Numerical Cognition Research Articles

Numerical cognition in children with cerebral palsy

Children with Cerebral Palsy (CP) often perform poorly in mathematics. It is not yet clear to what extent mathematics difficulties in this clinical condition are similar to those observed in developmental dyscalculia. To better elucidate this issue, we conducted an exploratory cross-sectional study with a sample of children and adolescents with congenital brain injuries and educational history of problems in Mathematics. Fifty students aged 7–15 years, of both genders (28 males) participated in the study, 31 with typical development (TD) and 19 of whom diagnosed with spastic CP. Nine had hemiplegia and ten diplegia. Assessment procedures included a neuropsychological battery covering numerical cognition (ZAREKI-R) and working memory (AWMA) skills, and a computerized task for comparing non-symbolic magnitudes as a measure of number sense. Despite average intelligence coefficient, participants with CP underperformed the TD in five of the 12 ZAREKI-R subtests, as well as in the number sense and working memory tasks. scores were lower among hemiplegic children compared to diplegic, numerical cognition was impaired in all CP group, unveiling a dyscalculia secondary to neurodevelopmental impairments. Therefore, we can consider that mathematical learning difficulties in CP as being heterogeneous and associated with the immaturity of neuropsychological functions, with consequences for the development of numerical cognition.

Numerical estimation strategies are correlated with math ability in school-aged children

Various laboratory-based numerical cognition measures are often found to be linked with standardized math scores, yet the exact nature of these links over development remains in question. In this study, we investigated how 0–100 number line estimation performance and dot numerosity discrimination explained variance in 6- to 8-year-olds’ standardized math ability scores, and we assessed links between tasks. Our primary goal was to assess the link between estimation performance and math ability from a theoretical perspective that models number line estimation as an instance of proportion estimation. As age increased, children’s proportion-based strategies for number line estimation became more sophisticated and estimation bias decreased. Children who spontaneously inferred and used a central reference point to guide their proportion estimates scored higher on the math ability measure (TEMA-3), demonstrating a relation between more advanced proportion estimation strategies and higher math abilities. Individual differences in estimation bias also made a lesser contribution to explaining variance in children’s math ability. Dot discrimination accuracy uniquely contributed to children’s math ability. Number line estimation performance and dot discrimination accuracy were not associated, consistent with prior research suggesting that number line estimation does not directly map onto mental numerical magnitude representations. Educational implications and future directions are discussed.

The relation between parietal GABA concentration and numerical skills

Several scientific, engineering, and medical advancements are based on breakthroughs made by people who excel in mathematics. Our current understanding of the underlying brain networks stems primarily from anatomical and functional investigations, but our knowledge of how neurotransmitters subserve numerical skills, the building block of mathematics, is scarce. Using 1H magnetic resonance spectroscopy (N = 54, 3T, semi-LASER sequence, TE = 32 ms, TR = 3.5 s), the study examined the relation between numerical skills and the brain’s major inhibitory (GABA) and excitatory (glutamate) neurotransmitters. A negative association was found between the performance in a number sequences task and the resting concentration of GABA within the left intraparietal sulcus (IPS), a key region supporting numeracy. The relation between GABA in the IPS and number sequences was specific to (1) parietal but not frontal regions and to (2) GABA but not glutamate. It was additionally found that the resting functional connectivity of the left IPS and the left superior frontal gyrus was positively associated with number sequences performance. However, resting GABA concentration within the IPS explained number sequences performance above and beyond the resting frontoparietal connectivity measure. Our findings further motivate the study of inhibition mechanisms in the human brain and significantly contribute to our current understanding of numerical cognition's biological bases.

