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Why numerical symbols count in the development of mathematical skills: Evidence from brain and behavior
... 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). ...
Numerical ability is developed in the early years of age and is the basis for later learning mathematics, as well as academic and work success in adulthood. Initial mathematics education has traditionally focused on the teaching of general processes, seeking the development of logical-mathematical thinking and mathematical language. This article seeks to think about the importance of training specific numerical cognitive processes, based on the findings in cognitive psychology. Thus, in this work the latest empirical evidence is reviewed, based on recent studies with behavioral approaches in numerical cognition, focused on the development of early numerical skills. For this, the main milestones of numerical development in relation to the acquisition of later arithmetic are reviewed, taking into account the intrinsic and extrinsic influences on the individual during the first years of age.
... 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 main goal of this study was to analyse, using structural equation modeling, the contribution of predictors of both domain-general (working memory, processing speed and receptive vocabulary) and domain-specific (estimation and magnitude comparison) processes to informal mathematical performance (numbering, comparison, calculation and understanding of concepts) in preschoolers. A total of 158 preschool students (ages ranging from 52 to 64 months) participated in the investigation. Students were assessed with informal tasks measuring mathematical thinking, numerical estimation, symbolic and non-symbolic comparison-making, coding, receptive vocabulary, and backward digit span. Results showed that a structural equation model for multiple indicators and several factors could explain informal mathematical thinking capacity in young children. The model reduced specific-domain factor effects such as magnitude comparison. In conclusion, the effect of working memory was found to be less than the straight impact of the general-domain predictors considered in the study.
... 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.
... 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. 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.
... Early emphasis on symbolic abilities may have long-term effects on the development of mathematics. Evidence from young toddlers, as well as neurological and behavioral data from school-aged kids, has shown significant connections between understanding of symbolic numbers and mathematical achievement (Merkley et al.,2016). ...
Mathematical skills are important for individuals with intellectual and developmental disabilities (IDDs) to function and be independent in their daily life. Learning math skills is more crucial since people with IDDs have weak social, practical, and conceptual skills. The main purpose of this quantitative study is to identify the problems faced by special education teachers in teaching mathematical skills to students with IDDs. Population of the study consisted of special education teachers working in Lahore. A sample of 80 special educationalists selected purposefully from different special education schools in Lahore. Researchers used self-developed questionnaire to take responses from respondents. Researchers analyzed the data by using parametric statistic. Frequency distributions of teacher’s responses were calculated. Major findings revealed that special education teachers face a little problem in teaching pre-number concepts but face a lot of problems in teaching division, subtraction and addition in mathematical operations. Conclusion was drawn and recommendations were given to parents and teachers to overcome the issues regarding mathematical skills of students with IDDs.
... Mathematics depends on an abstract symbol system, which represents a complex set of numerical and functional relations (Lyons et al., 2016;Merkley & Ansari, 2016;Núñez, 2017;Thompson & Saldanha, 2003;Xu et al., 2019). These relations can include, for example, cardinal knowledge (e.g., 2 is bigger than 1 and smaller than 3), ordinal knowledge (e.g., 3 follows 2 and comes after 1), and arithmetic knowledge (e.g., 2 + 1 = 3; 3 − 1 = 2; 2 × 3 = 6; 6 ÷ 2 = 3). ...
Mathematical competencies can be conceptualized as layers of knowledge, with numeracy skills as the foundational core and more complex mathematical skills as the additional layers over the core. In this study we tested an expanded Hierarchical Symbol Integration (HSI) model by examining the hierarchical relations among mathematical skills. Undergraduate students (N = 236) completed order judgment, simple arithmetic, fraction arithmetic, algebra, and verbal working memory tasks. In a series of hierarchical multiple regressions, we found support for the hierarchical model: Additive skills (i.e., addition and subtraction) predicted unique variance in multiplicative skills (i.e., multiplication and division); multiplicative skills predicted unique variance in fraction arithmetic; and fraction skills predicted unique variance in algebra. These results support the framework of the HSI model in which mathematical competencies are related hierarchically, capturing the increasing complexity of symbolic mathematical skills.
