The Cognitive Predictors of Computational Skill with Whole versus Rational Numbers: An Exploratory Study

The purpose of the present study was to explore the 3rd-grade cognitive predictors of 5th-grade computational skill with rational numbers and how those are similar to and different from the cognitive predictors of whole-number computational skill. Students (n = 688) were assessed on incoming whole-number calculation skill, language, nonverbal reasoning, concept formation, processing speed, and working memory in the fall of 3rd grade. Students were followed longitudinally and assessed on calculation skill with whole numbers and with rational numbers in the spring of 5th grade. The unique predictors of skill with whole-number computation were incoming whole-number calculation skill, nonverbal reasoning, concept formation, and working memory (numerical executive control). In addition to these cognitive abilities, language emerged as a unique predictor of rational-number computational skill.     

Sources of Individual Differences in Children's Understanding of Fractions

Longitudinal associations of domain‐general and numerical competencies with individual differences in children's understanding of fractions were investigated. Children (n = 163) were assessed at 6 years of age on domain‐general (nonverbal reasoning, language, attentive behavior, executive control, visual‐spatial memory) and numerical (number knowledge) competencies; at 7 years on whole‐number arithmetic computations and number line estimation; and at 10 years on fraction concepts. Mediation analyses controlling for general mathematics ability and general academic ability revealed that numerical and mathematical competencies were direct predictors of fraction concepts, whereas domain‐general competencies supported the acquisition of fraction concepts via whole‐number arithmetic computations or number line estimation. Results indicate multiple pathways to fraction competence.     

Language-related longitudinal predictors of arithmetic word problem solving: A structural equation modeling approach

We investigated the longitudinal relations between cognitive skills, specifically language-related skills, and word-problem solving in 340 children (6.10–9.02 years). We used structural equation modeling to examine whether word-problem solving, computation skill, working memory, nonverbal reasoning, oral language, and word reading fluency measured at second grade were associated with performance on measures of word-problem solving in fourth grade. Results indicated that prior word-problem solving, computation skill, nonverbal reasoning, and oral language were significantly associated with children’s later word-problem solving. Multi-group modeling suggested that these relations were not significantly different for boys versus girls. Implications of these findings are discussed.     

Covariation between reading and arithmetic skills from Grade 1 to Grade 7

This study examined the extent to which reading and arithmetic skills show covariation at Grade 1 and at Grade 7, to what extent this covariation is time-invariant or time-specific, and to what extent different antecedents will predict these time-invariant and time-specific portions of the covariation. The reading and arithmetic skills of a total of 1335 Finnish children were assessed at the end of Grade 1 and then again at the end of Grade 7. Phonological awareness, letter knowledge, rapid automatized naming (RAN), counting, and parental education levels were measured in kindergarten; working memory at Grade 1 and nonverbal reasoning at Grade 3. The results showed that reading and arithmetic had a substantial amount of covariation at grades 1 and 7, and that most of the covariation between these grades was time-invariant and could be predicted by RAN, counting, letter knowledge, working memory, and nonverbal reasoning. The time-specific portion of the covariation between reading and arithmetic in Grade 1 was predicted by phonological awareness, letter knowledge, and counting; while time-specific covariation in Grade 7 was predicted by parental education level and nonverbal reasoning.     

Investigating the Unique Predictors of Word-Problem Solving Using Meta-Analytic Structural Equation Modeling

The purpose of the present study is to clarify the contributions of cognitive skills (nonverbal reasoning, language comprehension, working memory, attention, processing speed) and academic skills (mathematics facts retrieval, mathematics computation, mathematics vocabulary, reading comprehension) in performing mathematics word problems among elementary school students. With the two-stage meta-analytic structural equation modeling approach, I synthesized 112 correlation matrices from 98 empirical studies (N?=?111,346) and fitted the hypothesized partial mediation model. Overall, path analysis indicated that language comprehension, working memory, attention, mathematics vocabulary, and mathematics computation were unique predictors of word-problem solving. Subgroup analysis demonstrated different unique predictors for younger and older students to perform word problems (K-2nd grades versus 3rd–5th grades). Implications, limitations, and future directions are discussed.

