Refining the quantitative pathway of the Pathways to Mathematics model

Numerical cognition across the lifespan: A selective review of key developmental stages and neural, cognitive, and affective underpinnings

Numerical cognition constitutes a set of hierarchically related skills and abilities that develop-and may subsequently begin to decline-over developmental time. An innate "number sense" has long been argued to provide a foundation for the development of increasingly complex and applied numerical cognition, such as symbolic numerical reference, arithmetic, and financial literacy. However, evidence for a direct link between basic perceptual mechanisms that allow us to determine numerical magnitude (e.g., "how many" objects are in front of us and whether some of these are of a "greater" or "lesser" quantity), and later symbolic applications for counting and mathematics, has recently been challenged. Understanding how one develops an increasingly precise sense of number and which neurocognitive mechanisms support arithmetic development and achievement is crucial for developing successful mathematics curricula, supporting individual financial literacy and decision-making, and designing appropriate intervention and remediation programs for mathematical learning disabilities as well as mathematics anxiety. The purpose of this review is to provide a broad overview of the cognitive, neural, and affective underpinnings of numerical cognition-spanning the earliest hours of infancy to senior adulthood-and highlight gaps in our knowledge that remain to be addressed.

Cognitive characteristics of children with high mathematics achievement before they start formal schooling

This 5-year longitudinal study examined whether high mathematics achievers in primary school had cognitive advantages before entering formal education. High mathematics achievement was defined as performing above Pc 90 in Grades 1 and 3. The predominantly White sample (M age in preschool: 64 months) included 31 high achievers (12 girls) and 114 average achievers (63 girls). We measured children's early numerical abilities, complex mathematical abilities, and general cognitive abilities in preschool (2017). High mathematics achievers had advantages on most tasks in preschool (ds > 0.62). Number order, numeral recognition, and proportional reasoning were unique predictors of belonging to the high-achieving group in primary school. This study shows that the cognitive advantages of high mathematics achievement are already observed in preschool.

Executive functioning profiles and mathematical and reading achievement in Grades 2, 6, and 10

Using a person-centered approach, we aimed to identify different executive functioning profiles to assess heterogeneity across individuals within the same school grade through latent profile analysis. A sample of 150 Grade 2 (7-8 years old), 150 Grade 6 (11-12 years old), and 150 Grade 10 (15-16 years old) children and adolescents were assessed on 11 different executive tasks representative of the three main executive functioning subcomponents (i.e., inhibition, cognitive flexibility, and working memory), fluid intelligence, processing speed, problem-solving, and reading comprehension. Three different executive functioning profiles of different patterns of interactions based on inhibition, cognitive flexibility, and working memory within and between grades were identified. Moreover, these profiles were differentially related to reading comprehension and mathematical achievement. Second, as expected, we did not find these profiles to be associated with sociodemographic variables such as chronological age or sex. Still, fluid intelligence and processing speed were differentially related to the different profiles at each grade. We also found that the executive functioning profiles interacted with each cognitive skill (i.e., fluid intelligence and processing speed) in predicting reading comprehension and math achievement. These findings provide valuable insights for developing preventive and intervention strategies in education.

The importance of domain-specific number abilities and domain-general cognitive abilities for early arithmetic achievement and development

BACKGROUND

Children's numerical and arithmetic skills differ greatly already at an early age. Although research focusing on accounting for these large individual differences clearly demonstrates that mathematical performance draws upon several cognitive abilities, our knowledge concerning key abilities underlying mathematical skill development is still limited.

AIMS

First, to identify key cognitive abilities contributing to children's development of early arithmetic skills. Second, to examine the extent to which early arithmetic performance and early arithmetic development rely on different or similar constellations of domain-specific number abilities and domain-general cognitive abilities.

SAMPLE

In all, 134 Swedish children (Mage  = 6 years and 4 months, SD = 3 months, 74 boys) participated in this study.

METHOD

Verbal and non-verbal logical reasoning, non-symbolic number comparison, counting knowledge, spatial processing, verbal working memory and arithmetic were assessed. Twelve months later, arithmetic skills were reassessed. A latent change score model was computed to determine whether any of the abilities accounted for variations in arithmetic development.

