No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Why numerical symbols count in the development of mathematical skills: Evidence from brain and behavior
... In the early years, curiosity about discovering the world around us draws attention to the environment, helping to develop informal mathematical reasoning from experiences with the close environment (Pfannkuch et al., 2018). Before beginning formal mathematical instruction upon entering the educational system, children need to handle a wide variety of skills (Merkley & Ansari, 2016). This set of informal knowledge may include skills such as numbering and counting, comparison of magnitudes, and the ability to solve simple arithmetic problems (Aguilar et al., 2015). ...
... As for the development of informal thinking in mathematics (Merkley & Ansari, 2016), the superior records collected in the two experimental groups versus the control group stand out. In the posttest assessment, the low-performance group equaled the control group in informal thinking. ...
... This new point of view is supported by studies such as the one conducted by Raghubar and Barnes (2017) considering working on both types of predictors could be the best option to enhance mathematical performance especially for students at risk of presenting difficulties in this subject (Aragón et al., 2019;Ramani et al., 2017). Consequently, the incorporation of this innovative tool would strengthen the transition from informal to formal achievements in numerical knowledge (Merkley & Ansari, 2016). ...
Introduction. The main goal of this study was to validate six educational apps developed for training the cognitive fundamentals associated with mathematical learning in Early Childhood Education students. Method. The mathematical performance of 193 Early Childhood Education students aged between 57 and 79 months (M=63.3, SD=3.7) from various schools was assessed using the TEMA-3. They were divided into three groups: two experimental groups and one control group. Students in the experimental groups underwent a training program in cognitive processes related to mathematical learning using Apps. A pre- and post-intervention comparative analysis was conducted. Results. The training program with apps showed significant gains in those students who had low scores in the pre-assessment tests. As expected, students at higher levels also improved with the training program. However, those students who started with a disadvantage in mathematical learning achieved a significant scores improvement, especially in measures of formal mathematical thinking. Discussion and Conclusion. The advantages derived from the use of these Apps, both in the school environment and at home, are discussed.
... Early numeracy skills are crucial for math development in preschool and kindergarten years (Merkley & Ansari, 2016;Peake et al., 2021). Three early mathematical skills have been identified as precursors for later math development: numbering, relations, and arithmetic operations (National Research Council, 2009;Purpura & Lonigan, 2013). ...
... Relations skills include understanding how quantities interrelate and the ability to transcode between various numerical formats, including oral (/seven/), written in Arabic (7), or verbally written (seven). Within this domain, number identification, ordinality, and number comparison hold particular importance as early numeracy predictors of math development (Merkley & Ansari, 2016). Number identification refers to the capacity to recognize and associate each Arabic symbol with its corresponding name and quantity. ...
... However, it has been suggested that arithmetic operation skills develop later than those in the numbering and relations domains (Litkowski et al., 2020). Actually, these skills include calculations involving symbols, what demands the knowledge of symbols, their meaning, their ordinal relations (Merkley & Ansari, 2016) and the counting list (Carey & Barner, 2019), skills conceptualized here as numbering or relations, according to the Purpura and Lonigan (2013) model. It is crucial at this point to clarify conceptual terms and to specify the main goal of this research by stating the meaning and differences of the terms numeracy and math or mathematics. ...
Home numeracy environment has been related to the development of early
numeracy skills. Previous research has shown the relation between dimensions
of the home numeracy environment with general or composite measures
of numeracy skills, but specific relations between types of activities at
home and the early development of specific numeracy skills are under
debate. This research aimed to study these specific relations based on two
recognized theoretical models in the literature of mathematical cognition:
the Home Numeracy Environment and the Informal Numeracy Skills Model.
Early numeracy skills were assessed in 189 Chilean prekindergartners (4.6
years on average, SD = 3 months; 50% girls) at the beginning (Time 1) and the
end (Time 2) of the school year, while their parents completed a home
numeracy environment questionnaire, which included questions about
their education, expectations, attitudes, and math activities. Research
Findings: Results from hierarchical multivariate multiple regression showed
specific relations depending on the testing time point, showing relations at
the beginning of the year and after some months of preschool. Parent’s
education was positively correlated with all children’s early math skills in
both Time 1 and Time 2. Practice or Policy: Educational implications and
future research directions are discussed.
... Mathematical development involves the integration of various sets of number associations wherein integration reflects the construction of a higher-level understanding of number through the process of combining subsets of associations into a single mental representation (Clements et al., 2023;Hiebert, 1988;Siegler & Chen, 2008;Xu et al., 2019). Within this mental representation, the development of whole number skills, such as learning to represent and manipulate whole numbers to characterize cardinal (e.g., 4 is smaller than 5 and bigger than 3), ordinal (e.g., 4 comes after 3 and before 5), and arithmetic (e.g., 2 + 2, 5 À 1, 2 Â 2, 8 Ä 2) associations (Lyons et al., 2016;Merkley & Ansari, 2016;, is fundamental. When solving mathematical problems, these associations need to be differentially activated, depending on the context (Ashcraft, 1982;Campbell, 1994;Siegler & Chen, 2008;Verguts & Fias, 2005;Xu et al., 2019). ...
... The development of mathematical skills is a hierarchical process wherein complex mathematical skills are dependent on strong foundational skills (Hiebert, 1988;Merkley & Ansari, 2016;Siegler & Lortie-Forgues, 2014). Associations among numbers within a single mental representation have been explored in studies of arithmetic (Ashcraft, 1982;Campbell, 1994Campbell, , 1995Rickard, 2005;Siegler, 1988;Verguts & Fias, 2005). ...
... Mathematical skills are ever evolving; the development of these skills is hierarchical, such that strong foundational skills are necessary for students to advance toward an understanding of more complex skills (Hiebert, 1988;Merkley & Ansari, 2016;Siegler & Lortie-Forgues, 2014). Among these skills, fractions receive a lot of attention in the mathematics learning literature; the required abstract Note. ...
