitudinal study measuring language and visuospatial skills in 358 preschoolers, and testing their mental calculation skills at the beginning of middle school. Language skills at 5.5 years significantly predicted multiplication, but not addition nor subtraction scores at 11.5 years. Conversely, early visuospatial skills predicted addition and subtraction, but not multiplication scores. These results provide strong support for the existence of a double dissociation in mental arithmetic operations, and demonstrate the existence of long-lasting links between language/visuospatial skills and specific calculation abilities. Using structural equation modelling, we analyzed longitudinal data from 358 children. Language skills in preschool significantly predicted multiplication, but not addition nor subtraction scores in middle school. The reverse was true for preschool visuo-spatial skills. Importantly, this pattern remained unchanged when we controlled addition and multiplication operations for overall difficulty and for the magnitude of the operands. These results provide strong support for the existence of a double dissociation in mental arithmetic operations. This article is protected by copyright. All rights reserved.

Operational momentum during children's approximate arithmetic relates to symbolic math skills and space-magnitude association

Operational momentum (OM) refers to the behavioral tendency to overestimate or underestimate the results of addition or subtraction, respectively. The cognitive mechanism of the OM effect and how it is related to the development of symbolic math abilities are not well understood. The current study examined whether individual differences in the OM effect are related to symbolic arithmetic abilities, number line estimation performance, and the space-magnitude association effect in young children. In this study, first-grade elementary school children manifested the OM effect during approximate addition and subtraction. Individual differences in the OM effect were not correlated with number line estimation error. Interestingly, children who showed a greater degree of the OM effect performed not worse, but better on the symbolic arithmetic task. In addition, the OM effect was correlated with the space-magnitude association (size congruity) effect measured with the Numerical Stroop task. More specifically, the OM bias was correlated with the ability to inhibit interference from competing information on the incongruent trials of the Numerical Stroop task. Our results suggest that the inaccuracy of numerical magnitude representations is not the source of the OM effect. Given that children with better math ability showed a greater OM bias, a stronger OM effect may reflect better intuition in arithmetic operations. Altogether, we carefully interpret these findings as suggesting that a greater OM effect reflects superior intuition or fundamental knowledge of arithmetic operations and a more adult-like maturation of the reorienting component of the attentional system.

Priming effects of arithmetic signs in 10- to 15-year-old children

In this research, 10- to 12- and 13- to 15-year-old children were presented with very simple addition and multiplication problems involving operands from 1 to 4. Critically, the arithmetic sign was presented before the operands in half of the trials, whereas it was presented at the same time as the operands in the other half. Our results indicate that presenting the 'x' sign before the operands of a multiplication problem does not speed up the solving process, irrespective of the age of children. In contrast, presenting the '+' sign before the operands of an addition problem facilitates the solving process, but only in 13 to 15-year-old children. Such priming effects of the arithmetic sign have been previously interpreted as the result of a pre-activation of an automated counting procedure, which can be applied as soon as the operands are presented. Therefore, our results echo previous conclusions of the literature that simple additions but not multiplications can be solved by fast counting procedures. More importantly, we show here that these procedures are possibly convoked automatically by children after the age of 13 years. At a more theoretical level, our results do not support the theory that simple additions are solved through retrieval of the answers from long-term memory by experts. Rather, the development of expertise for mental addition would consist in an acceleration of procedures until automatization.

The influence of locative expressions on context-dependency of endpoint control in aiming

It has been claimed that increased reliance on context, or allocentric information, develops when aiming movements are more consciously monitored and/or controlled. Since verbalizing target features requires strong conscious monitoring, we expected an increased reliance on allocentric information when verbalizing a target label (i.e. target number) during movement execution. We examined swiping actions towards a global array of targets embedded in different local array configurations on a tablet under no-verbalization and verbalization conditions. The global and local array configurations allowed separation of contextual-effects from any possible numerical magnitude biases triggered from calling out specific target numbers.The patterns of constant errors in the target directionwere used to assess differences between conditions. Variation in the target context configuration systematically biased movement endpoints in both the no-verbalization and verbalization conditions. Ultimately, our results do not support the assertion that calling out target numbers during movement execution increases the context-dependency of targeted actions.

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.