RESULTS

Arithmetic performance was supported by counting knowledge, verbal and non-verbal logical reasoning and spatial processing. Arithmetic skill development was only supported by spatial processing.

CONCLUSIONS

Results show that young children's early arithmetic performance and arithmetic development are supported by different cognitive processes. The findings regarding performance supported Fuchs et al.'s model (Dev Psychol, 46, 2010b, 1731) but the developmental findings did not. The developmental findings align partially to Geary et al.'s (J Educ Psychol, 109, 2017, 680) hypothesis stating that young children's early arithmetic development is more dependent on general cognitive abilities than number abilities.

Walking another pathway: The inclusion of patterning in the pathways to mathematics model

According to the Pathways to Mathematics model [LeFevre et al. (2010), Child Development, Vol. 81, pp. 1753-1767], children's cognitive skills in three domains-linguistic, attentional, and quantitative-predict concurrent and future mathematics achievement. We extended this model to include an additional cognitive skill, patterning, as measured by a non-numeric repeating patterning task. Chilean children who attended schools of low or high socioeconomic status (N = 98; 54% girls) completed cognitive measures in kindergarten (M_age = 71 months) and numeracy and mathematics outcomes 1 year later in Grade 1. Patterning and the original three pathways were correlated with the outcomes. Using Bayesian regressions, after including the original pathways and mother's education, we found that patterning skills predicted additional variability in applied problem solving and arithmetic fluency, but not number ordering, in Grade 1. Similarly, patterning skills were included in the best model for applied problem solving and arithmetic fluency, but not for number ordering, in Grade 1. In accord with the hypotheses of the original Pathways to Mathematics model, patterning varied in its unique and relative contributions to later mathematical performance, depending on the demands of the tasks. We conclude that patterning is a useful addition to the Pathways to Mathematics model, providing further insights into the range of cognitive precursors that are related to children's mathematical development.

Transcoding counts: Longitudinal contribution of number writing to arithmetic in different languages

Number writing involves transcoding from number words (e.g., "thirty-two") to written digit strings (32) and is an important unique predictor of arithmetic. The existing longitudinal evidence about the relation between transcoding and arithmetic is mostly language specific. In languages with number word inversion (e.g., German), the order of tens and units is transposed in spoken number words compared with Arabic numbers. This makes transcoding more challenging than in languages without number word inversion (e.g., English). In the current study, we aimed to understand whether the contribution of number writing to the development of arithmetic is similar in languages with and without number word inversion. German-speaking children (n = 166) and English-speaking children (n = 201) were followed over the first 3 years of primary school. In a series of multiple linear regressions, we tested whether number writing of multi-digit numbers was a significant unique predictor of arithmetic after controlling for well-known non-numerical predictors (nonverbal reasoning and working memory) and numerical predictors (symbolic and nonsymbolic magnitude comparison). Number writing in Grade 1 predicted arithmetic in Grades 1, 2, and 3 over and above the other predictors. Crucially, number writing performance was of comparable importance for arithmetic development in German- and English-speaking children. Our findings extend previous evidence by showing that transcoding predicts the development of arithmetic skills during the first 3 years of primary school in languages with and without number word inversion.

Predicting Math Ability Using Working Memory, Number Sense, and Neurophysiology in Children and Adults

Previous work has shown relations between domain-general processes, domain-specific processes, and mathematical ability. However, the underlying neurophysiological effects of mathematical ability are less clear. Recent evidence highlighted the potential role of beta oscillations in mathematical ability. Here we investigate whether domain-general (working memory) and domain-specific (number sense) processes mediate the relation between resting-state beta oscillations and mathematical ability, and how this may differ as a function of development (children vs. adults). We compared a traditional analysis method normally used in EEG studies with a more recently developed parameterization method that separates periodic from aperiodic activity. Regardless of methods chosen, we found no support for mediation of working memory and number sense, neither for children nor for adults. However, we found subtle differences between the methods. Additionally, we showed that the traditional EEG analysis method conflates periodic activity with aperiodic activity; in addition, the latter is strongly related to mathematical ability and this relation differs between children and adults. At the cognitive level, our findings do not support previous suggestions of a mediation of working memory and number sense. At the neurophysiological level our findings suggest that aperiodic, rather than periodic, activity is linked to mathematical ability as a function of development.

