The Calculating Brain.

The human brain possesses neural networks and mechanisms enabling the representation of numbers, basic arithmetic operations, and mathematical reasoning. Without the ability to represent numerical quantity and perform calculations, our scientifically and technically advanced culture would not exist. However, the origins of numerical abilities are grounded in an intuitive understanding of quantity deeply rooted in biology. Nevertheless, more advanced symbolic arithmetic skills necessitate a cultural background with formal mathematical education. In the past two decades, cognitive neuroscience has seen significant progress in understanding the workings of the calculating brain through various methods and model systems. This review begins by exploring the mental and neuronal representations of non-symbolic numerical quantity, then progresses to symbolic representations acquired in childhood. During arithmetic operations (addition, subtraction, multiplication, and division), these representations are processed and transformed according to arithmetic rules and principles, leveraging different mental strategies and types of arithmetic knowledge that can be dissociated in the brain. While it was once believed that number processing and calculation originated from the language faculty, it is now evident that mathematical and linguistic abilities are primarily processed independently in the brain. Understanding how the healthy brain processes numerical information is crucial for gaining insights into debilitating numerical disorders, including acquired conditions like acalculia and learning-related calculation disorders such as developmental dyscalculia.

Diverse mathematical knowledge among indigenous Amazonians.

We investigate number and arithmetic learning among a Bolivian indigenous people, the Tsimane', for whom formal schooling is comparatively recent in history and variable in both extent and consistency. We first present a large-scale meta-analysis on child number development involving over 800 Tsimane' children. The results emphasize the impact of formal schooling: Children are only found to be full counters when they have attended school, suggesting the importance of cultural support for early mathematics. We then test especially remote Tsimane' communities and document the development of specialized arithmetical knowledge in the absence of direct formal education. Specifically, we describe individuals who succeed on arithmetic problems involving the number five-which has a distinct role in the local economy-even though they do not succeed on some lower numbers. Some of these participants can perform multiplication with fives at greater accuracy than addition by one. These results highlight the importance of cultural factors in early mathematics and suggest that psychological theories of number where quantities are derived from lower numbers via repeated addition (e.g., a successor function) are unlikely to explain the diversity of human mathematical ability.

Distinct Contributions of the Cerebellum and Basal Ganglia to Arithmetic Procedures.

Humans exhibit complex mathematical skills attributed to the exceptional enlargement of neocortical regions throughout evolution. In the current work, we initiated a novel exploration of the ancient subcortical neural network essential for mathematical cognition. Using a neuropsychological approach, we report that degeneration of two subcortical structures, the cerebellum and basal ganglia, impairs performance in symbolic arithmetic. We identify distinct computational impairments in male and female participants with cerebellar degeneration (CD) or Parkinson's disease (PD). The CD group exhibited a disproportionate cost when the arithmetic sum increased, suggesting that the cerebellum is critical for iterative procedures required for calculations. The PD group showed a disproportionate cost for equations with increasing addends, suggesting that the basal ganglia are critical for chaining multiple operations. In Experiment 2, the two patient groups exhibited intact practice gains for repeated equations at odds with an alternative hypothesis that these impairments were related to memory retrieval. Notably, we discuss how the counting and chaining operations relate to cerebellar and basal ganglia function in other task domains (e.g., motor processes). Overall, we provide a novel perspective on how the cerebellum and basal ganglia contribute to symbolic arithmetic. Our studies demonstrate the constraints on the computational role of two subcortical regions in higher cognition.

Neural Circuitry of Reward Prediction Error.

Dopamine neurons facilitate learning by calculating reward prediction error, or the difference between expected and actual reward. Despite two decades of research, it remains unclear how dopamine neurons make this calculation. Here we review studies that tackle this problem from a diverse set of approaches, from anatomy to electrophysiology to computational modeling and behavior. Several patterns emerge from this synthesis: that dopamine neurons themselves calculate reward prediction error, rather than inherit it passively from upstream regions; that they combine multiple separate and redundant inputs, which are themselves interconnected in a dense recurrent network; and that despite the complexity of inputs, the output from dopamine neurons is remarkably homogeneous and robust. The more we study this simple arithmetic computation, the knottier it appears to be, suggesting a daunting (but stimulating) path ahead for neuroscience more generally.

