An Integrative Theory of Numerical Development

Knowing what they know: Preschool teachers' knowledge of math skills and its relation to instruction

Math experiences during the preschool years play an important role in children's later math learning. Preschool teachers exhibit considerable variability in the amount and types of mathematics activities they engage in with their students; one potentially important source of these individual differences is adults' knowledge of early math development. The current study aimed to describe preschool teachers' knowledge of numeracy, patterning, and spatial/geometric skills developed in preschool and its relation to their reported mathematics instruction. Participants (N = 83) completed a survey in which they judged whether particular early math skills could be observed in typically developing 4-year-olds in the United States and reported their frequency of engaging in different math instructional activities. Pre- and in-service preschool teachers' knowledge varied across the different domains (i.e., numeracy, patterning, and spatial/geometric) of mathematical thinking, but their reported frequency of instruction did not. Teachers who were found to be more accurate in their knowledge of early math development were more likely to report higher frequency of math instruction; looking specifically at the domains, the strength of association between knowledge and instruction was the strongest for numeracy. Such findings highlight the possibility that supporting preschool teachers' knowledge of the range of math skills their students can be developing may be one component of improving early math teaching and learning.

Exploring the impact of a fraction sense intervention in authentic school environments: An initial investigation

A solid understanding of fractions is the cornerstone for acquiring proficiency with rational numbers and paves the way for learning advanced mathematical concepts such as algebra. Fraction difficulties limit not only students' educational and vocational opportunities but also their ability to solve everyday problems. Students who exit sixth grade with inadequate understanding of fractions may experience far-reaching repercussions that lead to lifelong avoidance of mathematics. This article presents the results of a randomized controlled trial focusing on the first two cohorts of a larger efficacy investigation aimed at building fraction sense in students with mathematics difficulties. Teachers implemented an evidence-informed fraction sense intervention (FSI) within their sixth-grade intervention classrooms. The lessons draw from research in cognitive science as well as mathematics education research. Employing random assignment at the classroom level, multilevel modeling revealed a significant effect of the intervention on posttest fractions scores after controlling for pretest fractions scores, working memory, vocabulary, proportional reasoning, and classroom attentive behavior. Students in the FSI group outperformed their counterparts in the control group, with noteworthy effect sizes on most fraction measures. Challenges associated with carrying out school-based intervention research are addressed.

Frontoparietal and salience network synchronizations during nonsymbolic magnitude processing predict brain age and mathematical performance in youth

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.

Concepts of order: Why is ordinality processed slower and less accurately for non-consecutive sequences?

Both adults and children are slower at judging the ordinality of non-consecutive sequences (e.g., 1-3-5) than consecutive sequences (e.g., 1-2-3). It has been suggested that the processing of non-consecutive sequences is slower because it conflicts with the intuition that only count-list sequences are correctly ordered. An alternative explanation, however, may be that people simply find it difficult to switch between consecutive and non-consecutive concepts of order during order judgement tasks. Therefore, in adult participants, we tested whether presenting consecutive and non-consecutive sequences separately would eliminate this switching demand and thus improve performance. In contrast with this prediction, however, we observed similar patterns of response times independent of whether sequences were presented separately or together (Experiment 1). Furthermore, this pattern of results remained even when we doubled the number of trials and made participants explicitly aware when consecutive and non-consecutive sequences were presented separately (Experiment 2). Overall, these results suggest slower response times for non-consecutive sequences do not result from a cognitive demand of switching between consecutive and non-consecutive concepts of order, at least not in adults.

From integers to fractions: The role of analogy in transfer and long-term learning