     

Effect on development of proportional reasoning skill of physical experience and cognitive abilities associated with prefrontal lobe activity

The present study tested the hypothesis that maturing prefrontal lobes play a role in the development of proportional reasoning skill because the prefrontal lobes are involved in the inhibition of task‐irrelevant information and the representation of task‐relevant information. The hypothesis that reasoning development is in part dependent upon physical experience was also tested. Students (all males) who failed to solve a diagnostic proportions task were administered several tests of prefrontal lobe functions. The students were then randomly assigned to manipulative or verbal tutoring groups. Both groups received a series of individual testing, tutoring and testing sessions on proportional reasoning. As predicted, performance on the prefrontal lobe tasks (measures of inhibiting ability, planning ability, dissembedding ability, and working memory capacity) significantly predicted performance on proportional reasoning tasks following tutoring. Students' computational skills were not a significant predictor. Also, the manipulative group's proportional reasoning performance was significantly higher than that of the verbal tutoring group. Therefore, the present results provide support for the hypothesis that maturing prefrontal lobes and physical experience play roles in the development of proportional reasoning skill. © 2000 John Wiley & Sons, Inc. J Res Sci Teach 37: 1171–1182, 2000     

Is Word-Problem Solving a Form of Text Comprehension?

This study’s hypotheses were that (a) word-problem (WP) solving is a form of text comprehension that involves language comprehension processes, working memory, and reasoning, but (b) WP solving differs from other forms of text comprehension by requiring WP-specific language comprehension as well as general language comprehension. At the start of the 2nd grade, children (n = 206; on average, 7 years, 6 months) were assessed on general language comprehension, working memory, nonlinguistic reasoning, processing speed (a control variable), and foundational skill (arithmetic for WPs; word reading for text comprehension). In spring, they were assessed on WP-specific language comprehension, WPs, and text comprehension. Path analytic mediation analysis indicated that effects of general language comprehension on text comprehension were entirely direct, whereas effects of general language comprehension on WPs were partially mediated by WP-specific language. By contrast, effects of working memory and reasoning operated in parallel ways for both outcomes.     

Heterogeneity in children at risk of math learning difficulties

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

Text Comprehension and Oral Language as Predictors of Word-Problem Solving: Insights into Word-Problem Solving as a Form of Text Comprehension

This study was designed to deepen insights on whether word-problem (WP) solving is a form of text comprehension (TC) and on the role of language in WPs. A sample of 325 second graders, representing high, average, and low reading and math performance, was assessed on (a) start-of-year TC, WP skill, language, nonlinguistic reasoning, working memory, and foundational skill (word identification, arithmetic) and (b) year-end WP solving, WP-language processing (understanding WP statements, without calculation demands), and calculations. Multivariate, multilevel path analysis, accounting for classroom and school effects, indicated that TC was a significant and comparably strong predictor of all outcomes. Start-of-year language was a significantly stronger predictor of both year-end WP outcomes than of calculations, whereas start-of-year arithmetic was a significantly stronger predictor of calculations than of either WP measure. Implications are discussed in terms of WP solving as a form of TC and a theoretically coordinated approach, focused on language, for addressing TC and WP-solving instruction.     

Criterion‐related validity of two curriculum‐based measures of mathematical skill in relation to reading comprehension in secondary students

This study analyzed the relationship between benchmark scores from two curriculum‐based measurement probes in mathematics (M‐CBM) and student performance on a state‐mandated high‐stakes test. Participants were 298 students enrolled in grades 7 and 8 in a rural southeastern school. Specifically, we calculated the criterion‐related and predictive validity of benchmark scores from CBM probes measuring math computation and math reasoning skills. Results of this study suggest that math reasoning probes have strong concurrent and predictive validity. The study also provides evidence that calculation skills, while important, do not have strong predictive strength at the secondary level when a state math assessment is the criterion. When reading comprehension skill is taken into account, math reasoning scores explained the greatest amount of variance in the criterion measure. Computation scores explained less than 5% of the variance in the high‐stakes test, suggesting that it may have limitations as a universal screening measure for secondary students.     

