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

(Dis)similarities between non-symbolic and symbolic number representations: Insights from vector space models

Empirical evidence in support of a shared system for non-symbolic and symbolic number processing has been inconclusive. The current study aims to address this question in a novel way, specifically by testing whether the efficient coding principle based on co-occurrence of number symbols in natural language holds for both non-symbolic and symbolic number processing. The efficient coding principle postulates that perception is optimized when stimuli frequently co-occur in a natural environment. We hypothesized that both numerical ratios and co-occurrence frequencies of symbolic numbers would significantly influence participants' performance on a non-symbolic and symbolic number comparison task. To test this hypothesis, we employed latent semantic analysis on a TASA corpus to quantify number co-occurrence in natural language and calculate language similarity estimates. We engaged 73 native English speakers (mean age = 19.36, standard deviation = 1.83) with normal or corrected vision and no learning disorders in a number comparison task involving non-symbolic (dot arrays) and symbolic stimuli (Arabic numerals and English number words). Results showed that numerical ratios significantly predicted participants' performances across all number formats (ps < 0.001). Language similarity estimates derived from everyday language also predicted performance on the non-symbolic task and the symbolic task involving number words (ps < 0.007). Our results highlight the complex nature of numerical processing, pointing to the co-occurrence of number symbols in natural language as an auxiliary factor in understanding the shared characteristics between non-symbolic and symbolic number representations. Given that our study focused on a limited number range (5 to 16) and a specific task type, future studies should explore a wider range of tasks and numbers to further test the role of the efficient coding principle in number processing.

Neuroanatomical, transcriptomic, and molecular correlates of math ability and their prognostic value for predicting learning outcomes

Foundational mathematical abilities, acquired in early childhood, are essential for success in our technology-driven society. Yet, the neurobiological mechanisms underlying individual differences in children's mathematical abilities and learning outcomes remain largely unexplored. Leveraging one of the largest multicohort datasets from children at a pivotal stage of knowledge acquisition, we first establish a replicable mathematical ability-related imaging phenotype (MAIP). We then show that brain gene expression profiles enriched for candidate math ability-related genes, neuronal signaling, synaptic transmission, and voltage-gated potassium channel activity contributed to the MAIP. Furthermore, the similarity between MAIP gene expression signatures and brain structure, acquired before intervention, predicted learning outcomes in two independent math tutoring cohorts. These findings advance our knowledge of the interplay between neuroanatomical, transcriptomic, and molecular mechanisms underlying mathematical ability and reveal predictive biomarkers of learning. Our findings have implications for the development of personalized education and interventions.

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.

Is implicit memory associated with the hippocampus?

According to the traditional memory-systems view, the hippocampus is critical during explicit (conscious) long-term memory, whereas other brain regions support implicit (nonconscious) memory. In the last two decades, some fMRI studies have reported hippocampal activity during implicit memory tasks. The aim of the present discussion paper was to identify whether any implicit memory fMRI studies have provided convincing evidence that the hippocampus is associated with nonconscious processes without being confounded by conscious processes. Experimental protocol and analysis parameters included the stimulus type(s), task(s), measures of subjective awareness, explicit memory accuracy, the relevant fMRI contrast(s) or analysis, and confound(s). A systematic review was conducted to identify implicit memory studies that reported fMRI activity in the hippocampus. After applying exclusion criteria, 13 articles remained for analysis. We found that there were no implicit memory fMRI studies where subjective awareness was absent, explicit memory performance was at chance, and there were no confounds that could have driven the observed hippocampal activity. The confounds included explicit memory (including false memory), imbalanced attentional states between conditions (yielding activation of the default-mode network), imbalanced stimuli between conditions, and differential novelty. As such, not a single fMRI study provided convincing evidence that implicit memory was associated with the hippocampus. Neuropsychological evidence was also considered, and implicit memory deficits were caused by factors known to disrupt brain regions beyond the hippocampus, such that the behavioral effects could not be attributed to this region. The present results indicate that implicit memory is not associated with the hippocampus.

Eye movements reveal that young school children shift attention when solving additions and subtractions

Adults shift their attention to the right or to the left along a spatial continuum when solving additions and subtractions, respectively. Studies suggest that these shifts not only support the exact computation of the results but also anticipatively narrow down the range of plausible answers when processing the operands. However, little is known on when and how these attentional shifts arise in childhood during the acquisition of arithmetic. Here, an eye-tracker with high spatio-temporal resolution was used to measure spontaneous eye movements, used as a proxy for attentional shifts, while children of 2nd (8 y-o; N = 50) and 4th (10 y-o; N = 48) Grade solved simple additions (e.g., 4+3) and subtractions (e.g., 3-2). Gaze patterns revealed horizontal and vertical attentional shifts in both groups. Critically, horizontal eye movements were observed in 4th Graders as soon as the first operand and the operator were presented and thus before the beginning of the exact computation. In 2nd Graders, attentional shifts were only observed after the presentation of the second operand just before the response was made. This demonstrates that spatial attention is recruited when children solve arithmetic problems, even in the early stages of learning mathematics. The time course of these attentional shifts suggests that with practice in arithmetic children start to use spatial attention to anticipatively guide the search for the answer and facilitate the implementation of solving procedures. RESEARCH HIGHLIGHTS: Additions and subtractions are associated to right and left attentional shifts in adults, but it is unknown when these mechanisms arise in childhood. Children of 8-10 years old solved single-digit additions and subtractions while looking at a blank screen. Eye movements showed that children of 8 years old already show spatial biases possibly to represent the response when knowing both operands. Children of 10 years old shift attention before knowing the second operand to anticipatively guide the search for plausible answers.