Three number processing systems: Different features and parallel activation

<p id="p00010">Clarifying the relationship among mechanisms underlying nonsymbolic numerical cognition is one of the most critical questions in the numerical cognition field. From a novel perspective, the hypothesis that three-number processing systems exists provides a plausible explanation for this relationship. According to this hypothesis, distinct mechanisms are involved in the number processing of nonsymbolic stimuli within different number ranges. Small numbers (1~4) can be appraised rapidly and errorlessly based on the activation of subitizing system. Moderate numbers are proposed to be processed spontaneously with a constant error rate of about 20% due to the activity of numerosity system. Typically, Weber’s law of number perception is demonstrated in this number range. For large numbers, the stimulus number relationship is suggested to be inferred indirectly via density analysis, and number processing, which is fast and has an error rate proportional to the square root of the stimulus number, is mediated by the activity of density system. <p id="p00010">A series of studies have revealed different behavioral features, as well as distinct Event-related potentials (ERP) features, among number tasks based on these three systems. For subitizing and density systems, better processing efficiency is demonstrated by a shorter reaction time and a lower error rate, whereas a higher dependence on attention resources is also a characteristic of these two mechanisms. Neither subitizing nor density mechanisms are correlated with math ability for children in school. For the numerosity mechanism, on the contrary, lower efficiency and independence of attention are shown, and the accuracy of numerosity comparisons is suggested to be significantly correlated with math scores for school children. ERP features are also suggested to be distinguishable for these three mechanisms. An early component related to attention is typically found under a subitizing mechanism rather than under numerosity or density mechanisms. P2p showed a distance effect, namely, higher P2p amplitude correlates with more errors in comparison tasks, as the ratio of the numbers to be compared approaches “1”, under both numerosity and density mechanisms. In general, the amplitude of P2p (about 200 ms after the onset of stimuli) is larger for the density mechanism compared with that for the numerosity mechanism, whereas the P2p component is more sensitive to the change in number within the numerosity range than within the density range. <p id="p00010">There are two major challenges for the three-number processing hypotheses. First, as proposed by some researchers, even when the numerosity and density mechanisms are distinguished by different modes of Weber fractions, more direct evidence is still needed to demonstrate that the processing bases of these two mechanisms are numerosity and density, respectively. Second, the narrow range described in the original hypothesis may induce discontinuity between the three-number processing hypotheses and the classical hypothesis, for example, the approximate number system hypothesis or ANS hypothesis. To solve these problems, it is proposed that the cause for switching from numerosity to density should be reanalyzed and that the hypothesis that the numerosity mechanism is activated in a wider number range should be considered. <p id="p00010">Some experimental evidence indicates parallel activation between subitizing and numerosity systems as well as between density and numerosity systems. The parallel activation hypothesis is also noteworthy. Multiple number processing systems could be activated simultaneously, and cognition relies on the system whose processing result is more precise to achieve the best processing results.

Bidirectional Mapping Between the Symbolic Number System and the Approximate Number System.

Previous studies have discussed the symmetry of bidirectional mapping between approximate number system (ANS) and symbolic number system (SNS). However, these studies neglected the essential significance of bidirectional mapping in the development of numerical cognition. That is, with age, the connection strength between the ANS and SNS in ANS-SNS mapping could be higher than that in SNS-ANS mapping. Therefore, this study attempted to explore the symmetry of bidirectional mapping by examining whether the connection between the ANS and SNS is the same. Using two types of dot array materials (extensive and intensive) and sequence priming paradigms, this study found a stable negative priming effect in the ANS-SNS priming task, but no priming effect in the SNS-ANS priming task. In addition, although sensory cues (extensive and intensive) could affect performance in the ANS-SNS mapping task, these cues did not affect performance in the ANS-SNS priming task. In general, this study provides valuable insight into the symmetry of bidirectional mapping.

Incongruity in fraction verification elicits N270 and P300 ERP effects

Understanding how the numerical magnitudes of fractions are accessed is a topic of major interest in numerical cognition and mathematics education. Only a few studies have investigated fraction processing using EEG methods. In the present study, 24 adult participants completed a fraction magnitude verification task while EEGs were recorded. Similar to other arithmetic verification tasks, behavioral results show increased response times to validate mismatching magnitudes compared to matching ones. ERP results show an early frontal N270 component to mismatching trials and a late parietal P300 component during matching trials. These ERP results highlight that participants treat matching fractions as targets and suggest that additional cognitive resources are needed to process mismatching targets. These results provide evidence that fractions processing shares a similar neurocognitive process as those observed during the processing of arithmetic operations and open the door to further explore fraction processing using ERP methods.

Supplemental Material for Connections Between Mathematics and Reading Development: Numerical Cognition Mediates Relations Between Foundational Competencies and Later Academic Outcomes

Supplemental Material for Connections Between Mathematics and Reading Development: Numerical Cognition Mediates Relations Between Foundational Competencies and Later Academic Outcomes

The Neural Representation of Ordinal Information: Domain-Specific or Domain-General?

