The place-value concept is fundamental to understanding the symbolic number system. It dictates that the value of a digit in a number is based on its position or place within the number (e.g., the "5" in "510" is five units of 100, whereas the "5" in "51" is five units of 10). Place value is central to understanding multidigit numbers, performing arithmetic, and learning more complex math. Despite its significance, relatively little research has systematically examined the developmental trajectory and cognitive underpinnings of the place-value concept. In this article, we synthesize prior findings and propose a conceptual framework that delineates the core properties of the place-value concept and characterizes its developmental trajectory. We also identify key cognitive factors that may underpin individual differences in its acquisition. This framework can guide future research to understand how children acquire the place-value concept and how best to support this learning. It also has broad implications for understanding the cognitive architecture of human compositional symbol systems.

Developmental dyscalculia (DD), a learning disorder that affects one’s ability to work with numerical information and perform calculations, presents significant challenges for children acquiring mathematical skills. Recent research has provided a robust understanding of the behavioral and cognitive profile of DD, but its neurobiological underpinnings remain poorly understood. In this study, we use resting-state functional connectivity (rsFC) to investigate the neural differences between third graders with DD and typically achieving (TA) peers (DD = 30, TA = 37, mean age = 9.04 y). We employed two complementary analytical approaches: 1) a seed-based functional connectivity analysis to assess connectivity between a priori regions of interest (ROIs)—subregions of the intraparietal sulcus (IPS), angular gyrus, and the hippocampus—and the rest of the brain, and 2) a modified whole-brain, connectome-based predictive modeling approach to detect DD based on brain connectivity patterns. The seed-based connectivity analysis revealed greater functional connectivity for the TA group between the bilateral IPS and left hippocampal ROIs and frontal structures. Our whole-brain classification approach achieved a mean accuracy of 0.671 and an AUC of 0.816 and identified 14 brain connections that consistently classified the TA and DD groups. Our findings point to network-level differences underlying DD in brain regions previously implicated in mathematical cognition and offer a novel, data-driven approach to identifying differences in brain connectivity associated with DD.

Human infants are assumed to be born with an innate understanding of quantities—a number sense— that enables them to detect non-symbolic number differences, for example, between sets of objects. This early ability has been found to be a predictor of later math skills. The assumption of innateness is based largely on studies in infants at 6 months of age or older and using visual stimuli with large numerosities, or in studies lacking statistical power. Using EEG, we studied pre-attentive discrimination for auditorily presented non-verbal numerical and non-numerical magnitudes of sleeping newborns (N = 104) to investigate whether the newborn auditory system is discriminating the number of sequentially presented sounds. The results showed that the 2-part repeated standard sound was discriminated from 3-part deviating sound and from 1-part deviating sound. As also the responses to 3-part sound differed from the responses of the same stimulus presented alone as a control, these results suggest that the difference between 2-part repeated sound and 3-part deviant sound is not merely due to physical features of the stimuli but reflects processing of numerosity. These results suggest that newborn infants are able to pre-attentively process at least some differences in the number of stimuli supporting the view of innateness of numerical magnitude discrimination. As the earlier evidence for innateness in newborns is limited, our study utilising small numerosities and auditory modality instead of vision—given the confounds in visual paradigms—offers complementary results on the innate brain processing on numerical information.

The use of educational technology (EdTech) has grown rapidly, yet there remains a lack of synthesized evidence on how EdTech tools are evaluated for math and literacy learning. This study conducted a systematic review of how EdTech has been evaluated in K-8 education, based on 44 peer-reviewed studies published between 2013 and 2024. It explored key dimensions, including study methodology (e.g., design), technological features (e.g., device types), evaluation aspects (e.g., user experience), assessment tools (e.g., surveys), and math and literacy learning outcomes (e.g., counting, vocabulary). The analysis showed that majority of the studies employed quasi-experimental designs (57%), pre-posttest designs (66%), and featured medium sample sizes (55%). Twelve distinct EdTech platforms were identified, with educational games most common in math and web-based applications in literacy. Tablets and iPads were the most frequently used devices, and feedback was the most frequent educational feature. Performance assessments, particularly pre-posttests, were used in most studies (86%), and nearly half relied on platform-generated data (41%). Outcome measures focused largely on basic math skills and reading comprehension, with limited focus on integrated math and literacy learning outcomes. These findings provide a structured lens for evaluating EdTech in elementary education, therefore, guiding the creation of more comprehensive and effective methods.