... For example we count on our fingers as children, but use the same process as adults to mentally count on our fingers. Children learn the count sequence by rote before understanding the numerical meaning of number words and Arabic numerals [25]. ...
Data science is a relatively new requirement for business students. Historically, many business students shied away from business statistics and quantitative or operational research (OR) modules believing them to be boring and irrelevant. The high-profile use of analytics and modelling during the COVID pandemic has drawn awareness to the relevance of analytics. Greater availability of data and modelling tools afford business students an opportunity to re-engage with operational research and analytics and to enjoy the satisfaction of modelling and solving real-world problems, but the challenge of the mathematical modelling skills gap of business students remains. In this paper, we describe a learning pathway of modules in business analytics that can enhance business students’ confidence and capabilities in performing statistical and analytical business tasks. We recommend modelling tools and incremental innovative mathematical modelling teaching approaches that are pedagogically sound and suitable for business students with varying quantitative backgrounds.
... There are two ways of thinking about numbers: On the one hand, they describe magnitude (i.e., cardinality), and, on the other hand, they represent order (i.e., ordinality) (e.g., Merkley & Ansari, 2016). These two aspects have already been differentiated in the numerical cognition literature for a long time (Gelman & Gallistel, 1978). ...
While symbolic number processing is an important correlate for typical and low mathematics achievement, it remains to be determined whether children with high mathematics achievement also have excellent symbolic number processing abilities. We investigated this question in 64 children (aged 8 to 10), i.e., 32 children with persistent high achievement in mathematics (above the 90th percentile) and 32 average-achieving peers (between the 25th and 75th percentile). Children completed measures of symbolic number processing (comparison and order). We additionally investigated the roles of spatial visualization and working memory. High mathematics achievers were faster and more accurate in order processing compared to average achievers, but no differences were found in magnitude comparison. High mathematics achievers demonstrated better spatial visualization ability, while group differences in working memory were less clear. Spatial visualization ability was the only significant predictor of group membership. Our results therefore highlight the role of high spatial visualization ability in high mathematics achievement.
... Informally, one could say that when children grasp the cardinality principle, they gain a new conceptual understanding of what numbers are. This understanding is a prerequisite for acquiring the kindergarten and first-grade skills that predict math achievement in later years (Geary et al., 2018;Jordan et al., 2006Jordan et al., , 2007Merkley & Ansari, 2016;Moore et al., 2016). ...
The authors assessed a battery of number skills in a sample of over 500 preschoolers, including both monolingual and bilingual/multilingual learners from households at a range of socio-economic levels. Receptive vocabulary was measured in English for all children, and also in Spanish for those who spoke it. The first goal of the study was to describe entailment relations among numeracy skills by analyzing patterns of co-occurrence. Findings indicated that transitive and intransitive counting skills are jointly present when children show understanding of cardinality and that cardinality and knowledge of written number symbols are jointly present when children successfully use number lines. The study’s second goal was to describe relations between symbolic numeracy and language context (i.e., monolingual vs. bilingual contexts), separating these from well-documented socio-economic influences such as household income and parental education: Language context had only a modest effect on numeracy, with no differences detectable on most tasks. However, a difference did appear on the scaffolded number-line task, where bilingual learners performed slightly better than monolinguals. The third goal of the study was to find out whether symbolic number knowledge for one subset of children (Spanish/English bilingual learners from low-income households) differed when tested in their home language (Spanish) vs. their language of preschool instruction (English): Findings indicated that children performed as well or better in English than in Spanish for all measures, even when their receptive vocabulary scores in Spanish were higher than in English.
... For example we count on our fingers as children, but use the same process as adults to mentally counting on our fingers. Children learn the count sequence by rote before understanding the numerical meaning of number words and Arabic numerals [28]. ...
Data Science is a relatively new requirement for business students. Historically many business students shied away from business statistics and quantitative or Operational Research (OR) modules believing them to be boring and irrelevant. The high profile use of analytics and modelling during the Covid pandemic has drawn awareness to the relevance of analytics. Greater availability of data and modelling tools afford business students an opportunity to re-engage with Operational Research and Analytics and to enjoy the satisfaction of modelling and solving real world problems, but the challenge of the mathematical modelling skills gap of business students remains. In this paper we describe a learning pathway of modules in business analytics that can enhance business students’ confidence and capabilities in performing statistical and analytical business tasks. We recommend modelling tools and incremental innovative mathematical modelling teaching approaches that are pedagogically sound and suitable for business students with varying quantitative backgrounds.
