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

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

Perceptual subitizing performance in 3- and 4-year-olds: The impact of visual features of sets.

Perceptual subitizing is a pivotal skill in children's mathematical development. It is defined as the rapid identification of small numerosities. Previous studies pointed to the contribution of visual features of sets to perceptual subitizing performance in adults. Insights into the contribution of visual features to subitizing performance in the critical 3- to 4-year age range are scant. This study aimed to address this gap by investigating the impact of visual features on perceptual subitizing performance (accuracy and response time) in 3- and 4-year-olds. Participants (119 3- and 4-year-olds) were offered a subitizing task that incorporated pictures of sets of three to five objects. The pictures systematically varied across four visual features: (a) pictorial context (distractors present vs. absent), (b) set homogeneity (homogeneous vs. heterogeneous objects), (c) set arrangement (linearly vs. randomly arranged objects), and (d) set differentiation (distinct vs. overlapping objects). Pictures with distractors, heterogeneous objects, randomly arranged objects, or overlapping objects were associated with lower subitizing accuracy and longer response times compared with pictures without distractors, homogeneous objects, linearly arranged objects, or distinct objects, respectively. Pictures with randomly arranged or overlapping objects along with distractors were associated with even lower subitizing accuracy. Pictures featuring a simple visual design-without distractors and with homogeneous, linearly arranged, and distinct sets-yielded the best subitizing performance in terms of accuracy and response time. Our findings might be explained by the cognitive processes underlying 3- and 4-year-olds' subitizing performance. The findings offer building blocks for future research in the domain and preschool educational practice.

Subtraction by addition in young multi-digit subtraction learners: A choice/no-choice study.

When solving subtraction problems such as 83-46, children use the direct subtraction (DS) strategy (e.g., 83 - 40 = 43, 43 - 6 = 37) or the subtraction by addition (SBA) strategy (e.g., 46 + 4 = 50, 50 + 30 = 80, 80 + 3 = 83, so the answer is 4 + 30 + 3 = 37). This study is the first to use the choice/no-choice method to examine DS and SBA use in third-graders (8- and 9-year-olds) with varying mathematical achievement levels. All children (N = 66) solved a series of small difference subtractions (e.g., 72 - 64) and large difference subtractions (e.g., 94 - 8) in one choice condition (choice between DS and SBA) and two no-choice conditions (obligatory use of either DS or SBA). Results showed that, although only the DS strategy was taught to these children, in the choice condition nearly half the children already made use of the SBA strategy and that SBA was used on one in five subtractions. Whereas DS was the fastest strategy on large difference items, interestingly, children achieved a similar level of accuracy with the SBA strategy compared with DS. Finally, 1 in 5 children made problem-based adaptive strategy choices, and children were generally adaptive to their individual strategy speed. This study clearly demonstrates the nascent use of SBA in third-graders and provides evidence for the plea to introduce SBA as an alternative to DS already in the lower grades of elementary school.

Longitudinal pathways of numerical abilities in preschool: Cognitive and environmental correlates and relation to primary school mathematics achievement.

Children start preschool with large individual differences in their early numerical abilities. Little is known about the importance of heterogeneous patterns that exist within these individual differences. A person-centered analytic approach might be helpful to unravel these patterns and the cognitive and environmental factors that are associated with them. We applied a person-centered approach to a 5-year longitudinal study (N = 410, 213 boys) conducted in Belgium from preschool to grade 3. Preschoolers (Mage = 58.14 months, SDage = 3.51) were selected to represent the full range of socioeconomic backgrounds. We examined via Latent Profile Analysis the heterogeneous patterns that exist in preschoolers' early numerical development using measures of counting, numeral identification, comparison, ordering, and arithmetic abilities. We investigated the association between the derived numerical ability pathways, general cognitive factors (working memory, language, spatial ability) and the home math environment. We also evaluated the relation of these early numerical ability pathways to later mathematics achievement in grade 1 and 3. Four longitudinal pathways emerged: a low (15%), below-average (28%), above-average (44%), and a high numerical ability pathway (13%). Differences between the four pathways were mostly quantitative. Most of the general cognitive factors contributed to pathway membership, whereas the home math environment and socioeconomic status (SES) did not. The pathways differed in mathematics achievement in grade 1 and 3, and most of these differences remained when the covariates were considered. The results highlight the heterogeneity that is already present in preschoolers' numerical abilities and their predictive value for subsequent mathematics achievement. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Associations Between Repeating Patterning, Growing Patterning, and Numerical Ability: A Longitudinal Panel Study in 4- to 6-Year Olds.

