Mathematics anxiety and working memory: Longitudinal associations with mathematical performance in Chinese children

The link between mathematics anxiety and mathematical performance in young children remains inconclusive. The present study examined the longitudinal associations between mathematics anxiety and mathematical performance (calculation and story problem solving) in 246 Chinese children followed from second to third grade. Multiple regression analyses showed that mathematics anxiety made independent contributions to mathematical performance beyond non-verbal intelligence, working memory, number skills, general and test anxieties. However, mathematics anxiety does not affect all children and all kinds of mathematical performance equally. Mathematics anxiety has a more pronounced impact on mathematical problems that require more processing resources, as opposed to simple arithmetic problems and straightforward story problems and children who are higher in working memory are more vulnerable to its deleterious impacts.     

Individual differences in memory updating in relation to arithmetic problem solving

The study investigates the relationship between memory updating and arithmetic word problem solving. Two groups of 35 fourth graders with high and low memory-updating abilities were selected from a sample of 89 children on the basis of an updating task used by Palladino et al. [Memory & Cognition 29 (2002) 344]. The two groups were required to solve a set of arithmetic word problems and to recall relevant information from another set of problems. Several span tasks, a computation test, and the PMA verbal subtest were also administered. The group with a high memory-updating ability performed better in problem solving, recalling text problems, and in the computation test. The two groups did not differ in the PMA verbal subtest or in the digit and word spans. Results were interpreted as supporting the importance of updating ability in problem solving and of the substantial independence between memory updating and problem solving on one hand and verbal intelligence on the other.     

Pathways to word problem solving: The mediating roles of schema construction and mathematical vocabulary

When solving word problems, many children encounter difficulties in making sense of the information and integrate it into a meaningful schema. This is the fundamental phase on which subsequent problem solution depends. To better understand the processing underlying this fundamental phase, this study examined the roles of schema construction and knowledge of mathematical vocabularies in word problem solving. The participants were 139 Chinese third graders studying in Hong Kong. Path analysis showed that there were two kinds of pathways to word problem solving: language-related and number-related. In particular, reading fluency was related to word problem solving in two mediated language-related pathways: one via schema construction, the other via knowledge of mathematical vocabularies. In the number-related pathway, arithmetic concept was related to word problem solving via knowledge of mathematical vocabularies. These findings highlight the specific roles of schema construction and mathematical vocabulary in word problem solving, thereby providing useful implications of how best to support children in understanding and integrating the information from the problem.     

Mathematics anxiety in young children: Concurrent and longitudinal associations with mathematical performance

This study explored mathematics anxiety in a longitudinal sample of 113 children followed from second to third grade. We examined how mathematics anxiety related to different types of mathematical performance concurrently and longitudinally and whether the relations between mathematics anxiety and mathematical performance differed as a function of working memory. Concurrent analyses indicated that mathematics anxiety represents a unique source of individual differences in children’s calculation skills and mathematical applications, but not in children’s geometric reasoning. