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... Establecer la correspondencia entre los numerales y las cantidades que éstos representan es un proceso lento y complejo e implica establecer la relación entre la representación simbólica y aproximada de una cantidad. Una de las principales controversias actuales en relación con cómo se establece la correspondencia entre símbolos numéricos y las cantidades que representan es el llamado en ingles symbol grounding problem (Leibovich y Ansari, 2016;Merkley y Ansari, 2016;Szkudlarek y Brannon, 2017). Existen dos principales posturas para explicar la adquisición del sistema simbólico. ...

... Con todo, se ha visto que el conocimiento del número es un predictor robusto del logro matemático, ya que media la transición entre el aprendizaje informal y el formal de la habilidad matemática, por lo que debería ser introducido en actividades no formales en el hogar (Purpura et al., 2013). Específicamente, se ha propuesto que la cardinalidad, la ordinalidad, la identificación numérica y la función del sucesor son los componentes cognitivos clave que mediarán el aprendizaje de la matemática formal a partir del entrenamiento de habilidad numéricas no formales, como la lista de conteo (Carey y Barner, 2019;Merkley y Ansari, 2016). ...

... Los avances en cognición numérica han mostrado que ciertas habilidades de dominio específico sustentan la consolidación del símbolo numérico, como proceso cognitivo que nos permite manipular cantidades exactas y operar con ellas, y, por tanto, desarrollar el cálculo aritmético, que está en la base de conocimientos y procedimientos matemáticos complejos. Consecuentemente, se ha propuesto que el uso de la lista de conteo verbal, la identificación de símbolos numéricos, la cardinalidad, la ordinalidad y la función del sucesor, serían los procesos cognitivos clave que se deben trabajar durante la educación inicial, con el fin de consolidar la comprensión del sistema numérico (Carey y Barner, 2019;Merkley y Ansari, 2016). En este sentido, comprender la estructura ordinal del sistema numérico se ha mostrado como un predictor de impacto creciente para el desarrollo de la habilidad de cálculo a lo largo de Educación Básica (Lyons et al., 2014;Lyons y Ansari, 2015). ...

... The results were attributed to either a numerical accuracy sense weakening or immaturity. These data do not detract from the role of the approximate number sense (ANS), since it is considered that symbolic magnitude processing skills are built on the ability to represent quantities in a non-symbolic way (Bugden et al. 2016;Merkley and Ansari 2016). Symbolic magnitude processing skills start becoming evident when compulsory school begins (Mundy and Gilmore 2009;Siegler and Lortie-Forgues 2014). ...

... It is important to bear in mind that estimation roughly implies being aware of numerosity and its position in a number line oriented from left to right. This ability is significant in understanding the relationship of numerical symbols, since symbolic knowledge is built on non-symbolic knowledge (Bugden et al. 2016;Merkley and Ansari 2016). As students begin to improve their symbolic skills, its role diminishes. ...

... The complexity of basic symbolic number processing Symbolic numbers, particularly Arabic numerals, are the crux of mathematics and they have to be learned via education. This learning of Arabic numerals already begins early in development before the start of formal schooling (e.g., Bakker et al., 2019;Merkley & Ansari, 2016;Yuan et al., 2019) albeit in non-formal contexts, such as in preschool or the home environment. The acquisition of Arabic numerals has been extensively investigated in the field of numerical cognition, yet it also entails one of its most debated areas (e.g., Merkley & Ansari, 2016). ...

... This learning of Arabic numerals already begins early in development before the start of formal schooling (e.g., Bakker et al., 2019;Merkley & Ansari, 2016;Yuan et al., 2019) albeit in non-formal contexts, such as in preschool or the home environment. The acquisition of Arabic numerals has been extensively investigated in the field of numerical cognition, yet it also entails one of its most debated areas (e.g., Merkley & Ansari, 2016). One outstanding question is how these Arabic numerals acquire their semantic meaning, that is, the symbol-grounding problem (e.g., Leibovich & Ansari, 2016): Are these Arabic numerals grounded in children's understanding of (nonsymbolic) quantities, as is assumed by the majority of studies in the field? ...

This chapter reflects on the impact of education on numerical cognition. Mathematics is a symbolic activity, which must be learned via education, and this learning process will impact on how we process number. There are huge differences in the educational contexts around the globe in which this learning occurs, but these contexts are relatively underappreciated in studies on numerical cognition. I will illustrate the impact of the educational context on numerical cognition by showing that the processing of natural Arabic numerals might be changed by learning other symbolic representations (e.g., fractions), which children learn early in primary school, and by highlighting that the association between number processing and arithmetic depends on the methods children learn to calculate (e.g., mental arithmetic vs. algorithmic computation). I further demonstrate that the educational experience of learning to calculate by itself affects children’s numerical processing. There is a need for experimental studies that manipulate learning to test causal associations between numerical cognition and mathematics learning. The educational context can be used as a natural experiment, via the school cutoff design and cross-national comparisons, to obtain more direct evidence on the broad impact of education on numerical cognition. I end with the suggestion to take the educational context more seriously in future studies on numerical cognition, which will increase both their external and internal validity.

... For example, the implicit presence of Piagetian principles, such as classification or seriation, is included in the early childhood education curriculum (Piaget, 1965). Essential items regarding quantifiers, counting, cardinality and ordinality are also included, but there is no explicit reference to the evidence that supports the learning and development of these aspects at this age (e.g., Dolscheid, Winter, Ostrowski, & Penke, 2017;Merkley & Ansari, 2016). Nor are items such as succession (Carey & Barner, 2019), estimation or transcoding (Ebersbach, 2016) included, which favour number-quantity associations and early understanding of the Arabic symbol. ...

... Por ejemplo, en las bases curriculares de educación parvularia se aprecia la presencia implícita de los principios piagetianos, como la clasificación o la seriación (Piaget, 1965). También es posible distinguir elementos centrales de los cuantificadores, el conteo, la cardinalidad y la ordinalidad, pero no se hace referencia explícita a la evidencia que apoya el aprendizaje y desarrollo de estos aspectos en esta edad (e.g., Dolscheid, Winter, Ostrowski, & Penke, 2017;Merkley & Ansari, 2016). Tampoco se incorporan elementos como la sucesión (Carey & Barner, 2019), la estimación, o la transcodificación (Ebersbach, 2016), que favorecen las asociaciones número-cantidad y la comprensión temprana del símbolo arábigo. ...

Research on numerical cognition, an emerging area that has received increasing interest in Chile, can contribute to understanding how mathematical skills are developed and support education. This article reviews the studies published in the field of numerical cognition in Chile, indexed in the WoS, PubMed, Scopus and SciELO databases. The articles reviewed address mechanisms that underlie mathematical performance and strategies used by children, adolescents and adults to solve mathematical tasks, with a special emphasis on early skills, calculations and arithmetic, as well as factors associated with individual differences in the development of mathematical notions. This article also analyses the sources of funding that support research on numerical cognition and education in Chile and the projects awarded such funds in recent years. Finally, the contribution and challenges of numerical cognition research in Chile in and educational context are discussed.
La investigación en cognición numérica, un área emergente y que ha recibido creciente interés en Chile, puede contribuir a comprender el desarrollo de las habilidades matemáticas y aportar a la educación.Este artículo revisa las publicaciones indexadas en las bases de datos WoS, PubMed, Scopus y SciELO en el ámbito de cognición numérica
en Chile. Los artículos revisados abordan mecanismos que subyacen al desempeño matemático y estrategias utilizadas por niños, jóvenes y adultos para resolver tareas matemáticas, con especial énfasis en habilidades tempranas, cálculo y aritmética, así como factores asociados a las diferencias individuales en el desarrollo de las nociones matemáticas. Este artículo también analiza las fuentes de financiamiento que apoyan la investigación en cognición numérica y educación en Chile y los proyectos adjudicados en los últimos años. Finalmente, se discute el aporte y los desafíos de la investigación en cognición numérica en Chile en el ámbito educativo.

... The development of children's mathematical competence involves learning how written numerals (i.e., Arabic digits) are related to each other (Merkley & Ansari, 2016). Consider the written numerals 1, 2, and 3. ...

... The development of number symbol associations occurs slowly, starting as early as 2 years of age. Children's acquisition of number symbols starts with their ability to map quantitative representations to verbal labels and written numbers (e.g., verbally label the digit 5; Jiménez Lira, Carver, Douglas, & LeFevre, 2017;Merkley & Ansari, 2016). Continued interactions with number symbols help children to develop associations beyond the representational mappings and make judgments using the symbols directly, such as which number is larger (e.g., 3 vs. ...

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 .

... David and Oliver had also begun to write standard numerals. This is of particular interest since multiple researchers have identified relationships between children's early use of numerical symbols and subsequent longitudinal achievement (Merkley & Ansari, 2016). In her study in preschools, Munn (1995) identified a relationship between children's understandings and achievement in recognising numerals and letters: the progress made during their first year of primary school "strongly related to the understanding of symbols they had brought with them at school entry", suggesting, "that the important developments taking place concerned the children's understanding of symbols as communicative systems" (p. ...

... 120). Other researchers have identified relationships between children's early knowledge (recognition) of standard Arabic numerals and subsequent longitudinal achievement (e.g., Griffin et al., 1995;Habermann et al., 2020;Merkley & Ansari, 2016;Rubinsten et al., 2002). In connection with this, it became evident that those who used the greatest and most divergent range of graphical signs to communicate (Shereen and Elizabeth), also most frequently spontaneously wrote standard (Arabic) numerals. ...

The aims of this thesis are to investigate the evolution of young children’s graphical signs and texts, chosen and used freely by them to communicate ideas. Until now, no previous studies have been found that researched the very beginnings of young children’s signs and symbols in depth, in contexts that can be understood as mathematical, making this research unique. The research began by determining if the children explored aspects of mathematics in their pretend play linked to their funds of knowledge. To achieve this, the study documents children’s interest in exploring and communicating through their literacies, and the types of signs young children use to represent their thinking, including those to communicate their mathematical thinking. The main focus of the study is children’s use of their Mathematical Graphics. The thesis also identifies some of the processes that are involved in the course of children’s developing use and understandings of mathematical signs, children’s mathematical abstraction and the role of intertextuality and mathematisation. Rather than viewing young children’s mathematics from a single, subject-based discipline, this study takes the child’s perspective, mathematics seen within the context of all the child’s meaning-making and learning, children having considerable agency as active learners. The thesis reveals a number of interesting findings that are direct outcomes of the democratic culture and open ethos of the nursery setting, coupled with the teachers’ deep understandings of pretend play; of early mathematical development; graphicacy and emergent learning that together support children’s learning.

... In the current study, among the set of precursors assessed, symbolic comparison was the variable with the highest statistical weight in explaining mathematical performance, and was followed by non-symbolic comparison. However, symbolic magnitude processing skills are considered to be built on the ability to represent non-symbolic quantities (Bugden et al., 2016;Merkley & Ansari, 2016), and the key stage for building this knowledge base is the early school years (Siegler & Lortie-Forgues, 2014). These studies indicate a clear role of the Numerical Approximation System (NAS), a cognitive system that facilitates the manipulation and representation of information about numbers and quantities in an approximate manner, and is found not only in adults but also in infants and even in animals (Feigenson et al., 2013). ...