Fractions are the gatekeepers to advanced mathematics but are difficult to learn. One powerful learning mechanism is analogy, which builds fraction understanding on a pre-existing foundation of integer knowledge. Indeed, a short intervention that aligned fractions and integers on number lines improved children's estimates of fractions (Yu et al., 2022). The breadth and durability of such gains, however, are unknown, and analogies to other sources (such as percentages) may be equally powerful. To investigate this issue, we randomly assigned 109 fourth and fifth graders to one of three experimental conditions with different analogical sources (integers, percentages, or fractions) or a control condition. During training, children in the experimental conditions solved pairs of aligned fraction number line problems and proportionally-equivalent problems expressed in integers, percentages, or fractions (e.g., 3/8 on a 0-1 number line aligned with 3 on a 0-8 number line). Children in the control group solved fraction number-line problems sequentially. At pretest and a two-week delayed posttest, children completed a broad fraction knowledge battery, including estimation, comparison, categorization, ordering, and arithmetic. Results showed that aligning integers and fractions on number lines facilitated better estimation of fractional magnitudes, and the training effect transferred to novel fraction problems after two weeks. Similar gains were not observed for analogies using percentages. These findings highlight the importance of building new mathematical knowledge through analogies to familiar, similar sources.

The relation between number line performance and mathematics outcomes: Two meta-analyses

Understanding the magnitudes represented by numerals is a core component of early mathematical development and is often assessed by accuracy in situating numerals and fractions on a number line. Performance on these measures is consistently related to performance in other mathematics domains, but the strength of these relations may be overestimated because general cognitive ability has not been fully controlled in prior studies. The first of two meta-analyses (162 studies, 33,101 participants) confirmed a relation between performance on whole number (r = 0.33) and fractions number (r = 0.41) lines and overall mathematics performance. These relations were generally consistent across content domains (e.g., algebra and computation) and other moderators. The second (71 studies, 14,543 participants) used meta-analytic structural equation modeling to confirm these relations while controlling general cognitive ability (defined by IQ and working memory measures) and, in one analysis, general mathematics competence. The relation between number line performance and general mathematics competence remained significant but reduced (β = 0.13). Controlling general cognitive ability, whole number line performance consistently predicted competence with fractions but not performance on numeracy or computations measures. The results suggest an understanding of the magnitudes represented by whole numbers might be particularly important for students' fractions learning. RESEARCH HIGHLIGHTS: Two meta-analyses examined the link between the number line and mathematics performance. The first revealed significant relations across domains (e.g., algebra and computation). The second controlled for general cognitive ability and resulted in reduced but still significant relations. The relation between number line and fractions performance was stronger than relations to other domains.

From whole numbers to fractions to word problems: Hierarchical relations in mathematics knowledge for Chinese Grade 6 students

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.

Fraction Ball impact on student and teacher math talk and behavior

We assessed the impacts of Fraction Ball-a novel suite of games combining the benefits of embodied guided play for math learning-on the math language production and behavior of students and teachers. In the Pilot Experiment, 69 fifth and sixth graders were randomly assigned to play four different Fraction Ball games or attend normal physical education class. The Efficacy Experiment was implemented to test improvements made through co-design with teachers with 160 fourth through sixth graders. Researchers observed and coded for use of math language and behavior. Playing Fraction Ball resulted in consistent increases of students' and teachers' use of fraction (SDs = 0.98-2.42) and decimal (SDs = 0.65-1.64) language and number line arithmetic, but not in whole number, spatial language, counting, instructional gesturing, questioning, and planning. We present evidence of the math language production in physical education and value added by Fraction Ball to support rational number language and arithmetic through group collaboration.

Socio-Economic and Cultural Context in the Development of Early Mathematical Competencies: A Comparative Study of Specific Educational Contexts in Chile and Spain

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

Conceptual foundations of early numeracy: Evidence from infant brain data

Understanding the conceptual resources that children bring to mathematics learning is crucial for developing effective instruction and interventions. Despite the considerable number of studies examining the neural underpinnings of number representations in adults and the growing number of reports in children, very few studies have examined the neural correlates of the link between foundational resources related to numerical information and symbolic number representations in infants. There is currently an active debate about which foundational resources are critical for symbolic mathematics. Is early numerical discrimination best explained by a holistic and generalized sense of magnitude rather than a number sense? Does early number sense provide the conceptual basis for mapping numerical symbols to their meaning? Are foundational number systems marginal while children learn to count and perform symbolic arithmetic, and only later children map non symbolic representations of numerical magnitudes onto symbols? After describing the mainstream theories of numerical cognition and the sources of controversy, we review recent studies of the neural bases of human infants' numerical performance with the aim of clarifying the link between early conceptual resources and symbolic number systems as children's mathematical minds develop.