... No BP was observed for symbolic processing, likely because of a short pre-stimulus interval (less than 1000 ms) (Di Perri et al., 2014). Through visual inspection and reference to previous literature (Heine et al., 2013;Wang et al., 2022;Yao et al., 2015), the N1 was analyzed in the time window of 135-185 ms over the left parieto-occipital region of interest (ROI), which pooled the P3, P5 and PO3 electrodes by locating electrodes with maximal amplitude focusing on left parietal dominance in number processing (Merkley and Ansari, 2016). Similarly, the P1 was analyzed in the time window of 75-125 ms for symbolic processing over the left parietal ROI according to previous reports (Csépe et al., 2003;Koychev et al., 2010;Nikolaev et al., 2020), which pooled the P3 and P5 electrodes. ...
The neural markers for individual differences in mathematical achievement have been studied extensively using magnetic resonance imaging; however, high temporal resolution electrophysiological evidence for individual differences in mathematical achievement require further elucidation. This study evaluated the event-related potential (ERP) when 48 college students with high or low mathematical achievement (HA vs. LA) matched non-symbolic and symbolic rational numbers. Behavioral results indicated that HA students had better performance in the discretized non-symbolic matching, although the two groups showed similar performances in the continuous matching. ERP data revealed that even before non-symbolic stimulus presentation, HA students had greater Bereitschaftspotential (BP) amplitudes over posterior central electrodes. After the presentation of non-symbolic numbers, HA students had larger N1 amplitudes at 160 ms post-stimulus, over left-lateralized parieto-occipital electrodes. After the presentation of symbolic numbers, HA students displayed more profound P1 amplitudes at 100 ms post-stimulus, over left parietal electrodes. Furthermore, larger BP and N1 amplitudes were associated with the shorter reaction times, and larger P1 amplitudes corresponded to lower error rates. The BP effect could indicate preparation processing, and early left-lateralized N1 and P1 effects could reflect the non-symbolic and symbolic number processing along the dorsal neural pathways. These results suggest that the left-lateralized P1 and N1 components elicited by matching non-symbolic and symbolic rational numbers can be considered as neurocognitive markers for individual differences in mathematical achievement.
... The diagnostic test applied to the 40 students of the seventh year of Basic Media shows that a considerable percentage of the students are not clear about the processes of solving basic operations of numbers, basic operations of fractions, addition and subtraction with parentheses, percentage and measures of central tendency, which means that the population of students in Basic Media does not develop cognitive abilities (Hannula & Lehtinen, 2005;Vukovic et al., 2010;Cunska & Savicka, 2012). With the application of the diagnostic test, it was possible to verify the existence of an unsatisfactory cognitive level that requires strengthening the development of skills, to achieve full mastery of these, essentially in the resolution of multiplication and division of natural numbers, multiplication and division of fractions and measures of central tendency, considering that the seventh year of Middle School is where the established skills must be developed to move on to High School (Boaler, 1998;Chamoso et al., 2012;Merkley & Ansari, 2016). ...
The objective of the research is to diagnose cognitive abilities in the academic performance of students in Basic Middle School based on learning styles with a constructivist approach in the Cinco de Mayo Fiscomisional Educational Unit in the Basic average period 2022. The results of the research carried out through a diagnostic test are shown, as a way of projecting the national educational dimension in the development of mathematical skills in its various forms. The teaching-learning process presents difficulties that lead to low academic performance.A categorization of topics on knowledge achieved in the block of natural numbers, fractions and statistics was developed. was used as an instrument survey of students natural, fractional and statistical numbers, the qualitative, quantitative and documentary method used, in addition to the inductive and descriptive method. The technique was a structured base test to carry out the analysis and interpretation of the results obtained from the students, on the development of mathematical skills of Basic Media. The diagnosis made resulted in weaknesses in the approach and resolution of exercises that affect academic performance, so there is a need to improve this process through the application of methodological strategies that allow strengthening the teaching-learning process through ICTs.
