Sources of individual differences in children's understanding of fractions

Cross-notation knowledge of rational numbers predicts fraction arithmetic

BACKGROUND

Recent research showed that cross-notation magnitude knowledge of fractions and decimals was related to better performance in fraction arithmetic, but it remains unclear whether it made an independent contribution to fraction arithmetic longitudinally when other cognitive variables are considered.

AIMS

To examine the extent to which children's earlier knowledge of cross-notation magnitude predicted subsequent performance in fraction addition and subtraction as well as fraction multiplication and division longitudinally.

SAMPLE

Three hundred and fifty-four Chinese children (Mage = 112.1 months).

METHODS

During the first wave of assessment, a range of cognitive abilities of children were measured, including within-notation fraction and decimal magnitude comparisons, whole-number arithmetic fluency, non-verbal intelligence, attentive behaviours, counting recall, word-level reading, and phonological awareness. Twelve months later, the same children were assessed again with two tasks of fraction arithmetic: fraction addition and subtraction as well as fraction multiplication and division.

RESULTS AND CONCLUSIONS

Multiple linear regressions showed that within-notation fraction and decimal magnitude knowledge predicted fraction addition and subtraction longitudinally, after the effects of working memory, nonverbal intelligence, language skills, attentive behaviour, and whole-number arithmetic were controlled. Cross-notation magnitude knowledge made independent contributions to fraction addition and subtraction longitudinally beyond the influence of within-notation fraction and decimal magnitude knowledge and other covariates. However, within-notation fraction and decimal magnitude knowledge were not associated with fraction multiplication and division, whereas cross-notation magnitude knowledge remained a unique predictor. These findings suggest that it may be useful to incorporate cross-notation knowledge in the assessments of children's mathematics abilities and teaching.

A Role for Visual Memory in Vocabulary Development: A Systematic Review and Meta-Analysis

Although attention and early associative learning in preverbal children is predominantly driven by rapid eye-movements in response to moving visual stimuli and sounds/words (e.g., associating the word "bottle" with the object), the literature examining the role of visual attention and memory in ongoing vocabulary development across childhood is limited. Thus, this systematic review and meta-analysis examined the association between visual memory and vocabulary development, including moderators such as age and task selection, in neurotypical children aged 2-to-12 years, from the brain-based perspective of cognitive neuroscience. Visual memory tasks were classified according to the visual characteristics of the stimuli and the neural networks known to preferentially process such information, including consideration of the distinction between the ventral visual stream (processing more static visuo-perceptual details, such as form or colour) and the more dynamic dorsal visual stream (processing spatial temporal action-driven information). Final classifications included spatio-temporal span tasks, visuo-perceptual or spatial concurrent array tasks, and executive judgment tasks. Visuo-perceptual concurrent array tasks, reliant on ventral stream processing, were moderately associated with vocabulary, while tasks measuring spatio-temporal spans, associated with dorsal stream processing, and executive judgment tasks (central executive), showed only weak correlations with vocabulary. These findings have important implications for health professionals and researchers interested in language, as they advocate for the development of more targeted language learning interventions that include specific and relevant aspects of visual processing and memory, such as ventral stream visuo-perceptual details (i.e., shape or colour).

The interplay between math performances, spatial abilities, and affective factors: The role of task

Many studies have suggested that cognitive and affective abilities (such as math anxiety- MA and math self-efficacy) explain individual differences in math.

PURPOSE

The present study explores the interplay between MA, math self-efficacy, spatial anxiety and spatial abilities in explaining individual differences on two complex math tasks.

PROCEDURES

Ninety-three college students took part in the experiment and completed 3 emotional questionnaires, in addition to 2 math tasks and a mental rotation task.

FINDINGS

The interplay between math performances and cognitive and affective factors is related to task demand. MA and spatial abilities affected math performances directly, regardless of task. Spatial anxiety had only an indirect effect on math performances via MA, regardless of task.