Investigating the Dimensionality of Early Numeracy Using the Bifactor Exploratory Structural Equation Modeling Framework

The few studies that have analyzed the factorial structure of early number skills have mainly used confirmatory factor analysis (CFA) and have yielded inconsistent results, since early numeracy is considered to be unidimensional, multidimensional or even underpinned by a general factor. Recently, the bifactor exploratory structural equation modeling (bifactor-ESEM)-which has been proposed as a way to overcome the shortcomings of both the CFA and the exploratory structural equation modeling (ESEM)-proved to be valuable to account for the multidimensionality and the hierarchical nature of several psychological constructs. The present study is the first to investigate the dimensionality of early number skills measurement through the application of the bifactor-ESEM framework. Using data from 644 prekindergarten and kindergarten children (4 to 6 years old), several competing models were contrasted: the one-factor CFA model; the independent cluster model (ICM-CFA); the exploratory structural equation modeling (ESEM); and their bifactor counterpart (bifactor-CFA and bifactor-ESEM, respectively). Results indicated acceptable fit indexes for the one-factor CFA and the ICM-CFA models and excellent fit for the others. Among these, the bifactor-ESEM with one general factor and three specific factors (Counting, Relations, Arithmetic) not only showed the best model fit, but also the best coherent factor loadings structure and full measurement invariance across gender. The bifactor-ESEM appears relevant to help disentangle and account for general and specific factors of early numerical ability. While early numerical ability appears to be mainly underpinned by a general factor whose exact nature still has to be determined, this study highlights that specific latent dimensions with substantive value also exist. Identifying these specific facets is important in order to increase quality of early numerical ability measurement, predictive validity, and for practical implications.

Children's Knowledge of Symbolic Number in Grades 1 and 2: Integration of Associations

How do children develop associations among number symbols? For Grade 1 children (n = 66, M = 78 months), sequence knowledge (i.e., identify missing numbers) and number comparison (i.e., choose larger number) predicted addition, both concurrently and indirectly at the end of Grade 1. Number ordering (i.e., touch numbers in order) did not predict addition but was predicted by number comparison, suggesting that magnitude associations underlie ordering performance. In contrast, for Grade 2 children (n = 80, M = 90 months), number ordering predicted addition concurrently and at the end of Grade 2; number ordering was predicted by number comparison, sequencing, and inhibitory processing. Development of symbolic number competence involves the hierarchical integration of sequence, magnitude, order, and arithmetic associations.

Linguistic precursors of advanced math growth in first-language and second-language learners

BACKGROUND

Language plays an important role in the development of mathematics. Previous research has shown that both basic and advanced linguistic skills relate to fifth-grade advanced mathematics (i.e., geometry and fractions), but these effects have not yet been investigated longitudinally or in a linguistically diverse population.

AIMS

The present study first examined the differences between first-language and second-language learners in advanced mathematics. Second, we investigated the extent to which the basic and advanced linguistic skills of first-language and second-language learners directly and indirectly (through arithmetic) predict their growth in advanced mathematics from fifth to sixth grade.

METHODS AND PROCEDURES

Participants were 153 first-language and 80 second-language learners from 10 to 12 years of age. Classroom as well as individual measures were administered.

OUTCOMES AND RESULTS

First, the results showed lower scores for second-language learners on advanced mathematics. Second, for both groups of language learners, basic linguistic skills were found to indirectly predict the growth in advanced mathematics via arithmetic skills, whereas advanced linguistic skills directly predicted the growth in geometry and fractions.

CONCLUSIONS AND IMPLICATIONS

These results highlight the general need for opportunities to learn the basic and advanced linguistic skills associated with mathematics over individual native language background.