It is well established in the literature that fraction knowledge is important for learning more advanced mathematics, but the hierarchical relations among whole number arithmetic, fraction knowledge, and mathematics word problem-solving are not well understood. In the current study, Chinese Grade 6 students (N = 1160; 465 girls; Mage = 12.1 years, SD = 0.6) completed whole number arithmetic (addition, subtraction, multiplication, and division), fraction (mapping, equivalence, comparison, and arithmetic), and mathematics word problem-solving assessments. They also completed two control measures: number writing speed and nonverbal intelligence. Structural equation modeling was used to investigate the hierarchical relations among these assessments. Among the four fraction tasks, the correlations were low to moderate, suggesting that each task may tap into a unique aspect of fraction understanding. In the model, whole number arithmetic was directly related to all four fraction tasks, but was only indirectly related to mathematics word problem-solving, through fraction arithmetic. Only fraction arithmetic, the most advanced fraction skill, directly predicted mathematics word problem-solving. These findings are consistent with the view that students need to build these associations into their mathematics hierarchy to advance their mathematical competence.
... Early mathematical abilities are believed to include not only basic arithmetic skills but also word problem solving and applied math skills (De Smedt, 2022;Van Hoogmoed et al., 2022). Numeracy-specific cognitive skills that are associated with early math abilities encompass various domains of numerical cognition such as counting, symbolic and non-symbolic magnitude processing, mapping between quantities and numer-ical symbols, ordering, and estimation (De Smedt et al., 2013;Friso-van den Bos et al., 2015;Kolkman et al., 2013;Koponen et al., 2019;Merkley & Ansari, 2016). ...
... Our finding that counting and symbolic comparison skills are predictive of arithmetic abilities in kindergarten is in line with a multitude of earlier studies (Cahoon et al., 2021;De Smedt et al., 2013;Hawes et al., 2019;Kolkman et al., 2013;Merkley & Ansari, 2016). Furthermore, our finding that visuospatial STM, number line estimation, and symbolic comparison skills were the strongest predictors of arithmetic fluency scores within the precocious arithmetic learners offers an elaboration of the results obtained by Bakker and colleagues, who found that high mathematical achievement in 8-to 10-year-old children was associated with higher spatial visualization ability and visuospatial working memory (Bakker et al., 2023a;Bakker et al., 2022), but who did not specifically include measures of number line estimation ability. ...
... This skill allows one to understand that when counting the elements in a set, the last number mentioned represents the total quantity. Individuals must acquire cardinality and be able to identify numeral symbols to truly comprehend the meaning of numbers (Merkley & Ansari, 2016). ...
... On the other hand, Numeral identification is one component of symbolic numerical knowledge and involves associating the Arabic (written) symbol of the number with the word representing the numerical concept (Chan & Scalise, 2022;Merkley & Ansari, 2016). It comprises awareness that quantities are represented by numerals and it's an essential skill that enables a child to understand how the base-ten system works (Aunio & Räsänen, 2016). ...
... 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.
... Nevertheless, those researchers and practitioners focused on assessing only numerical skills might want to drop the dimension of spatial reasoning to obtain a shorter version of the TPM which might be more efficient to this aim. Using different frameworks would have suggested measuring other important precursors for arithmetic learning, such as verbal counting and ordinality (e.g., Merkley & Ansari, 2016). Still, our theoretical framework and the technology constraints prioritized measuring the seven tasks reported in this study. ...
Improving early mathematical competence is a major priority worldwide; thus, assessing early math abilities is critical. Although various international standardized instruments serve this purpose, their usage in underdeveloped countries is prohibitive due to their resource-intensive requirements. In this report, we explore the development of the “Test de Pensamiento Matemático" (TPM, Test of Preschool Mathematics), which is an automated, game-based, digital instrument for assessing early math abilities in 4-to-6-year-old children in accordance with international curricular standards. A confirmatory factor analysis shows an optimal fit for two dimensions: numerical thinking and visuospatial reasoning. By drawing on technology, the TPM can be applied to large groups of children, so it becomes an efficient tool for assessing performance, monitoring learning improvements, and screening children who need additional support to develop their math abilities at the same pace with their peers.
... En el contexto chileno, en el trabajo de Castro et al. (2017), se encontró que la comparación simbólica presenta mayor desarrollo que la comparación no simbólica en niños desde primero a sexto grado y en preescolares también fue un predictor significativo del pensamiento matemático informal (Cerda et al., 2021). Por otra parte, la comparación no simbólica quedó excluida del modelo, y en concordancia, investigaciones previas han señalado que esta variable tiene un rol más relevante en el desarrollo numérico de los preescolares ( Respecto a las variables predictoras de dominio general, a diferencia de estudios longitudinales previos, en los que se destaca una alta relación de la alternancia atencional con el aprendizaje de las matemáticas (Lau et al., 2021;Merkley & Ansari, 2016), en el presente estudio esta variable no mostró un alto peso relativo en la explicación de la variabilidad de la eficiencia en aritmética básica. Este hallazgo no indica una contradicción respecto a hallazgos previos sobre la relevancia de la alternancia atencional en el procesamiento numérico, sino que la preeminencia del precursor de dominio específico de comparación de magnitudes simbólicas probablemente ha atenuado el peso predictivo de las variables de dominio general. ...