... 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.
... 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.
... 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.
... 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). ...
The overall goal of the ISEE Assessment is to pool multi-disciplinary expertise on educational systems and reforms from a range of stakeholders in an open and inclusive manner, and to undertake a scientifically robust and evidence based assessment that can inform education policy-making at all levels and on all scales. Its aim is not to be policy prescriptive but to provide policy relevant information and recommendations to improve education systems and the way we organize learning in formal and non-formal settings. It is also meant to identify information gaps and priorities for future research in the field of education.
... 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.
... The existing problem is a situation that must be addressed urgently by those who educate at this level, in that mathematics is considered essential in children's learning since it will help them develop their reasoning and problem-solving skills. then if we intervene objectively to the reality of what is happening at the initial sub-level, then the children will learn in a pleasant and simple way with the appropriate material for this age, then it is necessary the accompaniment at home by the parents, these must constitute the ideal support in the development of their activities because children require accompaniment for their cognitive development (Passolunghi et al., 2007;Brown & Burton, 1978;Merkley & Ansari, 2016). ...
The study aimed to determine the development of mathematical logic skills and abilities of the Initial sublevel of the Jacinto Educational Unit Santos Verduga 2021-2022, which raised as a problem the causes that prevent the development of mathematical logic skills, due to the lack of stimulation of parents towards their children at an early age, due to ignorance and disorientation of the teacher towards parents of how to help them. Logical-mathematical knowledge is what the child builds by relating the experiences obtained in the manipulation of objects. The methodology used was based on a descriptive-explanatory level of research with a quantitative approach due to the relationship of the variable with the object of study, using the inductive-deductive, analytical, synthetic, and statistical method. applying various data collection instruments such as: the survey to teachers, questionnaires to parents and observation of students. It was obtained as a result that the boys and girls of the educational unit have little development in mathematical logic skills, which can trigger difficulties in the appropriation of mathematical concepts.
... (Ministry of Education of Ecuador, 2019), (MIEDUC, 2020). Probably one of the most outstanding dimensions in the teaching-learning process in the area of mathematics is the development of mathematical skills with performance criteria, proposed in the national curriculum, these are fundamental for the intellectual development of children, it helps to be logical, to reason in an orderly manner and to have a mind prepared for thought, criticism and abstraction (Merkley & Ansari, 2016;Alloway & Passolunghi, 2011). The last two years where Ecuador and the world faced a pandemic that forced the educational system prepared or not to undertake distance education and the ministry of education rethought a prioritized curriculum for the emergency with the essential skills in each subject and guidelines for a distance education process is synchronous or asynchronous. ...
The objective of the research was to evaluate the development of mathematical skills at the cognitive level of Basic General Education students of the Magaly Masson de Valle Carrera Educational Unit in the 2022, that in view of the learning results of the students and that according to national evaluations, the students did not reach the established average according to the evaluations carried out in Ecuador, they present a low cognitive level and a limited development of mathematical skills, for reasons of teacher training in technological and virtual tools, connectivity and the accompaniment of a significant adult at home. In the study, the theory of cognitive development is proposed, who explains how these are developed in the child and how their contribution helps to solve problems, which configure logical connections for the understanding of life situations. The level of the investigation was descriptive and explanatory; the quantitative approach, using scientific, inductive, deductive, analytical, synthetic, and statistical methods. The result was no relationship of mathematical skills in proportion to the cognitive level, in addition to the development of skills in children who had synchronous connectivity was high, who do not accept the virtual class system.
... 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. ...
... 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.
... 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.
... 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.
... 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.
... Emphasis on acquiring numerical skills in preschool years resulting in the prediction of later math achievement is documented in studies. This shows that teaching young children with strong mathematical skills such as ordinality, cardinality, and identification and also identifying the momentous role of cognitive and neural mechanisms (Merkley and Ansari 2016) are important. Between the symbolic numerical comparison and the non-symbolic magnitude comparison, the former was found to be an unswerving predictor of mathematics (De Smedt et al. 2013). ...
