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

It is well established in the literature that fraction knowledge is important for learning more advanced mathematics, but the hierarchical relations among whole number arithmetic, fraction knowledge, and mathematics word problem-solving are not well understood. In the current study, Chinese Grade 6 students (N = 1160; 465 girls; Mage = 12.1 years, SD = 0.6) completed whole number arithmetic (addition, subtraction, multiplication, and division), fraction (mapping, equivalence, comparison, and arithmetic), and mathematics word problem-solving assessments. They also completed two control measures: number writing speed and nonverbal intelligence. Structural equation modeling was used to investigate the hierarchical relations among these assessments. Among the four fraction tasks, the correlations were low to moderate, suggesting that each task may tap into a unique aspect of fraction understanding. In the model, whole number arithmetic was directly related to all four fraction tasks, but was only indirectly related to mathematics word problem-solving, through fraction arithmetic. Only fraction arithmetic, the most advanced fraction skill, directly predicted mathematics word problem-solving. These findings are consistent with the view that students need to build these associations into their mathematics hierarchy to advance their mathematical competence.

The role of working memory updating, inhibition, fluid intelligence, and reading comprehension in explaining differences between consistent and inconsistent arithmetic word-problem-solving performance.

Children's performance in arithmetic word problems (AWPs) predicts their academic success and their future employment and earnings in adulthood. Understanding the nature and difficulties of interpreting and solving AWPs is important for theoretical, educational, and social reasons. We investigated the relation between primary school children's performance in different types of AWPs and their basic cognitive abilities (reading comprehension, fluid intelligence, inhibition, and updating processes). The study involved 182 fourth- and fifth-graders. Participants were administered an AWP-solving task and other tasks assessing fluid intelligence, reading comprehension, inhibition, and updating. The AWP-solving task included comparison problems incorporating either the adverb more than or the adverb less than, which demand consistent or inconsistent operations of addition or subtraction. The results showed that consistent problems were easier than inconsistent problems. Efficiency in solving inconsistent problems is related to inhibition and updating. Moreover, our results seem to indicate that the consistency effect is related to updating processes' efficiency. Path analyses showed that reading comprehension was the most important predictor of AWP-solving accuracy. Moreover, both executive functions-updating and inhibition-had a distinct and significant effect on AWP accuracy. Fluid intelligence had both direct and indirect effects, mediated by reading comprehension, on the overall measure of AWP performance. These domain-general factors are important factors in explaining children's performance in solving consistent and inconsistent AWPs.

The Mathematical Performance of At-Risk First Graders as a Function of Limited English Proficiency Status.

The purpose of this study was to explore interactions between limited English proficiency (LEP) status, as a function of risk status (low math performance at the start of the school year), on computation and word-problem solving performance. Among 260 1st-grade students, classified as at-risk (AR) or not at-risk (NAR) for math disability, we compared the performance of LEP students to native English-speaking peers. A series of 2-way ANOVAs were conducted on computation and word-problem solving skill at 2 time points, fall and spring of 1st-grade. On fall computation measures, there was no main effect for LEP status and no interaction between LEP and risk status. On spring computation, a main effect for LEP status had emerged, but again no interaction. By contrast, on fall word-problem solving, there was an interaction between LEP and risk status; however, this interaction was no longer significant by spring. Results suggest that language proficiency is an important factor in the development of computation and word-problem solving skill. Implications for future research are discussed.

A set for relational reasoning: Facilitation of algebraic modeling by a fraction task.

Recent work has identified correlations between early mastery of fractions and later math achievement, especially in algebra. However, causal connections between aspects of reasoning with fractions and improved algebra performance have yet to be established. The current study investigated whether relational reasoning with fractions facilitates subsequent algebraic reasoning using both pre-algebra students and adult college students. Participants were first given either a relational reasoning fractions task or a fraction algebra procedures control task. Then, all participants solved word problems and constructed algebraic equations in either multiplication or division format. The word problems and the equation construction tasks involved simple multiplicative comparison statements such as "There are 4 times as many students as teachers in a classroom." Performance on the algebraic equation construction task was enhanced for participants who had previously completed the relational fractions task compared with those who completed the fraction algebra procedures task. This finding suggests that relational reasoning with fractions can establish a relational set that promotes students' tendency to model relations using algebraic expressions.

Toward self-regulated learning: effects of different types of data-driven feedback on pupils' mathematics word problem-solving performance.