la influencia de los predictores de dominio específico y general en el desarrollo de la aritmética básica en escolares chilenos. Rev. CES Psico, 18(1), 18-34. https://dx.doi.org/10.21615/cesp.7570 Resumen En el aprendizaje de la aritmética, en los primeros años de la educación formal, interactúan determinados procesos cognitivos, así como variables de tipo sociodemográfico. En este contexto, el objetivo del presente estudio fue analizar la contribución específica de los predictores de dominio específico (comparación simbólica y no simbólica) y de dominio general (memoria de trabajo verbal y visoespacial, alternancia atencional, control inhibitorio e inteligencia fluida) en la resolución de una tarea de aritmética básica en escolares chilenos. Es un estudio correlacional y predictivo, y la muestra estuvo conformada por 203 participantes con desarrollo normotípico, 94 niñas y 109 niños. El modelo de regresión lineal múltiple con pasos sucesivos explicó un 30.4% de la variabilidad en la aritmética básica, y la comparación simbólica fue la variable que tuvo mayor poder predictivo seguida de la alternancia atencional. Estos resultados destacan la implicación de la comparación simbólica y la alternancia atencional en la explicación de la variabilidad en el rendimiento en aritmética básica durante los primeros años de la educación formal, aspecto que destaca la importancia de la evaluación en edades tempranas de múltiples componentes cognitivos que se ha constatado que son predictores de la adquisición del pensamiento matemático, y no sólo centrar las evaluaciones en mediciones basadas en el currículo. Palabras clave: Cognición; aritmética; aprendizaje; educación formal; memoria de trabajo; habilidades matemáticas; aprendizaje matemático; logros matemáticos; niños en edad escolar. Abstract Certain cognitive processes interact in the learning of arithmetic, as well as sociodemographic variables in the first years of formal education. In this context, the present study is aimed to analyze the specific contribution of the specific domains (symbolic and non-symbolic comparison) and the general domains (verbal and visuospatial working memory, attentional shifting, inhibitory control and fluid intelligence) in the resolution of a basic arithmetic task. This is a correlational and predictive study, whose sample corresponded to 203 participants with norm typical development, totalizing 94 girls and 109 boys. The results of the multiple linear regression with successive steps explained 30.4% of the variability in basis arithmetic where the symbolic comparison had greater predictive power, followed by attentional shifting. These findings highlight the importance of symbolic comparison and attentional shifting in explaining variability in basic arithmetic performance during the early years of formal education, an aspect that underlines the importance of early assessment of multiple cognitive components that have been shown to predict the acquisition of mathematical thinking, rather than focusing assessments solely on curriculum-based measures.
... Kemampuan literasi numerasi dapat didefinisikan sebagai kemampuan untuk menganalisis dan memahami pernyataan yang terdapat dalam suatu aktivitas dengan memanipulasi simbol atau bahasa yang ditemukan dalam kehidupan sehari-hari. Selain itu, kemampuan ini juga mencakup pengungkapan pernyataan tersebut baik secara lisan maupun tulisan (Toll et al., 2011), (Raghubar & Barnes, 2017) (Göbel et al., 2014;Merkley & Ansari, 2016). Di samping itu, kemampuan literasi numerasi juga didefinisikan sebagai kemampuan untuk memahami dan memanipulasi angka, baik dalam bentuk simbolik maupun nyata (Ekowati et al., 2019). ...
Pembelajaran literasi numerasi menjadi salah satu bagian pokok dari kurikulum merdeka dan sebagai bekal keterampilan yang harus dimiliki siswa SD. Namun, dilapangan ditemukan data rendahnya literasi numerasi siswa berdasarkan AKM. Maka melalui pendekatan user requirement perlu dilaksanakan pelatihan dan pendampingan pengembangan kemampuan literasi numerasi kepada guru SD untuk mengatasi permasalahan tersebut. Tujuan dari kegiatan ini adalah untuk meningkatkan literasi numerasi guru yang dilaksanakan dengan berbasis digital. Kegiatan PKM ini dilaksanakan dengan memberikan pelatihan dan pendampingan kepada guru tentang literasi numerasi. Kegiatan ini dilaksanakan bagi seluruh guru SD Negeri di Kec. Lareh Sago Halaban, dengan jumlah peserta 35 orang yang merupakan perwakilan dari masing-masing sekolah. Kegiatan pengabdian ini terselenggara dengan baik. Peserta antusias dalam setiap aktivitas. Peserta pun menyatakan bahwa mereka mendapatkan tambahan pengetahuan dan wawasan baru dengan adanya kegiatan pelatihan ini. Hasil evaluasi menunjukkan bahwa terdapat respon positif peserta terhadap pelatihan dimana persentase capaian tentang efektivitas pelaksanaan pelatihan adalah 90,63%. Dari aspek kemampuan peserta pelatihan, terjadi peningkatan kemampuan peserta dari 80,58% menjadi 89.09%.
... The ability to recognize digits, another domain-specific precursor involving both symbolic number knowledge and understanding of the meaning of the number word, has also been found at preschool age to be predictive of later mathematics competence (for a review, see Merkley & Ansari, 2016). For instance, knowledge of Arabic numerals in six-yearolds was found to predict the longitudinal development of their arithmetic skills (Göbel et al., 2014), representing a key indicator of mathematical learning from preschool onwards (Cahoon et al., 2021). ...
Evidence has shown the importance of early numerical skills in sustaining future mathematical abilities. However , the literature has largely ignored the potential of educational videos to improve numerical abilities in children at risk of developing numeracy difficulties. The aim of the present study was to examine the effectiveness of a numerical video training on domain-specific precursors in first-year preschoolers (Mean age = 43.64 months) by comparing two intervention groups (i.e., at-risk of developing numeracy difficulties group; average intervention group) with an active control group, while controlling for domain-general precursors. Results revealed that the training was effective in enhancing counting skills in both the at-risk and average intervention groups. The findings also showed an enhancement of cardinality knowledge and digit recognition in the delayed post-test, but only for the group with average numerical abilities. Results will be discussed considering the implications for children who are at risk of experiencing numerical difficulties.
... (2020), pese a que el curriculum chileno en educación preescolar incluye algunos elementos clásicamente estudiados por la cognición numérica y la psicología del desarrollo, no se aprecian mayores conexiones con los hallazgos de las últimas décadas de la investigación internacional en este campo. En concreto, refieren que, si bien se observa de forma implícita la presencia de los principios piagetianos, como la clasificación o la seriación; y de elementos centrales del conteo, los cuantificadores, la cardinalidad y la ordinalidad; no se indica explícitamente la evidencia que respalda el aprendizaje y desarrollo de estos aspectos durante este nivel educativo (Merkley y Ansari, 2016). Tampoco se observan elementos como la estimación, la transcodificación numérica (Ebersbach, 2016) o la sucesión (Carey y Barner, 2019), que propician tanto la comprensión temprana del símbolo arábigo como las asociaciones número-cantidad. ...