Finger motor representation supports the autonomy in arithmetic: neuroimaging evidence from abacus training.

Researches have reported the close association between fingers and arithmetic. However, it remains unclear whether and how finger training can benefit arithmetic. To address this issue, we used the abacus-based mental calculation (AMC), which combines finger training and mental arithmetic learning, to explore the neural correlates underlying finger-related arithmetic training. A total of 147 Chinese children (75 M/72 F, mean age, 6.89 ± 0.46) were recruited and randomly assigned into AMC and control groups at primary school entry. The AMC group received 5 years of AMC training, and arithmetic abilities and resting-state functional magnetic resonance images data were collected from both groups at year 1/3/5. The connectome-based predictive modeling was used to find the arithmetic-related networks of each group. Compared to controls, the AMC's positively arithmetic-related network was less located in the control module, and the inter-module connections between somatomotor-default and somatomotor-control modules shifted to somatomotor-visual and somatomotor-dorsal attention modules. Furthermore, the positive network of the AMC group exhibited a segregated connectivity pattern, with more intra-module connections than the control group. Overall, our results suggested that finger motor representation with motor module involvement facilitated arithmetic-related network segregation, reflecting increased autonomy of AMC, thus reducing the dependency of arithmetic on higher-order cognitive functions.

Three world wars: Fiscal-monetary consequences.

Directed by a consolidated government budget constraint, we compare US monetary­fiscal responses to World Wars I and II and the War on COVID-19.

Early Gray Matter Structural Covariance Predicts Longitudinal Gain in Arithmetic Ability in Children.

Previous neuroimaging studies on arithmetic development have mainly focused on functional activation or functional connectivity between brain regions. It remains largely unknown how brain structures support arithmetic development. The present study investigated whether early gray matter structural covariance contributes to later gain in arithmetic ability in children. We used a public longitudinal sample comprising 63 typically developing children. The participants received structural magnetic resonance imaging scanning when they were 11 years old and were tested with a multiplication task at 11 years old (time 1) and 13 years old (time 2), respectively. Mean gray matter volumes were extracted from eight brain regions of interest to anchor salience network (SN), frontal-parietal network (FPN), motor network (MN), and default mode network (DMN) at time 1. We found that longitudinal gain in arithmetic ability was associated with stronger structural covariance of the SN seed with frontal and parietal regions and stronger structural covariance of the FPN seed with insula, but weaker structural covariance of the FPN seed with motor and temporal regions, weaker structural covariance of the MN seed with frontal and motor regions, and weaker structural covariance of the DMN seed with temporal region. However, we did not detect correlation between longitudinal gain in arithmetic ability and behavioral measure or regional gray matter volume at time 1. Our study provides novel evidence for a specific contribution of gray matter structural covariance to longitudinal gain in arithmetic ability in childhood.

A longitudinal study of the role of fingers in the development of early number and arithmetic skills in children with Apert syndrome.

This paper discusses a longitudinal study with children with Apert syndrome aged between 4 and 11 years. There has long been an interest in the role of fingers in the development of early number skills and arithmetic. As children with Apert syndrome are born with complex fusions of their fingers, they have to undergo several surgical procedures in order to obtain individuated fingers. This has implications for their finger mobility and finger awareness. It has been suggested that children with Apert syndrome have specific difficulties with early number and arithmetic activities. The findings from this study suggest that engaging children with Apert syndrome in activities that develop finger awareness (finger gnosis) and finger mobility (fine motor skills) may have a positive impact on their ability to engage with appropriate mathematics curricula at school. This is relevant to all those involved in the care of children with Apert syndrome and will be of particular relevance to those involved in early childhood and primary education. This study also provides new insights into the role of finger use in the development of skills and understanding in early number and arithmetic.

The role of formal schooling in the development of children's reading and arithmetic white matter networks.