The present study aimed to analyze the direction of the associations between repeating patterning, growing patterning, and numerical ability. Participants were 410 children who were annually assessed on their repeating patterning, growing patterning, and numerical ability, at ages 4, 5, and 6 years (i.e., spring 2017, 2018, and 2019). A cross-lagged panel model identified bidirectional associations between all three abilities from ages 4 to 5 years while taking into account spatial skills. From ages 5 to 6 years, both patterning abilities predicted later numerical ability, but the reverse was no longer true. Associations between performances on both pattern types also disappeared. Results highlight the unique associations between repeating patterning, growing patterning, and numerical ability, above spatial skills.

No Association Between the Home Math Environment and Numerical and Patterning Skills in a Large and Diverse Sample of 5- to 6-year-olds.

Selecting a large and diverse sample of 5-6-year-old preschool children (179 boys and 174 girls; M age = 70.03 months, SD age = 3.43), we aimed to extend previous findings on variability in children's home math environment (i.e., home math activities, parental expectations, and attitudes) and its association with children's mathematical skills. We operationalized mathematics in a broader way than in previous studies, by considering not only children's numerical skills but also their patterning skills as integral components of early mathematical development. We investigated the effects of children's gender and socioeconomic status (SES) on their home math environment, examined the associations between children's home math environment and their mathematical skills, and verified whether these associations were moderated by children's gender and/or SES. Parents of 353 children completed a home math environment questionnaire and all children completed measures of their numerical (e.g., object counting) and patterning skills (e.g., extending repeating patterns). Results indicated no effect of children's gender on their home math environment. There was no effect of SES on the performed home math activities, but small SES differences existed in parents' math-related expectations and their attitudes. We found no evidence for associations between children's home math environment and their mathematical skills. Furthermore, there were no moderating effects of gender or SES on these associations. One explanation for these findings might relate to the characteristics of the general preschool system in the country of the present study (Belgium). Future studies should consider the effect of the preschool learning environment because it might explain differences between studies and countries with regard to the home math environment and its association with mathematical skills.

Are preschoolers who spontaneously create patterns better in mathematics?

BACKGROUND: Early patterning competence has recently been identified as an important precursor of mathematical development. Whereas the focus of this research has been on children's ability regarding repeating patterns, children might also differ in their spontaneous attention to patterns. AIMS: The present study aimed to explore 4- to 5-year olds' Spontaneous Focusing On Patterns (SFOP) and its association with their patterning and mathematical ability. SAMPLE: Participants were 378 children (Mage  = 4 years 10 months; 191 boys) from 17 preschools. METHODS: Spontaneous Focusing On Patterns was measured with a construction task in which children had to build a tower with 15 blocks of three different colours. The constructions of the children were grouped into three categories (i.e., pattern, random, and sorting). We additionally administered tasks assessing their patterning ability, mathematical ability, spatial ability, and visuospatial working memory. RESULTS: When building a tower, 37% of the preschoolers spontaneously created a pattern, 49% made a random construction, and 14% sorted the blocks per colour. Preschoolers who spontaneously created a pattern had better patterning and mathematical ability than children in the random group. Group differences in patterning ability and spatial skills accounted for the difference in mathematical ability. CONCLUSIONS: The current data suggest that children's spontaneous attention to patterns is an important component of their mathematical ability. Children's spontaneous pattern constructions may provide opportunities to discuss and practice patterns in preschool settings or at home, but more research is required to further analyse the role of SFOP in early mathematical development.

Gender equality in 4- to 5-year-old preschoolers' early numerical competencies.

Numerical competencies acquired in preschool are foundational and predictive for children's later mathematical development. It remains to be determined whether there are gender differences in these early numerical competencies which could explain the often-reported gender differences in later mathematics and STEM-related abilities. Using a Bayesian approach, we quantified the evidence in favor of the alternative hypothesis of gender differences versus the null hypothesis of gender equality. Participants were 402 4- to 5-year-old children attending preschool in Flanders (Belgium). Children were selected via stratified cluster sampling to represent the full range of socioeconomic backgrounds. All children completed eight numerical tasks (verbal counting, object counting, numeral recognition, symbolic comparison, nonsymbolic comparison, nonverbal calculation, number order, dot enumeration). Results supported the gender equality hypothesis, and this evidence was substantial for seven of the eight numerical tasks. Preschoolers' early numerical competencies are characterized by gender equality. They probably do not explain later-reported gender differences.