Mathematics teaching is one of the fundamental pillars of early childhood education. Several studies have highlighted that adequate approach at early levels can positively influence future school performance and other cognitive domains. In Chile, mathematics education is an area of concern due to local evidence and observed gaps following the pandemic. In this context, educators and educational assistance professionals are challenged to provide equal teaching opportunities for all, including preschoolers with Developmental Language Disorder, who usually only receive support in language area. This review aims to update the literature on the math skills of students with Developmental Language Disorder and to establish their possible impact on the mathematics curriculum. The main findings include identifying barriers in counting and arithmetic tasks. Finally, the learning objectives that could require more attention for their approach were recorded, and studies were identified that highlighted the potential of visual and gestural representation as a way to favor the teaching-learning of these children. Thus, the approach to early mathematical skills can contribute to the inclusion and equalization of opportunities for these students.
... A pertinent issue concerning task specificity is the generalizability of our age prediction model to symbolic number tasks. A growing body of research emphasizes the significance of number representation for acquiring symbolic number processing skills and achieving proficiency in mathematics (De Smedt et al., 2013;Krajcsi et al., 2018;Krajcsi et al., 2023;Leibovich & Ansari, 2016;Lyons et al., 2015;Merkley & Ansari, 2016;Schneider et al., 2017 ...
The development and refinement of functional brain circuits crucial to human cognition is a continuous process that spans from childhood to adulthood. Research increasingly focuses on mapping these evolving configurations, with the aim to identify markers for functional impairments and atypical development. Among human cognitive systems, nonsymbolic magnitude representations serve as a foundational building block for future success in mathematical learning and achievement for individuals. Using task‐based frontoparietal (FPN) and salience network (SN) features during nonsymbolic magnitude processing alongside machine learning algorithms, we developed a framework to construct brain age prediction models for participants aged 7–30. Our study revealed differential developmental profiles in the synchronization within and between FPN and SN networks. Specifically, we observed a linear increase in FPN connectivity, concomitant with a decline in SN connectivity across the age span. A nonlinear U‐shaped trajectory in the connectivity between the FPN and SN was discerned, revealing reduced FPN‐SN synchronization among adolescents compared to both pediatric and adult cohorts. Leveraging the Gradient Boosting machine learning algorithm and nested fivefold stratified cross‐validation with independent training datasets, we demonstrated that functional connectivity measures of the FPN and SN nodes predict chronological age, with a correlation coefficient of .727 and a mean absolute error of 2.944 between actual and predicted ages. Notably, connectivity within the FPN emerged as the most contributing feature for age prediction. Critically, a more matured brain age estimate is associated with better arithmetic performance. Our findings shed light on the intricate developmental changes occurring in the neural networks supporting magnitude representations. We emphasize brain age estimation as a potent tool for understanding cognitive development and its relationship to mathematical abilities across the critical developmental period of youth.
Practitioner Points
This study investigated the prolonged changes in the brain's architecture across childhood, adolescence, and adulthood, with a focus on task‐state frontoparietal and salience networks.
Distinct developmental pathways were identified: frontoparietal synchronization strengthens consistently throughout development, while salience network connectivity diminishes with age. Furthermore, adolescents show a unique dip in connectivity between these networks.
Leveraging advanced machine learning methods, we accurately predicted individuals' ages based on these brain circuits, with a more mature estimated brain age correlating with better math skills.
... The overall aim has been to improve problem-solving performance in pupils at risk. Mathematical skills are, in fact, cumulative because they build on the knowledge acquired in previous years (Merkley & Ansari, 2016), making failure over the years all the more difficult to remedy. It is therefore important to intervene early. ...
Previous research has shown the importance of conducting early interventions in mathematics in disadvantaged children. Solving arithmetical word problems is a field in which children particularly fail. In this study, preschoolers from disadvantaged French public schools (n = 101; Mage = 5–6) were taught strategies for using fingers to solve arithmetic word problems and compared with a control group. The intervention consisted of collective rituals based on learning finger patterns and 7 sessions spread over 4 weeks, for about 20 min, focusing on explaining how to use the fingers to solve problems. The results showed that the intervention has a significant post-test impact on the targeted transformation problem-solving skill and that children with lower performances in problem-solving at the pre-test benefited more from the intervention. The intervention also indirectly benefited the other problem-solving skills. However, there was no intervention effect on the arithmetic addition task. Our research highlights that an intervention focused on the explicit teaching of finger strategies for problem-solving can be successfully implemented into ecological learning contexts, especially in disadvantaged areas.
... Despite the relevance of mapping skills in the development of the concept of number-mapping activities feature in many early childhood education (ECE) curriculums worldwide and ECE practitioners are typically encouraged to support young children's mapping skills (Merkley & Ansari, 2016)-few studies have directly analyzed young children's mapping skills during the preschool years and the order in which these mappings between different numerical representations are acquired (Benoit et al., 2013;Hurst et al., 2017;Jiménez Lira et al., 2017;Marinova et al., 2021). Benoit et al. (2013) assessed 3, 4, and 5-year-olds' mapping skills (n = 144) between number words, written digits, and non-symbolic quantities with small (1-3) and large numbers (4-6). ...
Mapping skills between different codes to represent numerical information, such as number symbols (i.e., verbal number words and written digits) and non‐symbolic quantities, are important in the development of the concept of number. The aim of the current study is to investigate children's mapping skills by incorporating another numerical code that emerges at early stages in development, finger patterns. Specifically, the study investigates (i) the order in which mapping skills develop and the association with young children's understanding of cardinality; and (ii) whether finger patterns are processed similarly to symbolic codes or rather as non‐symbolic quantities. Preschool children (3‐year‐olds, N = 113, M age = 40.8 months, SD age = 3.6 months; 4‐year‐olds, N = 103, M age = 52.9 months, SD age = 3.4 months) both cardinality knowers and subset‐knowers, were presented with twelve tasks that assessed the mappings between number words, Arabic digits, finger patterns, and quantities. The results showed that children's ability to map symbolic numbers precedes the understanding that such symbols reflect quantities, and that children recognize finger patterns above their cardinality knowledge, suggesting that finger patterns are symbolic in essence.
