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Research on mathematical cognition, learning, and instruction (MCLI) often takes cognition as its point of departure and considers instruction at a later point in the research cycle. In this article, we call for psychologists who study MCLI to reflect on the "status quo" of their research practices and to consider making instruction an earlier and more central aspect of their work. We encourage scholars of MCLI (a) to consider the needs of educators and schools when selecting research questions and developing interventions; (b) to compose research teams that are diverse in the personal, disciplinary, and occupational backgrounds of team members; (c) to make efforts to broaden participation in research and to conduct research in authentic settings; and (d) to communicate research in ways that are accessible to practitioners and to the general public. We argue that a more central consideration of instruction will lead to shifts that make research on MCLI more theoretically valuable, more actionable for educators, and more relevant to pressing societal challenges.

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The intersection of mathematical cognition, metacognition, and advanced technologies presents a frontier with profound implications for human learning and artificial intelligence. This paper traces the historical roots of these concepts from the Pythagoreans and Aristotle to modern cognitive science and explores their relevance to contemporary technological applications. We examine how the Pythagoreans' view of mathematics as fundamental to understanding the universe and Aristotle's contributions to logic and categorization have shaped our current understanding of mathematical cognition and metacognition. The paper investigates the role of Boolean logic in computational processes and its relationship to human logical reasoning, as well as the significance of Bayesian inference and fuzzy logic in modelling uncertainty in human cognition and decision-making. We also explore the emerging field of Chemical Artificial Intelligence and its potential applications. We argue for unifying mathematical metacognition with advanced technologies, including artificial intelligence and robotics, while identifying the multifaceted benefits and challenges of such unification. The present paper examines essential research directions for integrating cognitive sciences and advanced technologies, discussing applications in education, healthcare, and business management. We provide suggestions for developing cognitive robots using specific cognitive tasks and explore the ethical implications of these advancements. Our analysis underscores the need for interdisciplinary collaboration to realize the full potential of this integration while mitigating potential risks.

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There are three main types of number used in modern, industrialized societies. Cardinals count sets (e.g., people, objects) and quantify elements of conventional scales (e.g., money, distance), ordinals index positions in ordered sequences (e.g., years, pages), and nominals serve as unique identifiers (e.g., telephone numbers, player numbers). Many studies that have cited number frequencies in support of claims about numerical cognition and mathematical cognition hinge on the assumption that most numbers analyzed are cardinal. This paper is the first to investigate the relative frequencies of different number types, presenting a corpus analysis of morphologically unmarked numbers (not, e.g., "eighth" or "21st") in which we manually annotated 3,600 concordances in the Corpus of Contemporary American English. Overall, cardinals are dominant-both pure cardinals (sets) and measurements (scales)-except in the range 1,000-10,000, which is dominated by ordinal years, like 1996 and 2004. Ordinals occur less often overall, and nominals even less so. Only for cardinals do round numbers, associated with approximation, dominate overall and increase with magnitude. In comparison with other registers, academic writing contains a lower proportion of measurements as well as a higher proportion of ordinals and, to some extent, nominals. In writing, pure cardinals and measurements are usually represented as number words, but measurements-especially larger, unround ones-are more likely to be numerals. Ordinals and nominals are mostly represented as numerals. Altogether, this paper reveals how numbers are used in American English, establishing an initial baseline for any analyses of number frequencies and shedding new light on the cognitive and psychological study of number.

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Mathematics teaching strategies have a positive impact on learning. However, there is a lack of studies on non-traditional approaches to early mathematics education in the specialized scientific literature. In this theoretical framework, a study to connect teaching methodology with the various cognitive processes implicated in learning has been designed. A total of 114 primary school students aged 74 and 84 months who were taught mathematics either with the method called Open Algorithm Based on Numbers or with the more traditional Closed Algorithm Based on Ciphers, participated in the study. After conducting a thorough examination of cognitive processes and early math performance using well-established assessment instruments, a comparative analysis was undertaken to explore the relationship between cognitive predictors of mathematical performance, while considering the mathematics teaching strategies used. Students were distributed according to their level of mathematical competence and teaching methodology and the type of schools (Charter or Public). The results from the multivariate statistical test showed that the teaching strategy was inconclusive for most of the cognitive factors studied. Significant differences according to mathematical performance were found for fluid intelligence, verbal short-term memory, and visuospatial working memory. Finally, no significant differences were found in the cognitive variables studied when considering the interaction between the teaching approach, school characteristics, and mathematical achievement as a reference.