Children's white matter development is driven by experience, yet it remains poorly understood how it is shaped by attending formal education. A small number of studies compared children before and after the start of formal schooling to understand this, yet they do not allow to separate maturational effects from schooling-related effects. A clever way to (quasi-)experimentally address this issue is the longitudinal school cut-off design, which compares children who are similar in age but differ in schooling (because they are born right before or after the cut-off date for school entry). We used for the first time such a longitudinal school cut-off design to experimentally investigate the effect of schooling on children's white matter networks. We compared "young" first graders (schooling group, n = 34; Mage = 68 months; 20 girls) and "old" preschoolers (non-schooling group, n = 33; Mage = 66 months; 18 girls) that were similar in age but differed in the amount of formal instruction they received. Our study revealed that changes in fractional anisotropy and mean diffusivity in five a priori selected white matter tracts during the transition from preschool to primary school were predominantly driven by age-related maturation. We did not find specific schooling effects on white matter, despite their strong presence for early reading and early arithmetic skills. The present study is the first to disentangle the effects of age-related maturation and schooling on white matter within a longitudinal cohort of 5-year-old preschoolers. RESEARCH HIGHLIGHTS: White matter tracts that have been associated with reading and arithmetic may be susceptible to experience-dependent neuroplasticity when children learn to read and calculate. This longitudinal study used the school cut-off design to isolate schooling-induced from coinciding maturational influences on children's white matter development. White matter changes during the transition from preschool to primary school are predominantly driven by age-related maturation and not by schooling effects. Strong effects of schooling on behavior were shown for early reading and early arithmetic, but not for verbal ability and spatial ability.

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.

Short-term number sense training recapitulates long-term neurodevelopmental changes from childhood to adolescence.

Number sense is fundamental to the development of numerical problem-solving skills. In early childhood, children establish associations between non-symbolic (e.g., a set of dots) and symbolic (e.g., Arabic numerals) representations of quantity. The developmental estrangement theory proposes that the relationship between non-symbolic and symbolic representations of quantity evolves with age, with increased dissociation across development. Consistent with this theory, recent research suggests that cross-format neural representational similarity (NRS) between non-symbolic and symbolic quantities is correlated with arithmetic fluency in children but not in adolescents. However, it is not known if short-term training (STT) can induce similar changes as long-term development. In this study, children aged 7-10 years underwent a theoretically motivated 4-week number sense training. Using multivariate neural pattern analysis, we investigated whether short-term learning could modify the relation between cross-format NRS and arithmetic skills. Our results revealed a significant correlation between cross-format NRS and arithmetic fluency in distributed brain regions, including the parietal and prefrontal cortices, prior to training. However, this association was no longer observed after training, and multivariate predictive models confirmed these findings. Our findings provide evidence that intensive STT during early childhood can promote behavioral improvements and neural plasticity that resemble and recapitulate long-term neurodevelopmental changes that occur from childhood to adolescence. More generally, our study contributes to our understanding of the malleability of number sense and highlights the potential for targeted interventions to shape neurodevelopmental trajectories in early childhood. RESEARCH HIGHLIGHTS: We tested the hypothesis that short-term number sense training induces the dissociation of symbolic numbers from non-symbolic representations of quantity in children. We leveraged a theoretically motivated intervention and multivariate pattern analysis to determine training-induced neurocognitive changes in the relation between number sense and arithmetic problem-solving skills. Neural representational similarity between non-symbolic and symbolic quantity representations was correlated with arithmetic skills before training but not after training. Short-term training recapitulates long-term neurodevelopmental changes associated with numerical problem-solving from childhood to adolescence.

Parietal cortical alpha/beta suppression during prospective memory retrieval.

Prospective memory (PM) impairment is among the most frequent memory complaints, yet little is known about the underlying neural mechanisms. PM for a planned intention may be achieved through strategic monitoring of the environment for cues, involving ongoing attentional processes, or through spontaneous retrieval. We hypothesized that parietal spectral power modulation accompanies prospectively encoded intention retrieval, irrespective of PM retrieval approach. A cognitively engaging arithmetic-based ongoing task (OGT) was employed to encourage spontaneous retrieval, with a focal, internally generated PM cue to eliminate OGT/PM trial differentiation based on perceptual or conceptual PM cue features. Two PM repetition frequencies were used to vary the extent of strategic monitoring. We observed a transient parietal alpha/beta spectral power reduction directly preceding the response, which was distinguishable on a single trial basis, as revealed by an OGT/PM trial classification rate exceeding 70% using linear discriminant analysis. The alpha/beta idling rhythm reflects cortical inhibition. A disengagement of task-relevant neural assemblies from this rhythm, reflected in alpha/beta power reduction, is deemed to increase information content, facilitate information integration, and enable engagement of neural assemblies in task-related cortical networks. The observed power reduction is consistent with the Dual Pathways model, where PM strategies converge at the PM retrieval stage.