... These types of activities do not require specialized knowledge of mathematics (Bautista-Galeano et al., 2018) and may favor task recognition but restrict its use to activities such as orally repeating a number sequence or counting non-contextualized objects (Ormeño et al., 2013). The development of number sense requires an understanding of numerals and the notational system, as well as of the meaning of number words and magnitude in counting or comparison activities (Lee & Md-Yunus, 2015;Merkley & Ansari, 2016). ...

This research presents the findings of a comparative study of mathematical competence among 130 students (M = 54.08 months; SD = 2.57) from vulnerable school contexts in Chile and the Spanish public school system. The study analyses a set of general and specific domain precursors for which evidence of socioeconomic background exists. Using multivariate regression and discriminant analysis techniques, we calculated similarities and differences between groups by comparing these precursors. Significant differences were found between the Spanish and Chilean groups (p < .05); however, no differences were observed in non-symbolic comparison and receptive vocabulary. Possible reasons for the existence and extent of these differences are discussed in terms of socio-cultural and educational contexts.

... The first is subitizing, the ability to quickly recognize or name the number of a group arithmetic ability when they enter school (Geary et al., 2018). Several studies have shown that young schoolchildren's ability to compare symbolic quantities (quantities represented by numerals and number words) is one of the strongest predictors of their future mathematical development (Merkley and Ansari, 2016;Vanbinst et al., 2016). ...

... suggested that assessing basic numeracy skills (Jordan, Glenn and McGhie-Richmond, 2010;Merkley and Ansari, 2016;Bugden, Szkudlarek and Brannon, 2021) can improve the efficiency for early classification of maths learning disabilities, more work is needed to identify reliable assessment tools to identify dyscalculia. ...

The goal of this chapter is to assess research that can inform understandings of places and spaces of learning.The chapter assesses
evidence across three types of learning spaces: built spaces, digital spaces, and natural spaces. It looks at the role of these different kinds of spaces for learning, attainment, interpersonal relationships, skills development, wellbeing and behaviours ‒ across four pillars of learning to know, to be, to do and to live together. The chapter also explores how learning spaces can be actively shaped, felt and understood through practices and policies that occur within and around them.

... The first is subitizing, the ability to quickly recognize or name the number of a group arithmetic ability when they enter school (Geary et al., 2018). Several studies have shown that young schoolchildren's ability to compare symbolic quantities (quantities represented by numerals and number words) is one of the strongest predictors of their future mathematical development (Merkley and Ansari, 2016;Vanbinst et al., 2016). ...

... suggested that assessing basic numeracy skills (Jordan, Glenn and McGhie-Richmond, 2010;Merkley and Ansari, 2016;Bugden, Szkudlarek and Brannon, 2021) can improve the efficiency for early classification of maths learning disabilities, more work is needed to identify reliable assessment tools to identify dyscalculia. ...

This chapter assesses ways to identify and support children with learning disabilities.
Learning disabilities affect many students and are seldom attributable to a single cause. They arise through complex interactions between biological and environmental factors within individual developmental trajectories. Early identification of children at risk for learning disabilities as well as adequate identification of children with learning
disabilities are important for ensuring that children have access to the supports they need
in order to reach their full potential. Here, we discuss identifying children’s learning needs and providing educational support. Although many school systems recognize the need to provide inclusive education to support all learners, more work is needed to raise awareness and enable adequate evidence-based early identification of children with learning disabilities and support their learning trajectories and instructional needs
inside and outside of the classroom. It is also fundamental to acknowledge the importance of research on diverse populations that could inform identification and support in various countries and socio-cultural contexts.

... Early numeracy includes several skills which are important for later mathematics learning (Aunio & Räsänen, 2015;Merkley & Ansari, 2016). More specifically, understanding the mental number line and differences in magnitudes (Merkley & Ansari, 2016;Muldoon, Towse, Simms, Perra, & Menzies, 2013;LeFevre et al., 2010), recognition and naming of number symbols (Göbel, Watson, Lervåg, & Hulme, 2014;Pinto, Bigozzi, Tarchi, Vezzani, https://doi.org/10.1016/j.ecresq.2020.12.002 0885-2006/© 2020 The Author(s). ...

... Early numeracy includes several skills which are important for later mathematics learning (Aunio & Räsänen, 2015;Merkley & Ansari, 2016). More specifically, understanding the mental number line and differences in magnitudes (Merkley & Ansari, 2016;Muldoon, Towse, Simms, Perra, & Menzies, 2013;LeFevre et al., 2010), recognition and naming of number symbols (Göbel, Watson, Lervåg, & Hulme, 2014;Pinto, Bigozzi, Tarchi, Vezzani, https://doi.org/10.1016/j.ecresq.2020.12.002 0885-2006/© 2020 The Author(s). Published by Elsevier Inc. ...

The aim of this study was to investigate whether early numeracy skills of South African first graders who are at-risk for mathematical learning difficulties can be improved with an intervention program. The participants were 267 children from 17 classrooms in the greater Johannesburg area. In this quasi-experimental small group intervention study (15 sessions over 5 weeks) the outcome measure was early numeracy skills. Based on pretest early numeracy scores, the children were divided into an intervention group (N = 40), a low performing control group (N = 32), and an average performing control group (N = 195). The main result was that the intervention group had improved more in numerical relational skills, compared to low-controls; this effect remained statistically significant after controlling for executive functions, language skills and kindergarten attendance, and was also observable in the delayed post-measurement. Executive functions, language skills and kindergarten attendance all predicted the level of early numeracy skills at the beginning of the intervention, but only executive functions explained individual differences in counting skills development from pre-to delayed posttest.

... Together, these findings indicate that the learning of symbolic representations is much more complex than simply mapping quantities onto symbols (see the following references for a detailed discussion [35][36][37]64 ). They suggest that the construction and learning of symbolic numerical information are related to the integration of multiple knowledge dimensions 98 , such as numerical order and counting, all of which should be fostered through (mathematics) education. ...

The development of numerical and arithmetic abilities constitutes a crucial cornerstone in our modern and educated societies. Difficulties to acquire these central skills can lead to severe consequences for an individual’s well-being and nation’s economy. In the present review, we describe our current broad understanding of the functional and structural brain organization that supports the development of numbers and arithmetic. The existing evidence points towards a complex interaction among multiple domain-specific (e.g., representation of quantities and number symbols) and domain-general (e.g., working memory, visual–spatial abilities) cognitive processes, as well as a dynamic integration of several brain regions into functional networks that support these processes. These networks are mainly, but not exclusively, located in regions of the frontal and parietal cortex, and the functional and structural dynamics of these networks differ as a function of age and performance level. Distinctive brain activation patterns have also been shown for children with dyscalculia, a specific learning disability in the domain of mathematics. Although our knowledge about the developmental brain dynamics of number and arithmetic has greatly improved over the past years, many questions about the interaction and the causal involvement of the abovementioned functional brain networks remain. This review provides a broad and critical overview of the known developmental processes and what is yet to be discovered.

... The main aim of this study was to gain insight into the interactions between foundational domain-general and domain-specific cognitive skills in preschool children. This was done by assessing various domain-general (e.g., inhibition skills, attention and working memory) and domainspecific skills (e.g., cardinal number knowledge, counting, digit identification) that are known to be strong predictors of math achievement longitudinally (Lau et al., 2021;Merkley & Ansari, 2016). A longitudinal design allowed us to investigate the factor structure of domain-general and domain-specific skills in order to better understand the development of these inter-related foundational skills. ...

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.

... Regardless of the format in which number symbols are presented, they convey a sense of magnitude, which is the quantity of elements in a set (e.g., Gilmore et al., 2018;Piazza, 2010), and order, which is the sequential position or rank of a number in relation to other numbers (e.g., Goffin & Ansari, 2016;Jacob & Nieder, 2008;Lyons & Beilock, 2011;Lyons et al., 2014). Furthermore, how we process magnitude and order information for number symbols has been found to be related to more complex mathematical skills (e.g., De Smedt et al., 2013;Holloway & Ansari, 2009;Lyons et al., 2016;Merkley & Ansari, 2016;Morsanyi et al., 2017;Sasanguie et al., 2012;Schneider et al., 2016). For example, there is behavioural evidence that order processing of Arabic numerals is significantly related to mathematics achievement in adults (e.g., Goffin & Ansari, 2016;Lyons & Beilock, 2011;Morsanyi et al., 2017;Sasanguie, Lyons, et al., 2017;Vogel et al., 2017;Vos et al., 2017) and children (e.g., Attout & Majerus, 2018;Lyons et al., 2014). ...

This study probed the cognitive mechanisms that underlie order processing for number symbols, specifically the extent to which the direction and format in which number symbols are presented influence the processing of numerical order, as well as the extent to which the relationship between order processing and mathematical achievement is specific to Arabic numerals or generalisable to other notational formats. Seventy adults who were bilingual in English and Chinese completed a Numerical Ordinality Task, using number sequences of various directional conditions (i.e., ascending, descending, mixed) and notational formats (i.e., Arabic numerals, English number words, and Chinese number words). Order processing was found to occur for ascending and descending number sequences (i.e., ordered but not non-ordered trials), with the overall pattern of data supporting the theoretical perspective that the strength and closeness of associations between items in the number sequence could underlie numerical order processing. However, order processing was found to be independent of the notational format in which the stimuli were presented, suggesting that the psychological representations and processes associated with numerical order are abstract across different formats of number symbols. In addition, a relationship between the processing speed for numerical order and mathematical achievement was observed for Arabic numerals and Chinese number words, and to a weaker extent, English number words. Together, our findings have started to uncover the cognitive mechanisms that could underlie order processing for different formats of number symbols, and raise new questions about the generalisability of these findings to other notational formats.

... Recent findings in the field of numerical cognition, however, demonstrate that children's mathematical competence is more strongly associated with their symbolic rather than with their nonsymbolic numerical abilities, suggesting that symbolic numerical abilities, such as naming Arabic numerals and connecting Arabic numerals to quantities, play a pivotal role in mathematical development (De Smedt et al., 2013;Göbel et al., 2014;Merkley & Ansari, 2016;Schneider et al., 2017). Based on these findings, it may be assumed that also for the dispositional side of early mathematical development, children's tendency to spontaneously attend to Arabic number symbols may be key for their later mathematics learning and development. ...