Investigating the influence of body movements on children's mental arithmetic performance

Several lines of research have demonstrated spatial-numerical associations in both adults and children, which are thought to be based on a spatial representation of numerical information in the form of a mental number line. The acquisition of increasingly precise mental number line representations is assumed to support arithmetic learning in children. It is further suggested that sensorimotor experiences shape the development of number concepts and arithmetic learning, and that mental arithmetic can be characterized as "motion along a path" and might constitute shifts in attention along the mental number line. The present study investigated whether movements in physical space influence mental arithmetic in primary school children, and whether the expected effect depends on concurrency of body movements and mental arithmetic. After turning their body towards the left or right, 48 children aged 8 to 10 years solved simple subtraction and addition problems. Meanwhile, they either walked or stood still and looked towards the respective direction. We report a congruency effect between body orientation and operation type, i.e., higher performance for the combinations leftward orientation and subtraction and rightward orientation and addition. We found no significant difference between walking and looking conditions. The present results suggest that mental arithmetic in children is influenced by preceding sensorimotor cues and not necessarily by concurrent body movements.

Assessing Cognitive Skills in Early Childhood Education Using a Bilingual Early Language Learner Assessment Tool

In this article, we propose that basic cognitive skills may be fostered and assessed in early childhood educational (pre-K) settings using a technology-based approach to assessment. BELLA (Bilingual English Language Learner Assessment), designed for use with both monolingual (English or Spanish speaking) and bilingual (English and Spanish speaking) children, is designed to attend to cognitive skill development in addition to (pre-)academic knowledge. Specifically, BELLA assesses analytical, creative, and practical thinking in 3-5-year-old children through unique item content and delivery. BELLA is among the first tablet-based pre-K assessments designed to assess cognitive skills needed for the era of the Anthropocene.

Learning-induced reorganization of number neurons and emergence of numerical representations in a biologically inspired neural network

Number sense, the ability to decipher quantity, forms the foundation for mathematical cognition. How number sense emerges with learning is, however, not known. Here we use a biologically-inspired neural architecture comprising cortical layers V1, V2, V3, and intraparietal sulcus (IPS) to investigate how neural representations change with numerosity training. Learning dramatically reorganized neuronal tuning properties at both the single unit and population levels, resulting in the emergence of sharply-tuned representations of numerosity in the IPS layer. Ablation analysis revealed that spontaneous number neurons observed prior to learning were not critical to formation of number representations post-learning. Crucially, multidimensional scaling of population responses revealed the emergence of absolute and relative magnitude representations of quantity, including mid-point anchoring. These learnt representations may underlie changes from logarithmic to cyclic and linear mental number lines that are characteristic of number sense development in humans. Our findings elucidate mechanisms by which learning builds novel representations supporting number sense.

Spatial-Numerical Magnitude Estimation Mediates Early Sex Differences in the Use of Advanced Arithmetic Strategies

An accumulating body of literature points to a link between spatial reasoning and mathematics learning. The present study contributes to this line of research by investigating sex differences both in spatial representations of magnitude and in the use of arithmetic strategies, as well as the relation between the two. To test the hypothesis that sex differences in spatial-numerical magnitude knowledge mediate sex differences in the use of advanced strategies (retrieval and decomposition), two studies were conducted. Study 1 included 96 US first graders (53% girls); Study 2 included 210 Russian first graders (49% girls). All participants completed a number line estimation task (a spatially based measure of numerical magnitude knowledge) and an arithmetic strategy task (a measure of strategy choice). The studies showed parallel results: boys produced more accurate numerical magnitude estimates on the number line estimation task and used advanced strategies more frequently on the arithmetic task. Critically, both studies provide support for the mediation hypothesis (although there were some differences in the pattern obtained for the two strategies). The results are discussed in the context of broader research about the relation between spatial and mathematical skills.

Heterogeneity in children at risk of math learning difficulties

This study recruited 428 Singaporean children at risk of math learning difficulties (MLD; M_age  = 83.9 months, SD_age  = 4.35 months; 41% female). Using a factor mixture model that considered both quantitative and qualitative differences in math ability, two qualitatively different groups were identified: one with generalized difficulties across different math skills and the other with more focal difficulties in arithmetic fluency. Reading, working memory capacity, and numeracy (number line estimation skills and numerical discrimination) uniquely explained group membership. Children within each group differed in the extent of difficulties they exhibited, with numeracy variables differentially contributing to math ability in each group. Findings speak against a dimensional view of MLD and underscore the conceptual limitations of using basic numeracy performance to profile learning difficulties.