Mathematical Difficulties vs. High Achievement: An Analysis of Arithmetical Cognition in Elementary School

This study analyzed the contribution of cognitive processes (planning, attention, simultaneous and successive processing) and domain-specific skills (counting, number processing and conceptual comprehension) to the arithmetic performance achieved in the last three grades (4th, 5th, and 6th) of elementary school. Three groups of students with a different arithmetic achievement level were characterized. The predictive value of the cognitive processes and the math specific skills are explored through diverse covariance and discriminant analyses. Participants were 110 students (M = 10.5 years, SD = 1.17) classified in three groups: mathematical difficulties (MD; n = 26), high achieving (HA; n = 26), and typical achieving (TA; n = 58). Cognitive processes and domain-specific skills were evaluated in two individual sessions at the end of the school year. Nonverbal intelligence was assessed in a final collective session with each class. The mathematical difficulties group's achievement was deficient in simultaneous and successive processing, number processing, and conceptual comprehension compared to the typical achievement group. High achievement children obtained significantly better results than the typical achievement children in simultaneous processing, counting, number processing, and conceptual comprehension. Number processing and conceptual comprehension were the most consistent classifiers, although successive and simultaneous processing, respectively, also contributed to identifying students with mathematical difficulties and high achievement. These findings have practical implications for preventive and intervention proposals linked to the observed profiles.

How does mathematics anxiety impair mathematical abilities? Investigating the link between math anxiety, working memory, and number processing

In contemporary society, it is essential to have adequate mathematical skills. Being numerate has been linked to positive life outcomes and well-being in adults. It is also acknowledged that math anxiety (MA) hampers mathematical skills increasingly with age. Still, the mechanisms by which MA affect performance remain debated. Using structural equation modeling (SEM), we contrast the different ways in which MA has been suggested to interfere with math abilities. Our models indicate that MA may affect math performance through three pathways: (1) indirectly through working memory ability, giving support for the 'affective drop' hypothesis of MA's role in mathematical performance, (2) indirectly through symbolic number processing, corroborating the notion of domain-specific mechanisms pertaining to number, and (3) a direct effect of MA on math performance. Importantly, the pathways vary in terms of their relative strength depending on what type of mathematical problems are being solved. These findings shed light on the mechanisms by which MA may interfere with mathematical performance.

Innate or Acquired? - Disentangling Number Sense and Early Number Competencies

The clinical profile termed developmental dyscalculia (DD) is a fundamental disability affecting children already prior to arithmetic schooling, but the formal diagnosis is often only made during school years. The manifold associated deficits depend on age, education, developmental stage, and task requirements. Despite a large body of studies, the underlying mechanisms remain dubious. Conflicting findings have stimulated opposing theories, each presenting enough empirical support to remain a possible alternative. A so far unresolved question concerns the debate whether a putative innate number sense is required for successful arithmetic achievement as opposed to a pure reliance on domain-general cognitive factors. Here, we outline that the controversy arises due to ambiguous conceptualizations of the number sense. It is common practice to use early number competence as a proxy for innate magnitude processing, even though it requires knowledge of the number system. Therefore, such findings reflect the degree to which quantity is successfully transferred into symbols rather than informing about quantity representation per se. To solve this issue, we propose a three-factor account and incorporate it into the partly overlapping suggestions in the literature regarding the etiology of different DD profiles. The proposed view on DD is especially beneficial because it is applicable to more complex theories identifying a conglomerate of deficits as underlying cause of DD.

What counts in preschool number knowledge? A Bayes factor analytic approach toward theoretical model development

Preschool children vary tremendously in their numerical knowledge, and these individual differences strongly predict later mathematics achievement. To better understand the sources of these individual differences, we measured a variety of cognitive and linguistic abilities motivated by previous literature to be important and then analyzed which combination of these variables best explained individual differences in actual number knowledge. Through various data-driven Bayesian model comparison and selection strategies on competing multiple regression models, our analyses identified five variables of unique importance to explaining individual differences in preschool children's symbolic number knowledge: knowledge of the count list, nonverbal approximate numerical ability, working memory, executive conflict processing, and knowledge of letters and words. Furthermore, our analyses revealed that knowledge of the count list, likely a proxy for explicit practice or experience with numbers, and nonverbal approximate numerical ability were much more important to explaining individual differences in number knowledge than general cognitive and language abilities. These findings suggest that children use a diverse set of number-specific, general cognitive, and language abilities to learn about symbolic numbers, but the contribution of number-specific abilities may overshadow that of more general cognitive abilities in the learning process.