Children start formal schooling with large individual differences in their mathematical competence. While some children can already perform simple calculations, others are still learning how to count small numerosities. This large variety in mathematical competence at the start of formal schooling can be explained by the early mathematical abilities children use in explicit mathematical situations, but to some extent also by their tendencies to spontaneously focus on mathematical aspects in everyday situations. Previous research has shown the importance of these spontaneous attentional processes particularly for children’s spontaneous focusing on numerosity (SFON). The present dissertation aimed to contribute to this emerging field of spontaneous mathematical focusing tendencies by proposing a new construct of spontaneous focusing on Arabic number symbols (SFONS) different from SFON and investigating its role in early mathematical development in four related studies.
In a first study, we measured SFONS for the first time and explored its concurrent association with SFON, numerical abilities, and teacher ratings of mathematical competence in the first, second, and third year of kindergarten (Chapter 2). We found large individual differences in children’s SFONS and significant associations with their numerical abilities and teacher ratings of their mathematical competence. The second study further explored the validity of the novel SFONS construct by investigating its factor structure and exploring its unique contribution to numerical abilities and mathematics achievement in the second year of kindergarten (Chapter 3). We obtained empirical evidence for the hypothesized two-factor structure of SFON and SFONS and found that the latter uniquely predicted numerical abilities and mathematics achievement above age, parental education, spatial and verbal ability, and SFON. In a third study, we investigated the structure of children’s spontaneous number focusing tendencies and their longitudinal associations with numerical abilities and mathematics achievement from the second year of kindergarten until first grade (Chapter 4). Results again provided evidence for the distinctiveness of SFONS and SFON and revealed moderate associations with numerical abilities and mathematics achievement across development. In a final study, we explored the origins of individual differences in children’s spontaneous number focusing tendencies by relating SFON and SFONS to the home numeracy environment in the second and third year of kindergarten (Chapter 5). We found no significant associations between children’s spontaneous number focusing tendencies and the frequency of numeracy activities at home and their parents’ numeracy expectations.
These four studies are preceded by an introductory chapter (Chapter 1) and followed by a general discussion chapter wherein the main conclusions and the theoretical, methodological, and educational implications of these four studies are discussed (Chapter 6).

... Likewise, these skills acquired in preschool have also been linked to later mathematical achievement Merkley & Ansari, 2016). ...

Numerical skills encompass a variety of cognitive processes and are crucial for performance in today’s modern world but vary greatly between individuals. Several approaches of numerical cognition have been studied ranging from behavioral to neuroimaging studies. Transcranial electrical stimulation (tES) has become a promising tool to influence numerical cognition and is frequently used in clinical settings. The main aim of this chapter is to shed light onto current tES research as an intervention in this domain. We first provide a brief overview of the underlying neurocognitive mechanisms of basic and advanced mathematical skills, such as working memory, executive functions, and (non)symbolic number skills, since tES allows to intervene and further unravel the role of these mechanisms. In addition, we discuss the need for tES research to focus on the transfer of these skills to a similar numerical task to determine facilitation by means of neuroplasticity. Therefore we emphasize studies using tES as a numerical intervention and focus on transfer effects since these outcomes could contribute to implications for educational practice.

... Although the Woodcock-Johnson is a widely used assessment in the field, the types of skills it assesses may be limited, particularly for children at age 54 months. For example, the Woodcock-Johnson provides limited information about children's non-symbolic, symbolic, spatial, geometry and magnitude skills as discrete components; each of which is likely an important contributor to mathematical development (Merkley & Ansari, 2016;Purpura & Simms, 2018;Siegler, 2016). As a result, our analyses and discussion focus on numeracy, rather than mathematical skills more generally. ...

Using data from the Applied Problems subtest of the Woodcock-Johnson Tests of Achievement (Woodcock, McGrew, & Mather, 2001) administered to 1,364 children from the National Institute of Child Health and Human Development (NICHD) Study of Early Childcare and Youth Development (SECCYD), this study measures children’s mastery of three numeric competencies (counting, arithmetic operations using visual object representations, and abstract arithmetic operations) at 54 months of age. We find that, even after controlling for key demographic characteristics, the numeric competency that children master prior to school entry relates to important educational transitions in secondary and post-secondary education. Those children who showed low numeric competency prior to school entry enrolled in lower math track classes in high school and were less likely to attend college. Important numeracy competency differences at age 54 months related to socioeconomic inequalities were also found. These findings suggest that important indicators of long-term schooling success (i.e., advanced math courses, college attendance) are evident prior to schooling based on the levels of numeracy mastery.

... In the past two decades, symbolic numerical magnitude processing has gained importance as a precursor competence for higher numerical as well as math competencies (Merkley & Ansari, 2016). Performance in this ability is typically assessed through a number comparison task in which two single-digit Arabic numbers are presented simultaneously on a computer screen, and participants have to indicate as fast and accurately as possible, which of the two numbers is the larger one. ...

While the cognitive foundations for mathematical abilities have been investigated thoroughly in individuals with and without mathematical difficulties, our current knowledge about the cognitive abilities as well as the personality traits associated with mathematical expertise is still scarce. In this study we systematically investigated which domain-general (working memory [WM], patterning, visual statistical learning [VSL]) and domain-specific cognitive abilities (approximate number system [ANS], symbolic numerical magnitude comparison, ordinality, arithmetic), as well as personality traits (Big Five, need for cognition [NFC], attitudes towards mathematics), are related specifically to mathematical expertise. To this end, we compared 42 mathematicians with 42 non-mathematicians from fields with no to minimal mathematical content. In contrast to previous research, this study included not only mathematicians with lower expertise (Bachelor and Master students) but also mathematicians with higher expertise (faculty members of the institute of mathematics) to provide a more differentiated look at mathematical expertise. Mathematicians and non-mathematicians were matched for age, sex, educational level and, importantly, for general intelligence. All analyses were done with Bayesian statistics to investigate differences and similarities across these groups. After controlling for intelligence, the results showed that mathematicians and non-mathematicians had very similar profiles. They were comparable in WM capacity, VSL, and general patterning abilities; only in the patterning domain time did mathematicians solve more items. Both groups performed equally in ANS and the ordinality task. However, mathematicians had a more accurate mental representation of symbolic numbers and a better arithmetic fact knowledge. Similarities also emerged in NFC and the Big Five, except for openness where mathematicians were less open to experiences. Unsurprisingly, mathematicians had a more positive attitude towards mathematics than non-mathematicians. Comparing mathematicians with lower and higher expertise did not reveal differences in domain-general and domain-specific abilities. This also applied to the personality traits; the groups did not differ except for the motivation to do mathematics, in which the faculty members were more motivated than the students. Overall, these findings contribute to a deeper and more differentiated understanding of mathematical expertise.

... For example, children in Montessori classrooms systematically learn to read using multisensorial activities and a phonetic approach, consistent with the literature on both embodied cognition (Kontra, Goldin-Meadow, & Beilock, 2012;Pouw, van Gog, & Paas, 2014) and reading acquisition (Castles, Rastle, & Nation, 2018). The Montessori math curriculum also stresses the importance of understanding the correspondence between numerical symbols and quantities, in keeping with studies showing that early symbolic and arithmetic knowledge predict later math competence (Jordan, Kaplan, Ramineni, & Locuniak, 2009;Merkley & Ansari, 2016). Because Montessori classrooms are highly organized and involve a relatively strict set of rules and principles, it has also been argued that Montessori education may promote the growth of executive functions (Diamond & Lee, 2011;Lillard, 2019). ...

Previous research on Montessori preschool education is inconsistent and prone to analytic flexibility. In this preregistered study, disadvantaged preschoolers in a French public school were randomly assigned to either conventional or Montessori classrooms, with the latter being adapted to French public education. Adaptations included fewer materials, shorter work periods, and relatively limited Montessori teacher training. Cross-sectional analyses in kindergarten (N = 176; Mage = 5–6) and longitudinal analyses over the 3 years of preschool (N = 70; Mage = 3–6) showed that the adapted Montessori curriculum was associated with outcomes comparable to the conventional curriculum on math, executive functions, and social skills. However, disadvantaged kindergarteners from Montessori classrooms outperformed their peers on reading (d = 0.68). This performance was comparable to that of advantaged children from an accredited Montessori preschool.

... Beyond counting, children need to learn numerical symbols and varied representations of numbers in order to understand the concept of quantity (Leibovich & Ansari, 2016;Merkley & Ansari, 2016). Showing children different representations of the same number through Arabic numerals, number words, and sets of identical or related objects is often used to accomplish this goal. ...

... More specifically, efficient access to small quantities may facilitate the development of counting skills by supporting children's understanding of the cardinality principle (Cheung & Le Corre, 2018;Le Corre & Carey, 2007). Cardinality, that is, the understanding that the final counting word produced when enumerating a set of objects indicates the quantity of that set, is a critical precursor of number system knowledge more generally (Butterworth, 2005;Lyons & Ansari, 2015a;Merkley & Ansari, 2016). ...

What is the foundational knowledge that children rely on to provide meaning as they construct an exact symbolic number system? People and animals can quickly and accurately distinguish small exact quantities (i.e., 1 to 3). One possibility is that children’s ability to map small quantities to spoken number words supports their developing exact number system. To test this hypothesis, it is important to have valid and reliable measures of the efficiency of quantity-number word mapping. In the present study, we explored the reliability and validity of a measure for assessing the efficiency of mapping between small quantities and number words -- speeded naming of quantity. Study 1 (N = 128) with 5- and 6-year-old children and Study 2 (N = 182) with 3- and 4-year-old children show that the speeded naming of quantites is a simple and reliable measure that is correlated with individual differences in children’s developing numeracy knowledge. This measure could provide a useful tool for testing comprehensive theories of how children develop their symbolic number representations.

... For example, Thomas, Mulligan and Goldin (2002) write that it seems that the attentive processing of graphical images has an important role in children's developing understanding of numerals, counting and calculation. Moreover, Merkley and Ansari (2016) and others established that young children's knowledge of symbolic numbers is predictive of ensuing achievement in mathematics, although Worthington cautions against direct and narrow formal teaching of standard numerals. In her research, Papandreou (2019) investigated children's mathematical graphics, analysing four to six-year-old's data investigations and their own strategies as they employed a combination of writing, numerals, calculations or other formal 'written' mathematics. ...

Young children’s graphical sign lexicons and the emergence of mathematical symbols
Maulfry Worthington
ABSTRACT
Young children’s personal repertoires or lexicons of graphical signs comprise multiple and diverse signs and symbols. These signs support understanding and progress of the symbolic languages of the culturally established, alphanumerical systems, development evolving early in childhood. Investigating language and inscriptional systems - including those that are drawn, written and mathematical - this evidence-based position paper explores the extent to which children’s graphical sign lexicons support their emergent understandings, as they move from intuitive marks and informal signs to formal symbols. These inscriptions are indispensable in communicating ideas, and have significance for the study of young children’s understanding of the abstract symbolic language of mathematics.
KEYWORDS
Early childhood, graphical sign lexicons, repertoires, children’s mathematical graphics, emergent learners

... Using different criteria can lead to sampling children with divergent cognitive profiles and etiologies, thereby making it difficult to draw conclusions across studies about the core deficits leading to DD. Given that children's success in mathematics is scaffolded by early numerical magnitude processing skills (for reviews see De Smedt, Noël, Gilmore, & Ansari, 2013;Merkley & Ansari, 2016), researchers have been focusing on exploring numerical magnitude processing skills as a potential root cause of severe arithmetic deficits. ...

Developmental dyscalculia (DD) is a mathematical learning disability that occurs in around 5%–7% of the population. At present, there are only a handful of screening tools to identify children that might be at risk of developing DD. The present study evaluated the classification accuracy of one such tool: The Numeracy Screener, a 2‐min test of symbolic (Arabic numerals) and nonsymbolic (dot arrays) discrimination ability. A sample of 222 children who demonstrated persistent deficits (n = 55), inconsistent deficits (n = 51), or typical performance (n = 116) on standardized tests of math achievement over multiple observations was tested. The Numeracy Screener correctly classified children in all three groups. Notably, the symbolic condition has greater sensitivity in discriminating children with persistent DD from the other two groups. Screening tools that assess early numeracy skills may be promising for identifying children at risk for developing severe mathematical difficulties.