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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.

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There is an ongoing debate in the scientific community regarding the nature and role of the mental representations involved in solving arithmetic word problems. In this study, we took a closer look at the interplay between mental representations, drawing production, and strategy choice. We used dual-strategy isomorphic word problems sharing the same mathematical structure, but differing in the entities they mentioned in their problem statement. Due to the non-mathematical knowledge attached to these entities, some problems were believed to lead to a specific (cardinal) encoding compatible with one solving strategy, whereas other problems were thought to foster a different (ordinal) encoding compatible with the other solving strategy. We asked 59 children and 52 adults to solve 12 of those arithmetic word problems and to make a diagram of each problem. We hypothesized that the diagrams of both groups would display prototypical features indicating either a cardinal representation or an ordinal representation, depending on the entities mentioned in the problem statement. Joint analysis of the drawing task and the problem-solving task showed that the cardinal and ordinal features of the diagrams are linked with the hypothesized semantic properties of the problems and, crucially, with the choice of one solving strategy over another. We showed that regardless of their experience, participants' strategy use depends on their problem representation, which is influenced by the non-mathematical information in the problem statement, as revealed in their diagrams. We discuss the relevance of drawing tasks for investigating mental representations and fostering mathematical development in school.

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Many famous scientists have reported anecdotes where a new understanding occurred to them suddenly, in an unexpected flash. Do people generally experience such "Eureka" moments when learning science concepts? And if so, do these episodes truly vehicle sudden insights, or is this impression illusory? To address these questions, we developed a paradigm where participants were taught the mathematical concept of geodesic, which generalizes the common notion of straight line to straight trajectories drawn on curved surfaces. After studying lessons introducing this concept on the sphere, participants (N = 56) were tested on their understanding of geodesics on the sphere and on other surfaces. Our findings indicate that Eureka experiences are common when learning mathematics, with reports by 34 (61%) participants. Moreover, Eureka experiences proved an accurate description of participants' learning, in two respects. First, Eureka experiences were associated with learning and generalization: the participants who reported experiencing Eurekas performed better at identifying counterintuitive geodesics on new surfaces. Second, and in line with the firstperson experience of a sudden insight, our findings suggest that the learning mechanisms responsible for Eureka experiences are inaccessible to reflective introspection. Specifically, reports of Eureka experiences and of participants' confidence in their own understanding were associated with different profiles of performance, indicating that the mechanisms bringing about Eureka experiences and those informing reflective confidence were at least partially dissociated. Learning mathematical concepts thus appears to involve mechanisms that operate unconsciously, except when a key computational step is reached and a sudden insight breaks into consciousness.

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BACKGROUND: When mathematical knowledge is expressed in general language, it is called verbalized mathematics. Previous studies on verbalized mathematics typically paid attention to mathematical vocabulary or educational practice. However, these studies did not exclude the role of symbolic mathematics ability, and almost no research has focused on verbalized mathematical principles. AIMS: This study is aimed to investigate whether verbalized mathematics ability independently predicts mathematics achievement. The current study hypothesized that verbalized mathematics ability supports mathematics achievement independent of general language, related cognitive abilities and even symbolic mathematical ability. SAMPLE: A sample of 241 undergraduates (136 males, 105 females, mean age = 21.95, SD = 2.38) in Beijing, China. METHODS: A total of 12 tests were used, including a verbalized arithmetic principle test, a mathematics achievement test, and tests on general language (sentence completion test), symbolic mathematical ability (including symbolic arithmetic principles test, simple arithmetic computation and complex arithmetic computation), approximate number sense ability (numerosity comparison test) and several related cognitive covariates (including the non-verbal matrix reasoning, the syllogism reasoning, mental rotation, figure matching and choice reaction time). RESULTS: Results showed that the processing of verbalized arithmetic principles displayed a significant role in mathematics achievement after controlling for general language, related cognitive abilities, approximate number sense ability and symbolic mathematics ability. CONCLUSIONS: The results suggest that verbalized mathematics ability was an independent predictor and provided empirical evidence supporting the verbalized mathematics role on achievement as an independent component in three-component mathematics model.