CONCLUSIONS

These finding suggest that for math performances, contrary to MA, real spatial abilities rather than perceived spatial anxiety play a significant role in explaining individual differences.  Hence, the present result dissociates cognitive and emotional factors.

Beyond linearity: Using IRT-scaled level models to describe the relation between prior proportional reasoning skills and fraction learning outcomes

Previous research on the role of prior skills like proportional reasoning skills for the development of mathematical concepts offers conclusions such as "more (prior skills) is better (for later learning)." Insights, which prior skill level goes along with which level of learning outcomes, may advance the understanding of the development of mathematical concepts. An exploratory approach is presented based on level models to describe the relation between symbolic proportional reasoning skills and fraction outcomes beyond linearity. Analyses draw on samples of German fourth to sixth graders from a scaling (2017, N = 325, 54.8% female) and longitudinal study (2018/2019, N = 436, 42.7% female). Particularly mastering natural and internal rational ratios in proportional reasoning seems relevant for successful fraction learning.

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.

School Achievement in Early Adolescence Is Associated With Students' Self-Perceived Executive Functions

The primary aim of this study was to investigate the relation between self-perceived executive functions (EFs) and the school achievement of young adolescents (aged 10-12 years), while controlling for parental education and sex. We specifically focused on executive aspects of daily life behavior and the higher-order EFs, as measured with self-report, rather than on the more basic EFs which have been the primary focus of prior investigations. In two independent samples of sixth graders (N > 200 each), students evaluated their EFs on a self-report questionnaire, the Amsterdam Executive Functioning Inventory. School achievement in the domains of mathematics and reading comprehension were evaluated with nationally used, norm-based achievement tests. Results revealed that the self-perceived EFs of young adolescents were significantly correlated with their school achievement in both study samples. School achievement was also correlated with the level of parental education, but the factor sex did not have such influence. In study 1, self-perceived EFs explained additional variance in school achievement, while controlling for parental education and sex. In study 2, this was only the case for the most robust measure of school achievement, i.e., the end-of-primary-school final achievement test. Furthermore, besides the relation with achievement tests, we also found a relation between self-perceived EFs and teacher ratings behavioral problems in the classroom. Together, our findings imply that young students can properly reflect on the effectiveness and appropriateness of their EFs in a way that is relevant to their academic achievement and classroom behavior. The findings underscore the importance of considering the development of EFs and parental education in the evaluation of academic achievements in early adolescence.

Examining the impact of a tutoring program implemented with community support on math proficiency and growth

The current study evaluated the impact of a math tutoring program delivered in 20 schools to students in 4th through 8th grades by community members over one academic year. Students were randomly assigned to treatment and control groups. Multi-level linear and generalized linear mixed models were used to evaluate group differences in post-test scores and the probability of attaining the spring proficiency benchmark on two increasingly distal measures of math achievement. Intent-to-treat analyses identified higher achievement scores among students assigned to treatment on a measure of fact fluency and a computer adaptive measure of overall math achievement. Students assigned to treatment also had a higher probability of reaching grade-level benchmarks on the computer adaptive test. No statistically significant effects were observed on a state proficiency test. Implications for significant and null findings are discussed within the context of intervention content and delivery.

Predicting adaptive expertise with rational number arithmetic

Background

Adaptive expertise is a highly valued outcome of mathematics curricula. One aspect of adaptive expertise with rational numbers is adaptive rational number knowledge, which refers to the ability to integrate knowledge of numerical characteristics and relations in solving novel tasks. Even among students with strong conceptual and procedural knowledge of rational numbers, there are substantial individual differences in adaptive rational number knowledge.

Aims

We aimed to examine how a wide range of domain‐general and mathematically specific skills and knowledge predicted different aspects of rational number knowledge, including procedural, conceptual, and adaptive rational number knowledge.

Sample

173 6^th and 7^th grade students from a school in the southeastern US (51% female) participated in the study.