Research Highlights
Children are more accurate in mapping between finger patterns and symbols (number words and Arabic digits) than in mapping finger patterns and quantities, indicating that fingers are processed holistically as symbolic codes.
Children can map finger patterns to symbols above their corresponding cardinality level even in subset‐knowers.
Finger patterns may play a role in the process by which children learn to map symbols to quantities.
Fingers patterns’ use in the classroom context may be an adequate instructional and diagnostic tool.
... These graphical representations facilitate quick understanding of information. Ansari (2016) and Merkley & Ansari (2016) noted that a combination of clear structure, regularity, comparability, graphical representation, and statistical analysis, makes patterned numerical data clearer and easier to comprehend. For instance, data like Figure 2 depict the comparison of hydrogen gas amounts with the same concentration and weight of aluminum waste reacted with HCl and NaOH solutions. ...
This research aims to explore the impact of a computational chemistry-supported STEM Project-Based Learning (PjBL-STEM) application on fostering critical thinking competencies. The project focuses on green energy, utilizing waste cooking oil and aluminum to enhance students' critical thinking skills. A pre-experimental design was employed for this study, involving a sample of 22 students. Data were collected using the 21st-Century Skills Usage Scale, along with structured and semi-structured interview forms. Critical thinking was assessed through a descriptive test comprising 10 questions. The collected data were analyzed using the Wilcoxon Signed Rank Test and thematic analysis techniques. The analysis revealed a significant improvement in the students' 21st-century skills, such as autonomy, collaboration, environmental sensitivity, communication, problem-solving, creativity, responsibility, and IT literacy. Particularly notable was the students' ability to interpret structured and patterned graphical data, which they found easier to understand compared to image and narrative data. The N-gain test results indicated that the STEM-PjBL model had a positive impact on developing critical thinking abilities, with 50% of students achieving medium and high categories. Overall, the STEM-PjBL model positively influenced the development of critical thinking competencies.
... Understanding the cardinality principle is considered a crucial step in the mathematical development of young children. It reflects the idea that any quantity of items can be represented by a single word or symbol, an important step in the development of numerical understanding (Merkley & Ansari, 2016). ...
Educational neuroscience aims to merge neuroscience and education for better teaching and learning outcomes. Translating scientific findings into educator-friendly terms is crucial to empower teachers in applying these insights to teaching methods. Despite advancements in neuroeducation, a significant gap persists, and there is a lack of specific neuro-educational pedagogical knowledge, notably in early childhood math education. This exploratory review article outlines recent neuroscientific insights into numerical understanding in the human brain. It presents seven neuro-pedagogical principles essential for developing mathematical abilities in early childhood: (1) Mathematical abilities are distributed across various brain regions. (2) Innate mathematical abilities exist in young children. (3) Brain networks for perceiving quantities are vital for mathematical thinking and evolve with age and experience. (4) Distinguishing between small and large quantities (subitizing) can mediate between the innate ability to understand quantities and the later understanding of numbers and their meaning. (5) A mental number line maps numbers spatially. (6) Neurocognitive mechanisms perceive and manipulate numerical information through a triple code: quantity, numerals, and number words. (7) Finger gnosis, the ability to identify fingers without visual involvement, correlates with mathematical abilities. Each principle is discussed in terms of ongoing research, limitations of the current findings, and practical implications and examples suitable for early childhood educators.
... With the numeracy intervention and enhancement strategies discussed in the previous theme, there is an enhancement in the performance of the learners [32] . Thus, numeracy skills predict their achievement and success, indicating the need for a strong foundation in these skills [33,34] . The counterpart of this concept denotes that having these skills at an early stage is crucial for future academic success [35] . ...
Learning mathematics requires an effective and strategic teaching approach. This study aimed to assess themathematics performance of the learners with the implementation of the numeracy enhancement strategy QD2R (Questions, Drills, Repetition, and Recitation) and to propose a strategy implementation plan to elevate their performances. This study employed the use of a quasi-experimental research design, purposive sampling with 70 Grade 10 students of Lian National High School who were distributed equally to control and treatment groups. The pre-test and post-test results were statistically analyzed using independent and paired sample t-tests, and a survey questionnaire was examined by getting the mean and standard deviation. The results indicated that better performance was achieved by the students from the treatment group compared to the students from the control group, as revealed by the Mean Percentage Score (MPS) results, mean scores, and P values of their pre-test and post-test scores. The learners’ perception of the implementation of this strategy was to a great extent, wherein it was perceived to be more helpful in concepts related to understandingthe lesson compared to concepts related to developing their attitude and skills. Moreover, the proposed implementation plan of numeracy enhancement strategy QD2R had three expected outcomes: elevated understanding and performance in mathematics lessons; modified strategy to focus on the development of attitude and skills towards mathematics; and refined and well-implemented QD2R strategy in teaching mathematics. Relative to these expected outcomes, appropriate measures, timeframe, and resources of each were comprehensively formulated.
... From a psychological perspective, significant headway has been made in the last couple of decades in terms of isolating cognitive factors that underlie and scaffold mathematical learning, such as logical reasoning (e.g., de Jong & van der Leij, 1999;Geary, 2013), working memory capacity (e.g., Bull et al., 2008;Szűcs et al., 2014), and executive functions (e.g., Geary, 2010;Lefevre et al., 2013;Szűcs et al., 2014). In addition, extensive research has established that domain-specific number processing abilities are predictive of children's learning of mathematics, such as the ability to estimate and manipulate non-symbolic quantities (i.e., number sense; Decarli et al., 2023) and decode and understand the meaning of number symbols (i.e., Arabic digits; Merkley & Ansari, 2016). ...