The relevance of basic numerical skills for fraction processing: Evidence from cross-sectional data

Recent research indicated that fraction understanding is an important predictor of later mathematical achievement. In the current study we investigated associations between basic numerical skills and students' fraction processing. We analyzed data of 939 German secondary school students (age range = 11.92 to 18.00 years) and evaluated the determinants of fraction processing considering basic numerical skills as predictors (i.e., number line estimation, basic arithmetic operations, non-symbolic magnitude comparison, etc.). Additionally, we controlled for general cognitive ability, grade level, and sex. We found that multiplication, subtraction, conceptual knowledge, number line estimation, and basic geometry were significantly associated with fraction processing beyond significant associations of general cognitive ability and sex. Moreover, relative weight analysis revealed that addition and approximate arithmetic should also be considered as relevant predictors for fraction processing. The current results provide food for thought that further research should focus on investigating whether recapitulating basic numerical content in secondary school mathematics education can be beneficial for acquiring more complex mathematical concepts such as fractions.

Relating mathematical abilities to numerical skills and executive functions in informal and formal schooling

Background

The current evidence on an integrative role of the domain-specific early mathematical skills and number-specific executive functions (EFs) from informal to formal schooling and their effect on mathematical abilities is so far unclear. The main objectives of this study were to (i) compare the domain-specific early mathematics, the number-specific EFs, and the mathematical abilities between preschool and primary school children, and (ii) examine the relationship among the domain-specific early mathematics, the number-specific EFs, and the mathematical abilities among preschool and primary school children.

Methods

The current study recruited 6- and 7-year-old children (N_total = 505, n_6yrs = 238, and n_7yrs = 267). The domain-specific early mathematics as measured by symbolic and nonsymbolic tasks, number-specific EFs tasks, and mathematics tasks between these preschool and primary school children were compared. The relationship among domain-specific early mathematics, number-specific EFs, and mathematical abilities among preschool and primary school children was examined. MANOVA and structural equation modeling (SEM) were used to test research hypotheses.

Results

The current results showed using MANOVA that primary school children were superior to preschool children over more complex tests of the domain-specific early mathematics; number-specific EFs; mathematical abilities, particularly for more sophisticated numerical knowledge; and number-specific EF components. The SEM revealed that both the domain-specific early numerical and the number-specific EFs significantly related to the mathematical abilities across age groups. Nevertheless, the number comparison test and mental number line of the domain-specific early mathematics significantly correlated with the mathematical abilities of formal school children. These results show the benefits of both the domain-specific early mathematics and the number-specific EFs in mathematical development, especially at the key stages of formal schooling. Understanding the relationship between EFs and early mathematics in improving mathematical achievements could allow a more powerful approach in improving mathematical education at this developmental stage.

Incongruity in fraction verification elicits N270 and P300 ERP effects

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

Relations among spatial skills, number line estimation, and exact and approximate calculation in young children

Decades of research have established that spatial skills correlate with numerical skills. However, because both spatial and numerical skills are multidimensional, we sought to determine how specific spatial skills relate to specific numeracy skills. We used a cohort-sequential design, assessing a large diverse sample of students (N = 612, initially in pre-kindergarten [pre-K]-3rd grade, 4-9 years of age) at four time points spanning 2 years. We examined how initial levels of five spatial skills (visuospatial working memory [VSWM], mental transformation, mental rotation, proportional reasoning, and analog magnitude system [AMS] acuity) related to initial levels and growth rates in exact and approximate calculation skills, and we further investigated number line estimation as a potential mediator. We found unique patterns of relations between spatial skills and numeracy. Initial levels of mental rotation, proportional reasoning, and AMS acuity related to initial levels of exact calculation skill; initial levels of AMS acuity related to initial levels of approximate calculation; and initial levels of proportional reasoning related to initial levels of number line estimation. VSWM and mental transformation did not relate to numeracy skills after controlling for other spatial skills. Initial levels of number line estimation related to both exact and approximate calculation after controlling for spatial skills. Notably, neither spatial skills nor number line estimation predicted growth in exact or approximate calculation skills. These results indicate that there is specificity in the time-invariant relations between spatial skills and numeracy, and they suggest that researchers and educators should treat spatial skills and numeracy as multidimensional constructs with complex and unique interrelations.