Methods

At three time points across 1.5 years, we measured students’ domain‐general and domain‐specific skills and knowledge. We used multiple hierarchal regression analysis to examine how these predictors related to rational number knowledge at the third time point.

Result

Prior knowledge of rational numbers, general mathematical calculation knowledge, and spontaneous focusing on multiplicative relations (SFOR) tendency uniquely predicted adaptive rational number knowledge, after taking into account domain‐general and mathematically specific skills and knowledge. Although conceptual knowledge of rational numbers and general mathematical achievement also predicted later conceptual and procedural knowledge of rational numbers, SFOR tendency did not.

Conclusion

Results suggest expanding investigations of mathematical development to also explore different features of adaptive expertise as well as spontaneous mathematical focusing tendencies.

The cognitive profiles for different samples of mathematical learning difficulties and their similarity to typical development: Evidence from a longitudinal study

Several cognitive deficits have been suggested to induce mathematical learning difficulties (MLD), but it is unclear whether the cognitive profile for all children with MLD is the same and to what extent it differs from typically developing (TD) children. This study investigated whether such a profile could be distinguished when cognitive skills and math performance are compared between TD children and children with MLD. This was accomplished by employing two-way repeated-measures analyses of covariance in 276 10-year-old participants (60 with MLD) from fourth and fifth grades. In addition, we investigated whether more restrictive selection criteria for MLD result in different mathematical and cognitive profiles by means of independent-samples t tests. Results revealed that cognitive mechanisms for math development are mostly similar for children with MLD and TD children and that variability in sample selection criteria did not produce different mathematical or cognitive profiles. To conclude, the cognitive mechanisms for math development are broadly similar for children with MLD and their TD counterparts even when different MLD samples were selected. This strengthens our idea that MLD can be defined as the worst performance on a continuous scale rather than as a discrete disorder.

The association between visual attention and arithmetic competence: The mediating role of enumeration

The current study aimed at clarifying the nature of relation between visual attention and arithmetic competence. A group of 301 Chinese second graders was assessed. Children's visual attention was measured using two versions of a visual search task, with efficient visual search (the similarity between the target and the distractors is low) tapping automatic, stimulus-driven visual attention and inefficient visual search (the similarity between the target and the distractors is high) tapping effortful, goal-directed visual attention. Children's arithmetic competence, enumeration skills (assessed in about half of the participants), and other domain-general cognitive abilities were also assessed. The results suggest that only inefficient visual search significantly predicted children's arithmetic competence, and such a relation was mediated through their enumeration skills. The findings highlight the role of fundamental cognitive capacities in mathematics learning and provide insights into potential interventions for improving children's arithmetic competence.

Teaching students with ASD to solve fraction computations using a video modeling instructional package

BACKGROUND

With the surge of intervention research examining ways of supporting students with autism spectrum disorder (ASD) in inclusive settings, there remains a need to examine how technology supports could enhance students' learning by offering one size fits one instruction. Furthermore, intervention studies focused on teaching students with ASD how to solve fractions are scarce.

AIMS

The purpose of this research study was to examine the effects of providing instruction via video modeling (VM), concrete manipulatives, a self-monitoring checklist, and practice for comprehension check on the accuracy of fraction problem solving of three middle school students with ASD.

METHODS AND PROCEDURES

Through the use of single-case multiple probe across students experimental design, we examined whether a functional relation existed between the intervention and students' improved accuracy of solving simple proper fraction problems.

OUTCOMES AND RESULTS

All three students improved the accuracy of solving simple proper fraction problems from baseline to intervention sessions and two students generalized the skill to solving whole proper fraction problems.

CONCLUSIONS AND IMPLICATIONS

The intervention consisting of VM and concrete manipulatives along with additional behavioral strategies offers an option for teachers to accommodate diverse learning needs of students with ASD in a variety of settings.