Ordinal processing allows for the representation of the sequential relations between stimuli and is a fundamental aspect of different cognitive domains such as verbal working memory (WM), language and numerical cognition. Several studies suggest common ordinal coding mechanisms across these different domains but direct between-domain comparisons of ordinal coding are rare and have led to contradictory evidence. This fMRI study examined the commonality of ordinal representations across the WM, the number, and the letter domains by using a multivoxel pattern analysis approach and by focusing on triplet stimuli associated with robust ordinal distance effects. Neural patterns in fronto-parietal cortices distinguished ordinal distance in all domains. Critically, between-task predictions of ordinal distance in fronto-parietal cortices were robust between serial order WM, alphabetical order judgment but not when involving the numerical order judgment tasks. Moreover, frontal ROIs further supported between-task prediction of distance for the luminance judgment control task, the serial order WM, and the alphabetical tasks. These results suggest that common neural substrates characterize processing of ordinal information in WM and alphabetical but not numerical domains. This commonality, particularly in frontal cortices, may however reflect attentional control processes involved in judging ordinal distances rather than the intervention of domain-general ordinal codes.

Learning Numerosity Representations with Transformers: Number Generation Tasks and Out-of-Distribution Generalization.

One of the most rapidly advancing areas of deep learning research aims at creating models that learn to disentangle the latent factors of variation from a data distribution. However, modeling joint probability mass functions is usually prohibitive, which motivates the use of conditional models assuming that some information is given as input. In the domain of numerical cognition, deep learning architectures have successfully demonstrated that approximate numerosity representations can emerge in multi-layer networks that build latent representations of a set of images with a varying number of items. However, existing models have focused on tasks requiring to conditionally estimate numerosity information from a given image. Here, we focus on a set of much more challenging tasks, which require to conditionally generate synthetic images containing a given number of items. We show that attention-based architectures operating at the pixel level can learn to produce well-formed images approximately containing a specific number of items, even when the target numerosity was not present in the training distribution.

Domain-general and domain-specific influences on emerging numerical cognition: Contrasting uni-and bidirectional prediction models

Domain-general skills such as executive functions (EFs), and domain-specific skills such as non-symbolic number sense and symbolic understanding are often pitted against each other as predictors of emerging maths. Here we aimed to investigate early childhood relations between these foundational skills with a balanced, longitudinal design. One hundred and seventy 3- and 4-year-old-children were tested at two time points, 5 months apart, on four domain-general executive and five domain-specific numeracy tasks. A latent EF factor was a strong predictor of symbolic maths and of their growth. In addition, stronger symbolic maths at Time 1 was correlated with later stronger EF, but symbolic maths did not predict EF growth. Our findings provide novel insights into dynamic interplay between general and specific cognitive skills contributing to preschool maths.

Towards a standardization of non-symbolic numerical experiments: GeNEsIS, a flexible and user-friendly tool to generate controlled stimuli

Several studies have suggested that vertebrate and invertebrate species may possess a number sense, i.e. an ability to process in a non-symbolic and non-verbal way the numerousness of a set of items. However, this hypothesis has been challenged by the presence of other non-numerical continuous physical variables, which vary along with numerosity (i.e., any change in the number of visual physical elements in a set naturally involves a related change in visual features such as area, density, contour length and convex hull of the stimulus). It is therefore necessary to control and manipulate the continuous physical information when investigating the ability of humans and other animals to perceive numerousness. During decades of research, different methods have been implemented in order to address this issue, which has implications for experiment replicability and inter-species comparisons, since no general standardized procedure is currently being used. Here we present the ‘Generation of Numerical Elements Images Software’ (GeNEsIS) for the creation of non-symbolic numerical arrays in a standardized and user-friendly environment. The main aim of this tool is to provide researchers in the field of numerical cognition a manageable and precise instrument to produce visual numerical arrays controlled for all the continuous variables. Additionally, we implemented the ability to actively guide stimuli presentation during habituation/dishabituation and dual-choice comparison tasks used in human and comparative research.