... Most learners get their first introduction to the writing system of mathematics when they are introduced to the names of digits and the base-ten system for combining digits. Knowledge of digit symbols and the development of a network of associations among those symbols is related to success in school-based mathematics (Hawes et al., 2019;Marinova et al., 2020;Merkley & Ansari, 2016;Purpura et al., 2013;Yuan et al., 2019). Cognitive scientists have identified several key relations between digit knowledge and mathematical skills. ...

The written language of mathematics is dense with symbols and with conventions for combining those symbols to express mathematical ideas. For example, reading a factored polynomial function such as f(x) = x²(2x + 15) requires the knowledge that parenthesis can be used to signify function notation in one context and multiplication in another. Mathematical orthography is defined as orthographic knowledge of symbolic mathematics. It entails both knowledge of discrete mathematical symbols and the conventions for combining those symbols into expressions and equations. The ability to read text written in the base-ten system, comprised of digits and conventions for combining digits to express whole and rational quantities, is an important aspect of mathematical orthography. However, success in secondary and post-secondary programs requires more advanced mathematical orthography. The goal of this research was to determine if a simple and novel measure of mathematical orthography captures individual differences in adults' mathematical skills. Mathematical orthography was measured with a timed dichotomous symbol decision task. Adults (N = 58) discriminated between conventional and non-conventional combinations of mathematical symbols (e.g., x² vs. ²x; |y| vs. ||y). The mathematical symbol decision task uniquely predicted individual differences in whole-number arithmetic, fraction/algebra procedures, and word problem solving. These findings suggest that the symbol decision task is a useful index of symbol associations in mathematical development and, thus, provides a tool for understanding the role of mathematical orthography in individual differences in adults' mathematical skills.

... Early mathematics, or informal math skills, occurs over three stages during which children progress from non-numeric quantity discrimination (e.g., comparing piles of jelly beans), quantity to number-word linkage (e.g., counting the number of jelly beans in a pile), and then to symbolic and nonsymbolic quantity relations (e.g., adding or subtracting jelly beans across piles; Krajewski & Schneider, 2009;Siegler & Lortie-Forgues, 2014) beginning prior to school entry and continuing until from around first grade to third grade where more formal math skills begin to develop (e.g., Sarama & Clements, 2009). Formal math knowledge involves the mastery of number competencies, which include number recognition, number comparisons, and understanding number magnitudes (e.g., Mazzocco & Thompson, 2005;Jordan, Kaplan, Ramineni, & Locuniak, 2009;Merkley & Ansari, 2016). As these comparisons and magnitudes extend to increasingly larger numbers and number arrays, and nonsymbolic concepts are mapped onto increasingly more complex symbolic systems, mastery of formal math knowledge progresses (e.g., Lyons & Ansari, 2015;Siegler & Lortie-Forgues, 2014). ...

This study investigated developmental trajectories of reading and math using latent-growth-curve analyses across multiple academic skills, measures, and multiple time periods within a single sample. Reading-related growth was marked by significant individual differences during the early elementary-school period and non-significant individual differences during the late elementary-school period. For math-related skills, non-significant individual differences were present for early math growth and significant individual differences were present in late elementary-school. No clear pattern of cumulative, compensatory, or stable development emerged for either reading-related or math skills. These differing growth patterns highlight developmental complexities and suggest domain-specific differences in achievement growth that are potentially associated with contextual factors.

... These three tasks were selected as they represent basic numeracy skill development at this age (Purpura, Baroody, & Lonigan, 2013), and have been linked to trajectories in numeracy development after children start school (Merkley & Ansari, 2016;Nguyen et al., 2016;Purpura & Lonigan, 2015). If parents can rate these numeracy skills at young ages, it will be useful for identifying children who are at-risk and planning appropriate interventions. ...

This study examines whether parent ratings of children's numeracy skills are more related to direct assessments of corresponding skills, broad numeracy, or other cognitive skills, to inform how to best utilize parent ratings in research. Children in the sample (N = 129) ranged from 3.07 to 5.95 years old and 52.3% were male. Most (81%) of the children were White. Parents rated their children's counting, arithmetic, and numeral identification skills. Children were directly assessed on these skills, broad numeracy, and other cognitive skills (i.e., expressive vocabulary, executive function). Parent ratings of children's numeracy abilities varied in terms of whether they were more related to directly assessed corresponding skills or broad numeracy abilities. Aggregated parent ratings predicted broad numeracy abilities more than other cognitive skills, providing evidence for discriminant validity. Findings inform how parent ratings may be used when children cannot be directly assessed, such as when large-scale parent surveys are used.

... Research in cognitive science and education has identified knowledge of number symbols (i.e., count words and Arabic digits) as key foundational skills of mathematics (Merkley & Ansari, 2016;Purpura, Baroody, & Lonigan, 2013). The Give-A-Number (Give-N) task (Wynn, 1990) is widely used in developmental cognitive research to assess children's understanding of the cardinal principle, or that the last number word they say when they count represents how many items are in the set. ...

Research in cognitive development has highlighted that early numeracy skills are associated with later math achievement, suggesting that these skills should be targeted in early math education. Here we tested whether tools used by researchers to assess mathematical thinking could be useful in the classroom. This paper describes a collaborative project between cognitive scientists and school board researchers/educators implementing numeracy screeners with kindergarten students over the course of three school years. The Give‐N task (Wynn, 1990) was used with first‐year kindergarten students and the Numeracy Screener (Nosworthy, Bugden, Archibald, Evans, & Ansari, 2013) with second‐year kindergarten students. Results indicated that educators (N = 59) found the tools feasible to implement and helpful for exploring their students' thinking and targeting instruction. The educators' feedback also helped inform improvements to the implementation of the tools and future directions for both the schools and the researchers. This work emphasizes the importance of transdisciplinary collaboration to address the research‐practice gap. We investigated educators' experiences implementing numeracy assessment tools. Through a 3‐year collaboration between a research lab and school board, we learned that educators found the tools relatively easy to implement and useful for their teaching practice. Educator feedback was helpful for making improvements to the tools' implementation and future directions for the schools and the researchers. This work highlights the importance of collaboration between researchers and educators to address the gap between research and practice.

... e rst is subitizing, the ability to quickly recognize or name the number of a group arithmetic ability when they enter school (Geary et al., 2018). Several studies have shown that young schoolchildren's ability to compare symbolic quantities (quantities represented by numerals and number words) is one of the strongest predictors of their future mathematical development (Merkley and Ansari, 2016;Vanbinst et al., 2016). ...

Foundations of Academic Knowledge: This chapter assesses the acquisition of academic knowledge and skills in domains including literacy, numeracy, sciences, arts and physical education. It examines how learning trajectories arise from complex interactions between individual brain development and sociocultural environments. Teaching literacy and numeracy to all students is a goal of most school systems. While there are some fundamental skills children should grasp to succeed in these domains, the best way to support each student's learning varies depending on their individual development, language, culture and prior knowledge. Here we explore considerations for instruction and assessment in different academic domains. To accommodate the ourishing of all children, exibility must be built into education systems, which need to acknowledge the diverse ways in which children can progress through learning trajectories and demonstrate their knowledge.

... The contribution of the verbal and non-verbal components on learning mathematics is still an important research focus. Studies on the effect of each component of WM on learning arithmetic can contribute to a better understanding of the model and validation of the dissociated nature of its components, as well as to properly target cognitive training aimed at improving WM performance, with transfer of cognitive acquisitions to the learning process and school performance (Merkley & Ansari, 2016;Ofen, Yu, & Chen, 2016;Swanson, 2016). ...

Working memory (WM) is a predictor of school learning. This study aimed to investigate the predictive power of verbal and non-verbal working memory (WM) on students’ performance in arithmetic. 126 children between 6 and 11 years old participated in the research. The instruments were: School Performance Test, Raven’s Colored Progressive Matrices, Corsi Block-tapping Test, and Digits Subtest. The results showed strong and positive correlations of school performance with fluid intelligence r = 0.64, with verbal WM and non-verbal WM, both with r = 0.51 (p < 0.001). After multiple linear regression, it was found that the performance in visuospatial WM was a strong predictor for arithmetic, an effect not found for reading. The regression showed that WM explains 38% of the variance for arithmetic. It is concluded that WM has an expressive contribution to school performance, being more specific the contributions of visuospatial WM for arithmetic.

... The concept of mathematical competence is based on numerical knowledge and is currently understood as the ability to identify Arabic numbers and connect them with their respective quantities (Mou, Berteletti, & Hyde, 2018;Purpura, Baroody, & Lonigan, 2013). Mathematical competence has been divided into both formal and informal patterns of thought (Merkley & Ansari, 2016). The study of mathematical cognition has increased each year as educators and others have recognized the personal and social consequences associated with difficulties in learning mathematics . ...

https://rodin.uca.es/handle/10498/26368
There has been a substantial increase in research focused on numerical cognition in recent years. Although the notable growth in scientific productivity worldwide has focused on methods that might improve mathematical learning, mathematics is not yet perceived by all students as an accessible and enjoyable discipline. The study of mathematics is critical to academic success and can have a major impact on adaptation to everyday life. Given the relevance of early education and its impact on future development, advances in this research topic should be addressed using several different approaches. It is certainly essential to explore the cognitive profiles of students who are beginning in mathematical learning, as the intellectual development predicted by these cognitive processes may lead to improved methods of instruction. Studies that focus on variables that influence learning are also important. Among these variables, students' sociodemographics and/or attitudes towards mathematics may be associated with their mathematical development. However, while taking into account the cognitive basis of these findings, it will also be critical to encourage the development of instruments that promote student motivation and improved mathematical learning. The use of state-of-the-art technological devices that operate via the use of touch screens is an influential, accessible, and familiar means of interaction with students in their daily lives and provides an attractive option for the teaching of mathematics. The increased use of technology-mediated methodologies in both educational and domestic fields has encouraged the design of new and effective tools that may be used to improve student learning. This has also led to new methods for instruction on the appropriate use of these technologies and devices by young children in their homes. With this as a background, this study aimed to develop computer applications (APPs) that focused on student training based on our understanding of the cognitive basis of mathematical learning. These APPs were intended to be both didactic and enjoyable tools that can be used in early childhood education. Our goal was to promote the transfer of scientific research on mathematical learning via the development of new tools and to generate synergies with the children's entertainment industry.
The results of this study are consistent with findings that document the relevance of the general foundations on which mathematical learning is based and highlighted specific aspects of mathematics as needed to obtain adequate development. The results of our study reveal that touch screen devices and their APPs can be used to develop programs for early childhood education that are focused on the cognitive bases of numerical learning and that cognitive predictors can be used to introduce the appropriate ways of employing these devices in early childhood.
This technology can be applied both in the classroom and at home. The goal of these efforts is to improve mathematical competence among students regardless of their initial academic achievement.

