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.

The structure of the notation system in adults' number line estimation: An eye-tracking study.

Research on rational numbers suggests that adults experience more difficulties in understanding the numerical magnitude of rational than natural numbers. Within rational numbers, the numerical magnitude of fractions has been found to be more difficult to understand than that of decimals. Using a number line estimation (NLE) task, the current study investigated two sources of difficulty in adults' numerical magnitude understanding: number type (natural vs rational) and structure of the notation system (place-value-based vs non-place-value-based). This within-subjects design led to four conditions: natural numbers (natural/place-value-based), decimals (rational/place-value-based), fractions (rational/non-place-value-based), and separated fractions (natural/non-place-value-based). In addition to percentage absolute error (PAE) and response times, we collected eye-tracking data. Results showed that participants estimated natural and place-value-based notations more accurately than rational and non-place-value-based notations, respectively. Participants were also slower to respond to fractions compared with the three other notations. Consistent with the response time data, eye-tracking data showed that participants spent more time encoding fractions and re-visited them more often than the other notations. Moreover, in general, participants spent more time positioning non-place-value-based than place-value-based notations on the number line. Overall, the present study contends that when both sources of difficulty are present in a notation (i.e., both rational and non-place-value-based), adults understand its numerical magnitude less well than when there is only one source of difficulty present (i.e., either rational or non-place-value-based). When no sources of difficulty are present in a notation (i.e., both natural and place-value-based), adults have the strongest understanding of its numerical magnitude.

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).

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.

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.

Intuitive errors in learners' fraction understanding: A dual-process perspective on the natural number bias.

Although a good rational number understanding is very important, many learners struggle to understand fractions. Recent research attributes many of these difficulties to the natural number bias - the tendency to apply natural number features in rational number tasks where this is inappropriate. Previous correlational dual process studies found evidence for the intuitive nature of the natural number bias in learners' response latencies. However, the reported correlations do not ascertain the causality that is assumed in this ascription. In the present study we therefore experimentally elicited intuitive responses in a fraction comparison task in educated adults by restricting reaction time. Results show that the natural number bias has an intuitive character. Findings also indicate that the detection of conflict between the natural number-based answer and the correct answer seems to work at an intuitive level.

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.

Developmental change in number line estimation: A strategy-based perspective.

The current study investigated developmental changes in children's benchmark-based strategy use in number line estimation. Third and fifth graders solved a 0-1,000 number line estimation task in one of three conditions. In the control condition, only the origin and endpoint were specified, the midpoint condition included an additional benchmark at 50%, and the quartile condition contained three additional benchmarks at 25%, 50%, and 75%. Trial-by-trial verbal strategy reports revealed that fifth graders, in comparison to third graders, spontaneously applied quartile-based strategies more frequently while they used strategies based on the origin, midpoint, and endpoint about equally often. Finally, children's mathematics achievement was positively related to the variety and frequency of benchmark-based strategies as well as their number line estimation performance. We conclude that developmental changes in number line estimation performance can at least partially be attributed by a refinement in children's benchmark-based strategy use. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

Disentangling the Mechanisms of Symbolic Number Processing in Adults' Mathematics and Arithmetic Achievement.

A growing body of research has shown that symbolic number processing relates to individual differences in mathematics. However, it remains unclear which mechanisms of symbolic number processing are crucial-accessing underlying magnitude representation of symbols (i.e., symbol-magnitude associations), processing relative order of symbols (i.e., symbol-symbol associations), or processing of symbols per se. To address this question, in this study adult participants performed a dots-number word matching task-thought to be a measure of symbol-magnitude associations (numerical magnitude processing)-a numeral-ordering task that focuses on symbol-symbol associations (numerical order processing), and a digit-number word matching task targeting symbolic processing per se. Results showed that both numerical magnitude and order processing were uniquely related to arithmetic achievement, beyond the effects of domain-general factors (intellectual ability, working memory, inhibitory control, and non-numerical ordering). Importantly, results were different when a general measure of mathematics achievement was considered. Those mechanisms of symbolic number processing did not contribute to math achievement. Furthermore, a path analysis revealed that numerical magnitude and order processing might draw on a common mechanism. Each process explained a portion of the relation of the other with arithmetic (but not with a general measure of math achievement). These findings are consistent with the notion that adults' arithmetic skills build upon symbol-magnitude associations, and they highlight the effects that different math measures have in the study of numerical cognition.

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.

Benchmark-based strategy use in atypical number lines.

Previous studies have indicated that the presence of atypical end points (e.g., 1,639 and 2,897) on a number line has a negative effect on number line estimation (NLE) performance (Booth & Newton, 2012; Hurst, Leigh Monahan, Heller, & Cordes, 2014). In the present study, we investigated whether this effect could be attributed to a disruption in the ease with which benchmarks on the number line can be determined and whether this possible disruption changes with age. Hence, we asked 5th graders and adults to perform a NLE task in a typical and an atypical condition with a number line ranging from 0 to 1,000 and from 367 to 1,367, respectively. Results showed that participants' overall estimates were less accurate in the atypical condition compared to the typical condition. A similar pattern of results was observed for the estimates around the self-created benchmarks at 50% in children and at 25%, 50%, and 75% in adults. Moreover, in both age groups, we found a shift from a larger proportion of estimation patterns' being best fit by more complex power models in the typical condition toward a larger proportion of estimation patterns best described by simpler power models in the atypical condition. Of importance, we also found evidence that adults' estimation patterns in the typical condition are best described by a 4-cycle power model, hereby extending previous modeling results. Overall, our findings indicate that hindering the application of benchmark-based strategies negatively affects children's and adults' NLE performance. (PsycINFO Database Record (c) 2018 APA, all rights reserved).