Associations of Number Line Estimation With Mathematical Competence: A Meta-analysis.

The number line estimation task is widely used to investigate mathematical learning and development. The present meta-analysis statistically synthesized the extensive evidence on the correlation between number line estimation and broader mathematical competence. Averaged over 263 effect sizes with 10,576 participants with sample mean ages from 4 to 14 years, this correlation was r = .443. The correlation increased with age, mainly because it was higher for fractions than for whole numbers. The correlation remained stable across a wide range of task variants and mathematical competence measures (i.e., counting, arithmetic, school achievement). These findings demonstrate that the task is a robust tool for diagnosing and predicting broader mathematical competence and should be further investigated in developmental and experimental training studies.

Grade-related differences in strategy use in multidigit division in two instructional settings.

We aimed to investigate upper elementary children's strategy use in the domain of multidigit division in two instructional settings: the Netherlands and Flanders (Belgium). A cross-sectional sample of 119 Dutch and 122 Flemish fourth to sixth graders solved a varied set of multidigit division problems. With latent class analysis, three distinct strategy profiles were identified: children consistently using number-based strategies, children combining the use of column-based and number-based strategies, and children combining the use of digit-based and number-based strategies. The relation between children's strategy profiles and their instructional setting (country) and grade were generally in line with instructional differences, but large individual differences remained. Furthermore, Dutch children more frequently made adaptive strategy choices and realistic solutions than their Flemish peers. These results complement and refine previous findings on children's strategy use in relation to mathematics instruction. Statement of contribution What is already known? Mathematics education reform emphasizes variety, adaptivity, and insight in arithmetic strategies. Countries have different instructional trajectories for multidigit division. Mixed results on the impact of instruction on children's strategy use in multidigit division. What does this study add? Latent class analysis identified three meaningful strategy profiles in children from grades 4-6. These strategy profiles substantially differed between children. Dutch and Flemish children's strategy use is related to their instructional trajectory.

Verbal and action-based measures of kindergartners' SFON and their associations with number-related utterances during picture book reading.

BACKGROUND: Young children's spontaneous focusing on numerosity (SFON) as measured by experimental tasks is related to their mathematics achievement. This association is hypothetically explained by children's self-initiated practice in number recognition during everyday activities. As such, experimentally measured SFON should be associated with SFON exhibited during everyday activities and play. However, prior studies investigating this assumed association provided inconsistent findings. AIMS: We aimed to address this issue by investigating the association between kindergartners' SFON as measured by two different experimental tasks and the frequency of their number-related utterances during a typical picture book reading activity. SAMPLE: Participants were 65 4- to 6-year-olds in kindergarten (before the start of formal education). METHODS: Kindergartners individually participated in two sessions. First, they completed an action-based SFON Imitation task and a verbal SFON Picture task, with a short visuo-motor task in between. Next, children were invited to spontaneously comment on the pictures of a picture book during a typical picture book reading activity. RESULTS: Results revealed a positive association between children's SFON as measured by the Picture task and the frequency of their number-related utterances during typical picture book reading, but no such association for the Imitation task. CONCLUSIONS: Our findings indicate that children with higher SFON as measured by a verbal experimental task also tend to focus more frequently on number during verbal everyday activities, such as picture book reading. In view of the divergent associations between our SFON measures under study with everyday number activities, the current data suggest that SFON may not be a unitary construct and/or might be task-dependent.

Unique contribution of Ecuadorian kindergartners' spontaneous focusing on numerosity to their early numerical abilities.

Recent evidence indicates that young children's spontaneous focusing on numerosity (SFON) uniquely contributes to their early numerical abilities. This study complements previous findings by validating the relation between young children's SFON and their early numerical abilities in a developing country, namely Ecuador. We analysed 355 Ecuadorian 5- to 6-year-olds' SFON in relation to their early numerical abilities at the start of kindergarten, controlling for children's socio-demographic (socio-economic status, age) and general cognitive (working memory, intelligence) characteristics. Our results evidence the unique contribution of Ecuadorian kindergartners' SFON to their early numerical abilities, controlling for children's working memory, intelligence, socio-economic status, and age. Our findings support the validity of previous findings on the unique contribution of SFON to young children's early numerical abilities in developed countries for developing countries. Additionally, they raise timely questions for further theoretical and methodological studies on young children's numerical development worldwide, in developing and developed countries. Statement of contribution What is already known? SFON uniquely contributes to early numerical abilities. However, this is only documented in Finnish samples. Previously only limited control for domain-general cognitive and socio-demographic characteristics. What does the study add? SFON uniquely contributes to early numerical abilities in Ecuadorian 5- to 6-year-olds. This unique contribution remains after controlling for WM, IQ, SES, and age. Evidence for the universal nature of the association between SFON and early numerical ability.