One important factor that hampers children’s learning of mathematics is math anxiety (MA). Still, the mechanisms by which MA affects performance remain debated. The current study investigated the relationship between MA, basic number processing abilities (i.e., cardinality and ordinality processing), and executive functions in school children enrolled in grades 4–7 ( N = 127). Children were divided into a high math anxiety group ( N = 29) and a low math anxiety group ( N = 31) based on the lowest quartile and the highest quartile. Using a series of analyses of variances, we find that highly math-anxious students do not perform worse on cardinality processing tasks (i.e., digit comparison and non-symbolic number sense), but that they perform worse on numerical and non-numerical ordinality processing tasks. We demonstrate that children with high MA show poorer performance on a specific aspect of executive functions—shifting ability. Our models indicate that shifting ability is tied to performance on both the numerical and non-numerical ordinality processing tasks. A central factor seems to be the involvement of executive processes during ordinality judgements, and executive functions may constitute the driving force behind these delays in numerical competence in math-anxious children.
... Kemampuan simbol angka awal terdiri atas belajar untuk berhitung secara urut dan memahami makna dari setiap simbolnya. Pengetahuan siswa tentang simbol angka berpengaruh terhadap penguasaan keterampilan matematika pada tahap berikutnya (Göbel et al., 2014;Merkley & Ansari, 2016). Sedangkan untuk kemampuan nonsimbolik berhubungan dengan kemampuan untuk mengoperasikan bilangan langsung dengan obyeknya. ...
Pembelajaran literasi numerasi dan karakter menjadi salah satu bagian pokok dari kurikulum merdeka. Kedua aspek ini menjadi salah satu bekal keterampilan yang harus dimiliki siswa sekolah dasar. Tujuan penelitian ini adalah menganalisis permasalahan pembelajaran literasi numerasi dan karakter yang dapat dijadikan sebagai kebutuhan pengembangan sebuah produk pembelajaran. Metode penelitian ini adalah kualitatif dengan teknik wawancara dan observasi. Partisipan yang terlibat yakni guru dan siswa kelas IV sekolah dasar. Hasil analisis data menunjukkan bahwa terdapat tiga tema utama yang menjadi pembahasan dalam penelitian ini. Tiga tema tersebut adalah bahan ajar yang terbatas untuk menumbuhkan kemampuan literasi numerasi, minimnya latihan soal literasi numerasi, dan permasalahan perilaku yang berhubungan dengan karakter. Implikasi hasil penelitian ini dapat digunakan bagi peneliti selanjutnya untuk analisis kebutuhan dalam pengembangkan model pembelajaran, modul pembelajaran untuk siswa, dan buku panduan untuk guru yang dapat mengembangkan kemampuan literasi numerasi dan karakter siswa
... By age five, these proportions are 95.7% and 40.9% respectively, suggesting rapid progress with age. These developments coincide with neurological changes suggestive of increasing specialisation in the left intraparietal sulcus (Merkley & Ansari, 2016) as well as alongside growth in other informal skills (particularly counting, cardinality and one-to-one correspondence (Litkowski et al., 2020). Children begin to understand that some single digits such as 1-5, represent smaller quantities than others such as 6-9, sometimes independently of precise meaning and magnitude quantification (Le Corre & Carey, 2007). ...
... 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). ...
Les compétences mathématiques peuvent être conceptualisées comme des couches de connaissances, les compétences en numératie constituant le noyau de base et les compétences mathématiques plus complexes étant les couches supplémentaires enveloppant le noyau. Dans cette étude, nous avons testé un modèle élargi d’intégration hiérarchique des symboles (HSI) en examinant les relations hiérarchiques entre les compétences mathématiques. Des étudiants de premier cycle (N = 236) ont effectué des tâches de jugement d’ordre, d’arithmétique simple, d’arithmétique fractionnaire, d’algèbre et de mémoire de travail verbale. Une série de régressions multiples hiérarchiques a permis de confirmer le modèle hiérarchique : les compétences additives (c’est-à-dire l’addition et la soustraction) prédisent une variance unique dans les compétences multiplicatives (c’est-à-dire la multiplication et la division); les compétences multiplicatives prédisent une variance unique dans l’arithmétique fractionnaire; et les compétences fractionnaires prédisent une variance unique dans l’algèbre. Ces résultats soutiennent le cadre du modèle HSI dans lequel les compétences mathématiques sont liées de manière hiérarchique, ce qui permet de saisir la complexité croissante des compétences mathématiques symboliques.
... 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.
... 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. ...
... 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.
Public preschool boosts academic skills in kindergarten, but little is known about whether that boost lasts to third grade because many studies stop directly assessing children after kindergarten. The current study tests for sustained associations between preschool attendance and an array of repeatedly measured, directly assessed language and math skills; we do this separately for public pre-K and Head Start, the two major publicly funded preschool programs. We draw on a large, racially diverse sample of children from families with low incomes in Tulsa, OK (N = 689, M age at 3rd = 8.5 years). Using propensity score weighting, we compare children who attended school-based pre-K or Head Start to those who did not attend preschool. Both school-based pre-K and Head Start attenders outperformed preschool nonattenders on numeracy in third grade. There was weaker evidence of a sustained preschool advantage on language and literacy skills, and no evidence that associations differed by preschool program.
Mastery motivation predicts achievement, but intricacies amongst pre-schoolers are unclear. In keeping with the Specificity Principle, school-age, and adolescent research demonstrates the importance of considering the setting conditions in which mastery motivation is observed. Here, Singaporean 4-year-olds’ (N = 63) mastery-motivation-related behaviour (MMRB) (e.g. signs of persistence, focus, and pleasure) in mathematical and non-mathematical activities were observed. Relations between numeracy and MMRB during a mathematical game (outcome relevant setting) were determined, controlling for MMRB in other activities (outcome irrelevant settings). Association between MMRB during the mathematical game and receptive language (outcome irrelevant setting) was also examined. Consistent with the Specificity Principle, MMRB during the mathematical game was (i) associated with numeracy, after controlling for MMRB in other activities and (ii) did not predict language. Enhancing preschoolers’ experiences, especially when implemented in contexts related to areas targeted for improvement, may benefit outcomes. These skills acquired in early life can become important predictors of future ability.