Probing the mechanisms underlying numerosity-to-numeral mappings and their relation to math competence

Numerosity estimation performance (e.g., how accurate, consistent, or proportionally spaced (linear) numerosity-numeral mappings are) has previously been associated with math competence. However, the specific mechanisms that underlie such a relation is unknown. One possible mechanism is the mapping process between numerical sets and symbolic numbers (e.g., Arabic numerals). The current study examined two hypothesized mechanisms of numerosity-numeral mappings (item-based "associative" and holistic "structural" mapping) and their roles in the estimation-and-math relation. Specifically, mappings for small numbers (e.g., 1-10) are thought to be associative and resistant to calibration (e.g., feedback on accuracy of estimates), whereas holistic "structural" mapping for larger numbers (e.g., beyond 10) may be supported by flexibly aligning a numeral "response grid" (akin to a ruler) to an analog "mental number line" upon calibration. In 57 adults, we used pre- and post-calibration estimates to measure the range of continuous associative mappings among small numbers (e.g., a base range of associative mappings from 1 to 10), and obtained measures of math competence and delayed multiple-choice strategy reports. Consistent with previous research, uncalibrated estimation performance correlated with calculation competence, controlling for reading fluency and working memory. However, having a higher base range of associative mappings was not related to estimation performance or any math competence measures. Critically, discontinuity in calibration effects was typical at the individual level, which calls into question the nature of "holistic structural mapping". A parsimonious explanation to integrate previous and current findings is that estimation performance is likely optimized by dynamically constructing numerosity-numeral mappings through the use of multiple strategies from trial to trial.

A comparison of simple movement behaviors across three different devices

Reaching trajectories have provided a unique tool to observe changes in internal cognitive decisions. Furthermore, technological advances have made devices for measuring reach movements more accessible and researchers have recognized that various populations including children, elderly populations, and non-human primates can easily execute simple movements as responses. As a result, devices such as a three-dimensional (3D) reach tracker, a stylus, or a computer-mouse have been increasingly utilized to study cognitive processes. However, although the specific type of tracking device that a researcher uses may impact behavior due to the constraints it places on movements, most researchers in these fields are unaware of this potential issue. Here, we examined the potential behavioral impact of using each of these three devices. To induce re-directed movements that mimic the movements that often occur following changes in cognitive states, we used a double-step task in which displacement of an initial target location requires participants to quickly re-direct their movement. We found that reach movement parameters were largely comparable across the three devices. However, hand movements measured by a 3D reach tracker showed earlier reach initiation latencies (relative to stylus movements) and more curved movement trajectories (relative to both mouse and stylus movements). Reach movements were also re-directed following target displacement more rapidly. Thus, 3D reach trackers may be ideal for observing fast, subtle changes in internal decision-making processes compared to other devices. Taken together, this study provides a useful reference for comparing and implementing reaching studies to examine human cognition.

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.

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.

Brain Mechanisms of Arithmetic: A Crucial Role for Ventral Temporal Cortex

Elementary arithmetic requires a complex interplay between several brain regions. The classical view, arising from fMRI, is that the intraparietal sulcus (IPS) and the superior parietal lobe (SPL) are the main hubs for arithmetic calculations. However, recent studies using intracranial electroencephalography have discovered a specific site, within the posterior inferior temporal cortex (pITG), that activates during visual perception of numerals, with widespread adjacent responses when numerals are used in calculation. Here, we reexamined the contribution of the IPS, SPL, and pITG to arithmetic by recording intracranial electroencephalography signals while participants solved addition problems. Behavioral results showed a classical problem size effect: RTs increased with the size of the operands. We then examined how high-frequency broadband (HFB) activity is modulated by problem size. As expected from previous fMRI findings, we showed that the total HFB activity in IPS and SPL sites increased with problem size. More surprisingly, pITG sites showed an initial burst of HFB activity that decreased as the operands got larger, yet with a constant integral over the whole trial, thus making these signals invisible to slow fMRI. Although parietal sites appear to have a more sustained function in arithmetic computations, the pITG may have a role of early identification of the problem difficulty, beyond merely digit recognition. Our results ask for a reevaluation of the current models of numerical cognition and reveal that the ventral temporal cortex contains regions specifically engaged in mathematical processing.

Decoding the processing stages of mental arithmetic with magnetoencephalography

Elementary arithmetic is highly prevalent in our daily lives. However, despite decades of research, we are only beginning to understand how the brain solves simple calculations. Here, we applied machine learning techniques to magnetoencephalography (MEG) signals in an effort to decompose the successive processing stages and mental transformations underlying elementary arithmetic. Adults subjects verified single-digit addition and subtraction problems such as 3 + 2 = 9 in which each successive symbol was presented sequentially. MEG signals revealed a cascade of partially overlapping brain states. While the first operand could be transiently decoded above chance level, primarily based on its visual properties, the decoding of the second operand was more accurate and lasted longer. Representational similarity analyses suggested that this decoding rested on both visual and magnitude codes. We were also able to decode the operation type (additions vs. subtraction) during practically the entire trial after the presentation of the operation sign. At the decision stage, MEG indicated a fast and highly overlapping temporal dynamics for (1) identifying the proposed result, (2) judging whether it was correct or incorrect, and (3) pressing the response button. Surprisingly, however, the internally computed result could not be decoded. Our results provide a first comprehensive picture of the unfolding processing stages underlying arithmetic calculations at a single-trial level, and suggest that externally and internally generated neural codes may have different neural substrates.