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).
... 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.
... 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.
... 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).
This study aims to determine the skills that children should possess during early childhood and to interpret the problems that are obstacles, as well as efforts that can be taken as solutions. The method used in this research is a literature study. The survey conducted showed that children's classification and resultative counting skills are still relatively low. The results of the study found that children's skills in early childhood mathematics can predict how successful they will be in their mathematics learning achievement and in developing other skills. There are 4 skills that need to be developed, which is concept of comparison, classification, resultative counting dan general understanding of number. Teachers are expected to be able to prepare high-quality lesson plans to help develop children's abilities and can act as facilitators in learning for children. Keywords: Early childhood; Mathematics; Numerical; Skills Abstrak Penelitian ini bertujuan untuk mengetahui kemampuan apa saja yang harus dimiliki anak pada masa early childhood dan menginterpretasi apa saja masalah yang menjadi hambatan, serta usaha yang dapat diambil sebagai solusinya. Metode yang digunakan dalam penelitian ini adalah studi literatur. Survei yang dilakukan, menunjukkan bahwa kemampuan classification dan resultative counting anak masih tergolong rendah. Hasil dari penelitian ditemukan bahwa kemampuan anak dalam early childhood mathematics dapat memprediksi bagaimana anak tersebut dapat sukses dalam pencapaian pembelajaran matematika dan juga dalam mengembangkan kemampuan lainnya. Terdapat 4 kemampuan yang perlu dikembangkan, yaitu concept of comparison, classification, resultative counting dan general understanding of number. Guru diharapkan dapat mempersiapkan rencana pembelajaran yang berkualitas tinggi untuk membantu mengembangkan kemampuan anak dan dapat berperan sebagai fasilitator di dalam pembelajaran bagi anak.
The study used Bayesian and Frequentist methods to investigate whether the roles of linguistic, quantitative, and spatial attention skills are distinct in children's acquisition of reading and math. A sample of 175 Chinese kindergarteners was tested with measures of linguistic skills (phonological awareness and phonological memory), quantitative knowledge (number line task, symbolic digit comparison, and non‐symbolic number estimation), spatial attention skills (visual span, mental rotation, and visual search), word reading, and calculation. After statistically controlling for age and nonverbal intelligence, phonological awareness and digit comparison performance explained unique variance in both math and reading. Moreover, number line estimation was specifically important for math, while phonological memory was specifically essential for reading. These findings highlight the possibility of developing early screening tools with different cognitive measures for children at risk of learning disabilities in reading and/or math.
Written for pre-service and in-service educators, as well as parents of children in preschool through grade five, this book connects research in cognitive development and math education to offer an accessibly written and practical introduction to the science of elementary math learning. Structured according to children's mathematical development, How Children Learn Math systematically reviews and synthesizes the latest developmental research on mathematical cognition into accessible sections that explain both the scientific evidence available and its practical classroom application. Written by an author team with decades of collective experience in cognitive learning research, clinical learning evaluations, and classroom experience working with both teachers and children, this amply illustrated text offers a powerful resource for understanding children's mathematical development, from quantitative intuition to word problems, and helps readers understand and identify math learning difficulties that may emerge in later grades. Aimed at pre-service and in-service teachers and educators with little background in cognitive development, the book distills important findings in cognitive development into clear, accessible language and practical suggestions. The book therefore serves as an ideal text for pre-service early childhood, elementary, and special education teachers, as well as early career researchers, or as a professional development resource for in-service teachers, supervisors and administrators, school psychologists, homeschool parents, and other educators.
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.
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.
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.
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.
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
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.
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).
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.
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)
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.
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.
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.
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.
This study assessed whether a sample of two hundred seven 3- to 7-year-olds could interpret multidigit numerals using simple identification and comparison tasks. Contrary to the view that young children do not understand place value, even 3-year-olds demonstrated some competence on these tasks. Ceiling was reached by first grade. When training was provided, there were significant gains, suggesting that children can improve their partial understandings with input. Findings add to what is known about the processes of symbolic development and the incidental learning that occurs prior to schooling, as well as specifying more precisely what place value misconceptions remain as children enter the educational system.