Neurofunctional plasticity in fraction learning: An fMRI training study

BACKGROUND

Fractions are known to be difficult for children and adults. Behavioral studies suggest that magnitude processing of fractions can be improved via number line estimation (NLE) trainings, but little is known about the neural correlates of fraction learning.

METHOD

To examine the neuro-cognitive foundations of fraction learning, behavioral performance and neural correlates were measured before and after a five-day NLE training.

RESULTS

In all evaluation tasks behavioral performance increased after training. We observed a fronto-parietal network associated with number magnitude processing to be recruited in all tasks as indicated by a numerical distance effect. For symbolic fractions, the distance effect on intraparietal activation was only observed after training.

CONCLUSION

The absence of a distance effect of symbolic fractions before the training could indicate an initially less automatic access to their overall magnitude. NLE training facilitates processing of overall fraction magnitude as indicated by the distance effect in neural activation.

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

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

Finding patterns in objects and numbers: Repeating patterning in pre-K predicts kindergarten mathematics knowledge

Both recent evidence and research-based early mathematics curricula indicate that repeating patterns-predictable sequences that follow a rule-are a topic of major importance for mathematics development. The purpose of the current study was to help build a theory for how early repeating patterning knowledge contributes to early math development, focusing on development in children aged 4-6 years. The current study examined the relation between 65 preschool children's repeating patterning knowledge (via a fast, teacher-friendly measure) and their end-of-kindergarten broad math and numeracy knowledge, controlling for verbal and visual-spatial working memory (WM) skills as well as end-of-pre-K (pre-kindergarten) broad math knowledge. Relations were also examined between repeating patterning and specific aspects of numeracy knowledge-knowledge of the count sequence to 100 and the successor principle. Children's repeating patterning knowledge was significantly predictive of their broad math and general numeracy knowledge, as well as one specific aspect of their numeracy knowledge (counting to 100), even after controlling for verbal and visual-spatial WM skills. Further, repeating patterning knowledge remained a unique predictor of general numeracy knowledge and counting to 100 after controlling for end-of-pre-K broad math knowledge. The relation between repeating patterning and mathematics may be explained by the central role that identifying predictable sequences based on underlying rules plays in both. Theories of math development and early math instruction standards should thus give even greater attention to the role of children's repeating patterning knowledge.

Putting a Finger on Numerical Development - Reviewing the Contributions of Kindergarten Finger Gnosis and Fine Motor Skills to Numerical Abilities

The well-documented association between fingers and numbers is not only based on the observation that most children use their fingers for counting and initial calculation, but also on extensive behavioral and neuro-functional evidence. In this article, we critically review developmental studies evaluating the association between finger sensorimotor skills (i.e., finger gnosis and fine motor skills) and numerical abilities. In sum, reviewed studies were found to provide evidential value and indicated that both finger gnosis and fine motor skills predict measures of counting, number system knowledge, number magnitude processing, and calculation ability. Therefore, specific and unique contributions of both finger gnosis and fine motor skills to the development of numerical skills seem to be substantiated. Through critical consideration of the reviewed evidence, we suggest that the association of finger gnosis and fine motor skills with numerical abilities may emerge from a combination of functional and redeployment mechanisms, in which the early use of finger-based numerical strategies during childhood might be the developmental process by which number representations become intertwined with the finger sensorimotor system, which carries an innate predisposition for said association to unfold. Further research is nonetheless necessary to clarify the causal mechanisms underlying this association.