Children's understanding of the addition/subtraction complement principle.

BACKGROUND: In the last decades, children's understanding of mathematical principles has become an important research topic. Different from the commutativity and inversion principles, only few studies have focused on children's understanding of the addition/subtraction complement principle (if a - b = c, then c + b = a), mainly relying on verbal techniques. AIM: This contribution aimed at deepening our understanding of children's knowledge of the addition/subtraction complement principle, combining verbal and non-verbal techniques. SAMPLE: Participants were 67 third and fourth graders (9- to 10-year-olds). METHODS: Children solved two tasks in which verbal reports as well as accuracy and speed data were collected. These two tasks differed only in the order of the problems and the instructions. In the looking-back task, children were told that sometimes the preceding problem might help to answer the next problem. In the baseline task, no helpful preceding items were offered. The looking-back task included 10 trigger-target problem pairs on the complement relation. RESULTS: Children verbally reported looking back on about 40% of all target problems in the looking-back task; the target problems were also solved faster and more accurately than in the baseline task. These results suggest that children used their understanding of the complement principle. The verbal and non-verbal data were highly correlated. DISCUSSION: This study complements previous work on children's understanding of mathematical principles by highlighting interindividual differences in 9- to 10-year-olds' understanding of the complement principle and indicating the potential of combining verbal and non-verbal techniques to investigate (the acquisition of) this understanding.

The association between symbolic and nonsymbolic numerical magnitude processing and mental versus algorithmic subtraction in adults.

There are two well-known computation methods for solving multi-digit subtraction items, namely mental and algorithmic computation. It has been contended that mental and algorithmic computation differentially rely on numerical magnitude processing, an assumption that has already been examined in children, but not yet in adults. Therefore, in this study, we examined how numerical magnitude processing was associated with mental and algorithmic computation, and whether this association with numerical magnitude processing was different for mental versus algorithmic computation. We also investigated whether the association between numerical magnitude processing and mental and algorithmic computation differed for measures of symbolic versus nonsymbolic numerical magnitude processing. Results showed that symbolic, and not nonsymbolic, numerical magnitude processing was associated with mental computation, but not with algorithmic computation. Additional analyses showed, however, that the size of this association with symbolic numerical magnitude processing was not significantly different for mental and algorithmic computation. We also tried to further clarify the association between numerical magnitude processing and complex calculation by also including relevant arithmetical subskills, i.e. arithmetic facts, needed for complex calculation that are also known to be dependent on numerical magnitude processing. Results showed that the associations between symbolic numerical magnitude processing and mental and algorithmic computation were fully explained by individual differences in elementary arithmetic fact knowledge.

Children's use of addition to solve two-digit subtraction problems.

Subtraction problems of the type M - S = ? can be solved with various mental calculation strategies. We investigated fourth- to sixth-graders' use of the subtraction by addition strategy, first by fitting regression models to the reaction times of 32 two-digit subtractions. These models represented three different strategy use patterns: the use of direct subtraction, subtraction by addition, and switching between the two strategies based on the magnitude of the subtrahend. Additionally, we compared performance on problems presented in two presentation formats, i.e., a subtraction format (81 - 37 = .) and an addition format (37 + . = 81). Both methods converged to the conclusion that children of all three grades switched between direct subtraction and subtraction by addition based on the combination of two features of the subtrahend: If the subtrahend was smaller than the difference, direct subtraction was the dominant strategy; if the subtrahend was larger than the difference, subtraction by addition was mainly used. However, this performance pattern was only observed when the numerical distance between subtrahend and difference was large. These findings indicate that theoretical models of children's strategy choices in subtraction should include the nature of the subtrahend as an important factor in strategy selection.

Use of indirect addition in adults' mental subtraction in the number domain up to 1,000.