... In symbolic representation, it is necessary to have the ability to make a correct and immediate identification of each of the numerical symbols represented. They must then contrast the quantities and decide if the number is larger or smaller (Merkley and Ansari, 2016). ...

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.

... From a theoretical point of view, the current study also adds to the understanding of the development of numerical cognition, an area of (neuro)cognitive research that has boomed in the last decade (e.g., Merkley & Ansari, 2016;Schneider et al., 2017) but that has relatively ignored the specific impact of preschool education (De Smedt, 2021), for a discussion). The existing developmental studies in numerical cognition did not separate the effects of preschool from effects of age in this developmental window. ...

There are massive developments in children’s early number skills in the ages 4- to 6-year old during which they attend preschool education and before they transition to formal school. We investigated to which extent these developments can be explained by children’ schooling experiences during preschool or by chronological age related maturational changes. In a secondary data-analysis of an existing longitudinal dataset, we compared children who were similar in age but different in the amount of preschool education (Old Year 2, n = 104, Mage = 62 months SDage 0.9 months vs. Young Year 3, n = 71, Mage = 65 months, SDage = 1.5 months) as well as children who were similar in the amount of preschool experience but differed in age (Young Year 3, n = 71, Mage = 65 months, SDage = 1.5 months vs. Old Year 3, n = 104, Mage = 74 months, SDage = 1.1 months). All children completed measures of numbering (verbal counting, dot enumeration, object counting), relations (number order, numeral identification, symbolic comparison, nonsymbolic comparison) and arithmetic operations (nonverbal calculation). We observed effects of preschool on object counting, numeral recognition and number order. There were also effects of chronological age on verbal counting, number order, numeral recognition and nonverbal calculation. The current data highlight which early number skills may be particularly malleable through schooling. They provide a more careful characterization of the potential factors that contribute to children’s early numerical competencies.

... At school entry, children already have quantitative competencies that are the foundations of further mathematical development ( Merkley & Ansari, 2016 ). For instance, they show a basic understanding of number symbols and the quantities represented by both number words and Arabic numerals, as well as their relations (e.g., more, less). ...

The acquisition of cardinal numbers represents a crucial milestone in the development of early numerical skills and more advanced math abilities. However, relatively few studies have investigated how children's grasping of the cardinality principle can be supported. It has been suggested that the richness of number inputs children receive influences the acquisition of cardinal numbers. The present study was designed to investigate whether canonical finger patterns representing numbers may contribute to this acquisition. Fifty-one 3-year-olds were randomly assigned to 1 of 2 training conditions: (a) a condition that involved counting and labeling, which has shown efficacy to support the acquisition of cardinality, and (b) a condition in which counting and labeling were enriched with finger patterns. Crucially, we aimed at providing evidence of both training programs in a real-life learning environment where teachers incorporated the training as a group-based activity into their regular schedule of daily activities. Children assigned to the finger-based condition outperformed those who received the counting-and-label training. Findings suggest that finger patterns may have a role in children's cardinality understanding. Furthermore, our study shows that instructional approaches for improving cardinality understanding can be easily and successfully implemented into real-life learning settings.

... Symbolic number skills are associated with the development of counting skills and the development of numeracy skills. Early symbolic number skills include counting sequence, numerical meanings of numbers, and the last number indicates the number of objects in the group when counting a group of objects (Gobel et al., 2014;Merkley & Ansari, 2016). Studies have concluded that the acquisition of early symbolic number skills in the preschool period significantly affects mathematics achievement in the first grade of primary school (Gobel et al., 2014;Jordan et al., 2009;Jordan et al., 2007). ...

Starting from the preschool period, children need to grow up as individuals with high academic skills, academic enablers and respond positively to their social situations. Academic skills and academic enablers together constitute academic competence. The positive reaction of children to the problems they face constitutes social information processing. This study aimed to examine the relationship between the academic competencies of 60–72-month-old children and their social information processing. The study was designed with the relational survey method. The study group consisted of 132 children aged 60–72 months with normal development who attend preschool education. The data collection tools of the study are as follows: Personal Information Form, The Social Information Processing Interview–Preschool Version, and Teacher Rating Scales of Early Academic Competence. Spearman's rank-order correlation test was used to evaluate the relationship between the scales. The findings of the study revealed that there is a relationship between the interpretation of cues and response decision, which are subdimensions of the social information processing model, academic skills (numeracy, early literacy, thinking skills, and comprehension) and academic enablers (social-emotional competence, approaches to learning, and communication).

... Numerical skills have shown to be strong predictors of attention, literacy, and decision-making (Merkley & Ansari, 2016), as well as of socioeconomic status and planning skills (Fernandez & Liu, 2019). Therefore, for being able to identify an individual's performance on numerical skills -and consequently other cognitive skills and abilities -it is important to reliably track processes connected to the development of numerical skills and their related performance. ...

... Second, intervention programs for dyscalculia should be tailored to the individual neurocognitive profile highlighted by proper in-depth assessment. For instance, a study suggests to incorporate numerical symbols into informal play activities at an earlier age to promote the numerical development and mediate between informal and formal mathematical competences (Merkley & Ansari, 2016). However, a child with deficits in calculations would not benefit from a training designed to strengthen the connection between the concept of magnitude and the symbolic representation of number (e.g., exercises on the number line) (Woods et al., 2018). ...

Developmental dyscalculia (DD) is an heterogenous neurodevelopmental learning disability that manifests as persistent difficulties in learning mathematics. DD can occur in isolation but is often diagnosed as a co-occurring difficulty in children with language-based learning disabilities. Basic cognitive and neuroimaging findings suggest different subtypes of dyscalculia exist. However, a comprehensive theoretical framework that provides accepted terminology and clinical criteria to design appropriate interventions is still lacking. We developed a comprehensive battery of cognitive tests, the UCSF Dyscalculia Subtyping Battery (DSB), aiming at identifying deficits in four distinct mathematical domains: number processing, arithmetical procedures, arithmetic facts retrieval, and geometrical abilities. The mathematical abilities of a cohort of 75 children aged 7 to 16, referred to the UCSF Dyslexia Center for a language-based neurodevelopmental disorder, were initially evaluated using a behavioral neurology approach. A team of professional clinicians classified children with difficulties in mathematics in four groups, depending on their parents’ and teachers’ reported symptoms and clinical history, in one of the following domains: number processing, arithmetical procedures, arithmetic facts retrieval and geometrical abilities. The 75 children and 18 typically developing control children were then evaluated with the DSB to identify which subtests of the battery better represented each group. We describe the detailed profiles of four cases, each of them representative of deficits in one of the four domains, and report the pattern of impairment in the overall cohort. Our results show that a neuroscience-based DD evaluation battery enables identification of subtypes acknowledging the multidimensional nature of the disorder. If corroborated in large samples, these findings can pave the way for novel diagnostic approaches, consistent subtype classification, and ultimately personalized interventions.

... Learning mathematics is a hierarchical process in which the acquisition of basic concepts are the building blocks for more advanced concepts and new concepts logically follow from prior ones (Hiebert, 1988;Núñez, 2017;Siegler & Lortie-Forgues, 2014;Xu, Gu, Newman, & LeFevre, 2019). Students initially learn to use numerals to represent cardinal, ordinal, and arithmetic associations (Lyons, Vogel, & Ansari, 2016;Merkley & Ansari, 2016;Sasanguie & Vos, 2018;Xu & LeFevre, 2021). As they add to their hierarchy of mathematical symbol knowledge, students integrate more advanced and abstract associations such as rational numbers (Booth & Newton, 2012;Douglas, Headley, Hadden, & LeFevre, 2020). ...

How do whole number arithmetic skills support students’ understanding of fraction magnitude during the emerging stages of fraction learning? Chinese students in Grade 4 (N = 1038; Mage = 9.9 years; 55.6% boys) completed assessments of whole number arithmetic skills (i.e., addition, subtraction, multiplication, and division), fraction mapping (i.e., connecting visual fraction representations to fraction notations), and fraction comparison (i.e., comparing magnitudes of fraction symbols). We found that division skills uniquely differentiated students who had a basic understanding of fraction notation (mappers) from students with no understanding of fraction notation (non-mappers). Furthermore, we found that division mediated the relations between all three other arithmetic operations (i.e., addition, subtraction, and multiplication) and fraction mapping performance for the mappers. For fraction comparison, there was evidence of the whole number bias for the majority of students. The current results highlight the importance of the mastery of division skills and its dominance in predicting individual differences in fraction mapping for Chinese students in Grade 4.

... Mathematical development is hierarchical, with more basic numerical associations leading to the development of more complex mathematical concepts (Merkley & Ansari, 2016;Núñez, 2017;. Within this hierarchy, quantitative skills are foundational for numerical tasks. ...

Canadian students enrolled in either French-immersion or English-instruction programs were followed from Grades 2 to 3 (Mage = 7.8 years to 8.9 years; N = 244; 55% girls). In each grade, students completed two mathematical tasks that required oral language processing (i.e., word-problem solving and number transcoding from dictation) and two that did not (i.e., arithmetic fluency and number line estimation). Students in both English-instruction (n = 92) and French-immersion programs (n = 152) completed tasks in English. Students in French-immersion programs also completed word-problem solving and transcoding tasks in French. The models were framed within the Pathways to Mathematics model, with a focus on the linguistic pathways for students in English-instruction and French-immersion programs. For tasks with oral language processing, performance in Grade 3 was predicted by students’ English receptive vocabulary for both English-instruction and French-immersion students, even when
French-immersion students were tested in French, controlling for performance in Grade 2. In contrast, for tasks without oral language processing, receptive vocabulary in either English or French did not predict performance in Grade 3, controlling for performance in Grade 2. These results have implications for teaching mathematics within the context of immersion education.

... However, other work suggests that children acquire the meaning of verbal number knowledge by learning relations between words in the count list and that mappings to an ANS occurs later (e.g., Carey & Barner, 2019;Le Corre & Carey, 2007). Additional recent theoretical work, building on these ideas, suggests that learning number words may involve bidirectional processes rather than a unidirectional mapping of symbols onto pre-existing nonsymbolic representations of number (Barner, 2017;Merkley & Ansari, 2016). Specifically, learning number words, perhaps through relations between number words, might direct children's attention to discrete numerosity as the relevant dimension of sets (Merkley et al., 2017;Mix et al., 2016). ...

Which dimension of a set of objects is more salient to young children: number or size? The 'Build-A-Train' task was developed and used to examine whether children spontaneously use a number or physical size approach on an un-cued matching task. In the Build-A-Train task, an experimenter assembles a train using one to five blocks of a particular length and asks the child to build the same train. The child's blocks differ in length from the experimenter's blocks, causing the child to build a train that matches based on either the number of blocks or length of the train, as it is not possible to match on both. One hundred and nineteen children between 2 years 2 months and 6 years 0 months of age (M = 4.05, SD = 0.84) completed the Build-A-Train task, and the Give-a-Number task, a classic task used to assess children's conceptual knowledge of verbal number words. Across train lengths and verbal number knowledge levels, children used a number approach more than a size approach on the Build-A-Train task. However, children were especially likely to use a number approach over a size approach when they knew the verbal number word that corresponded to the quantity of blocks in the train, particularly for quantities smaller than four. Therefore, children's attention to number relates to their knowledge of verbal number words. The Build-A-Train task and findings from the current study set a foundation for future longitudinal research to investigate the causal relationship between children's acquisition of symbolic mathematical concepts and attention to number.