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.
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.
This study examined numerosity comparison in 3-year-old children. Predictions derived from the analog numerical model and the object-file model were contrasted by testing the effects of size and ratio between numerosities to be compared. Different perceptual controls were also introduced to evaluate the hypothesis that comparison by preschoolers is based on correlated perceptual variables rather than on number per se. Finally, the relation between comparison performance and verbal counting knowledge was investigated. Results showed no evidence that preschoolers use an analog number magnitude or an object-file mechanism to compare numerosities. Rather, their inability to compare sets controlled for surface area suggests that they rely on perceptual cues. Furthermore, the development of numerosity-based representations seems to be related to some understanding of the cardinality concept.
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.
In this chapter, I put together the first elements of a mathematical theory relating neuro- biological observations to psychological laws in the domain of numerical cognition. The starting point is the postulate of a neuronal code whereby numerosity—the cardinal of a set of objects—is represented approximately by the firing of a population of numerosity detectors. Each of these neurons fires to a certain preferred numerosity, with a tuning curve which is a Gaussian function of the logarithm of numerosity. From this log- Gaussian code, decisions are taken using Bayesian mechanisms of log-likelihood compu- tation and accumulation. The resulting equations for response times and errors in classical tasks of number comparison and same-different judgments are shown to tightly fit behavioral and neural data. Two more speculative issues are discussed. First, new chronometric evidence is presented supporting the hypothesis that the acquisition of number symbols changes the mental number line, both by increasing its precision and by changing its coding scheme from logarithmic to linear. Second, I examine how symbolic and nonsymbolic representations of numbers affect performance in arithmetic compu- tations such as addition and subtraction.
Shows how children learn, do, and understand elementary mathematics, especially arithmetic, and demonstrates how such knowledge can be used to improve mathematics education and to resolve children's difficulties in learning. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
The aims of the learning sciences (LS) are to understand the nature of learning from a broad range of perspectives, and to shape the ways that learning environments and resources are designed and used. LS incorporates both systemic and elemental approaches to investigating questions about learning, as a complement to the primarily elemental approach emphasized in cognitive science research. Thus, its greatest potential is in the integration of systemic and elemental perspectives. Four major themes are central. First, research in LS attempts to bridge the divide between research and practice. Second, research in LS is motivated by limitations of theories of learning and cognition for specifying instruction. Third, research in LS embraces the importance of analyzing and assessing complex interventions through both experimental and design-based research. Fourth, research in LS emphasizes the learning and behavior of the individual in interaction with the physical, social, and cultural world, as well as with semiotic and technical resources. Research in LS can be conceptualized along a continuum of time scales, from the more microscopic to the more macroscopic. The time-scale framework illustrates how disparate research traditions and research methods can function within a unifying framework for the study of learning and complex behavior. The effort to 'scale-up' from more elemental findings to more complex, authentic settings has been generative for LS, but faces serious challenges. There is an alternate route to establishing a cumulative scientific knowledge base, namely, 'scaling down' from more complex, ecologically valid levels to more elemental levels. Studies of basic learning processes, framed in the context of the larger system, are well positioned to support impact in authentic settings. Copyright © 2010 John Wiley & Sons, Ltd. For further resources related to this article, please visit the WIREs website.
Copyright © 2010 John Wiley & Sons, Ltd.
In this study we test the hypothesis that the functional connectivity of the frontal and parietal regions that children recruit during a basic numerical task (matching Arabic numerals to arrays of dots) is predictive of their math test scores (TEMA-3; Ginsburg, 2003). Specifically, we tested 4-11-year-old children on a matching task during fMRI to localize a fronto-parietal network that responds more strongly during numerical matching than matching faces, words, or shapes. We then tested the functional connectivity between those regions during an independent task: natural viewing of an educational video that included math topics. Using this novel natural viewing method, we found that the connectivity between frontal and parietal regions during task-independent free-viewing of educational material is correlated with children's basic number matching ability, as well as their scores on the standardized test of mathematical ability (the TEMA). The correlation between children's mathematics scores and fronto-parietal connectivity is math-specific in the sense that it is independent of children's verbal IQ scores. Moreover, a control network, selective for faces, showed no correlation with mathematics performance. Finally, brain regions that correlate with subjects' overall response times in the matching task do not account for our number- and math-related effects. We suggest that the functional intersection of number-related frontal and parietal regions is math-specific.
The neural foundations of arithmetic learning are not well understood. While behavioral studies have revealed relationships between symbolic number processing and individual differences in children's arithmetic performance, the neurocognitive mechanisms that bind symbolic number processing and arithmetic are unknown. The current fMRI study investigated the relationship between children's brain activation during symbolic number comparison (Arabic digits) and individual differences in arithmetic fluency. A significant correlation was found between the numerical ratio effect on reaction times and accuracy and children's arithmetic scores. Furthermore, children with a stronger neural ratio effect in the left intraparietal sulcus (IPS) during symbolic number processing exhibited higher arithmetic scores. Previous research has demonstrated that activation of the IPS during numerical magnitude processing increases over the course of development, and that the left IPS plays an important role in symbolic number processing. The present findings extend this knowledge to show that children with more mature response modulation of the IPS during symbolic number processing exhibit higher arithmetic competence. These results suggest that the left IPS is a key neural substrate for the relationship between the relative of precision of the representation of numerical magnitude and school-level arithmetic competence.