This study examined adults' use of indirect addition and direct subtraction strategies on multi-digit subtractions in the number domain up to 1,000. Seventy students who differed in their level of arithmetic ability solved multi-digit subtractions in one choice and two no-choice conditions. Against the background of recent findings in elementary subtraction, we manipulated the size of the subtrahend compared to the difference and only selected items with large distances between these two integers. Results revealed that adults frequently and efficiently apply indirect addition on multi-digit subtractions, yet adults with higher arithmetic ability performed more efficiently than those with lower arithmetic ability. In both groups, indirect addition was more efficient than direct subtraction both on subtractions with a subtrahend much larger than the difference (e.g., 713 - 695) and on subtractions with a subtrahend much smaller than the difference (e.g., 613 - 67). Unexpectedly, only adults with lower arithmetic ability fitted their strategy choices to their individual strategy performance skills. Results are interpreted in terms of mathematical and cognitive perspectives on strategy efficiency and adaptiveness.

Adults' use of subtraction by addition.

The present study investigates adults' use of addition to solve two-digit subtractions. Inspired by research on single-digit arithmetic, we first examined regression models in which different problem characteristics predicted participants' reaction times. Second, we compared performance on two-digit subtractions presented in 2 presentation formats, i.e., the standard subtraction format (81-37=.) and an addition format (37+.=81). Both methods lead to the conclusion that the participants switched between direct subtraction and subtraction by addition depending on the relative size of the subtrahend: If the subtrahend was smaller than the difference, direct subtraction was mainly used; if the subtrahend was larger than the difference, subtraction by addition was the dominant strategy. However, this performance pattern was only observed when the distance between the subtrahend and the difference was large; when the subtrahend and the difference were close to each other, there was no subtrahend-dependent selection of direct subtraction vs. subtraction by addition. These data indicate that theoretical models of people's strategy choices in subtraction should include the relative size of the subtrahend as an important factor in the strategy selection process.

The numerical stroop effect in primary school children: a comparison of low, normal, and high achievers.

Sixty-six primary school children were selected, of which 21 scored low on a standardized math achievement test, 23 were normal, and 22 high achievers. In a numerical Stroop experiment, children were asked to make numerical and physical size comparisons on digit pairs. The effects of congruity and numerical distance were determined. All children exhibited congruity and distance effects in the numerical comparison. In the physical comparison, children of all performance groups showed Stroop effects when the numerical distance between the digits was large but failed to show them when the distance was small. Numerical distance effects depended on the congruity condition, with a typical effect of distance in the congruent, and a reversed distance effect in the incongruent condition. Our results are hard to reconcile with theories that suggest that deficits in the automaticity of numerical processing can be related to differential math achievement levels. Immaturity in the precision of mappings between numbers and their numerical magnitudes might be better suited to explain the Stroop effects in children. However, as the results for the high achievers demonstrate, in addition to numerical processing capacity per se, domain-general functions might play a crucial role in Stroop performance, too.

Using addition to solve large subtractions in the number domain up to 20.

This study examined 25 university students' use of addition to solve large single-digit subtractions by contrasting performance in the standard subtraction format (12-9=.) and in the addition format (9+.=12). In particular, we investigated the effect of the relative size of the subtrahend on performance in both formats. We found a significant interaction between format, the magnitude of the subtrahend (S) compared to the difference (D) (S>D vs. S

Strategy development in children with mathematical disabilities: insights from the choice/no-choice method and the chronological-age/ ability-level-match design.

This study investigated the strategy characteristics and development of children with mathematical disabilities (MD) in the domain of simple addition and subtraction, in terms of Lemaire and Siegler's model of strategic change, using the choice/no-choice method and the combined chronological-age (CA)/ability-level (AL)-match design. Four groups of children, matched on either CA or mathematical AL, solved a series of 36 problems with the bridge over 10 in four conditions. In the choice condition, children could choose between retrieval, decomposition to 10, and counting on to solve each problem. In the retrieval, decomposition, and counting conditions, they had to answer the same 36 problems using retrieval, decomposition to 10, and counting on, respectively. The results revealed clear differences in the frequency, efficiency, and adaptiveness with which the CA-matched children applied the available strategies. In contrast, we observed no differences in strategy frequency, efficiency, and adaptiveness between the AL-matched children. These results support the hypothesis that the strategy development of children with MD is marked by a delay rather than a specific deficit. Moreover, this study further documents the value of the methodology used to study children's strategy use and development in the domain of simple arithmetic.