Introduction: Mathematical word problems refer to word problems where the information that is presented needs to be integrated, typically into a mathematical formula, to arrive at a solution to the problem. When solving mathematics word problems, elementary school students often have difficulties improving their performance due to a lack of self-regulated learning (SRL). However, SRL can be developed by adopting an appropriate teaching approach which offers quantitative feedback or learning prompts. With the sophistication of interactive and data-driven feedback technology, it is possible to provide timely and personalized strategies for promoting students' SRL. Methods: In this study, an interactive e-book editing platform was used to design self-regulation-level-based feedback(SRLF) and task-level-based feedback(TLF) teaching models, which were respectively conducted in two similar fifth-grade classes for the mathematics word problem solving lessons. Results: Using ANCOVA and repeated ANOVA, this study found that (1) the SRLF had a remarkably greater impact on elementary school students' mathematics word problem-solving performance than the TLF, with a partial η 2-value of .107; (2) In the short period of time, there was no significant difference between the two kinds of feedback on the learners' SRL. The TLF was slightly superior to the SRLF, especially in terms of total self-regulated learning scores and cognitive strategies; (3) The TLF had a significant interaction effect on self-regulated learning and cognitive strategies, respectively with a partial η 2-value of .059 and .056.

The contribution of general cognitive abilities and number abilities to different aspects of mathematics in children.

This study examined the relative contributions of general cognitive abilities and number abilities to word problem solving, calculation, and arithmetic fact retrieval in a sample of 134 children aged 10 to 13 years. The following tasks were administered: listening span, visual matrix span, verbal fluency, color naming, Raven's Progressive Matrices, enumeration, number line estimation, and digit comparison. Hierarchical multiple regressions demonstrated that number abilities provided an independent contribution to fact retrieval and word problem solving. General cognitive abilities contributed to problem solving and calculation. All three number tasks accounted for a similar amount of variance in fact retrieval, whereas only the number line estimation task contributed unique variance in word problem solving. Verbal fluency and Raven's matrices accounted for an equal amount of variance in problem solving and calculation. The current findings demonstrate, in accordance with Fuchs and colleagues' developmental model of mathematical learning (Developmental Psychology, 2010, Vol. 46, pp. 1731-1746), that both number abilities and general cognitive abilities underlie 10- to 13-year-olds' proficiency in problem solving, whereas only number abilities underlie arithmetic fact retrieval. Thus, the amount and type of cognitive contribution to arithmetic proficiency varies between the different aspects of arithmetic. Furthermore, how closely linked a specific aspect of arithmetic is to the whole number representation systems is not the only factor determining the amount and type of cognitive contribution in 10- to 13-year-olds. In addition, the mathematical complexity of the task appears to influence the amount and type of cognitive support.

The Effects of Visual Cueing on Students with and without Math Learning Difficulties in Online Problem Solving: Evidence from Eye Movement.

This study investigated the impact of visual cueing on attention guidance, deep-thinking promotion, and performance optimization in arithmetic word problem solving for students with mathematical learning difficulties (MLD). The participants included eight students with MLD and twenty students without MLD who attempted to solve mathematical word problems with and without visual cueing. Eye movements were recorded during the tasks. A repeated-measure design and nonparametric tests were applied to enhance the statistical power of the study. The data analysis results indicated that visual cueing effectively guided and sustained the attention of students with MLD, reducing their off-task duration. However, it showed limited influence in facilitating deep thinking and performance improvement for these students. There were no significant attention-guidance or performance-improvement effects observed in the problem-solving processes of students without MLD, who initially demonstrated better concentration levels and performance. The potential explanations for these findings are further discussed in this paper.

Retrieval practice may not benefit mathematical word-problem solving.

The retrieval practice effect refers to the fact that one or even multiple retrievals of memory content during the same period are more effective than repeated studying to promote future memory retention. It is effective for numerous declarative knowledge learning materials. However, studies have demonstrated that retrieval practice does not benefit problem-solving skill learning. This study used worked examples from math word problem tasks as learning materials, considering the retrieval difficulty as the main factor. Experiment 1 explored the effect of retrieval practice on acquiring problem-solving skills under different initial testing difficulties. Experiment 2 manipulated the difficulty of materials as a variable to ascertain the effect of retrieval practice on problem-solving skills under different material difficulty levels. Experiment 3 introduced feedback variables to facilitate the generation of the retrieval practice effect and examined the effects of various difficulty feedback levels on problem-solving skills learning. Results showed that, compared with restudying examples (SSSS), the example-problem pairs (STST) did not promote delayed test performance. As for the retrieval practice effect, as no differences or advantages were found in the repeated study group on the immediate test, the retrieval practice group generally outperformed the repeated study group on the delayed test. However, across the three experiments, we found no evidence of retrieval practice affecting results during an enhanced delayed test. Therefore, there may be no retrieval practice effect on acquiring problem-solving skills from worked examples.