On-line confidence monitoring during decision making

Humans can readily assess their degree of confidence in their decisions. Two models of confidence computation have been proposed: post hoc computation using post-decision variables and heuristics, versus online computation using continuous assessment of evidence throughout the decision-making process. Here, we arbitrate between these theories by continuously monitoring finger movements during a manual sequential decision-making task. Analysis of finger kinematics indicated that subjects kept separate online records of evidence and confidence: finger deviation continuously reflected the ongoing accumulation of evidence, whereas finger speed continuously reflected the momentary degree of confidence. Furthermore, end-of-trial finger speed predicted the post-decisional subjective confidence rating. These data indicate that confidence is computed on-line, throughout the decision process. Speed-confidence correlations were previously interpreted as a post-decision heuristics, whereby slow decisions decrease subjective confidence, but our results suggest an adaptive mechanism that involves the opposite causality: by slowing down when unconfident, participants gain time to improve their decisions.

Act on Numbers: Numerical Magnitude Influences Selection and Kinematics of Finger Movement

In the past decade hand kinematics has been reliably adopted for investigating cognitive processes and disentangling debated topics. One of the most controversial issues in numerical cognition literature regards the origin - cultural vs. genetically driven - of the mental number line (MNL), oriented from left (small numbers) to right (large numbers). To date, the majority of studies have investigated this effect by means of response times, whereas studies considering more culturally unbiased measures such as kinematic parameters are rare. Here, we present a new paradigm that combines a "free response" task with the kinematic analysis of movement. Participants were seated in front of two little soccer goals placed on a table, one on the left and one on the right side. They were presented with left- or right-directed arrows and they were instructed to kick a small ball with their right index toward the goal indicated by the arrow. In a few test trials participants were presented also with a small (2) or a large (8) number, and they were allowed to choose the kicking direction. Participants performed more left responses with the small number and more right responses with the large number. The whole kicking movement was segmented in two temporal phases in order to make a hand kinematics' fine-grained analysis. The Kick Preparation and Kick Finalization phases were selected on the basis of peak trajectory deviation from the virtual midline between the two goals. Results show an effect of both small and large numbers on action execution timing. Participants were faster to finalize the action when responding to small numbers toward the left and to large number toward the right. Here, we provide the first experimental demonstration which highlights how numerical processing affects action execution in a new and not-overlearned context. The employment of this innovative and unbiased paradigm will permit to disentangle the role of nature and culture in shaping the direction of MNL and the role of finger in the acquisition of numerical skills. Last but not least, similar paradigms will allow to determine how cognition can influence action execution.

Hippocampal spatial mechanisms relate to the development of arithmetic symbol processing in children

Understanding the meaning of abstract mathematical symbols is a cornerstone of arithmetic learning in children. Studies have long focused on the role of spatial intuitions in the processing of numerals. However, it has been argued that such intuitions may also underlie symbols that convey fundamental arithmetic concepts, such as arithmetic operators. In the present cross-sectional study, we used fMRI to investigate how and when associations between arithmetic operators and brain regions processing spatial information emerge in children from 3^rd to 10^th grade. We found that the mere perception of a ‘+’ sign elicited grade-related increases of spatial activity in the right hippocampus. That is, merely perceiving ‘+’ signs – without any operands – elicited enhanced hippocampal activity after around 7^th grade (12–13 years old). In these children, hippocampal activity in response to a ‘+’ sign was further correlated with the degree to which calculation performance was facilitated by the preview of that sign before an addition problem, an effect termed operator-priming. Grade-related increases of hippocampal spatial activity were operation-specific because they were not observed with ‘×’ signs, which might evoke rote retrieval rather than numerical manipulation. Our study raises the possibility that hippocampal spatial mechanisms help build associations between some arithmetic operators and space throughout age and/or education.

Finger Tracking Reveals the Covert Stages of Mental Arithmetic

We introduce a novel method capable of dissecting the succession of processing stages underlying mental arithmetic, thus revealing how two numbers are transformed into a third. We asked adults to point to the result of single-digit additions and subtractions on a number line, while their finger trajectory was constantly monitored. We found that the two operands are processed serially: the finger first points toward the larger operand, then slowly veers toward the correct result. This slow deviation unfolds proportionally to the size of the smaller operand, in both additions and subtractions. We also observed a transient operator effect: a plus sign attracted the finger to the right and a minus sign to the left and a transient activation of the absolute value of the subtrahend. These findings support a model whereby addition and subtraction are computed by a stepwise displacement on the mental number line, starting with the larger number and incrementally adding or subtracting the smaller number.