Time Reading in Middle and Secondary School Students: The Influence of Basic-Numerical Abilities

Time reading skills are central for the management of personal and professional life. However, little is known about the differential influence of basic numerical abilities on analog and digital time reading in general and in middle and secondary school students in particular. The present study investigated the influence of basic numerical skills separately for analog and digital time reading in N = 709 students from 5th to 8th grade. The present findings suggest that the development of time reading skills is not completed by the end of primary school. Results indicated that aspects of magnitude manipulation and arithmetic fact knowledge predicted analog time reading significantly over and above the influence of age. Furthermore, results showed that spatial representations of number magnitude, magnitude manipulation, arithmetic fact knowledge, and conceptual knowledge were significant predictors of digital time reading beyond general cognitive ability and sex. To the best of our knowledge, the present study is the first to show differential effects of basic numerical abilities on analog and digital time reading skills in middle and secondary school students. As time readings skills are crucial for everyday life, these results are highly relevant to better understand basic numerical processes underlying time reading.

Preschoolers' broad mathematics experiences with parents during play

The current study broadens our understanding of preschoolers' early math experiences with parents, recognizing that math knowledge and experiences are inclusive of numeracy as well as non-numeracy domains. Parents and preschoolers (N = 45) were observed exploring three domains of early mathematics knowledge (i.e., number, space, and pattern) during play in three activities (playing cards, building with blocks, and stringing beads, all with activity suggestions). Children were administered a broad math and numeracy measure and individual measures of spatial and patterning skills concurrently and 7 months later. Dyads explored math broadly across most activities but emphasized number more than space or patterning. In addition, there was more overall math exploration during card and bead play than during block play, with the greatest parent support during card play. Parent support was not linked to children's skills, although children's exploration of space and patterns related moderately to their concurrent spatial and pattern skills. Overall, parents and young children explored a variety of early math domains in guided play contexts, with an emphasis on numeracy. Future work should aim to increase the breadth and rigor of individual concepts that parents and preschoolers explore during play.

Numerical Estimation and Mathematical Learning Methodology in Preschoolers

One of the means for representing quantities/magnitudes is the mental number line. It is still a nonsolved question as to whether the method of learning mathematics in the early years could improve this type of estimating. A total of 233 students, aged four and five years, who learned mathematics with a new method called Open Algorithm Based on Number or the more traditional Closed Based on Ciphers approach, were evaluated with a reliable estimation on the number line test. Results revealed significant differences in participants' estimation functions based on the learning method used. Students who learned mathematics through the Open Algorithm Based on Number method used a linear representation more efficiently than those who were taught with the Closed Based on Ciphers methodology. This group exhibited a logarithmic function in their approach. We discussed whether these differences can be attributed to a recurrent practice in estimation tasks at school, characterized by the Open Algorithm Based on Number methodology.

Number line unidimensionality is a critical feature for promoting fraction magnitude concepts

Children's ability to estimate fractions on a number line is strongly related to algebra and overall high school math achievement, and number line training leads to better fraction magnitude comparisons compared with area model training. Here, we asked whether unidimensionality is necessary for the number line to promote fraction magnitude concepts and whether left-to-right orientation and labeled endpoints are sufficient. We randomly assigned second- and third-graders (N = 148) to one of four 15-min one-on-one, experimenter-led trainings. Three number line trainings had identical scripts, where the experimenter taught children to segment and shade the number line along the horizontal dimension. The number line conditions varied only in the vertical dimension of the training number line: pure unidimensional number line (17.5 cm horizontal line), hybrid unidimensional number line (17.5 × 0.6 cm rectangle), and square number line (17.5 × 17.5 cm). In the area model condition, children were taught to segment and shade a square (17.5 × 17.5 cm) along both dimensions. The conditions significantly differed in posttest fraction magnitude comparison accuracy (a transfer task), controlling for pretest accuracy, reading achievement, and age. In preregistered analyses, the hybrid unidimensional number line condition significantly outperformed the square area model condition and the square number line condition. In exploratory analyses accounting for training protocol fidelity, these results held and the pure unidimensional number line also outperformed the area model condition on fraction magnitude comparisons. We argue that unidimensionality is a critical feature of the number line for promoting fraction magnitude concepts because it aligns with a key concept-that real numbers, including fractions, can be ordered along a single dimension.