Neural evidence for procedural automatization during cognitive development: Intraparietal response to changes in very-small addition problem-size increases with age

Cognitive development is often thought to depend on qualitative changes in problem-solving strategies, with early developing algorithmic procedures (e.g., counting when adding numbers) considered being replaced by retrieval of associations (e.g., between operands and answers of addition problems) in adults. However, algorithmic procedures might also become automatized with practice. In a large cross-sectional fMRI study from age 8 to adulthood (n = 128), we evaluate this hypothesis by measuring neural changes associated with age-related reductions in a behavioral hallmark of mental addition, the problem-size effect (an increase in solving time as problem sum increases). We found that age-related decreases in problem-size effect were paralleled by age-related increases of activity in a region of the intraparietal sulcus that already supported the problem-size effect in 8- to 9-year-olds, at an age the effect is at least partly due to explicit counting. This developmental effect, which was also observed in the basal ganglia and prefrontal cortex, was restricted to problems with operands ≤ 4. These findings are consistent with a model positing that very-small arithmetic problems-and not larger problems-might rely on an automatization of counting procedures rather than a shift towards retrieval, and suggest a neural automatization of procedural knowledge during cognitive development.

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.

Bayesian interaction selection model for multimodal neuroimaging data analysis

Multimodality or multiconstruct data arise increasingly in functional neuroimaging studies to characterize brain activity under different cognitive states. Relying on those high-resolution imaging collections, it is of great interest to identify predictive imaging markers and intermodality interactions with respect to behavior outcomes. Currently, most of the existing variable selection models do not consider predictive effects from interactions, and the desired higher-order terms can only be included in the predictive mechanism following a two-step procedure, suffering from potential misspecification. In this paper, we propose a unified Bayesian prior model to simultaneously identify main effect features and intermodality interactions within the same inference platform in the presence of high-dimensional data. To accommodate the brain topological information and correlation between modalities, our prior is designed by compiling the intermediate selection status of sequential partitions in light of the data structure and brain anatomical architecture, so that we can improve posterior inference and enhance biological plausibility. Through extensive simulations, we show the superiority of our approach in main and interaction effects selection, and prediction under multimodality data. Applying the method to the Adolescent Brain Cognitive Development (ABCD) study, we characterize the brain functional underpinnings with respect to general cognitive ability under different memory load conditions.

Causal dynamics and information flow in parietal-temporal-hippocampal circuits during mental arithmetic revealed by high-temporal resolution human intracranial EEG

Mental arithmetic involves distributed brain regions spanning parietal and temporal cortices, yet little is known about the neural dynamics of causal functional circuits that link them. Here we use high-temporal resolution (1000 Hz sampling rate) intracranial EEG from 35 participants, 362 electrodes, and 1727 electrode pairs, to investigate dynamic causal circuits linking posterior parietal cortex (PPC) with ventral temporal-occipital cortex and hippocampal regions which constitute the perceptual, visuospatial, and mnemonic building blocks of mental arithmetic. Nonlinear phase transfer entropy measures capable of capturing information flow identified dorsal PPC as a causal inflow hub during mental arithmetic, with strong causal influences from fusiform gyrus in ventral temporal-occipital cortex as well as the hippocampus. Net causal inflow into dorsal PPC was significantly higher during mental arithmetic, compared to both resting-state and verbal memory recall. Our analysis also revealed functional heterogeneity of casual signaling in the PPC, with greater net causal inflow into the dorsal PCC, compared to ventral PPC. Additionally, the strength of causal influences was significantly higher on dorsal, compared to ventral, PPC from the hippocampus, and ventral temporal-occipital cortex during mental arithmetic, when compared to both resting-state and verbal memory recall. Our findings provide novel insights into dynamic neural circuits and hubs underlying numerical problem solving and reveal neurophysiological circuit mechanisms by which both the visual number form processing and declarative memory systems dynamically engage the PPC during mental arithmetic.