The integration of symbolic and non-symbolic representations of exact quantity in preschool children

Preschoolers (n=62) completed tasks that tapped their knowledge of symbolic and non-symbolic exact quantities, their ability to translate among different representations of exact quantity (i.e., digits, number words, and non-symbolic quantities), and their non-symbolic, digit, and spoken number comparison skills (e.g., which is larger, 2 or 4?). As hypothesized, children's knowledge about non-symbolic exact quantities, spoken number words, and digits predicted their ability to map between symbolic and non-symbolic exact quantities. Further, their knowledge of the mappings between digits and non-symbolic quantities predicted symbolic number comparison (i.e., of spoken number words or written digits). Mappings between written digits and non-symbolic exact quantities developed later than the other mappings. These results support a model of early number knowledge in which integration across symbolic and non-symbolic representations of exact quantity underlies the development of children's number comparison skills.

A longitudinal study on predictors of early calculation development among young children at risk for learning difficulties

The purpose of this study was to explore domain-general cognitive skills, domain-specific academic skills, and demographic characteristics that are associated with calculation development from first grade to third grade among young children with learning difficulties. Participants were 176 children identified with reading and mathematics difficulties at the beginning of first grade. Data were collected on working memory, language, nonverbal reasoning, processing speed, decoding, numerical competence, incoming calculations, socioeconomic status, and gender at the beginning of first grade and on calculation performance at four time points: the beginning of first grade, the end of first grade, the end of second grade, and the end of third grade. Latent growth modeling analysis showed that numerical competence, incoming calculation, processing speed, and decoding skills significantly explained the variance in calculation performance at the beginning of first grade. Numerical competence and processing speed significantly explained the variance in calculation performance at the end of third grade. However, numerical competence was the only significant predictor of calculation development from the beginning of first grade to the end of third grade. Implications of these findings for early calculation instructions among young at-risk children are discussed.

Comorbidity of Arithmetic and Reading Disorder

The aim of the present study was to investigate the cognitive profiles of primary school children (age 82-133 months) on a battery of basic number processing and calculation tasks. The sample consisted of four groups matched for age and IQ: arithmetic disorder only (AD; n = 20), reading disorder only (RD; n = 40), a comorbid group ( n = 27), and an unimpaired control group ( n = 40). Multiple 2 (RD vs. No RD) × 2 (AD vs. No AD) factorial ANCOVAs showed that children with RD had selective impairments in counting and number transcoding efficiency. In contrast, children with AD performed poorly in most tasks, including symbolic and nonsymbolic magnitude comparisons, subitizing, number line estimation, number sets, number transcoding accuracy, and calculation. These findings provide further support that AD is characterized by multiple, heterogeneous underlying deficits. In contrast, RD is associated with specific number processing impairments only if tasks require verbal processing. Taken together, the results fully support the assumption of comorbid additivity of AD and RD.

Early Literacy and Numeracy Skills in Bilingual Minority Children: Toward a Relative Independence of Linguistic and Numerical Processing

Many studies have suggested that the concept of "number" is relatively independent from linguistic skills, although an increasing number of studies suggest that language abilities may play a pivotal role in the development of arithmetic skills. The condition of bilingualism can offer a unique perspective into the role of linguistic competence in numerical development. The present study was aimed at evaluating the relationship between language skills and early numeracy through a multilevel investigation in monolingual and bilingual minority children attending preschool. The sample included 156 preschool children. Of these, 77 were bilingual minority children (mean age = 58.27 ± 5.90), and 79 were monolinguals (mean age = 58.45 ± 6.03). The study focused on three levels of analysis: group differences in language and number skills, concurrent linguistic predictors of early numeracy and, finally, profile analysis of linguistic skills in children with impaired vs. adequate numeracy skills. The results showed that, apart from the expected differences in linguistic measures, bilinguals differed from monolinguals in numerical skills with a verbal component, such as semantic knowledge of digits, but they did not differ in a pure non-verbal component such as quantity comparison. The multigroup structural equation model indicated that letter knowledge was a significant predictor of the verbal component of numeracy for both groups. Phonological awareness was a significant predictor of numeracy skills only in the monolingual group. Profile analysis showed that children with a selective weakness in the non-verbal component of numeracy had fully adequate verbal skills. Results from the present study suggest that only some specific components of language competence predict numerical processing, although linguistic proficiency may not be a prerequisite for developing adequate early numeracy skills.