Embedding Self-Regulation Instruction Within Fractions Intervention for Third Graders With Mathematics Difficulties

The purpose of this study was to explore the efficacy of fractions intervention with and without an embedded self-regulation (SR) component for third-grade students at risk for mathematics disabilities. Fractions intervention focused on magnitude understanding and word problems. Embedded SR was designed to support a growth mindset (fostering belief that intellectual and academic abilities can be developed) along with SR processes in which students set goals, self-monitor, and use strategies to engage motivationally, metacognitively, and behaviorally through challenging tasks. Students (n = 69) were randomly assigned to business-as-usual control and the two versions of fractions intervention. Multilevel models, accounting for the nested structure of the data, identified a moderation effect on fraction word problems: For students receiving fractions intervention with embedded SR, response to intervention was robust across the continuum of students' pretest word problem skill; by contrast, without SR, response to fractions intervention depended on students' pretest word problem skill. On the remaining outcomes, results reflected stronger outcomes when fractions intervention embedded SR instruction without moderation.

Benefits of Playing Numerical Card Games on Head Start Children's Mathematical Skills

Low-income preschoolers have lower average performance on measures of early numerical skills than middle-income children. The present study examined the effectiveness of numerical card games in improving children's numerical and executive functioning skills. Low-income preschoolers (N=76) were randomly assigned to play a numerical magnitude comparison card game, a numerical memory and matching card game, or a shape and color matching card game across four 15-minute sessions. Child who played either of the numerical games improved their numeral identification skills, while only children who played the numerical magnitude comparison game improved their symbolic magnitude comparison skills. These improvements were maintained eight weeks later. The results suggest that a brief, low-cost intervention can successfully improve the numerical skills of low-income children.

Individual differences in basic numerical skills: The role of executive functions and motor skills

The aim of the current study was to explore individual differences in basic numerical skills in a normative sample of 151 kindergarteners (mean age = 6.45 years). Whereas previous research claims a substantial link between executive functions and basic numerical skills, motor abilities have been put forward to explain variance in numerical skills. Regarding the current study, these two assumptions have been combined, revealing interesting results. Namely, executive functions (inhibition, switching, and visuospatial working memory) were found to relate to symbolic numerical skills, and motor skills (gross and fine motor skills) showed a significant correlation to nonsymbolic numerical skills. Suggesting that motor skills and executive functions are associated with basic numerical skills could lead to potential avenues for interventions in certain disorders or disabilities such as nonverbal learning disability, developmental dyscalculia, and developmental coordination disorder.

Identifying Fraction Measures as Screeners of Mathematics Risk Status

This study investigated the accuracy of three fraction measures (i.e., fraction number line estimation accuracy, general fraction concepts, and fraction arithmetic) for screening fourth graders who might be at risk for mathematics difficulties. Receiver operating characteristic (ROC) curve analyses assessed diagnostic accuracy of the fraction measures for predicting which students would not meet state standards on the state mathematics test in fourth grade (n = 411), fifth grade (n = 362), and sixth grade (n = 304). A combined measure consisting primarily of fraction number line estimation items and general fraction concept items was the most accurate screener of risk status in fourth, fifth, and sixth grades (area under the curve [AUC] = .84, .81, and .85, respectively). To maximize efficiency for classroom use, the length of the combined screener was reduced using best subset automatic linear modeling. The study highlights the importance of fraction knowledge for predicting mathematics achievement more generally and validates an effective and practical screening tool for the intermediate grades.

The Role of Approximate Number System in Different Mathematics Skills Across Grades

Although approximate number system (ANS) has been found to predict mathematics ability, it remains unclear if both aspects of ANS (symbolic and non-symbolic estimation) contribute equally well to mathematics performance and if their contribution varies as a function of the mathematics outcome and grade level. Thus, in this study, we examined the effects of both aspects of ANS on different mathematics skills across three grade levels. Three hundred eleven children (100 children from kindergarten, 107 children from Grade 2, and 104 children from Grade 4) from two kindergartens and three elementary schools in Shanghai, China, were assessed on measures of ANS (dot estimation and number line estimation), general cognitive ability (nonverbal intelligence, inhibition, and working memory), and mathematics abilities (numerical operations and mathematical problem solving in all grades, early mathematical skills in kindergarten, and calculation fluency in Grades 2 and 4). Results of hierarchical regression analyses showed that, in kindergarten, non-symbolic estimation predicted all mathematics skills even after controlling for age, gender, and general cognitive ability. In Grades 2 and 4, symbolic estimation accounted for unique variance in mathematical problem solving, but not in calculation fluency. Symbolic estimation also predicted numerical operations in Grade 4. Taken together, these findings suggest that in the early phases of mathematics development different aspects of ANS contribute to different mathematics skills.