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In 1974, Roger Sperry, based on his seminal studies on the split-brain condition, concluded that math was almost exclusively sustained by the language dominant left hemisphere. The right hemisphere could perform additions up to sums less than 20, the only exception to a complete left hemisphere dominance. Studies on lateralized focal lesions came to a similar conclusion, except for written complex calculation, where spatial abilities are needed to display digits in the right location according to the specific requirements of calculation procedures. Fifty years later, the contribution of new theoretical and instrumental tools lead to a much more complex picture, whereby, while left hemisphere dominance for math in the right-handed is confirmed for most functions, several math related tasks seem to be carried out in the right hemisphere. The developmental trajectory in the lateralization of math functions has also been clarified. This corpus of knowledge is reviewed here. The right hemisphere does not simply offer its support when calculation requires generic space processing, but its role can be very specific. For example, the right parietal lobe seems to store the operation-specific spatial layout required for complex arithmetical procedures and areas like the right insula are necessary in parsing complex numbers containing zero. Evidence is found for a complex orchestration between the two hemispheres even for simple tasks: each hemisphere has its specific role, concurring to the correct result. As for development, data point to right dominance for basic numerical processes. The picture that emerges at school age is a bilateral pattern with a significantly greater involvement of the right-hemisphere, particularly in non-symbolic tasks. The intraparietal sulcus shows a left hemisphere preponderance in response to symbolic stimuli at this age.

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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.

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Young children with limited knowledge of formal mathematics can intuitively perform basic arithmetic-like operations over nonsymbolic, approximate representations of quantity. However, the algorithmic rules that guide such nonsymbolic operations are not entirely clear. We asked whether nonsymbolic arithmetic operations have a function-like structure, like symbolic arithmetic. Children (n = 74 4- to -8-year-olds in Experiment 1; n = 52 7- to 8-year-olds in Experiment 2) first solved two nonsymbolic arithmetic problems. We then showed children two unequal sets of objects, and asked children which of the two derived solutions should be added to the smaller of the two sets to make them "about the same." We hypothesized that, if nonsymbolic arithmetic follows similar function rules to symbolic arithmetic, then children should be able to use the solutions of nonsymbolic computations as inputs into another nonsymbolic problem. Contrary to this hypothesis, we found that children were unable to reliably do so, suggesting that these solutions may not operate as independent representations that can be used inputs into other nonsymbolic computations. These results suggest that nonsymbolic and symbolic arithmetic computations are algorithmically distinct, which may limit the extent to which children can leverage nonsymbolic arithmetic intuitions to acquire formal mathematics knowledge.

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Mathematics skills are associated with future employment, well-being, and quality of life. However, many adults and children fail to learn the mathematics skills they require. To improve this situation, we need to have a better understanding of the processes of learning and performing mathematics. Over the past two decades, there has been a substantial growth in psychological research focusing on mathematics. However, to make further progress, we need to pay greater attention to the nature of, and multiple elements involved in, mathematical cognition. Mathematics is not a single construct; rather, overall mathematics achievement is comprised of proficiency with specific components of mathematics (e.g., number fact knowledge, algebraic thinking), which in turn recruit basic mathematical processes (e.g., magnitude comparison, pattern recognition). General cognitive skills and different learning experiences influence the development of each component of mathematics as well as the links between them. Here, I propose and provide evidence for a framework that structures how these components of mathematics fit together. This framework allows us to make sense of the proliferation of empirical findings concerning influences on mathematical cognition and can guide the questions we ask, identifying where we are missing both research evidence and models of specific mechanisms.

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Several studies have shown that the number line can be a useful tool to support early numeracy development. Here, we conducted a school-based training study to evaluate the effectiveness of the software "The Number Line" ("La Linea Dei Numeri"; Tressoldi and Peroni, 2013) in improving children's mathematical skills. We randomly allocated 10 classes of first, second and third graders (N=183) to one of three experimental groups: one group played with The Number Line; the second group played with Labyrinth, a computerized game designed to train attention skills; the third group had no intervention (business-as-usual). At the end of the first training phase, children in The Number Line group completed another training phase playing with Labyrinth, whereas the other two groups played with The Number Line. After playing with The Number Line, all groups displayed more accuracy when placing numbers in the number line task. However, we observed no evident improvement in other mathematical skills. These results suggest that specific training effects emerge even in the school context, although transfer to other numerical skills may be harder to achieve.