Development of early domain-specific and domain-general cognitive precursors of high and low math achievers in grade 6

This study investigated from a longitudinal retrospective perspective what characterizes and predicts 6th graders (M_age = 12.95, SD = 0.27) with low (LMA) or high (HMA) math achievement concerning the development of early domain-specific and domain-general cognitive abilities. They were examined and compared to average achievers (n = 88) at four-time points from kindergarten (M_age = 6.58, SD = 0.36) to third grade (M_age = 9.53, SD = 0.33). The LMA (n = 27) or HMA (n = 41) children exhibited persistent multi-weakness and multi-strength profiles, respectively, present already prior to formal schooling. The cognitive profiles of the two groups, and their development, were mostly qualitatively similar, but there were also important qualitative differences. Logistic regression analyzes showed that superior verbal arithmetic, logical reasoning, and executive functions are vital for developing superior mathematical skills while inferior verbal arithmetic, logical reasoning, and spatial processing ability constitute unique potential risk factors for low mathematical skills.

The growth rates of dot enumeration ability predict mathematics achievements: A 5-year longitudinal study

BACKGROUND

Dot enumeration is the basic mathematics competency in young children and a significant indicator of later mathematics achievement.

AIM

The present study focused on (1) how children's dot enumeration ability changed as they progressed from late kindergarten years (K3) to the second year of primary school in Hong Kong (P2), and (2) the extent to which such changes are associated with students' mathematics outcomes assessed at the fourth grade, including standardized mathematics achievement, whole number magnitude understanding, and rational number concept.

SAMPLE(S)

Two hundred and eleven Hong Kong kindergarteners were recruited.

METHODS

The participants' dot enumeration was assessed from K3 to P2. Their mathematics outcomes were assessed at P4, including standardized mathematics achievement, whole number magnitude understanding, and rational number concept.

RESULTS

The changes in their dot enumeration speed reflected a linear growth pattern. Further, both the initial level and growth rate of dot enumeration predicted standardized mathematics achievement and whole number magnitude understanding 2 years later while only the latter predicted rational number concept.

CONCLUSIONS

The results indicate the importance of focusing on children's growth in a specific mathematics skill, in addition to their status at one single time point. Practical implications are discussed in this article.

The nature of the association between number line and mathematical performance: An international twin study

BACKGROUND

The number line task assesses the ability to estimate numerical magnitudes. People vary greatly in this ability, and this variability has been previously associated with mathematical skills. However, the sources of individual differences in number line estimation and its association with mathematics are not fully understood.

AIMS

This large-scale genetically sensitive study uses a twin design to estimate the magnitude of the effects of genes and environments on: (1) individual variation in number line estimation and (2) the covariation of number line estimation with mathematics.

SAMPLES

We used over 3,000 8- to 16-year-old twins from the United States, Canada, the United Kingdom, and Russia, and a sample of 1,456 8- to 18-year-old singleton Russian students.

METHODS

Twins were assessed on: (1) estimation of numerical magnitudes using a number line task and (2) two mathematics components: fluency and problem-solving.

RESULTS

Results suggest that environments largely drive individual differences in number line estimation. Both genes and environments contribute to different extents to the number line estimation and mathematics correlation, depending on the sample and mathematics component.

CONCLUSIONS

Taken together, the results suggest that in more heterogeneous school settings, environments may be more important in driving variation in number line estimation and its association with mathematics, whereas in more homogeneous school settings, genetic effects drive the covariation between number line estimation and mathematics. These results are discussed in the light of development and educational settings.

The roles of place-value understanding and non-symbolic ratio processing system in symbolic rational number processing

BACKGROUND

While it has been widely demonstrated that children's and adolescents' understanding of rational number plays an important role in their mathematics achievement, we have limited knowledge about the cognitive correlates of this understanding.

AIMS

The current study aimed at examining whether children's non-symbolic ratio processing and their understanding of place-value structure of whole numbers play a role in their understanding of fractions and decimals and whether their roles are different for fractions versus decimal understanding.

SAMPLE

A sample of 124 fourth graders was tested.