Reading direction and spatial effects in parity and arithmetic tasks

This study investigated the relationship between numerical and spatial processing and reading direction, conducting conceptual replications of the Shaki et al. (Psychonomic Bulletin & Review 16(2): 328-331, 2009) parity task and the Mathieu et al. (Cognition 146: 229-239, 2016, Experiment 1) simple addition (e.g., 3 + 2) and subtraction (e.g., 3 - 2) task. Twenty-four left-to-right readers (LTR) and 24 right-to-left readers (RTL) were tested. The response time (RT) analysis of the parity task presented a robust spatial-numerical association of response codes (SNARC) effect (left-side response advantage for smaller numbers and right-side advantage for larger numbers) for LTR but not RTL readers. In the arithmetic task, the three problem elements (e.g., 3 + 4) were presented sequentially with the second operand displaced slightly to the left or right of fixation. RTL but not LTR readers presented a RT advantage for subtraction relative to addition with a right-shifted second operand compared to it being left-shifted. This is consistent with a spatial bias linked to native reading direction. For both reading-direction groups, effects of the left vs. right side manipulation in the arithmetic or parity task did not correspond to parallel effects in the other task. The results imply that the parity-based SNARC effects and side-related effects in cognitive arithmetic are not equivalent measures of space-related processes in cognitive number processing and likely reflect distinct mechanisms.

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.

How visual is the « number sense »? Insights from the blind

Is vision a necessary building block for the foundations of mathematical cognition? A straightforward model to test the causal role visual experience plays in the development of numerical abilities is to study people born without sight. In this review we will demonstrate that congenitally blind people can develop numerical abilities that equal or even surpass those of sighted individuals, despite representing numbers using a qualitatively different representational format. We will also show that numerical thinking in blind people maps onto regions typically involved in visuo-spatial processing in the sighted, highlighting how intrinsic computational biases may constrain the reorganization of numerical networks in case of early visual deprivation. More generally, we will illustrate how the study of arithmetic abilities in congenitally blind people represents a compelling model to understand how sensory experience scaffolds the development of higher-level cognitive representations.

Are small additions solved by direct retrieval from memory or automated counting procedures? A rejoinder to Chen and Campbell (2018)

Contrary to the longstanding and consensual hypothesis that adults mainly solve small single-digit additions by directly retrieving their answer from long-term memory, it has been recently argued that adults could solve small additions through fast automated counting procedures. In a recent article, Chen and Campbell (Psychonomic Bulletin & Review, 25, 739-753, 2018) reviewed the main empirical evidence on which this alternative hypothesis is based, and concluded that there is no reason to jettison the retrieval hypothesis. In the present paper, we pinpoint the fact that Chen and Campbell reached some of their conclusions by excluding some of the problems that need to be considered for a proper argumentation against the automated counting procedure theory. We also explain why, contrary to Chen and Campbell's assumption, the network interference model proposed by Campbell (Mathematical Cognition, 1, 121-164, 1995) cannot account for our data. Finally, we clarify a theoretical point of our model.

Neural representations of transitive relations predict current and future math calculation skills in children

A large body of evidence suggests that math learning in children is built upon innate mechanisms for representing numerical quantities in the intraparietal sulcus (IPS). Learning math, however, is about more than processing quantitative information. It is also about understanding relations between quantities and making inferences based on these relations. Consistent with this idea, recent behavioral studies suggest that the ability to process transitive relations (A > B, B > C, therefore A > C) may contribute to math skills in children. Here we used fMRI coupled with a longitudinal design to determine whether the neural processing of transitive relations in children could predict their current and future math skills. At baseline (T₁), children (n = 31) processed transitive relations in an MRI scanner. Math skills were measured at T₁ and again 1.5 years later (T₂). Using a machine learning approach with cross-validation, we found that activity associated with the representation of transitive relations in the IPS predicted math calculation skills at both T₁ and T₂. Our study highlights the potential of neurobiological measures of transitive reasoning for forecasting math skills in children, providing additional evidence for a link between this type of reasoning and math learning.

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.

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.

Magnitude or Multitude - What Counts?

Recent studies revealed an association of low or high numbers (e.g., 1 vs. 9) and word semantics referring to entities typically found in upper or lower space (e.g., roof vs. root) indicating overlapping spatial representations. Another line of research revealed a similar association of grammatical number as a syntactic aspect of language and physical space: singular words were associated with left and plural words with right - resembling spatial-numerical associations of low numbers with left and high numbers with right. The present study aimed at integrating these lines of research by evaluating both types of spatial relations in one experiment. In a lexical decision task, pairs of a numerical cue and a subsequent plural noun were presented. For word with spatial associations (e.g., roofs vs. roots) number magnitude was expected to serve as a spatial cue. For spatially neutral words (e.g., tables) numbers were expected to cue multitude. Results showed the expected congruency-effect between the numbers and words with spatial associations (i.e., small numbers facilitate responses to down-words and high numbers to up-words). However, no effect was found for numbers and spatially neutral words. This seems to indicate that spatial aspects of word meaning may be related more closely to the magnitude of numbers than grammatical number is to the multitude reflected by numbers - at least in the current experimental setting, where only plural words were presented.