Teachers' Spatial Anxiety Relates to 1st‐ and 2nd‐Graders' Spatial Learning

Teachers' anxiety about an academic domain, such as math, can impact students' learning in that domain. We asked whether this relation held in the domain of spatial skill, given the importance of spatial skill for success in math and science and its malleability at a young age. We measured 1st‐ and 2nd‐grade teachers' spatial anxiety (N = 19) and students' spatial skill (N = 132). Teachers' spatial anxiety significantly predicted students' end‐of‐year spatial skill, even after accounting for students' beginning‐of‐year spatial skill, phonological working memory, grade level, and teachers' math anxiety. Since spatial skill is not a stand‐alone part of the curriculum like math or reading, teachers with high levels of spatial anxiety may simply avoid incorporating spatial activities in the classroom. Results suggest that addressing teachers' spatial anxieties may improve spatial learning in early elementary school.     

A structural model of algebra achievement: computational fluency and spatial visualisation as mediators of the effect of working memory on algebra achievement

The goal of this study was to develop and evaluate a structural model of the relations among cognitive abilities and arithmetic skills and college students’ algebra achievement. The model of algebra achievement was compared to a model of performance on the Scholastic Assessment in Mathematics (SAT‐M) to determine whether the pattern of relations is similar for different types of higher level maths achievement. Structural equation modelling was used to test the effects of working memory, 3D spatial ability, and computational fluency on both types of higher order maths achievement. Computational fluency had the strongest effect on algebra achievement, with 3D spatial ability and working memory showing moderate effects. In contrast, 3D spatial ability had a stronger effect on SAT‐M scores than did computational fluency. Computational fluency and 3D spatial ability completely mediated the effect of working memory for both algebra and SAT‐M achievement.     

Cognitive Difficulties in Struggling Comprehenders and Their Relation to Reading Comprehension: A Comparison of Group Selection and Regression-Based Models

ABSTRACT

Difficulties suppressing previously encountered but currently irrelevant information from working memory characterize less skilled comprehenders in studies in which they are matched to skilled comprehenders on word decoding and nonverbal IQ. These “extreme” group designs are associated with several methodological issues. When sample size permits, regression approaches permit a more accurate estimation of effects. Using data for students in Grades 6 through 12 (n = 766), regression techniques assessed the significance and size of the relation of suppression to reading comprehension across the distribution of comprehension skill. After accounting for decoding efficiency and nonverbal IQ, suppression, measured by performance on a verbal proactive interference task, accounted for a small amount of significant unique variance in comprehension (less than 1%). A comparison of suppression in less skilled comprehenders matched to more skilled comprehenders (48 per group) on age, word reading efficiency, and nonverbal IQ did not show significant group differences in suppression. The implications of the findings for theories of reading comprehension and for informing comprehension assessment and intervention are discussed.     

Do the processes engaged during mathematical word-problem solving differ along the distribution of word-problem competence?

Conventional research methods for understanding sources of individual differences in word-problem solving (WPS) only permit estimation of average relations between component processes and outcomes. The purpose of the present study was instead to examine whether and if so how the component processes engaged in WPS differ along the spectrum of WPS performance. Second graders (N = 1130) from 126 classrooms in 17 schools were assessed on component processes (reasoning, in-class attentive behavior, working memory, language comprehension, calculation fluency, word reading) and WPS. Multilevel, unconditional quantile multiple regression indicated that 3 component processes, calculation fluency, language comprehension, and working memory, are engaged in WPS differentially depending on students’ overall word-problem skill. The role of calculation fluency and language comprehension was stronger with more competent word-problem solving ability. By contrast, the role of working memory was stronger with intermediate-level than for strong problem solving. Results deepen insight into the role of these processes in WPS and provide the basis for hypothesizing how instructional strategies may be differentiated depending on students’ overall level of WPS competence.     

The effects of computer-assisted instruction on number combination skill in at-risk first graders

The purpose of this pilot study was to assess the potential for computer-assisted instruction (CAI) to enhance number combination skill among children with concurrent risk for math disability and reading disability. A secondary purpose was to examine the effects of CAI on spelling. At-risk students were assigned randomly to math or spelling CAI, which they received in 50 sessions over 18 weeks. Acquisition and transfer effects were assessed. The results indicated that math CAI was effective in promoting addition but not subtraction number combination skill and that transfer to arithmetic story problems did not occur. Spelling CAI effects were reliable on acquisition and transfer spelling measures, with small to moderate effect sizes on transfer to reading measures. These results provide the basis for additional work with larger samples.     