A connectome-based neuromarker of nonverbal number acuity and arithmetic skills.

The approximate number system (ANS) is vital for survival and reproduction in animals and is crucial for constructing abstract mathematical abilities in humans. Most previous neuroimaging studies focused on identifying discrete brain regions responsible for the ANS and characterizing their functions in numerosity perception. However, a neuromarker to characterize an individual's ANS acuity is lacking, especially one based on whole-brain functional connectivity (FC). Here, based on the resting-state functional magnetic resonance imaging (rs-fMRI) data obtained from a large sample, we identified a distributed brain network (i.e. a numerosity network) using a connectome-based predictive modeling (CPM) analysis. The summed FC strength within the numerosity network reliably predicted individual differences in ANS acuity regarding behavior, as measured using a nonsymbolic number-comparison task. Furthermore, in an independent dataset of the Human Connectome Project (HCP), we found that the summed FC strength within the numerosity network also specifically predicted individual differences in arithmetic skills, but not domain-general cognitive abilities. Therefore, our findings revealed that the identified numerosity network could serve as an applicable neuroimaging-based biomarker of nonverbal number acuity and arithmetic skills.

From whole numbers to fractions to word problems: Hierarchical relations in mathematics knowledge for Chinese Grade 6 students.

It is well established in the literature that fraction knowledge is important for learning more advanced mathematics, but the hierarchical relations among whole number arithmetic, fraction knowledge, and mathematics word problem-solving are not well understood. In the current study, Chinese Grade 6 students (N = 1160; 465 girls; Mage = 12.1 years, SD = 0.6) completed whole number arithmetic (addition, subtraction, multiplication, and division), fraction (mapping, equivalence, comparison, and arithmetic), and mathematics word problem-solving assessments. They also completed two control measures: number writing speed and nonverbal intelligence. Structural equation modeling was used to investigate the hierarchical relations among these assessments. Among the four fraction tasks, the correlations were low to moderate, suggesting that each task may tap into a unique aspect of fraction understanding. In the model, whole number arithmetic was directly related to all four fraction tasks, but was only indirectly related to mathematics word problem-solving, through fraction arithmetic. Only fraction arithmetic, the most advanced fraction skill, directly predicted mathematics word problem-solving. These findings are consistent with the view that students need to build these associations into their mathematics hierarchy to advance their mathematical competence.

Impacts of number lines and circle visual displays on caregivers' fraction understanding.

Playful fraction picture books, together with math instructional content called "back matter," may promote fraction learning, which is crucial because fractions are difficult and often disliked content. However, open questions remain regarding how different types of back matter may affect caregivers' ability to use fraction picture books as a teaching tool. The current study offers a novel investigation into how back matter affects caregivers' (N = 160) fraction understanding (i.e., equivalence and arithmetic) and subjective beliefs about math using a pretest/posttest design. We contrasted existing back matter text with research-informed back matter text crossed with either circle area or number line visual displays. Caregivers' performance improved from pretest to posttest in the Researcher-Generated + Circles condition (fraction equivalence) and in the Existing + Circles, Researcher-Generated + Circles, and Researcher-Generated + Number Lines conditions (fraction arithmetic). In addition, caregivers were aware of their learning; they predicted improvements in their fraction arithmetic performance over time. These findings suggest that brief interventions, such as back matter in children's picture books, may improve adults' fraction understanding.

The efficacy of manipulatives versus fingers in supporting young children's addition skills.

Recent empirical investigations have revealed that finger counting is a strategy associated with good arithmetic performance in young children. Fingers could have a special status during development because they operate as external support that provide sensory-motor and kinesthetic affordances in addition to visual input. However, it was unknown whether fingers are more helpful than manipulatives such as tokens during arithmetic problem solving. To address this question, we conducted a study with 93 Vietnamese children (48 girls) aged 4 and 5 years (mean = 58 months, range = 47-63) with high arithmetic and counting skills from families with relatively high socioeconomic status. Their behaviors were observed as they solved addition problems with manipulatives at their disposal. We found that children spontaneously used both manipulatives and fingers to solve the problems. Crucially, their performance was not higher when fingers rather than manipulatives were used (i.e., 70% vs. 81% correct answers, respectively). Therefore, at the beginning of learning, it is possible that, at least for children with high numerical skills, fingers are not the only gateway to efficient arithmetic development and manipulatives might also lead to proficient arithmetic.