Relative magnitudes, such as ratios and proportions, are crucial to everyday life. According to the ratio processing system (RPS) theory, ratios across different formats share a common neural code. Prior neuroimaging studies have shown that ratios across different visual formats are processed by overlapping brain areas, thus providing support for this hypothesis. However, overlapping activation found from univariate analyses does not necessarily imply common representations. The aim of this study was to probe the key prediction of the RPS theory by investigating the neural representation of ratios depicted in discrete and continuous magnitudes using multivariate fMRI analyses. Thirty participants completed a delayed ratio comparison task on discrete (sets of dots) and continuous (line lengths) magnitudes while in the MRI scanner. Using representational similarity analysis (RSA), we examined the similarity between ratio representations across high-level variations, such as magnitude type (length and numerosity), and low-level variations, such as variations in line orientation and dot size, throughout the brain, notably in parietal areas. Results did not support a common ratio representation across discrete and continuous magnitudes. Instead, ratios in numerosity and length seemed to be encoded in qualitatively distinct ways. Altogether, results suggest that ratios are not represented in a format-independent way in the brain, thus challenging the RPS theory.

Standardized measures of achievement and teacher-assigned grades (henceforth grades) are both supposed to be measures of achievement. However, there might be concerns that grades are more subjective than standardized tests. The purpose of this study was to examine the relationship between standardized tests and grades in primary students. We collected data from 218 Grade 1 students and 124 were re-tested in Grade 2. Students completed subtests from the Woodcock-Johnson III Tests of Achievement (henceforth test scores) and grades were collected. Separate factor analyses were conducted on the test scores and grades, both yielding a 3-factor model with separate language, reading, and mathematics domains. We then examined the relationship between test scores and grades across domains, grade-level, genders, and schools. Structural equation models found that test scores explained half of the variance in grades across domains, except Grade 2 language. Bayesian paired t-test found that minimal discrepancies between test scores and grades existed across genders and schools. When a difference did exist, grades tended to be higher than test scores; discrepancies typically occurred in verbal subjects but attenuated by Grade 2; there was a female advantage for grades and male advantage for test scores; and, school differences depended on grade and domain. Understanding the degree of alignment between measures could bolster the use of teacher assessment as a valid index of performance in young children.

Robust behavioral evidence suggests an association between reading and math performance. Moreover, previous neuroimaging evidence suggests that arithmetic fact retrieval is supported by similar areas along the perisylvian language network as those typically involved in phonological processing. However, the neural correlates of these abilities have been mostly studied in isolation, and therefore remains unclear whether these abilities recruit functionally overlapping brain areas. We addressed this question by using functional magnetic resonance imaging to measure brain activity during an arithmetic and a word rhyming task. We then used both a test of univariate overlap and a rigorous pattern similarity analysis to provide a more nuanced assessment of brain-level associations across both domains. We identified clusters of significant overlap along the left inferior frontal gyrus, the left inferior temporal gyrus, and the right posterior cerebellum in adults; as well as multiple clusters along the left frontal gyrus in children. Moreover, we found significant similarity between the patterns corresponding to both abilities along the clusters of overlap. However, contrary to our expectations, we observed higher similarity between phonological processing and large problems than small problems, which grants the need for further research about the role of arithmetic strategies in this relationship. Our findings represent a contribution to the literature examining the potential links between the brain regions supporting arithmetic and word reading by providing direct, within-participant statistical evidence of the long-hypothesized overlap between these processes at the neural level.

The ability to understand and compare non-symbolic (e.g., dot arrays) and symbolic (e.g., Arabic numerals) magnitudes is a critical foundation for learning math. A meta-analysis has revealed that symbolic magnitude processing is a stronger predictor of math performance than non-symbolic, but the evidence base is restricted almost entirely to countries in the Minority World. It is unclear how the strength of the associations between symbolic and non-symbolic magnitude processing and math performance varies across contexts. An examination of cross-national similarities and differences in foundational numeracy skills is sorely needed. In the present study, we examine the predictive nature of symbolic and non-symbolic magnitude processing in school-aged children from Ghana (n = 350) and Côte d'Ivoire (CIV; n = 342), two West African countries in the Majority World. Contrary to prior studies from countries in the Minority World, we found that non-symbolic magnitude processing was a significant and unique predictor of math performance in 5- to 13-year-olds from Ghana. The strong association remains significant when controlling for symbolic magnitude processing, literacy, executive functioning, and socioemotional skills. A second preregistered study with participants from Côte d'Ivoire revealed the same pattern of results. These associations diverged from those that have been found in the Minority World and underscore the importance of taking a global perspective for understanding the cognitive precursors for math development. The data also highlight the potential use of the Numeracy Screener to measure children's understanding of numerical magnitude in classrooms around the world.