METHODS

Participants were tested on their symbolic rational number processing, non-symbolic ratio processing, place-value understanding of whole numbers, mathematics achievement, as well as a series of domain-general and domain-specific cognitive skills related to symbolic rational number processing.

RESULTS

The findings suggest that, while the understanding of place value of whole numbers significantly predicted the understanding of both fractions and decimals, non-symbolic ratio processing specifically predicted the understanding of fractions, but not decimals.

CONCLUSIONS

The findings highlight the roles of place-value understanding and non-symbolic ratio processing in the acquisition of symbolic rational numbers.

The Recruitment of Shifting and Inhibition in On-line Science and Mathematics Tasks

Prior research has investigated the recruitment of inhibition in the use of science/mathematics concepts in tasks that require the rejection of a conflicting, nonscientific initial concept. The present research examines if inhibition is the only EF skill recruited in such tasks and investigates whether shifting is also involved. It also investigates whether inhibition and/or shifting are recruited in tasks in which the use of science/mathematics concepts does not require the rejection of an initial concept, or which require only the use of initial concepts. One hundred and thirty-three third- and fifth-grade children participated in two inhibition and shifting tasks and two science and mathematics conceptual understanding and conceptual change (CU&C) tasks. All the tasks were on-line, and performance was measured in accuracy and RTs. The CU&C tasks involved the use of initial concepts and of science/mathematics concepts which required conceptual changes for their initial formation. Only in one of the tasks the use of the science/mathematics concepts required the concurrent rejection of an initial concept. The results confirmed that in this task inhibition was recruited and also showed that the speed of shifting was a significant predictor of performance. Shifting was a significant predictor of performance in all the tasks, regardless of whether they involved science/mathematics or initial concepts. It is argued that shifting is likely to be recruited in complex tasks that require multiple comparisons of stimuli and the entertainment of different perspectives. Inhibition seems to be a more selective cognitive skill likely to be recruited when the use of science/mathematics concepts requires the rejection of a conflicting initial concept.

Does Initial Learning about the Meaning of Fractions Present Similar Challenges for Students with and without Adequate Whole-Number Skill?

The purposes of this study were to (a) explore whether early fractions understanding at 4^th grade is differentially challenging for students with versus without adequate whole-number competence and (b) identify specific whole-number skill associated with difficulty in fractions understanding. Based on initial whole-number competence, 1,108 4^th graders were classified as having (a) adequate whole-number competence (n = 775), (b) less severe whole-number difficulty (n = 201), and (c) severe whole-number difficulty (n = 132). At the end of 4^th grade, they were assessed on fractions understanding and further classified as with versus without difficulty in fractions understanding. Multi-level logistic regression indicated that compared to students with adequate whole-number competence, those with less severe whole-number difficulty were almost 5 times as likely to experience difficulty with fractions understanding whereas those with severe whole-number difficulty were about 32 times as likely to experience difficulty with fractions understanding. Students with severe whole-number difficulty were about 7 times as likely to experience difficulty with fractions understanding compared to those with less severe whole-number difficulty. Among students with adequate whole-number competence, the pretest whole-number skill distinguishing those with versus without difficulty in fractions understanding was basic division facts (i.e., 2-digit dividend ÷ 1-digit divisor) and simple multiplication (i.e., 3-digit × 1-digit without regrouping). The role of whole-number competence in developing initial fractions understanding and implications for instruction are discussed.

Cognitive Profiles Associated With Responsiveness to Fraction Intervention

This study examined differences in cognitive processing between 4^th-grade students who respond adequately, as opposed to inadequately, to intervention on 3 fraction outcomes: number-line estimation, calculation, and word problems. Students were assessed on 7 cognitive processes and on the 3 fraction outcomes. Students were grouped as adequate or inadequate responders, using as the cut-point the control-group mean on pre-to-post improvement on the relevant measure. Between-group differences identified reasoning, concept formation, and listening comprehension related to all 3 fraction outcomes. On the number-line outcome, within-group profile analysis indicated that inadequate responders experienced low reasoning ability relative to their other forms of cognitive processing.