Enculturation and the historical origins of number words and concepts

In the literature on enculturation—the thesis according to which higher cognitive capacities result from transformations in the brain driven by culture—numerical cognition is often cited as an example. A consequence of the enculturation account for numerical cognition is that individuals cannot acquire numerical competence if a symbolic system for numbers is not available in their cultural environment. This poses a problem for the explanation of the historical origins of numerical concepts and symbols. When a numeral system had not been created yet, people did not have the opportunity to acquire number concepts. But, if people did not have number concepts, how could they ever create a symbolic system for numbers? Here I propose an account of the invention of symbolic systems for numbers by anumeric people in the remote past that is compatible with the enculturation thesis. I suggest that symbols for numbers and number concepts may have emerged at the same time through the re-semantification of words whose meanings were originally non-numerical.

Integrating numerical cognition research and mathematics education to strengthen the teaching and learning of early number.

Research into numerical cognition has contributed to a large body of knowledge on how children learn and perform mathematics. This knowledge has the potential to inform mathematics education. Unfortunately, numerical cognition research and mathematics education remain disconnected from one another, lacking the proper infrastructure to allow for productive and meaningful exchange between disciplines. The present study was designed to address this gap. This study reports on the design, implementation, and effects of a 16-week (25-hour) mathematics Professional Development (PD) model for Kindergarten to Grade 3 educators and their students. A central goal of the PD was to better integrate numerical cognition research and mathematics education. A total of 45 K-3 educators and 180 of their students participated. To test the reproducibility and replicability of the model, the study was carried out across two different sites, over a two-year period, and involved a combination of two different study designs: a quasi-experimental pre-post-research design and a within-group crossover intervention design. The results of the first implementation (Year 1), indicated that compared to a control group, both teachers and students benefited from the intervention. Teachers demonstrated gains on both a self-report measure and a test of numerical cognition knowledge, while students demonstrated gains in number line estimation, arithmetic, and numeration. In Year 2, teachers in the intervention group demonstrated greater improvements than the control group on the self-report measure, but not the test of numerical cognition knowledge. At the student level, there was some evidence of gains in numeration. The current PD model is a promising approach to better integrate research and practice. However, more research is needed to determine in which school contexts the model is most effective.

Emerging neurodevelopmental perspectives on mathematical learning

Strong foundational skills in mathematical problem solving, acquired in early childhood, are critical not only for success in the science, technology, engineering, and mathematical (STEM) fields but also for quantitative reasoning in everyday life. The acquisition of mathematical skills relies on protracted interactive specialization of functional brain networks across development. Using a systems neuroscience approach, this review synthesizes emerging perspectives on neurodevelopmental pathways of mathematical learning, highlighting the functional brain architecture that supports these processes and sources of heterogeneity in mathematical skill acquisition. We identify the core neural building blocks of numerical cognition, anchored in the posterior parietal and ventral temporal-occipital cortices, and describe how memory and cognitive control systems, anchored in the medial temporal lobe and prefrontal cortex, help scaffold mathematical skill development. We highlight how interactive specialization of functional circuits influences mathematical learning across different stages of development. Functional and structural brain integrity and plasticity associated with math learning can be examined using an individual differences approach to better understand sources of heterogeneity in learning, including cognitive, affective, motivational, and sociocultural factors. Our review emphasizes the dynamic role of neurodevelopmental processes in mathematical learning and cognitive development more generally.

JEEDMA: a utilização de um jogo eletrônico educacional gamificado para o desenvolvimento da cognição numérica

Os jogos eletrônicos educacionais podem ser utilizados nas mais diversas formas de ensino, tanto em um contexto tradicional quanto com alunos com necessidades especiais, visto seu potencial para auxiliar no desenvolvimento da Cognição Numérica. Este trabalho tem como objetivo geral apresentar um jogo eletrônico educacional gamificado, denominado de Jogo Eletrônico Educacional em Matemática – JEEDMA, para auxiliar o desenvolvimento da Cognição Numérica. O encaminhamento metodológico utilizado foi o qualitativo para analisar a aplicação do jogo em crianças dos anos iniciais, a fim de analisar o desenvolvimento da Cognição Numérica. Diante das análises, se torna possível identificar que o JEEDMA contribuiu para o desenvolvimento da Cognição Numérica dos indivíduos dos anos iniciais, uma vez que o tempo gasto para conclusão das tarefas e a quantidade de erros para completá-las diminuíram ao longo do tempo de sua aplicação.