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.

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.

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.

The natural number bias and its role in rational number understanding in children with dyscalculia. Delay or deficit?

BACKGROUND: Previous research indicated that in several cases learners' errors on rational number tasks can be attributed to learners' tendency to (wrongly) apply natural number properties. There exists a large body of literature both on learners' struggle with understanding the rational number system and on the role of the natural number bias in this struggle. However, little is known about this phenomenon in learners with dyscalculia. AIMS: We investigated the rational number understanding of learners with dyscalculia and compared it with the rational number understanding of learners without dyscalculia. METHOD: Three groups of learners were included: sixth graders with dyscalculia, a chronological age match group, and an ability match group. RESULTS: The results showed that the rational number understanding of learners with dyscalculia is significantly lower than that of typically developing peers, but not significantly different from younger learners, even after statistically controlling for mathematics achievement. CONCLUSION: Next to a delay in their mathematics achievement, learners with dyscalculia seem to have an extra delay in their rational number understanding, compared with peers. This is especially the case in those rational number tasks where one has to inhibit natural number knowledge to come to the right answer.

Evaluating the Effect of Labeled Benchmarks on Children's Number Line Estimation Performance and Strategy Use.

Some authors argue that age-related improvements in number line estimation (NLE) performance result from changes in strategy use. More specifically, children's strategy use develops from only using the origin of the number line, to using the origin and the endpoint, to eventually also relying on the midpoint of the number line. Recently, Peeters et al. (unpublished) investigated whether the provision of additional unlabeled benchmarks at 25, 50, and 75% of the number line, positively affects third and fifth graders' NLE performance and benchmark-based strategy use. It was found that only the older children benefitted from the presence of these benchmarks at the quartiles of the number line (i.e., 25 and 75%), as they made more use of these benchmarks, leading to more accurate estimates. A possible explanation for this lack of improvement in third graders might be their inability to correctly link the presented benchmarks with their corresponding numerical values. In the present study, we investigated whether labeling these benchmarks with their corresponding numerical values, would have a positive effect on younger children's NLE performance and quartile-based strategy use as well. Third and sixth graders were assigned to one of three conditions: (a) a control condition with an empty number line bounded by 0 at the origin and 1,000 at the endpoint, (b) an unlabeled condition with three additional external benchmarks without numerical labels at 25, 50, and 75% of the number line, and (c) a labeled condition in which these benchmarks were labeled with 250, 500, and 750, respectively. Results indicated that labeling the benchmarks has a positive effect on third graders' NLE performance and quartile-based strategy use, whereas sixth graders already benefited from the mere provision of unlabeled benchmarks. These findings imply that children's benchmark-based strategy use can be stimulated by adding additional externally provided benchmarks on the number line, but that, depending on children's age and familiarity with the number range, these additional external benchmarks might need to be labeled.

Benchmark-based strategies in whole number line estimation.

In this study, we used verbal protocols to identify whether adults spontaneously apply quartile-based strategies or whether they need additional external support to use these strategies when solving a 0-1,000 number line estimation (NLE) task. Participants were assigned to one of three conditions based on the number of external benchmarks provided on the number line. In the bounded condition only the origin and endpoint were indicated, the mid-point condition included an additional external benchmark at 50%, and in the quartile condition three additional external benchmarks at 25%, 50%, and 75% were specified. Firstly, participants in the bounded condition reported to spontaneously apply quartile-based strategies to calibrate their estimates. Moreover, participants frequently relied on the external benchmarks for creating internal benchmarks at the mid-point, quartiles, and even octiles of the number line. Secondly, overall estimation accuracy improved as the number of external benchmarks increased, and target numbers close to external benchmarks were estimated more accurately and with less variability. Thirdly, the use of a larger variety in benchmark-based strategies was positively related to NLE accuracy. In summary, this study provides evidence that the NLE task induces more sophisticated strategy use in participants than initially anticipated.

Number sense in the transition from natural to rational numbers.

BACKGROUND: Rational numbers are of critical importance both in mathematics and in other fields of science. However, they form a stumbling block for learners. One widely known source of the difficulty learners have with rational numbers is the natural number bias, that is the tendency to (inappropriately) apply natural number properties in rational number tasks. Still, it has been shown that a good understanding of natural numbers is highly predictive for mathematics achievement in general, and for performance on rational number tasks in particular. AIMS: In this study, we further investigated the relation between learners' natural and rational number knowledge, specifically in cases where a natural number bias may lead to errors. SAMPLE: Participants were 140 sixth graders from six different primary schools. METHOD: Participants completed a symbolic and a non-symbolic natural number comparison task, a number line estimation task, and a rational number sense test. RESULTS: Learners' natural number knowledge was found to be a good predictor of their rational number knowledge. However, after first controlling for learners' general mathematics achievement, their natural number knowledge only predicted the subaspect of operations with rational numbers. CONCLUSION: The results of this study suggest that the relation between learners' natural and rational number knowledge can largely be explained by their relation with learners' general mathematics achievement.