Individual differences in working memory, nonverbal IQ, and mathematics achievement and brain mechanisms associated with symbolic and nonsymbolic number processing

Individual differences in mathematics performance may stem from domain-general factors like working memory and intelligence. Parietal and frontal brain areas have been implicated in number processing, but the influence of such cognitive factors on brain activity during mathematics processing is not known. The relationship between brain mechanisms during symbolic and nonsymbolic number comparison and individual differences in working memory, nonverbal IQ, and mathematics achievement were investigated to determine possible associations with behavior and brain function. A number of brain areas showed correlations with working memory and number processing. For symbolic digits, working memory showed a positive relationship with brain activity in a network of bilateral parietal, temporal, and right frontal regions. For nonsymbolic dot arrays, working memory showed a negative relationship with several parietal and frontal brain areas. This relationship indicates differences between behavioral and brain function measures and points to the importance of working memory and basic number processing.     

The Cognitive and Behavioral Characteristics of Children With Low Working Memory

This study explored the cognitive and behavioral profiles of children with working memory impairments. In an initial screening of 3,189 five- to eleven-year-olds, 308 were identified as having very low working memory scores. Cognitive skills (IQ, vocabulary, reading, and math), classroom behavior, and self-esteem were assessed. The majority of the children struggled in the learning measures and verbal ability. They also obtained atypically high ratings of cognitive problems/inattentive symptoms and were judged to have short attention spans, high levels of distractibility, problems in monitoring the quality of their work, and difficulties in generating new solutions to problems. These data provide rich new information on the cognitive and behavioral profiles that characterize children with low working memory.     

Remediating Computational Deficits at Third Grade: A Randomized Field Trial

Abstract

The major purposes of this study were to assess the efficacy of tutoring to remediate 3rd-grade computational deficits and to explore whether remediation is differentially efficacious depending on whether students experience mathematics difficulty alone or concomitantly with reading difficulty. At 2 sites, 127 students were stratified on mathematics difficulty status and randomly assigned to 4 conditions: word recognition (control) tutoring or 1 of 3 computation tutoring conditions: fact retrieval, procedural computation and computational estimation, and combined (fact retrieval + procedural computation and computational estimation). Results revealed that fact retrieval tutoring enhanced fact retrieval skill, and procedural computation and computational estimation tutoring (whether in isolation or combined with fact retrieval tutoring) enhanced computational estimation skill. Remediation was not differentially efficacious as a function of students' mathematics difficulty status.     

Sequential pattern learning and its associations with inattentive and hyperactive concerns in college students

College students frequently experience inattentive and hyperactive concerns. In multiple independent samples and three randomised experiments, we examined multiple versions of a short performance‐based measure translated from basic research on how organisms learn sequential stimuli patterns when such patterns are interleaved with information that is irrelevant to the pattern being learned. In one experiment, performance was uniquely related to self‐reported inattentive and hyperactive concerns, taking into account performance on a widely used inhibitory control task (Study 1, n = 20). In the two other experiments, randomly assigned variants of this measure demonstrated that: (a) relations among performance and inattentive and hyperactive concerns could be identified regardless of irrelevant stimuli positioning (Study 2, n = 60), and (b) one could reverse the relation between performance and inattentive and hyperactive concerns by visually enhancing distinguishing features of irrelevant stimuli (Study 3, n = 20). The findings have significant implications for multi‐method assessments of inattentive and hyperactive concerns in college settings.     

Mathematics learning disabilities in girls with fragile X or Turner syndrome during late elementary school

The present study focuses on math and related skills among 32 girls with fragile X (n = 14) or Turner (n = 18) syndrome during late elementary school. Performance in each syndrome group was assessed relative to Full Scale IQ-matched comparison groups of girls from the general population (n = 32 and n = 89 for fragile X syndrome and Turner syndrome, respectively). Differences between girls with fragile X and their comparison group emerged on untimed arithmetic calculations, mastery of counting skills, and arithmetic problem verification accuracy. Relative to girls in the comparison group, girls with Turner syndrome did not differ on untimed arithmetic calculations or problem verification accuracy, but they had limited mastery of counting skills and longer response times to complete the problem verification task. Girls with fragile X or Turner syndrome also differed from their respective comparison groups on math-related abilities, including visual-spatial, working memory, and reading skills, and the associations between math and those related skills. Together, these findings support the notion that difficulty with math and related skills among girls with fragile X or Turner syndrome continues into late elementary school and that the profile of math and related skill difficulty distinguishes the two syndrome groups from each other.