This paper presents systematic survey of empirical studies that implement neurocognitive tools to study mathematical processing, learning and problem solving. The survey comprised three stages: identification, screening, and analysis. The search was restricted to English-language papers published in research journals. Of a total of 35,692 records that were identified initially, 598 papers were found eligible for precise data analysis through screening procedure. The bibliometric analysis focused on publication years, journals and authors as well as on collaboration between the researchers. In the content analysis, along with the analysis of neurocognitive tools used in the studies, we screened the papers for the groups of research participants; mathematical topics, concepts and skills examined in the studies. We found that there has been tremendous growth in the past decade in the use of neurocognitive tools to research mathematics learning. The most commonly used tools are the fMRI, EEG, and eye tracking, while use of tools such as GSR and fNIRS remains highly uncommon. There is a strong focus on studying arithmetic, and a recent trend toward examining problem-solving skills, but higher mathematics learning and equation solving remain under-researched. Finally, we found that despite the immense growth in neuroscience research relevant to mathematics education, few studies of this type are published in mathematics education journals.

High-quality mathematics education not only improves life outcomes for individuals but also drives innovation and progress across society. But what exactly constitutes high-quality mathematics education? In this article, we contribute to this discussion by focusing on arithmetic fluency. The debate over how best to teach arithmetic has been long and fierce. Should we emphasize memorization techniques such as flashcards and timed drills or promote "thinking strategies" via play and authentic problem solving? Too often, recommendations for a "balanced" approach lack the depth and specificity needed to effectively guide educators or inform public understanding. Here, we draw on developmental cognitive science, particularly Sfard's process-object duality and Karmiloff-Smith's implicit-explicit knowledge continuum, to present memorization and thinking strategies not as opposing methods but as complementary forces. This framework enables us to offer specific recommendations for fostering arithmetic fluency based on the science of learning. We define arithmetic fluency, provide evidence on its importance, describe the cognitive structures and processes supporting it, and share evidence-based guidance for promoting it. Our recommendations include progress monitoring for early numeracy, providing explicit instruction to teach important strategies and concepts, implementing well-structured retrieval practice, introducing time-limited practice only after students demonstrate accuracy, and allocating sufficient time for discussion and cognitive reflection. By blending theory, evidence, and practical advice, we equip educators and policymakers with the knowledge needed to ensure all children have access to the opportunities needed to achieve arithmetic fluency.

The present chapter explores the processes of acquisition and development of arithmetic problem-solving strategies in school-age children, focusing on how the interplay of individual and contextual factors fosters proficiency and flexibility. Individual factors include domain-specific competencies such as number knowledge and arithmetic fluency, alongside domain-general processes like working memory and metacognition, which underpin strategy selection, planning, and execution. These factors are analyzed from a developmental perspective to illustrate their impact at different age periods. The contextual influences on arithmetic development are framed through Bronfenbrenner's Ecological Systems Theory, emphasizing interactions within family, school, and broader societal environments. Within the microsystem level, we highlight the roles of parents, siblings, teachers, and peers, with special attention to how socioeconomic factors shape access to resources and educational opportunities. At the macrosystem level, this chapter examines curricular approaches, particularly contrasting traditional efficiency-based instructions with flexible strategy-based methods. Emerging practices, such as Number Talks, are discussed as instructional practices aimed to promote conceptual understanding and diverse problem-solving strategies. Combining insights from cognitive psychology, neuroscience, and educational practices, the chapter underscores the complexity of arithmetic strategy development and the need for interdisciplinary research. By identifying key influences and providing examples of effective instructional approaches, this chapter aims to systematize the available evidence regarding how arithmetic skills emerge and develop, bridging gaps between theory and practice to inform educational policies.

Mathematical and spatial abilities are positively related at both the behavioral and neural levels. Much of the evidence illuminating this relationship comes from classic laboratory-based experimental methods focused on cognitive performance despite most individuals also experiencing math and space in other contexts, such as in conversations or lectures. To broaden our understanding of math-space integration in these more commonplace situations, we used an auditory memory-encoding task with stimuli whose content evoked a range of educational and everyday settings related to math or spatial thinking. We used a multivariate approach to directly assess the extent of neural similarity between activity patterns elicited by these math and spatial stimuli. Results from whole-brain searchlight analysis revealed a highly specific positive relation between math and spatial activity patterns in bilateral anterior hippocampi. Examining individual variation in math-space similarity, we found that greater math-space similarity in bilateral anterior hippocampi was associated with poorer math skills and higher anxiety about math. Integration of neural responses to mathematical and spatial content may not always portend positive outcomes. We suggest that episodic simulation of quotidian contexts may link everyday experiences with math and spatial thinking—and the strength of this link is predictive of math in a manner that diverges from math-space associations derived from more lab-based tasks. On a methodological level, this work points to the value of considering a wider range of experimental paradigms, and of the value of combining multivariate fMRI analysis with behavioral data to better contextualize interpretations of brain data.