Comorbidity in Reading Comprehension and Word-Problem Solving Difficulties: Exploring Shared Risk Factors and Their Impact on Language Minority Learners.

The purpose of this study was threefold: to examine unique and shared risk factors of comorbidity for reading comprehension and word-problem solving difficulties, to explore whether language minority (LM) learners are at increased risk of what we refer to as higher order comorbidity (reading comprehension and word-problem solving difficulties), and to examine the profiles of at-risk LM learners compared with at-risk non-LM learners. At-risk (LM n = 70; non-LM n = 89) and not-at-risk (LM n = 44; non-LM n = 114) students were evaluated on foundational academic (word reading, calculation), behavioral (behavioral attention), cognitive (working memory, processing speed, nonverbal reasoning), and language (vocabulary, listening comprehension) measures in English. Results indicated listening comprehension was the only shared risk factor for higher order comorbidity. Furthermore, LM learners were 3 times more likely to be identified as at risk compared with non-LM learners. Finally, among at-risk learners, no differences were found on cognitive dimensions by language status, but LM learners had lower reading and listening comprehension skills than non-LM learners, with a relative advantage in behavioral attention. Results have implications for understanding higher order comorbidity and for developing methods to identify and intervene with higher order comorbidity among the growing population of LM learners.

Spatial skills and number skills in preschool children: The moderating role of spatial anxiety.

Spatial ability is a strong and stable predictor of mathematical performance. However, of the three key components of spatial ability, spatial perception and spatial visualization have received less attention than mental rotation in relation to specific mathematical competencies of young children. Even less is known about the role of spatial anxiety in this relationship. This study examined the longitudinal relations of spatial perception and spatial visualization to three number skills (i.e., number line estimation, subitizing, and word problem-solving) among 190 preschool children, and whether these relations varied as a function of spatial anxiety. The results showed that children's spatial perception and spatial visualization skills, measured in the third preschool year (Time 1 [T1]), were positively associated with their word problem-solving six months later (Time 2 [T2]). Children's T1 spatial perception was also positively associated with their T2 subitizing and number line skills. In addition, T1 spatial anxiety moderated the relation between T1 spatial perception and T2 subitizing: the relation between the two was stronger for children with low levels of spatial anxiety than it was for those with moderate or high levels. The findings offer valuable insights into how spatial cognition and affect jointly relate to children's early number skills.

Problem-appropriate diagram instruction for improving mathematical word problem solving.

The use of diagrams can be effective in solving mathematical word problems solving. However, students worldwide do not construct diagrams unprompted or have trouble using them. In the present study, the effects of problem-appropriate diagram use instruction were investigated with an adaptation of the multiple baseline design method. The instruction for using line diagrams, tables, and graphs was provided to 67 junior high school students in a staggered manner and the effects on problem solving of three different types of problems was examined. The results showed that use of problem-appropriate diagrams increased and persisted over time. More importantly, the instruction led to increases in problem solving performance and to decreases in perceived cognitive load. These findings support the argument that effective diagram use depends on the acquisition not only of declarative knowledge, but also sufficient procedural and conditional knowledge.

Arithmetic Word-Problem Solving as Cognitive Marker of Progression in Pre-Manifest and Manifest Huntington's Disease.

BACKGROUND: Arithmetic word-problem solving depends on the interaction of several cognitive processes that may be affected early in the disease in gene-mutation carriers for Huntington's disease (HD). OBJECTIVE: Our goal was to examine the pattern of performance of arithmetic tasks in premanifest and manifest HD, and to examine correlations between arithmetic task performance and other neuropsychological tasks. METHODS: We collected data from a multicenter cohort of 165 HD gene-mutation carriers. The sample consisted of 31 premanifest participants: 16 far-from (>12 years estimated time to diagnosis; preHD-A) and 15 close-to (≤12 years estimated time to diagnosis; preHD-B), 134 symptomatic patients (early-mild HD), and 37 healthy controls (HC). We compared performance between groups and explored the associations between arithmetic word-problem solving and neuropsychological and clinical variables. RESULTS: Total arithmetic word-problem solving scores were lower in preHD-B patients than in preHD-A (p < 0.05) patients and HC (p < 0.01). Early-mild HD patients had lower scores than preHD patients (p < 0.001) and HC (p < 0.001). Compared to HC, preHD and early-mild HD participants made more errors as trial complexity increased. Moreover, arithmetic word-problem solving scores were significantly associated with measures of global cognition (p < 0.001), frontal-executive functions (p < 0.001), attention (p < 0.001) visual working memory (p < 0.001), mental rotation (p < 0.001), and confrontation naming (p < 0.05). CONCLUSION: Arithmetic word-problem solving is affected early in the course of HD and is related to deficient processes in frontal-executive and mentalizing-related processes.