A probabilistic approach for quantifying children’s subitizing span

The development of enumeration skills over childhood is thought to reflect improvements in both subitizing (for small sets) and serial counting (for larger sets). However, investigations into the contribution of subitizing to advancing mathematics ability are limited by challenges in measuring subitizing capacity across developmental populations. Subitizing capacity in adults is traditionally assessed by calculating the bilinear inflection point for reaction times or accuracy across set sizes, but in children greater variability and dramatic improvements in counting ability introduce problems with this approach. This study demonstrates this limitation in a sample of elementary school children and proposes a novel probabilistic approach to measuring subitizing capacity. This metric captures well-established trends in the development of children’s subitizing. Furthermore, the proposed metric predicts unique variance in symbolic arithmetic ability, corroborating previous research that suggests a foundational role for subitizing in the development of numerical cognition. Findings demonstrate the advantages of a probabilistic approach to determining subitizing capacity in young children and suggest that it may be practically and theoretically well-suited for investigating subitizing and its role in mathematics development.

Turn the beat around: Commentary on \u201cSlow and fast beat sequences are represented differently through space" (De Tommaso & Prpic, 2020, in Attention, Perception, & Psychophysics)

There has been increasing interest in the spatial mapping of various perceptual and cognitive magnitudes, such as expanding the spatial-numerical association of response codes (SNARC) effect into domains outside of numerical cognition. Recently, De Tommaso and Prpic (Attention, Perception, & Psychophysics, 82, 2765–2773, 2020) reported in this journal that only fast tempos over 104 beats per minute have spatial associations, with more right-sided associations and faster responses for faster tempos. After discussing the role of perceived loudness and possible response strategies, we propose and recommend methodological improvements for further research.

A Database for Learning Numbers by Visual Finger Recognition in Developmental Neuro-Robotics.

Numerical cognition is a fundamental component of human intelligence that has not been fully understood yet. Indeed, it is a subject of research in many disciplines, e.g., neuroscience, education, cognitive and developmental psychology, philosophy of mathematics, linguistics. In Artificial Intelligence, aspects of numerical cognition have been modelled through neural networks to replicate and analytically study children behaviours. However, artificial models need to incorporate realistic sensory-motor information from the body to fully mimic the children's learning behaviours, e.g., the use of fingers to learn and manipulate numbers. To this end, this article presents a database of images, focused on number representation with fingers using both human and robot hands, which can constitute the base for building new realistic models of numerical cognition in humanoid robots, enabling a grounded learning approach in developmental autonomous agents. The article provides a benchmark analysis of the datasets in the database that are used to train, validate, and test five state-of-the art deep neural networks, which are compared for classification accuracy together with an analysis of the computational requirements of each network. The discussion highlights the trade-off between speed and precision in the detection, which is required for realistic applications in robotics.

Non-numerical strategies used by bees to solve numerical cognition tasks.

We examined how bees solve a visual discrimination task with stimuli commonly used in numerical cognition studies. Bees performed well on the task, but additional tests showed that they had learned continuous (non-numerical) cues. A network model using biologically plausible visual feature filtering and a simple associative rule was capable of learning the task using only continuous cues inherent in the training stimuli, with no numerical processing. This model was also able to reproduce behaviours that have been considered in other studies indicative of numerical cognition. Our results support the idea that a sense of magnitude may be more primitive and basic than a sense of number. Our findings highlight how problematic inadvertent continuous cues can be for studies of numerical cognition. This remains a deep issue within the field that requires increased vigilance and cleverness from the experimenter. We suggest ways of better assessing numerical cognition in non-speaking animals, including assessing the use of all alternative cues in one test, using cross-modal cues, analysing behavioural responses to detect underlying strategies, and finding the neural substrate.

Aphasia and Math: Deficits with Basic Number Comprehension and in Numerical Activities of Daily Living.

In the present study, we explored numerical problems in individuals with aphasia. We investigate whether numerical deficits, usually accompanying aphasia, can be observed on number comprehension tasks that do not necessarily require an oral response. Individuals with aphasia were classified into anterior, posterior, and global subgroups according to the lesion type. To investigate numerical cognition, we used a relatively recent tool, the Numerical Activities of Daily Living (NADL). The results showed that individuals with aphasia have problems with tasks of basic number comprehension as well as in most NADL. In the formal part of the NADL, anterior aphasic patients made comparatively more errors than the posterior aphasic patients. Global aphasic patients presented an invariably poor performance on almost all tasks. The results provide insight into how numerical deficits may impair an individual with aphasia in activities of daily living. This study is a preliminary attempt to start the validation process of the NADL for the Greek population.