Recently, cross-domain research has shown that some early cognitive precursors of language, reading, and mathematics overlap and predict one another. This study investigated how early cognitive predictors across domains could predict future academic skills across domains using data from 563 students in kindergarten to second grade (ages 5 to 8; 288 males; largely monolingual English). The roles of verbal, symbolic, and magnitude comparison skills as predictors of later academic grades for various language and math subjects were examined. Results found that Grade 1 marks were predicted by kindergarten verbal and symbolic skills, while Grade 2 marks were predicted by verbal skills and Grade 1 as well as indirectly by symbolic skills via Grade 1. Results are discussed in light of the overlapping relationships between language, reading, and mathematics.

Measuring progress in students is an important consideration when making decisions in education, clinical practice, and research. However, change due to learning over time at the group level cannot be applied to interpret individual change. Therefore, the current study compared four methods for measuring individual change: reliable change index controlling for practice effects (RCI), standardized individual difference (SID), estimated standardized regression-based (SRB) change, and a normalization approach. Participants included 157 children (4 years initially and 5 years at follow-up) who completed measures of language, reading, and mathematics and were tested 1 year apart. We measured individual differences in children as they developed academic-relevant competencies. The RCI and SID indices yielded the same results. While group-based statistics did not find a change overall, the RCI/SID and SRB methods identified 7.64% and 8.28% of students as having changed, respectively. Further, in a subgroup of 54 low scorers, the RCI/SID and SRB methods indicated that 14.81% and 16.67% of students changed, respectively, whereas the normalization method identified a higher rate at 24.07%. The RCI, SID, and SRB methods showed similar results, whereas the normalization method differed from the others. Finally, a practical tool (Excel-based Growth Calculator) is provided to assist practitioners in evaluating individual change. Overall, these methods provide starting points for measuring change in individuals.

A robust association exists between math anxiety and math achievement, with higher levels of anxiety correlating with lower achievement. Understanding this relationship is crucial due to the importance of math proficiency at individual and societal levels. In this review, we explore two prominent theories: Reduced Competency Theory, which suggests that initial low math achievement leads to math anxiety, and Processing Efficiency Theory, which suggests that math anxiety impairs performance by diverting cognitive resources. While these theories are supported by empirical evidence, they do not fully explain the mediators linking math anxiety and achievement. We propose ‘math avoidance’ as a critical mediator, suggesting that avoidance behaviors, formed through conditioning, create a feedback loop that exacerbates math anxiety and reduces proficiency.

The cardinal meanings of the first few number words are often assessed with the Give-N Task, and children’s knowledge of number words can be represented by “knower-level”, with N-knower representing children who have acquired cardinal knowledge of number words up to N. In the current study, we sought converging evidence for knower-levels by examining the correspondence of knower-level classifications between the Give-N Task and the Point-to-X Task. In Study 1, we tested 69 preschool-aged children and found that children at the higher knower-levels often did not receive the same classification across tasks. Further, children who were classified as having acquired the cardinal principle on the Give-N Task were more likely than subset-knowers to count on the Point-to-X Task, but they did not count very frequently. In Study 2, we conducted secondary data analysis on an existing study that included a Point-to-X Task with different stimuli and more trial types. We again found that children’s knower-level assessed with Give-N was not always associated with above chance performance for that number on the Point-To-X Task. These data suggest that knower-levels may adequately capture cardinal number knowledge for very small numbers but not for higher numbers. We discuss the practical implications of these findings and highlight a need for future studies to adopt a multi-method approach to establish a robust pattern of children’s number word acquisition.