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Many teaching websites, such as the Khan Academy, propose vivid videos illustrating a mathematical concept. Using functional magnetic resonance imaging, we asked whether watching such a video suffices to rapidly change the brain networks for mathematical knowledge. We capitalized on the finding that, when judging the truth of short spoken statements, distinct semantic regions activate depending on whether the statements bear on mathematical knowledge or on other domains of semantic knowledge. Here, participants answered such questions before and after watching a lively 5-min video, which taught them the rudiments of a new domain. During the video, a distinct math-responsive network, comprising anterior intraparietal and inferior temporal nodes, showed intersubject synchrony when viewing mathematics course rather than control courses in biology or law. However, this experience led to minimal subsequent changes in the activity of those domain-specific areas when answering questions on the same topics a few minutes later. All taught facts, whether mathematical or not, led to domain-general repetition enhancement, particularly prominent in the cuneus, posterior cingulate, and posterior parietal cortices. We conclude that short videos do not suffice to induce a meaningful lasting change in the brain's math-responsive network, but merely engage domain-general regions possibly involved in episodic short-term memory.

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Children with cerebral palsy (CP) are at greater risk of mathematical learning disabilities due to associated motor and cognitive limitations. However, there is currently little evidence on how to support the development of arithmetic skills within such a specific profile. The aim of this single-case study was to assess the effectiveness of a neuropsychological rehabilitation of arithmetic skills in NG, a 9-year-old boy with CP who experienced math learning disability and cumulated motor and short-term memory impairments. This issue was explored combining multiple-baseline and changing-criterion designs. The intervention consisted of training NG to solve complex additions applying calculation procedures with a tailor-made computation tool. Based on NG's strengths, in accordance with evidence-based practice in psychology, the intervention was the result of a co-construction process involving N, his NG's parents and professionals (therapist and researchers). Results were analyzed by combining graph visual inspections with non-parametric statistics for single-case designs (NAP-scores). Analyses showed a specific improvement in NG's ability to solve complex additions, which maintained for up to 3 weeks after intervention. The training effect did not generalize to his ability to perform mental additions, and to process the symbolic magnitude.

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Subtle visual manipulations to the presentation of mathematical notation influence the way that students perceive and solve problems. While there is a consistent impact of perceptual cues on students' problem-solving, other cognitive skills such as inhibitory control may interact with perceptual cues to affect students' arithmetic problem-solving performance. We present an online experiment in which college students completed a version of the Stroop task followed by arithmetic problems in which the spacing between numbers and operators was either congruent (e.g., 2 + 3×4) or incongruent (e.g., 2+3 × 4) to the order of precedence. We found that students were comparably accurate between problem types but might have spent longer responding to problems with congruent than incongruent spacing. There was no main effect of inhibitory control on students' performance on these problems. However, an exploratory analysis on a combined performance measure of accuracy and response time revealed an interaction between problem type and inhibitory control. Students with higher inhibitory control performed better on congruent versus incongruent problems, whereas students with lower inhibitory control performed worse on congruent versus incongruent problems. Together, these results suggest that the relation between inhibitory control and arithmetic performance may not be straightforward. Furthermore, this work advances perceptual learning theory and contributes new findings on the contexts in which perceptual cues, such as spacing, influence arithmetic performance.

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The role of angular gyrus (AG) in arithmetic processing remains a subject of debate. In the present study, we recorded from the AG, supramarginal gyrus (SMG), intraparietal sulcus (IPS), and superior parietal lobule (SPL) across 467 sites in 30 subjects performing addition or multiplication with digits or number words. We measured the power of high-frequency-broadband (HFB) signal, a surrogate marker for regional cortical engagement, and used single-subject anatomical boundaries to define the location of each recording site. Our recordings revealed the lowest proportion of sites with activation or deactivation within the AG compared to other subregions of the inferior parietal cortex during arithmetic processing. The few activated AG sites were mostly located at the border zones between AG and IPS, or AG and SMG. Additionally, we found that AG sites were more deactivated in trials with fast compared to slow response times. The increase or decrease of HFB within specific AG sites was the same when arithmetic trials were presented with number words versus digits and during multiplication as well as addition trials. Based on our findings, we conclude that the prior neuroimaging findings of so-called activations in the AG during arithmetic processing could have been due to group-based analyses that might have blurred the individual anatomical boundaries of AG or the subtractive nature of the neuroimaging methods in which lesser deactivations compared to the control condition have been interpreted as "activations". Our findings offer a new perspective with electrophysiological data about the engagement of AG during arithmetic processing.