... Although the Woodcock-Johnson is a widely used assessment in the field, the types of skills it assesses may be limited, particularly for children at age 54 months. For example, the Woodcock-Johnson provides limited information about children's non-symbolic, symbolic, spatial, geometry and magnitude skills as discrete components; each of which is likely an important contributor to mathematical development (Merkley & Ansari, 2016;Purpura & Simms, 2018;Siegler, 2016). As a result, our analyses and discussion focus on numeracy, rather than mathematical skills more generally. ...

Using data from the Applied Problems subtest of the Woodcock‐Johnson Tests of Achievement (Woodcock & Johnson, 1989/1990, Woodcock‐Johnson psycho‐educational battery‐revised. Allen, TX: DLM Teaching Resources) administered to 1,364 children from the National Institute of Child Health and Human Development (NICHD) Study of Early Childcare and Youth Development (SECCYD), this study measures children's mastery of three numeric competencies (counting, concrete representational arithmetic and abstract arithmetic operations) at 54 months of age. We find that, even after controlling for key demographic characteristics, the numeric competency that children master prior to school entry relates to important educational transitions in secondary and post‐secondary education. Those children who showed low numeric competency prior to school entry enrolled in lower math track classes in high school and were less likely to enrol in college. Important numeracy competency differences at age 54 months related to socioeconomic inequalities were also found. These findings suggest that important indicators of long‐term schooling success (i.e., advanced math courses, college enrollment) are evident prior to schooling based on the levels of numeracy mastery.

... In other studies with preschool participants, ANS measures correlate significantly with tasks assessing numeral knowledge (e.g., rs = .16 -.36; Merkley & Ansari, 2016;Mussolin et al., 2012;vanMarle et al., 2014). Indeed, one previous study found that 3-to-5-year-old children's numeral knowledge (along with verbal counting and cardinality understanding) mediated the relation between their fall ANS and spring mathematics achievement (vanMarle et al., 2014). ...

... For word-problem solving, the results of the present research supported the patterns of relations in the Pathways to Mathematics model (LeFevre et al., 2010): As shown in Table 6, all of the cognitive predictors were related to word-problem solving for both groups of learners. Quantitative skills draw on a student's ability to develop associative networks of mathematical knowledge that are involved in calculation (Merkley & Ansari, 2016;Núñez, 2017;Xu & LeFevre, 2021). Working memory may be important for extracting the meaning of the text and for maintaining intermediate calculations during problem solving (Fuchs et al., 2006;Raghubar et al., 2010;Swanson, 2011). ...

Language skills play an important role in mathematics development. Students (7 to 10 years of age) learning school mathematics either in the same language used at home (first-language learners; n = 103) or in a different language (second-language learners; n = 57) participated in the study. Relations among cognitive skills (i.e., receptive vocabulary, working memory, quantitative skills), domain-specific language skills (i.e., mathematical vocabulary, mathematical orthography), word-problem solving, arithmetic fluency, and word reading were investigated. Second-language learners had lower scores on measures with strong language components (i.e., receptive vocabulary, subitizing, and word-problem solving) than first-language learners, whereas they performed equally well on other tasks. Mathematical vocabulary and receptive vocabulary contributed to word-problem solving success for first-language learners, whereas only receptive vocabulary in the language of instruction related to mathematical outcomes for second-language learners. Mathematical vocabulary was related to arithmetic fluency for both groups, but mathematical orthography was not. For both groups, students’ word reading was predicted by receptive vocabulary but not by quantitative skills, highlighting the domain-specific nature of these skills. These findings have implications for supporting mathematical learning in second-language students.

... There is influential evidence that a strong foundation in the early years can help to promote children's mathematical development in the subsequent years (Claessens and Engels, 2013;Garon-Carrier et al., 2018;Göbel, Watson, Lervåg and Hulme, 2014;Nguyen et al., 2016;Rittle-Johnson, Fyfe, Hofer and Farran, 2017;Watts, Duncan, Siegler and Davis-Kean, 2014). Of the early math skills, the number skills are acknowledged as the most significant factor which affects math achievement in the forthcoming grades (Aunio and Niemivirta, 2010;Chu, vanMarle, Rouder and Geary, 2018;Garon-Carrier et al., 2018;Hawes, Nosworthy, Archibald and Ansari, 2019;Jordan, Glutting and Ramineni, 2010;Jordan, Kaplan, Ramineni and Locuniak, 2009;Marcelino, de Sousa and Lopes, 2017;Merkley and Ansari, 2016). ...

It is acknowledged that, of the math skills, the number skills are the most significant factor which affects math achievement in the forthcoming grades. Thus, it is critical to support the development of number skills early in life. The present study examines whether supplementing a global curriculum with the Big Math for Little Kids (BMLK) affects the growth of children’s number skills. Pretest-posttest experimental design was used. Seventy-seven kindergarten children (38 experimental; 39 comparison) participated in the study. For six weeks, children in the experimental group were exposed to the Ministry of National Education (MoNE) program plus the BMLK while those in the comparison group only experienced the MoNE program. Children’s number skills were measured by the Anatolian Early Childhood Mathematics Skills Scale (ANOMAT). Findings indicated that children who were exposed to the global MoNE curriculum supplemented with the BMLK had greater gains than did those who experienced only the MoNE curriculum. The results indicate that a global curriculum supplemented with a skill-based curriculum has a positive impact on children’s number skills.

This study examines the longitudinal relationships between home learning experiences and early number skills. The counting, number transcoding and calculation skills of 274 children were assessed in the penultimate term of preschool (Mage=4:0). Prior to these assessments, parents completed questionnaires that surveyed the frequency of the children's home learning experiences. Three types of experiences were indexed: code-focused home literacy experiences that focus on the phonological and orthographic features of language, meaning-focused home literacy experiences that focus on sharing the meaning of language and text, and home number experiences. The children's language abilities (phonological awareness and vocabulary) and nonverbal abilities (inhibitory control and nonverbal reasoning) were assessed in the final term of preschool (Mage=4:3). Their number skills were reassessed in the final term of the first year of primary school (Mage=5:3). Home letter-sound interaction experiences (interactive code-focused literacy experiences) had significant longitudinal relationships with counting and number transcoding that were independent of language and nonverbal abilities. The relationship between letter-sound interaction experiences and later counting was also independent of the autoregressive influence of baseline counting ability. We extend previous findings by demonstrating that interactive code-focused home literacy experiences in the preschool period predict growth in counting skills even when a broad range of language and cognitive abilities are controlled.

How do non-experts single out numbers for reference? Linnebo has argued that they do so using a criterion of identity based on the ordinal properties of numerals. Neo-logicists, on the other hand, claim that cardinal properties are the basis of individuation, when they invoke Hume’s Principle. I discuss empirical data from cognitive science and linguistics to answer how non-experts individuate numbers better in practice. I use those findings to develop an alternative account that mixes ordinal and cardinal properties to provide a detailed (though not conclusively proven) answer to the question: how do we in fact semantically individuate numbers?

Mathematics skills relate to lifelong career, health and financial outcomes. Individuals’ cognitive abilities predict mathematics performance and there is growing recognition that environmental influences, including differences in culture and variability in mathematics engagement, also affect mathematics performance. In this Review, we summarize evidence indicating that differences between languages, exposure to maths-focused language, socioeconomic status, attitudes and beliefs about mathematics, and engagement with mathematics activities influence young children’s mathematics performance. These influences play out at the community and individual levels. However, research on the role of these environmental influences for foundational number skills, including understanding of number words, is limited. Future research is needed to understand individual differences in the development of early emerging mathematics skills such as number word skills, examining to what extent different types of environmental input are necessary and how children’s cognitive abilities shape the impact of environmental input. Children’s individual abilities and environment influence their mathematics skills. In this Review, Silver and Libertus examine how language, socioeconomic status and other environmental factors influence mathematics skills across childhood, with a focus on number word acquisition.

Preschool number sense can be operationalized as three interconnected strands — number, number relations, and number operations. These strands involve key ideas that are foundational to mathematics education. Recent cognitive and behavioral research refines and extends our understanding of the early number sense framework in the following ways: (1) Although number sense can be viewed as a single construct, each strand predicts achievement when controlling for the others, and the strands appear to reinforce each other during development; (2) Level of representation (i.e. nonsymbolic versus symbolic) and set size affect children’s competencies and development within and across strands and should be considered in intervention research; (3) There are substantial individual differences in preschoolers’ number sense knowledge. We argue that instruction must weave together these number sense strands from the start of preschool to prepare children for success in formal mathematics.

Neurocognitive factors, including information integration and executive functioning, contribute significantly to a child’s early success in math achievement, even though the significance of home and school environments cannot be ignored. There are only a few studies that have systematically examined how information integration and executive function skills impact different aspects of learning math and math achievement. Using a comprehensive tool such as the brain-Based Intelligence Test (BBIT), a brain-based comprehensive approach to the understanding of cognition, for the assessment of information integration and executive function skills can have significant implications for mathematical education and remediation (brain plasticity).

In this article, we tested, using a 1-year longitudinal design, whether symbolic numerical magnitude processing or children's numerical representation of Arabic digits, is as important to arithmetic as phonological awareness is to reading. Children completed measures of symbolic comparison, phonological awareness, arithmetic, reading at the start of third grade and the latter two were retested at the start of fourth grade. Cross-sectional and longitudinal correlations indicated that symbolic comparison was a powerful domain-specific predictor of arithmetic and that phonological awareness was a unique predictor of reading. Crucially, the strength of these independent associations was not significantly different. This indicates that symbolic numerical magnitude processing is as important to arithmetic development as phonological awareness is to reading and suggests that symbolic numerical magnitude processing is a good candidate for screening children at risk for developing mathematical difficulties.

Children show individual differences in their tendency to focus on the numerical aspects of their environment. These individual differences in 'Spontaneous Focusing on Numerosity' (SFON) have been shown to predict both current numerical skills and later mathematics success. Here we investigated possible factors which may explain the positive relationship between SFON and symbolic number development. Children aged 4-5 years (N = 130) completed a battery of tasks designed to assess SFON and a range of mathematical skills. Results showed that SFON was positively associated with children's symbolic numerical processing skills and their performance on a standardised test of arithmetic. Hierarchical regression analyses demonstrated that the relationship between SFON and symbolic mathematics achievement can be explained, in part, by individual differences in children's nonsymbolic numerical processing skills and their ability to map between nonsymbolic and symbolic representations of number.