ABSTRACT How do young children develop numerical and mathematical skills? In this paper, I review what we have learnt over the past few decades about the foundational skills that underpin children’s numerical and mathematical development. I discuss the importance of learning the meaning of numerical symbols and review studies that show that early number symbol knowledge predicts later math skills and that training numerical symbol knowledge can enhance children’s early math skills. I close by discussing future directions. RESUMEN Introducción. Mucho se ha descrito sobre el aprendizaje en general, y más aún en campos tan complejos como las matemáticas. Sin embargo, la pregunta permanece: ¿Cómo desarrollan los niños pequeños las habilidades numéricas y matemáticas? Objetivo. En este documento se revisa lo que hemos aprendido en las últimas décadas sobre las habilidades fundamentales que sustentan el desarrollo numérico y matemático de los niños. Temas de reflexión. Algunos de los temas revisados son: Comprensión de las asociaciones entre palabras numéricas, dígitos y cantidades, ¿Cómo se construye la enseñanza formal de las matemáticas en el conocimiento temprano de los números en los niños?, Las capacidades de comparación simbólica predicen el rendimiento en matemáticas y, Futuras líneas de investigación. Conclusiones. Se discute la importancia de aprender el significado de los símbolos numéricos y se revisan estudios que muestran que el conocimiento temprano de los símbolos numéricos predice habilidades matemáticas posteriores y que entrenar el conocimiento de los símbolos numéricos puede mejorar las habilidades matemáticas tempranas de los niños. Por último, se concluye discutiendo las direcciones futuras. RESUMO Introdução. Muito já foi descrito sobre a aprendizagem em geral, e ainda mais em áreas tão complexas como a matemática. No entanto, a questão permanece: como as crianças pequenas desenvolvem habilidades numéricas e matemáticas? Objetivo. Este documento analisa o que aprendemos nas últimas décadas sobre as habilidades fundamentais que sustentam o desenvolvimento numérico e matemático das crianças. Tópicos para reflexão. Alguns dos tópicos revisados incluem: Compreensão das associações entre palavras numéricas, dígitos e quantidades; Como o ensino formal de matemática se baseia no conhecimento numérico inicial das crianças?; As habilidades de comparação simbólica predizem o desempenho em matemática?; e Linhas futuras de pesquisa. Conclusões. A importância de aprender o significado dos símbolos numéricos é discutida, e são revisados estudos que demonstram que o conhecimento precoce de símbolos numéricos prediz habilidades matemáticas futuras e que o treinamento no conhecimento de símbolos numéricos pode aprimorar as habilidades matemáticas iniciais das crianças. Por fim, conclui-se com uma discussão sobre as direções futuras.

Mastery motivation predicts achievement, but intricacies amongst pre-schoolers are unclear. In keeping with the Specificity Principle, school-age, and adolescent research demonstrates the importance of considering the setting conditions in which mastery motivation is observed. Here, Singaporean 4-year-olds’ ( N = 63) mastery-motivation-related behaviour (MMRB) (e.g. signs of persistence, focus, and pleasure) in mathematical and non-mathematical activities were observed . Relations between numeracy and MMRB during a mathematical game (outcome relevant setting) were determined, controlling for MMRB in other activities (outcome irrelevant settings). Association between MMRB during the mathematical game and receptive language (outcome irrelevant setting) was also examined. Consistent with the Specificity Principle, MMRB during the mathematical game was (i) associated with numeracy, after controlling for MMRB in other activities and (ii) did not predict language. Enhancing preschoolers’ experiences, especially when implemented in contexts related to areas targeted for improvement, may benefit outcomes. These skills acquired in early life can become important predictors of future ability.

The thalamus is increasingly considered a key component of the brain's reading network, although its precise role is less well understood. Here, we apply a network neuroscience approach to understanding how the thalamus is embedded into the reading network by examining how its wiring pattern is associated with different component aspects of reading skills in children. Using a connectome framework, we show that a local efficiency metric of the left thalamus is negatively correlated with rapid automatized naming, while its transmission cost measure is positively correlated with phonemic decoding. Connections between temporal areas of cortex and two thalamic nuclei, the pulvinar and mediodorsal thalamic nucleus, were also negatively correlated with phonemic decoding. Our result thus provides clearer specificity about how the thalamus supports reading by coordinating communications across reading network areas.

Numerous studies have shown that number word learning is a protracted process. One challenge facing children learning the meaning of number word such as “one”, “two”, or “three” is that number words refer to a property of a set and not to individual objects. In this study, we focused on a sample of children who have not learned the meaning of small number words such as “two” and “three” and tested whether children could learn number words from examples of sets that help them focus on set size. Specifically, the experimental training condition included examples that highlight a common relational structure between sets through varying object properties in the sets (e.g., three yellow stars and three red hearts are both “three”), whereas the control condition did not vary object properties(e.g., two sets of three yellow stars with different spatial arrangement). We trained two- and three-knowers (N = 65) on the next number (i.e., three or four) and assessed their learning with a Two-Alternative-Forced-Choice task and Give-a-Number task. Overall, we found weak effects of training. We discuss our findings in the broader literature on number word learning and explore the possibility of analogical reasoning as a mechanism of number word learning.