Preschoolers' selective sustained attention and numeracy skills and knowledge

This brief report addresses preschoolers' selective sustained attention (SSA) and early numeracy skills and knowledge. Past research indicates that children's attention and early numeracy are positively associated, yet some concerns have emerged about the age appropriateness of tools used to measure preschoolers' SSA. This study used a new measure-the Track-It Task-that demonstrates strong psychometric properties. In total, 31 at-risk preschoolers (M_age = 46.6 months) participated and were assessed on SSA, nonsymbolic quantity discrimination, and symbolic quantitative skills and knowledge. The ability to sustain attention in the face of distractions was positively correlated with preschoolers' verbal counting and one-to-one correspondence, Arabic numeral recognition, and cardinal principle knowledge. SSA was not significantly associated with child age or performance on a memory task. This study provides preliminary evidence that SSA may facilitate the process whereby young children become reliable counters and learn that the symbol system of numbers represents specific quantities.

Narrowing the Early Mathematics Gap: A Play-Based Intervention to Promote Low-Income Preschoolers' Number Skills

Preschoolers from low-income households lag behind preschoolers from middle-income households on numerical skills that underlie later mathematics achievement. However, it is unknown whether these gaps exist on parallel measures of symbolic and non-symbolic numerical skills. Experiment 1 indicated preschoolers from low-income backgrounds were less accurate than peers from middle-income backgrounds on a measure of symbolic magnitude comparison, but they performed equivalently on a measure of non-symbolic magnitude comparison. This suggests activities linking non-symbolic and symbolic number representations may be used to support children's numerical knowledge. Experiment 2 randomly assigned low-income preschoolers (M _age = 4.7 years) to play either a numerical magnitude comparison or a numerical matching card game across four 15 min sessions over a 3-week period. The magnitude comparison card game led to significant improvements in participants' symbolic magnitude comparison skills in an immediate posttest assessment. Following the intervention, low-income participants performed equivalently to an age- and gender-matched sample of middle-income preschoolers in symbolic magnitude comparison. These results suggest a brief intervention that combines non-symbolic and symbolic magnitude representations can support low-income preschoolers' early numerical knowledge.

Magnitude, numerosity, and development of number: Implications for mathematics disabilities

Leibovich et al. challenge the prevailing view that non-symbolic number sense (e.g., sensing number the same way one might sense color) is innate, that detection of numerosity is distinct from detection of continuous magnitude. In the present commentary, the authors' viewpoint is discussed in light of the integrative theory of numerical development along with implications for understanding mathematics disabilities.

Children's understanding of fraction and decimal symbols and the notation-specific relation to pre-algebra ability

Fraction and decimal concepts are notoriously difficult for children to learn yet are a major component of elementary and middle school math curriculum and an important prerequisite for higher order mathematics (i.e., algebra). Thus, recently there has been a push to understand how children think about rational number magnitudes in order to understand how to promote rational number understanding. However, prior work investigating these questions has focused almost exclusively on fraction notation, overlooking the open questions of how children integrate rational number magnitudes presented in distinct notations (i.e., fractions, decimals, and whole numbers) and whether understanding of these distinct notations may independently contribute to pre-algebra ability. In the current study, we investigated rational number magnitude and arithmetic performance in both fraction and decimal notation in fourth- to seventh-grade children. We then explored how these measures of rational number ability predicted pre-algebra ability. Results reveal that children do represent the magnitudes of fractions and decimals as falling within a single numerical continuum and that, despite greater experience with fraction notation, children are more accurate when processing decimal notation than when processing fraction notation. Regression analyses revealed that both magnitude and arithmetic performance predicted pre-algebra ability, but magnitude understanding may be particularly unique and depend on notation. The educational implications of differences between children in the current study and previous work with adults are discussed.