This article focuses on how young children acquire concepts for exact, cardinal numbers (e.g., three, seven, two hundred, etc.). I believe that exact numbers are a conceptual structure that was invented by people, and that most children acquire gradually, over a period of months or years during early childhood. This article reviews studies that explore children’s number knowledge at various points during this acquisition process. Most of these studies were done in my own lab, and assume the theoretical framework proposed by Carey (The origin of concepts, 2009). In this framework, the counting list (‘one,’ ‘two,’ ‘three,’ etc.) and the counting routine (i.e., reciting the list and pointing to objects, one at a time) form a placeholder structure. Over time, the placeholder structure is gradually filled in with meaning to become a conceptual structure that allows the child to represent exact numbers (e.g., There are 24 children in my class, so I need to bring 24 cupcakes for the party.) A number system is a socially shared, structured set of symbols that pose a learning challenge for children. But once children have acquired a number system, it allows them to represent information (i.e., large, exact cardinal values) that they had no way of representing before.

Many studies have investigated the association between numerical magnitude processing skills, as assessed by the numerical magnitude comparison task, and broader mathematical competence, e.g. counting, arithmetic, or algebra. Most correlations were positive but varied considerably in their strengths. It remains unclear whether and to what extent the strength of these associations differs systematically between non-symbolic and symbolic magnitude comparison tasks and whether age, magnitude comparison measures or mathematical competence measures are additional moderators. We investigated these questions by means of a meta-analysis. The literature search yielded 45 articles reporting 284 effect sizes found with 17.201 participants. Effect sizes were combined by means of a two-level random-effects regression model. The effect size was significantly higher for the symbolic (r = .302, 95% CI [.243, .361]) than for the non-symbolic (r = .241, 95% CI [.198, .284]) magnitude comparison task and decreased very slightly with age. The correlation was higher for solution rates and Weber fractions than for alternative measures of comparison proficiency. It was higher for mathematical competencies that rely more heavily on the processing of magnitudes (i.e. mental arithmetic and early mathematical abilities) than for others. The results support the view that magnitude processing is reliably associated with mathematical competence over the lifespan in a wide range of tasks, measures and mathematical subdomains. The association is stronger for symbolic than for non-symbolic numerical magnitude processing. So symbolic magnitude processing might be a more eligible candidate to be targeted by diagnostic screening instruments and interventions for school aged children and adults.

This seven-year longitudinal study examined how children’s spontaneous focusing on numerosity (SFON), subitizing based enumeration, and counting skills assessed at five or six years predict their school mathematics achievement at 12 years. The participants were 36 Finnish children without diagnosed neurological disorders. The results, based on partial least squares modeling, demonstrate that SFON and verbal counting skills before school age predict mathematical performance on a standardized test for typical school mathematics in Grade 5. After controlling for nonverbal IQ, only SFON predict school mathematics. Subitizing-based enumeration skills have an indirect effect via number sequence skills and SFON on mathematical performance at 12 years. Early mathematic skills do not predict reading skills at 12 years. Children’s early numerical skills, including SFON, before school age are important contributors to substantially later success in school mathematics.

The way the human brain constructs representations of numerical symbols is poorly understood. While increasing evidence from neuroimaging studies has indicated that the intraparietal sulcus (IPS) becomes increasingly specialized for symbolic numerical mag-nitude representation over developmental time, the extent to which these changes are associated with age-related differences in symbolic numerical magnitude representation or with developmental changes in non-numerical processes, such as response selection, remains to be uncovered. To address these outstanding questions we investigated devel-opmental changes in the cortical representation of symbolic numerical magnitude in 6-to 14-year-old children using a passive functional magnetic resonance imaging adapta-tion design, thereby mitigating the influence of response selection. A single-digit Arabic numeral was repeatedly presented on a computer screen and interspersed with the pre-sentation of novel digits deviating as a function of numerical ratio (smaller/larger number). Results demonstrated a correlation between age and numerical ratio in the left IPS, sug-gesting an age-related increase in the extent to which numerical symbols are represented in the left IPS. Brain activation of the right IPS was modulated by numerical ratio but did not correlate with age, indicating hemispheric differences in IPS engagement during the development of symbolic numerical representation. article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).

Although everyone perceives approximate numerosities, some people make more accurate estimates than others. The accuracy of this estimation is called approximate number system (ANS) acuity. Recently, several studies have reported that individual differences in young children's ANS acuity are correlated with their knowledge of exact numbers such as the word ‘six’ (Mussolin et al., 2012, Trends Neurosci. Educ., 1, 21; Shusterman et al., 2011, Connecting early number word knowledge and approximate number system acuity; Wagner & Johnson, 2011, Cognition, 119, 10; see also Abreu-Mendoza et al., 2013, Front. Psychol., 4, 1). This study argues that this correlation should not be trusted. It seems to be an artefact of the procedure used to assess ANS acuity in children. The correlation arises because (1) some experimental designs inadvertently allow children to answer correctly based on the size (rather than the number) of dots in the display and/or (2) young children with little exact-number knowledge may not understand the phrase ‘more dots’ to mean numerically more. When the task is modified to make sure that children respond on the basis of numerosity, the correlation between ANS acuity and exact-number knowledge in normally developing children disappears.

An ongoing debate in research on numerical cognition concerns the extent to which the approximate number system and symbolic number knowledge influence each other during development. The current study aims at establishing the direction of the developmental association between these two kinds of abilities at an early age. Fifty-seven children of 3-4 years performed two assessments at 7 months interval. In each assessment, children's precision in discriminating numerosities as well as their capacity to manipulate number words and Arabic digits was measured. By comparing relationships between pairs of measures across the two time points, we were able to assess the predictive direction of the link. Our data indicate that both cardinality proficiency and symbolic number knowledge predict later accuracy in numerosity comparison whereas the reverse links are not significant. The present findings are the first to provide longitudinal evidence that the early acquisition of symbolic numbers is an important precursor in the developmental refinement of the approximate number representation system.

The study assessed the relations among acuity of the inherent approximate number system (ANS), performance on measures of symbolic quantitative knowledge, and mathematics achievement for a sample of 138 (64 boys) preschoolers. The Weber fraction (a measure of ANS acuity) and associated task accuracy were significantly correlated with mathematics achievement following one year of preschool, and predicted performance on measures of children's explicit knowledge of Arabic numerals, number words, and cardinal value, controlling for age, sex, parental education, intelligence, executive control, and preliteracy knowledge. The relation between ANS acuity, as measured by the Weber fraction and task accuracy, and mathematics achievement was fully mediated by children's performance on the symbolic quantitative tasks, with knowledge of cardinal value emerging as a particularly important mediator. The overall pattern suggests that ANS acuity facilitates the early learning of symbolic quantitative knowledge and indirectly influences mathematics achievement through this knowledge.

We examined whether a theoretically based number board game could be translated into a practical classroom activity that improves Head Start children's numerical knowledge. Playing the number board game as a small group learning activity promoted low-income children's number line estimation, magnitude comparison, numeral identification, and counting. Improvements were also found when a paraprofessional from the children's classroom played the game with the children. Observations of the game-playing sessions revealed that paraprofessionals adapted the feedback they provided to individual children's improving numerical knowledge over the game-playing sessions and that children remained engaged in the board game play after multiple sessions. These findings suggest that the linear number board game can be used effectively in the classroom context.

The present study assessed the relationships between approximate and exact number abilities in children with little formal instruction to ask (1) whether individual differences in acuity of the approximate system are related to basic abilities with symbolic numbers; and (2) whether the link between non-symbolic and symbolic number performance changes over the development. To address these questions, four different age groups of 3- to 6-year-old children were asked to compare pairs of train wagons varying on numerical ratio, as well as to complete exact tasks including number words or Arabic numbers. When correlation analyses were conducted across age groups, results indicated that performance in numerosity comparison was associated with mastery of symbolic numbers, even when short-term memory, IQ and age were controlled for. Separate analyses by age group revealed that the precision in numerosity discrimination was related to both number word and Arabic number knowledge but differently across the development.

This study investigates the influence of aspects of home and preschool environments upon literacy and numeracy achievement at school entry and at the end of the 3rd year of school. Individuals with unexpected performance pathways (by forming demographically adjusted groups: overachieving, average, and underachieving) were identified in order to explore the effects of the home learning environment and preschool variables on child development. Multilevel models applied to hierarchical data allow the groups that differ with regard to expected performance to be created at the child and preschool center levels. These multilevel analyses indicate powerful effects for the home learning environment and important effects of specific preschool centers at school entry. Although reduced, such effects remain several years later.

Two studies were conducted to investigate, firstly, children's focusing on the aspect of numerosity in utilizing enumeration in action, and, secondly, whether children's Spontaneous FOcusing on Numerosity (SFON) is related to their counting development. The longitudinal data of 39 children from the age of 3.5 to 6 years showed individual differences in SFON, as well as stability in children's SFON across tasks during the follow-up. Path analyses indicated a reciprocal relationship between SFON and counting development. The results were confirmed by a cross-sectional study of 183 6.5-year-old children when the effects of non-verbal IQ, verbal comprehension and lacking enumeration and procedural skills were controlled.

Preschool and primary grade children have the capacity to learn substantial mathematics, but many children lack opportunities
to do so. Too many children not only start behind their more advantaged peers, but also begin a negative trajectory in mathematics.
Interventions designed to facilitate their mathematical learning during ages 3 to 5 years have a strong positive effect on
these children’s lives for many years thereafter.

Using 6 longitudinal data sets, the authors estimate links between three key elements of school readiness--school-entry academic, attention, and socioemotional skills--and later school reading and math achievement. In an effort to isolate the effects of these school-entry skills, the authors ensured that most of their regression models control for cognitive, attention, and socioemotional skills measured prior to school entry, as well as a host of family background measures. Across all 6 studies, the strongest predictors of later achievement are school-entry math, reading, and attention skills. A meta-analysis of the results shows that early math skills have the greatest predictive power, followed by reading and then attention skills. By contrast, measures of socioemotional behaviors, including internalizing and externalizing problems and social skills, were generally insignificant predictors of later academic performance, even among children with relatively high levels of problem behavior. Patterns of association were similar for boys and girls and for children from high and low socioeconomic backgrounds.

Research Findings: Big Math for Little Kids (BMLK) is a mathematics curriculum developed for use with 4- and 5-year-old children. To investigate the BMLK curriculum's effect on children's mathematics knowledge, this cluster-randomized controlled trial randomly assigned child care centers to provide mathematics instruction to children, using either the BMLK mathematics curriculum or the centers’ business-as-usual curriculum, over a 2-year period when children were in prekindergarten and kindergarten. Participants in the study were 762 children and their teachers at 16 publicly subsidized child care centers. The study assessed children's mathematics knowledge using the Early Childhood Longitudinal Study, Birth Cohort (ECLS-B), Direct Mathematics Assessment, a measure of young children's mathematics knowledge that is not aligned with the curriculum. The ECLS-B scores of children in the BMLK group increased significantly more than did those of children in the comparison group. The study also included exploratory analyses to examine whether children in the BMLK group demonstrated evidence of improved mathematical language. Practice or Policy: These results indicate that the BMLK curriculum, which is designed to help teachers use play-based, developmentally appropriate mathematics instruction, has a positive impact on young children's mathematics knowledge as measured by a general mathematics assessment that is not aligned with the curriculum.