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Objective:To study the characteristics of mathematical cognitive function in children with sleep-disordered breathing(SDB) by event-related potential (ERP).Methods:From October 2020 to October 2022, totally 22 cases of SDB children and 22 cases of normal children aged 8-11 were selected.All subjects performed mathematic tasks including calculating and deciding.The EEG changes and behavioral data of children with SDB and normal children were recorded.The latency and amplitude of N1, P2, N2, P3 in leads Fz were measured and compared by Matlab. SPSS 23.0 software was used for statistical analysis, and t test or Mann Whitney U test were used for two independent sample data. Results:(1)Behavior test: the interaction effect between group and type, the group main effect, and the type main effect in accuracy between SDB group and normal group were not significant ( F=0.470, 3.590, 0.003, all P>0.05). The group main effect and interaction effect between group and type in reaction time between SDB group and normal group were not significant ( F=0.465, 1.991, both P>0.05), while the type main effect was significant ( F=18.010, P<0.05). (2)ERP test: the N2, P3 latencies for addition in children with SDB were longer than those in normal group(N2: (371.38±34.23)ms vs (348.12±26.34)ms; P3: (610.72±64.78)ms vs (529.05±30.25)ms)( t=2.526, 5.358, both P<0.05). There was no significant difference between SDB group and normal group in ERP latency and amplitude for subtraction(both P>0.05). The N2, P3 latencies for multiplication in children with SDB were longer than those in normal group(N2: (439.20±24.28)ms vs (351.14±25.26)ms; P3: (531.71±35.42)ms vs (415.23±19.01)ms)( t=11.792, 13.590, both P<0.05). The P3 amplitudes in children with SDB was higher than that in normal group(P3: (3.16±4.78)μV vs (0.38±3.27)μV)( t=2.248, P<0.05). The P3 latency for correct judgment in children with SDB was longer than that in normal group(P3: (420.38±34.79)ms vs (398.54±33.71)ms)( t=2.115, P<0.05). The P3 latency for wrong judgment in children with SDB was longer than that in normal group(P3: (475.25±51.11)ms vs (414.88±27.53)ms)( t=4.878, P<0.05). Conclusion:The latency of N2 and P3 in ERP of SDB children is prolonged, and P3 latency is more sensitive than N2, indicating that SDB children have impairment of mathematical cognitive function.The latency changes of N2 and P3 occurs earlier than the behavioral changes (reaction time and accuracy), which can be used as one of the electrophysiological indicators for early assessment of mathematical cognitive impairment in SDB children.

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How can artificial neural networks capture the advanced cognitive abilities of pioneering scientists? I suggest they must learn to exploit human-invented tools of thought and human-like ways of using them, and must engage in explicit goal-directed problem solving as exemplified in the activities of scientists and mathematicians and taught in advanced educational settings.

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Educational interventions are necessary to develop mathematical competence at early ages and prevent widespread mathematics learning failure in the education system as indicated by the results of European reports. Numerous studies agree that domain-specific predictors related to mathematics are symbolic and non-symbolic magnitude comparison, as well as, number line estimation. The goal of this study was to design 4 digital learning app games to train specific cognitive bases of mathematical learning in order to create resources and promote the use of these technologies in the educational community and to promote effective scientific transfer and increase the research visibility. This study involved 193 preschoolers aged 57-79 months. A quasi-experimental design was carried out with 3 groups created after scores were obtained in a standardised mathematical competence assessment test, i.e., low-performance group (N = 49), high-performance group (N = 21), and control group (N = 123). The results show that training with the 4 digital learning app games focusing on magnitude, subitizing, number facts, and estimation tasks improved the numerical skills of the experimental groups, compared to the control group. The implications of the study were, on the one hand, provided verified technological tools for teaching early mathematical competence. On the other hand, this study supports other studies on the importance of cognitive precursors in mathematics performance.