The Role of Domain-General Cognitive Abilities and Decimal Labels in At-Risk Fourth-Grade Students' Decimal Magnitude Understanding

The purpose of the study was to determine whether individual differences in at-risk 4^th graders' language comprehension, nonverbal reasoning, concept formation, working memory, and use of decimal labels (i.e., place value, point, incorrect place value, incorrect fraction, or whole number) are related to their decimal magnitude understanding. Students (n = 127) completed 6 cognitive assessments, a decimal labeling assessment, and 3 measures of decimal magnitude understanding (i.e., comparing decimals to the fraction [Formula: see text] benchmark task, estimating where decimals belong on a 0-1 number line, and identifying fraction and decimal equivalencies). Each of the domain-general cognitive abilities predicted students' decimal magnitude understanding. Using place value labels was positively correlated with students' decimal magnitude understanding, whereas using whole-number labels was negatively correlated with students' decimal magnitude understanding. Language comprehension, nonverbal reasoning, and concept formation were positively correlated with students' use of place value labels. By contrast, language comprehension and nonverbal reasoning were negatively correlated with students' use of whole number labels. Implications for the development of decimal magnitude understanding and design of effective instruction for at-risk students are discussed.

Fraction magnitude understanding and its unique role in predicting general mathematics achievement at two early stages of fraction instruction

BACKGROUND

Prior studies on fraction magnitude understanding focused mainly on students with relatively sufficient formal instruction on fractions whose fraction magnitude understanding is relatively mature.

AIM

This study fills a research gap by investigating fraction magnitude understanding in the early stages of fraction instruction. It extends previous findings to children with limited and primary formal fraction instruction.

SAMPLE(S)

Thirty-five fourth graders with limited fraction instruction and forty fourth graders with primary fraction instruction were recruited from a Chinese primary school.

METHODS

Children's fraction magnitude understanding was assessed with a fraction number line estimation task. Approximate number system (ANS) acuity was assessed with a dot discrimination task. Whole number knowledge was assessed with a whole number line estimation task. General reading and mathematics achievements were collected concurrently and 1 year later.

RESULTS

In children with limited fraction instruction, fraction representation was linear and fraction magnitude understanding was concurrently related to both ANS and whole number knowledge. In children with primary fraction instruction, fraction magnitude understanding appeared to (marginally) significantly predict general mathematics achievement 1 year later.

CONCLUSIONS

Fraction magnitude understanding emerged early during formal instruction of fractions. ANS and whole number knowledge were related to fraction magnitude understanding when children first began to learn about fractions in school. The predictive value of fraction magnitude understanding is likely constrained by its sophistication level.

Error Patterns with Fraction Calculations at Fourth Grade as a Function of Students' Mathematics Achievement Status

The goal of the present study was to describe fraction-calculation errors among 4^th-grade students and determine whether error patterns differed as a function of problem type (addition vs. subtraction; like vs. unlike denominators), orientation (horizontal vs. vertical), or mathematics-achievement status (low- vs. average- vs. high-achieving). We specifically addressed whether mathematics-achievement status was related to students' tendency to operate with whole number bias. We extended this focus by comparing low-performing students' errors in two instructional settings that focused on two different types of fraction understandings: core instruction that focused on part-whole understanding vs. small-group tutoring that focused on magnitude understanding. Results showed students across the sample were more likely to operate with whole number bias on problems with unlike denominators. Students with low or average achievement (who only participated in core instruction) were more likely to operate with whole number bias than students with low achievement who participated in small-group tutoring. We suggest instruction should emphasize magnitude understanding to sufficiently increase fraction understanding for all students in the upper elementary grades.