Fraction Ball impact on student and teacher math talk and behavior.

We assessed the impacts of Fraction Ball-a novel suite of games combining the benefits of embodied guided play for math learning-on the math language production and behavior of students and teachers. In the Pilot Experiment, 69 fifth and sixth graders were randomly assigned to play four different Fraction Ball games or attend normal physical education class. The Efficacy Experiment was implemented to test improvements made through co-design with teachers with 160 fourth through sixth graders. Researchers observed and coded for use of math language and behavior. Playing Fraction Ball resulted in consistent increases of students' and teachers' use of fraction (SDs = 0.98-2.42) and decimal (SDs = 0.65-1.64) language and number line arithmetic, but not in whole number, spatial language, counting, instructional gesturing, questioning, and planning. We present evidence of the math language production in physical education and value added by Fraction Ball to support rational number language and arithmetic through group collaboration.

Exploring relative strengths in people with Down syndrome: Spatial thinking and its role in mathematics.

There is convincing evidence that training spatial abilities leads to improved mathematics performance in typically developing (TD) children. However, a lack of information on mathematical development and spatial-mathematical associations in people with Down syndrome (DS) hinders the translation of these interventions. Here, we established developmental trajectories of mathematics and explored whether spatial ability predicts attainment on different mathematics measures in individuals with DS. Participants with DS (n = 36; ages 9-35 years) and TD children (n = 132; ages 4-11 years) completed three groups of tasks: spatial tasks assessing different subdomains of spatial thinking; mathematics tasks assessing early mathematics skills, mathematical reasoning, arithmetic, and geometry; and IQ tasks. The developmental trajectories of mathematics performance against mental age revealed similar starting points of the trajectories and similar rates of development for DS and TD groups. Furthermore, after controlling for verbal skills, spatial skills explained 5.8% to 18.1% of the variation in mathematical performance across different mathematics tasks, and the pattern of spatial-mathematical relations was similar for DS and mental age-matched TD groups. This shows that mathematical development in DS groups appears to mirror that in TD children, indicative of delay only. Strong spatial-mathematical relations were observed for individuals with DS, like those seen for TD participants. This is the vital preliminary knowledge needed to support the design and use of spatial intervention for improving mathematics in individuals with DS.

Age-related changes in how 5- to 8-year-old children use and execute finger-based strategies in arithmetic.

To determine how young children use and execute finger-based strategies, 5- to 8-year-olds were asked to solve simple addition problems under a choice condition (i.e., they could choose finger-based or non-finger strategies on each problem) and under two no-choice conditions (one in which they needed to use finger-based strategies on all problems and one in which they could not use finger-based strategies). Results showed that children (a) used both finger-based and non-finger strategies to solve simple addition problems in all age groups, (b) used fingers less and less often as they grew older, especially while solving smaller problems, (c) calibrated their use of finger-based strategies to both problem features and strategy performance, and (d) improved efficiency of both finger-based and non-finger strategy execution. Moreover, (e) strategy performance was the best predictor of strategy selection in all age groups, and (f) when they had the possibility to use fingers, children of all age groups obtained better performance relative to when they could not use fingers, especially on larger problems.

You can count on your fingers: Finger-based intervention improves first-graders' arithmetic learning.

The question of whether finger use should be encouraged or discouraged in early mathematics instruction remains a topic of debate. Scientific evidence on this matter is scarce due to the limited number of systematic intervention studies. Accordingly, we conducted an intervention study in which first-graders (Mage = 6.48 years, SD = 0.35) completed a finger-based training (18 sessions of âˆ¼ 30 min each) over the course of the first school year. The training was integrated into standard mathematics instruction in schools and compared with business-as-usual curriculum teaching. At the end of first grade and in a follow-up test 9 months later in second grade, children who received the finger training (n = 119) outperformed the control group (n = 123) in written addition and subtraction. No group differences were observed for number line estimation tasks. These results suggest that finger-based numerical strategies can enhance arithmetic learning, supporting the idea of an embodied representation of numbers, and challenge the prevailing skepticism about finger use in primary mathematics education.