How do numerical symbols, such as number words, acquire semantic meaning? This question, also referred to as the "symbol-grounding problem," is a central problem in the field of numerical cognition. Present theories suggest that symbols acquire their meaning by being mapped onto an approximate system for the nonsymbolic representation of number (Approximate Number System or ANS). In the present literature review, we first asked to which extent current behavioural and neuroimaging data support this theory, and second, to which extent the ANS, upon which symbolic numbers are assumed to be grounded, is numerical in nature. We conclude that (a) current evidence that has examined the association between the ANS and number symbols does not support the notion that number symbols are grounded in the ANS and (b) given the strong correlation between numerosity and continuous variables in nonsymbolic number processing tasks, it is next to impossible to measure the pure association between symbolic and nonsymbolic numerosity. Instead, it is clear that significant cognitive control resources are required to disambiguate numerical from continuous variables during nonsymbolic number processing. Thus, if there exists any mapping between the ANS and symbolic number, then this process of association must be mediated by cognitive control. Taken together, we suggest that studying the role of both cognitive control and continuous variables in numerosity comparison tasks will provide a more complete picture of the symbol-grounding problem. (PsycINFO Database Record

Recent work has demonstrated that how we process the relative order-ordinality-of numbers may be key to understanding how we represent numbers symbolically, and has proven to be a robust predictor of more sophisticated math skills in both children and adults. However, it remains unclear whether numerical ordinality is primarily a by-product of other numerical processes, such as familiarity with overlearned count sequence, or is in fact a fundamental property of symbolic number processing. In a sample of nearly 1,500 children, we show that the reversed distance effect-a hallmark of symbolic ordinal processing-obtains in children as young as first grade, and is larger for less familiar sets of numbers. Furthermore, we show that the children's efficiency in evaluating the simplest ordered sequences (e.g., 2-3-4, 6-7-8) captures more unique variance in mental arithmetic than any other type of numerical sequence, and that this result cannot be accounted for by counting ability. Indeed, performance on just five such trials captured more unique mental arithmetic variance than any of several other numerical tasks assessed here. In sum, our results are consistent with the notion that ordinality is a fundamental property of how children process numerical symbols, that this property helps underpin more complex math processing, and that it shapes numerical processing even at the earliest stages of elementary education. © 2015 International Mind, Brain, and Education Society and Blackwell Publishing, Inc.

Humans are born with the ability to mentally represent the approximate numerosity of a set of objects, but little is known about the brain systems that sub-serve this ability early in life and their relation to the brain systems underlying symbolic number and mathematics later in development. Here we investigate processing of numerical magnitudes before the acquisition of a symbolic numerical system or even spoken language, by measuring the brain response to numerosity changes in pre-verbal infants using functional near-infrared spectroscopy (fNIRS). To do this, we presented infants with two types of numerical stimulus blocks: number change blocks that presented dot arrays alternating in numerosity and no change blocks that presented dot arrays all with the same number. Images were carefully constructed to rule out the possibility that responses to number changes could be due to non-numerical stimulus properties that tend to co-vary with number. Interleaved with the two types of numerical blocks were audio-visual animations designed to increase attention. We observed that number change blocks evoked an increase in oxygenated hemoglobin over a focal right parietal region that was greater than that observed during no change blocks and during audio-visual attention blocks. The location of this effect was consistent with intra-parietal activity seen in older children and adults for both symbolic and non-symbolic numerical tasks. A distinct set of bilateral occipital and middle parietal channels responded more to the attention-grabbing animations than to either of the types of numerical stimuli, further dissociating the specific right parietal response to number from a more general bilateral visual or attentional response. These results provide the strongest evidence to date that the right parietal cortex is specialized for numerical processing in infancy, as the response to number is dissociated from visual change processing and general attentional processing.

Numerical ratio effects are a hallmark of numerical comparison tasks. Moreover, ratio effects have been used to draw strong conclusions about the nature of numerical representations, how these representations develop, and the degree to which they generalize across stimulus formats. Here, we compute ratio effects for 1,719 children from Grades K-6 for each individual separately by computing not just the average ratio effect for each person, but also the variability and statistical magnitude (effect-size) of their ratio effect. We find that individuals' ratio effect-sizes in fact increase over development, calling into question the view that decreasing ratio effects over development indicate increasing representational precision. Our data also strongly caution against the use of ratio effects in inferring the nature of symbolic number representation. While 75% of children showed a statistically significant ratio effect for nonsymbolic comparisons, only 30% did so for symbolic comparisons. Furthermore, whether a child's nonsymbolic ratio effect was significant did not predict whether the same was true of their symbolic ratio effect. These results undercut the notions (a) that individuals' ratio effects are indicative of representational precision in symbolic numbers, and (b) that a common process generates ratio effects in symbolic and nonsymbolic formats. Finally, for both formats, it was the variability of an individual child's ratio effect (not its slope or even effect-size) that correlated with arithmetic ability. Taken together, these results call into question many of the long-held tenets regarding the interpretation of ratio effects-especially with respect to symbolic numbers. (PsycINFO Database Record
(c) 2015 APA, all rights reserved).

When children learn to count, they map newly acquired symbolic representations of number onto preexisting nonsymbolic representations. The nature and timing of this mapping is currently unclear. Some researchers have suggested this mapping process helps children understand the cardinal principle of counting, while other evidence suggests that this mapping only occurs once children have cardinality understanding. One difficulty with the current literature is that studies have employed tasks that only indirectly assess children’s nonsymbolic-symbolic mappings. We introduce a task in which preschoolers made magnitude comparisons across representation formats (e.g., dot arrays vs. verbal number), allowing a direct assessment of mapping. We gave this task to 60 children aged 2;7-4;10, together with counting and Give-a-Number tasks. We found that some children could map between nonsymbolic quantities and the number words of which they understood the cardinal meaning, even if they had yet to grasp the general cardinality principle of counting.

This is an Accepted Manuscript of an article published by Taylor & Francis Group in Mathematical Thinking and Learning on 7/05/2015, available online: http://www.tandfonline.com/10.1080/10986065.2015.1016810.

Research Findings: Big Math for Little Kids (BMLK) is a mathematics curriculum developed for use with 4- and 5-year-old children. To investigate the BMLK curriculum's effect on children's mathematics knowledge, this cluster-randomized controlled trial randomly assigned child care centers to provide mathematics instruction to children, using either the BMLK mathematics curriculum or the centers’ business-as-usual curriculum, over a 2-year period when children were in prekindergarten and kindergarten. Participants in the study were 762 children and their teachers at 16 publicly subsidized child care centers. The study assessed children's mathematics knowledge using the Early Childhood Longitudinal Study, Birth Cohort (ECLS-B), Direct Mathematics Assessment, a measure of young children's mathematics knowledge that is not aligned with the curriculum. The ECLS-B scores of children in the BMLK group increased significantly more than did those of children in the comparison group. The study also included exploratory analyses to examine whether children in the BMLK group demonstrated evidence of improved mathematical language. Practice or Policy: These results indicate that the BMLK curriculum, which is designed to help teachers use play-based, developmentally appropriate mathematics instruction, has a positive impact on young children's mathematics knowledge as measured by a general mathematics assessment that is not aligned with the curriculum.

The present study observed and coded instruction in 65 preschool classrooms to examine (a) overall amounts and (b) types of mathematics and science learning opportunities experienced by preschool children as well as (c) the extent to which these opportunities were associated with classroom and program characteristics. Results indicated that children were afforded an average of 24 and 26 minutes of mathematics and science learning opportunities, respectively, corresponding to spending approximately 25% of total instructional time in each domain. Considerable variability existed, however, in the amounts and types of mathematics and science opportunities provided to children in their classrooms; to some extent, this variability was associated with teachers' years of experience, teachers' levels of education, and the socioeconomic status of children served in the program.
Although results suggest greater integration of mathematics and science in preschool classrooms than previously established, there was considerable diversity in the amounts and types of learning opportunities provided in preschool classrooms. Affording mathematics and science experiences to all preschool children, as outlined in professional and state standards, may require additional professional development aimed at increasing preschool teachers' understanding and implementation of learning opportunities in these two domains in their classrooms.

Human children possess the ability to approximate numerical quantity nonverbally from a young age. Over the course of early childhood, children develop increasingly precise representations of numerical values, including a symbolic number system that allows them to conceive of numerical information as Arabic numerals or number words. Functional brain imaging studies of adults report that activity in bilateral regions of the intraparietal sulcus (IPS) represents a key neural correlate of numerical cognition. Developmental neuroimaging studies indicate that the right IPS develops its number-related neural response profile more rapidly than the left IPS during early childhood. One prediction that can be derived from previous findings is that there is longitudinal continuity in the number-related neural responses of the right IPS over development while the development of the left IPS depends on the acquisition of numerical skills. We tested this hypothesis using fMRI in a longitudinal design with children ages 4 to 9. We found that neural responses in the right IPS are correlated over a 1–2-year period in young children whereas left IPS responses change systematically as a function of children's numerical discrimination acuity. The data are consistent with the hypothesis that functional properties of the right IPS in numerical processing are stable over early childhood whereas the functions of the left IPS are dynamically modulated by the development of numerical skills.

The purpose of the present study was to determine if numeral knowledge—the ability to identify Arabic numerals and connect Arabic numerals to their respective quantities—mediates the relation between informal and formal mathematical knowledge. A total of 206 3- to 5-year-old preschool children were assessed on 6 informal mathematics tasks and 2 numeral knowledge tasks. A year later, these children were assessed on 2 measures of formal mathematical knowledge, namely, the Woodcock-Johnson III Calculation Subtest and a formal number combinations task. Mediation analyses revealed that the relation between informal and formal mathematical knowledge is fully mediated by numeral knowledge, but only when both the skill of numeral identification and an understanding of numeral to quantity relations are considered. (PsycINFO Database Record (c) 2013 APA, all rights reserved)

Math relies on mastery and integration of a wide range of simpler numerical processes and concepts. Recent work has identified several numerical competencies that predict variation in math ability. We examined the unique relations between eight basic numerical skills and early arithmetic ability in a large sample (N = 1391) of children across grades 1-6. In grades 1-2, children's ability to judge the relative magnitude of numerical symbols was most predictive of early arithmetic skills. The unique contribution of children's ability to assess ordinality in numerical symbols steadily increased across grades, overtaking all other predictors by grade 6. We found no evidence that children's ability to judge the relative magnitude of approximate, nonsymbolic numbers was uniquely predictive of arithmetic ability at any grade. Overall, symbolic number processing was more predictive of arithmetic ability than nonsymbolic number processing, though the relative importance of symbolic number ability appears to shift from cardinal to ordinal processing.

In this article, we present the results of an 11-month longitudinal study (beginning when children were 6 years old) focusing on measures of the approximate number sense (ANS) and knowledge of the Arabic numeral system as possible influences on the development of arithmetic skills. Multiple measures of symbolic and nonsymbolic magnitude judgment were shown to define a unitary factor that appears to index the efficiency of an ANS system, which is a strong longitudinal correlate of arithmetic skills. However, path models revealed that knowledge of Arabic numerals at 6 years was a powerful longitudinal predictor of the growth in arithmetic skills, whereas variations in magnitude-comparison ability played no additional role in predicting variations in arithmetic skills. These results suggest that verbal processes concerned with learning the labels for Arabic numerals, and the ability to translate between Arabic numerals and verbal codes, place critical constraints on arithmetic development.