Child-Level Predictors of Responsiveness to Evidence-Based Mathematics Intervention

We identified child-level predictors of responsiveness to 2 types of mathematics (calculation and word-problem) intervention among 2nd-grade children with mathematics difficulty. Participants were 250 children in 107 classrooms in 23 schools pretested on mathematics and general cognitive measures and posttested on mathematics measures. We assigned classrooms randomly assigned to calculation intervention, word-problem intervention, or business-as-usual control. Intervention lasted 17 weeks. Path analyses indicated that scores on working memory and language comprehension assessments moderated responsiveness to calculation intervention. No moderators were identified for responsiveness to word-problem intervention. Across both intervention groups and the control group, attentive behavior predicted both outcomes. Initial calculation skill predicted the calculation outcome, and initial language comprehension predicted word-problem outcomes. These results indicate that screening for calculation intervention should include a focus on working memory, language comprehension, attentive behavior, and calculations. Screening for word-problem intervention should focus on attentive behavior and word problems.

A systematic investigation of the link between rational number processing and algebra ability

Recent research suggests that fraction understanding is predictive of algebra ability; however, the relative contributions of various aspects of rational number knowledge are unclear. Furthermore, whether this relationship is notation-dependent or rather relies upon a general understanding of rational numbers (independent of notation) is an open question. In this study, college students completed a rational number magnitude task, procedural arithmetic tasks in fraction and decimal notation, and an algebra assessment. Using these tasks, we measured three different aspects of rational number ability in both fraction and decimal notation: (1) acuity of underlying magnitude representations, (2) fluency with which symbols are mapped to the underlying magnitudes, and (3) fluency with arithmetic procedures. Analyses reveal that when looking at the measures of magnitude understanding, the relationship between adults' rational number magnitude performance and algebra ability is dependent upon notation. However, once performance on arithmetic measures is included in the relationship, individual measures of magnitude understanding are no longer unique predictors of algebra performance. Furthermore, when including all measures simultaneously, results revealed that arithmetic fluency in both fraction and decimal notation each uniquely predicted algebra ability. Findings are the first to demonstrate a relationship between rational number understanding and algebra ability in adults while providing a clearer picture of the nature of this relationship.

Developmental Foundations of Children's Fraction Magnitude Knowledge

The conceptual insight that fractions represent magnitudes is a critical yet daunting step in children's mathematical development, and the knowledge of fraction magnitudes influences children's later mathematics learning including algebra. In this study, longitudinal data were analyzed to identify the mathematical knowledge and domain-general competencies that predicted 8^th and 9^th graders' (n=122) knowledge of fraction magnitudes and its cross-grade gains. Performance on the fraction magnitude measures predicted 9^th grade algebra achievement. Understanding and fluently identifying the numerator-denominator relation in 7^th grade emerged as the key predictor of later fraction magnitudes knowledge in both 8^th and 9^th grades. Competence at using fraction procedures, knowledge of whole number magnitudes, and the central executive contributed to 9^th but not 8^th graders' fraction magnitude knowledge, and knowledge of whole number magnitude contributed to cross-grade gains. The key results suggest fluent processing of numerator-denominator relations presages students' understanding of fractions as magnitudes and that the integration of whole number and fraction magnitudes occurs gradually.

Delaware Longitudinal Study of Fraction Learning: Implications for Helping Children With Mathematics Difficulties

The goal of the present article is to synthesize findings to date from the Delaware Longitudinal Study of Fraction Learning. The study followed a large cohort of children ( N = 536) between Grades 3 and 6. The findings showed that many students, especially those with diagnosed learning disabilities, made minimal growth in fraction knowledge and that some showed only a basic grasp of the meaning of a fraction even after several years of instruction. Children with low growth in fraction knowledge during the intermediate grades were much more likely to fail to meet state standards on a broad mathematics measure at the end of Grade 6. Although a range of general and mathematics-specific competencies predicted fraction outcomes, the ability to estimate numerical magnitudes on a number line was a uniquely important marker of fraction success. Many children with mathematics difficulties have deep-seated problems related to whole number magnitude representations that are complicated by the introduction of fractions into the curriculum. Implications for helping students with mathematics difficulties are discussed.