The preschool years are a time of great advances in children's numerical thinking, most notably as they master verbal counting. The present research assessed the relation between analog magnitude representations and cardinal number knowledge in preschool-aged children to ask two questions: (1) Is there a relationship between acuity in the analog magnitude system and cardinality proficiency? (2) Can evidence of the analog magnitude system be found within mappings of number words children have not successfully mastered? To address the first question, Study 1 asked three- to five-year-old children to discriminate side-by-side dot arrays with varying differences in numerical ratio, as well as to complete an assessment of cardinality. Consistent with the analog magnitude system, children became less accurate at discriminating dot arrays as the ratio between the two numbers approached one. Further, contrary to prior work with preschoolers, a significant correlation was found between cardinal number knowledge and non-symbolic numerical discrimination. Study 2 aimed to look for evidence of the analog magnitude system in mappings to the words in preschoolers' verbal counting list. Based on a modified give-a-number task (Wynn, 1990, 1992), three- to five-year-old children were asked to give quantities between 1 and 10 as many times as possible in order to assess analog magnitude variability within their developing cardinality understanding. In this task, even children who have not yet induced the cardinality principle showed signs of analog representations in their understanding of the verbal count list. Implications for the contribution of analog magnitude representations towards mastery of the verbal count list are discussed in light of the present work.
A model of the relations among cognitive precursors, early numeracy skill, and mathematical outcomes was tested for 182 children from 4.5 to 7.5 years of age. The model integrates research from neuroimaging, clinical populations, and normal development in children and adults. It includes 3 precursor pathways: quantitative, linguistic, and spatial attention. These pathways (a) contributed independently to early numeracy skills during preschool and kindergarten and (b) related differentially to performance on a variety of mathematical outcomes 2 years later. The success of the model in accounting for performance highlights the need to understand the fundamental underlying skills that contribute to diverse forms of mathematical competence.
This study investigated individual differences in different aspects of early number concepts in preschoolers. Eighty 4-year-olds from Oxford nursery classes took part. They were tested on accuracy of counting sets of objects; the cardinal word principle; the order irrelevance principle; and predicting the results of repeated addition and subtraction by 1 from a set of objects. There were marked individual differences for most tasks. Most children were reasonably proficient at counting and 70% understood the cardinal word principle. Based on the results of a repeated addition and subtraction by 1 task, the children were divided into three approximately equal groups: those who were already able to use an internalized counting sequence for the simplest forms of addition and subtraction; those who relied on a repeated 'counting-all' procedure for such tasks; and those who were as yet unable to cope with such tasks. In each group, significant relationships between some, but not all, of the numerical tasks were found. However, for almost any two tasks, it was possible to find individuals who could carry out either one of the tasks but not the other. Thus, even before formal instruction, arithmetical cognition is not unitary but is made up of many components.
This study examines the abstractness of children's mental representation of counting, and their understanding that the last number word used in a count tells how many items there are (the cardinal word principle). In the first experiment, twenty-four 2- and 3-year-olds counted objects, actions, and sounds. Children counted objects best, but most showed some ability to generalize their counting to actions and sounds, suggesting that at a very young age, children begin to develop an abstract, generalizable mental representation of the counting routine. However, when asked "how many" following counting, only older children (mean age 3.6) gave the last number word used in the count a majority of the time, suggesting that the younger children did not understand the cardinal word principle. In the second experiment (the "give-a-number" task), the same children were asked to give a puppet one, two, three, five, and six items from a pile. The older children counted the items, showing a clear understanding of the cardinal word principle. The younger children succeeded only at giving one and sometimes two items, and never used counting to solve the task. A comparison of individual children's performance across the "how-many" and "give-a-number" tasks shows strong within-child consistency, indicating that children learn the cardinal word principle at roughly 3 1/2 years of age. In the third experiment, 18 2- and 3-year-olds were asked several times for one, two, three, five, and six items, to determine the largest numerosity at which each child could succeed consistently. Results indicate that children learn the meanings of smaller number words before larger ones within their counting range, up to the number three or four. They then learn the cardinal word principle at roughly 3 1/2 years of age, and perform a general induction over this knowledge to acquire the meanings of all the number words within their counting range.
Since the publication of [Gelman, R., & Gallistel, C. R. (1978). The child's understanding of number. Cambridge, MA: Harvard University Press.] seminal work on the development of verbal counting as a representation of number, the nature of the ontogenetic sources of the verbal counting principles has been intensely debated. The present experiments explore proposals according to which the verbal counting principles are acquired by mapping numerals in the count list onto systems of numerical representation for which there is evidence in infancy, namely, analog magnitudes, parallel individuation, and set-based quantification. By asking 3- and 4-year-olds to estimate the number of elements in sets without counting, we investigate whether the numerals that are assigned cardinal meaning as part of the acquisition process display the signatures of what we call "enriched parallel individuation" (which combines properties of parallel individuation and of set-based quantification) or analog magnitudes. Two experiments demonstrate that while "one" to "four" are mapped onto core representations of small sets prior to the acquisition of the counting principles, numerals beyond "four" are only mapped onto analog magnitudes about six months after the acquisition of the counting principles. Moreover, we show that children's numerical estimates of sets from 1 to 4 elements fail to show the signature of numeral use based on analog magnitudes - namely, scalar variability. We conclude that, while representations of small sets provided by parallel individuation, enriched by the resources of set-based quantification are recruited in the acquisition process to provide the first numerical meanings for "one" to "four", analog magnitudes play no role in this process.
Theoretical analyses of the development of numerical representations suggest that playing linear number board games should enhance young children's numerical knowledge. Consistent with this prediction, playing such a game for roughly 1 hr increased low-income preschoolers' (mean age = 5.4 years) proficiency on 4 diverse numerical tasks: numerical magnitude comparison, number line estimation, counting, and numeral identification. The gains remained 9 weeks later. Classmates who played an identical game, except for the squares varying in color rather than number, did not improve on any measure. Also as predicted, home experience playing number board games correlated positively with numerical knowledge. Thus, playing number board games with children from low-income backgrounds may increase their numerical knowledge at the outset of school.