Difficulties of children with ADHD symptoms in solving mathematical problems when information must be updated.

It has been hypothesized that ADHD is associated both with difficulties in mathematical problem solving and in updating information in working memory. However, the relationship between updating and performance on mathematical word problems has never been studied for children with ADHD. The present study examined these issues comparing the performance of solving mathematical word problems (with no updating request vs high updating request) in a group of 11-12year old children with ADHD compared to a matched control group with typical development (TD). Results showed that children with ADHD solved fewer problems correctly than typically-developing children; moreover they made more errors in solving problems with updating requirements than those without updating requirements. In contrast, typically-developing children did not show any differences in problems performance on problems with and without updating requirements. Fine grained analyses of children's problem solving processes showed that children with ADHD found more difficult to select the appropriate data prior to calculation and to choose and execute the correct solution than typically-developing children. The difficulty to select the appropriate data results more severe in problems with updating requirements. Overall, these results support the hypothesis that the learning difficulties of children with ADHD are related to their executive dysfunctions, that negatively affect complex tasks requiring updating of to-be-processed information.

Calculation and word problem-solving skills in primary grades - Impact of cognitive abilities and longitudinal interrelations with task-persistent behaviour.

BACKGROUND: Primary school math skills form a basis for academic success down the road. Different math skills have different antecedents and there is a reason to believe that more complex math tasks require better self-regulation. AIMS: The study aimed to investigate longitudinal interrelations of calculation and problem-solving skills, and task-persistent behaviour in Grade 1 and Grade 3, and the effect of non-verbal intelligence, linguistic abilities, and executive functioning on math skills and task persistence. SAMPLE: Participants were 864 students (52.3% boys) from 33 different schools in Estonia. METHODS: Students were tested twice - at the end of Grade1 and at the end of Grade 3. Calculation and problem-solving skills, and teacher-rated task-persistent behaviour were measured at both time points. Non-verbal intelligence, linguistic abilities, and executive functioning were measured in Grade 1. RESULTS: Cross-lagged structural equation modelling indicated that calculation skills depend on previous math skills and linguistic abilities, while problem-solving skills require also non-verbal intelligence, executive functioning, and task persistence. Task-persistent behaviour in Grade 3 was predicted by previous problem-solving skills, linguistic abilities, and executive functioning. Gender and mother's educational level were added as covariates. CONCLUSIONS: The findings indicate that math skills and self-regulation are strongly related in primary grades and that solving complex tasks requires executive functioning and task persistence from children. Findings support the idea that instructional practices might benefit from supporting self-regulation in order to gain domain-specific, complex skill achievement.

Word Problem Solving in Contemporary Math Education: A Plea for Reading Comprehension Skills Training.

Successfully solving mathematical word problems requires both mental representation skills and reading comprehension skills. In Realistic Math Education (RME), however, students primarily learn to apply the first of these skills (i.e., representational skills) in the context of word problem solving. Given this, it seems legitimate to assume that students from a RME curriculum experience difficulties when asked to solve semantically complex word problems. We investigated this assumption under 80 sixth grade students who were classified as successful and less successful word problem solvers based on a standardized mathematics test. To this end, students completed word problems that ask for both mental representation skills and reading comprehension skills. The results showed that even successful word problem solvers had a low performance on semantically complex word problems, despite adequate performance on semantically less complex word problems. Based on this study, we concluded that reading comprehension skills should be given a (more) prominent role during word problem solving instruction in RME.

The relation between children's constructive play activities, spatial ability, and mathematical word problem-solving performance: a mediation analysis in sixth-grade students.

The scientific literature shows that constructive play activities are positively related to children's spatial ability. Likewise, a close positive relation is found between spatial ability and mathematical word problem-solving performances. The relation between children's constructive play and their performance on mathematical word problems is, however, not reported yet. The aim of the present study was to investigate whether spatial ability acted as a mediator in the relation between constructive play and mathematical word problem-solving performance in 128 sixth-grade elementary school children. This mediating role of spatial ability was tested by utilizing the current mediation approaches suggested by Preacher and Hayes (2008). Results showed that 38.16% of the variance in mathematical word problem-solving performance is explained by children's constructive play activities and spatial ability. More specifically, spatial ability acted as a partial mediator, explaining 31.58% of the relation between constructive play and mathematical word problem-solving performance.