The Interaction of Learning Disability Status and Student Demographic Characteristics on Mathematics Growth

This study examined mathematics achievement growth of students without disabilities (SWoD) and students with learning disabilities (LD) and tested whether growth and LD status interacted with student demographic characteristics. Growth was estimated in a statewide sample of 79,554 students over Grades 3 to 7. The LD group was significantly lower in achievement in each grade and had less growth than the SWoD group. We also found that student demographic characteristics were significantly related to mathematics growth, but only three demographic characteristics were statistically significant as interactions. We found that LD-SWoD differences at Grade 3 were moderated by student sex, while Black race/ethnicity and free or reduced lunch (FRL) status moderated LD-SWoD differences at all grades. These results provide practitioners and policy makers with more specific information about which particular LD students show faster or slower growth in mathematics. Our results show that simply including predictors in a regression equation may produce different results than direct testing of interactions and achievement gaps may be larger for some LD subgroups of students than previously reported.

Cognitive and numerosity predictors of mathematical skills in middle school

There is a strong research base on the underlying concomitants of early developing math skills. Fewer studies have focused on later developing skills. Here, we focused on direct and indirect contributions of cognitive measures (e.g., language, spatial skills, working memory) and numerosity measures, as well as arithmetic proficiency, on key outcomes of fraction performance, proportional reasoning, and broad mathematics achievement at sixth grade (N=162) via path analysis. We expected a hierarchy of skill development, with predominantly indirect effects of cognitive factors via number and arithmetic. Results controlling for age showed that the combination of cognitive, number, and arithmetic variables cumulatively accounted for 38% to 44% of the variance in fractions, proportional reasoning, and broad mathematics. There was consistency across outcomes, with more proximal skills providing direct effects and with the effects of cognitive skills being mediated by number and by more proximal skills. Results support a hierarchical progression from domain-general cognitive processes through numerosity and arithmetic skills to proportional reasoning to broad mathematics achievement.

A Matter of Balance: Motor Control is Related to Children's Spatial and Proportional Reasoning Skills

Recent research has shown close links between spatial and mathematical thinking and between spatial abilities and motor skills. However, longitudinal research examining the relations between motor, spatial, and mathematical skills is rare, and the nature of these relations remains unclear. The present study thus investigated the relation between children's motor control and their spatial and proportional reasoning. We measured 6-year-olds' spatial scaling (i.e., the ability to reason about different-sized spaces), their mental transformation skills, and their ability to balance on one leg as an index for motor control. One year later (N = 126), we tested the same children's understanding of proportions. We also assessed several control variables (verbal IQ and socio-economic status) as well as inhibitory control, visuo-spatial and verbal working memory. Stepwise hierarchical regressions showed that, after accounting for effects of control variables, children's balance skills significantly increased the explained variance in their spatial performance and proportional reasoning. Our results suggest specific relations between balance skills and spatial as well as proportional reasoning skills that cannot be explained by general differences in executive functioning or intelligence.

Sources of Individual Differences in Emerging Competence With Numeration Understanding Versus Multidigit Calculation Skill

This study investigated contributions of general cognitive abilities and foundational mathematical competencies to numeration understanding (i.e., base-10 structure) versus multidigit calculation skill. Children (n = 394, M = 6.5 years) were assessed on general cognitive abilities and foundational numerical competencies at start of 1st grade; on the same numerical competencies, multidigit calculation skill, and numeration understanding at end of 2nd grade; and on multidigit calculation skill and numeration understanding at end of 3rd grade. Path-analytic mediation analysis revealed that general cognitive predictors exerted more direct and more substantial effects on numeration understanding than on multidigit calculations. Foundational mathematics competencies contributed to both outcomes, but largely via 2nd-grade mathematics achievement, and results suggest a mutually supportive role between numeration understanding and multidigit calculations.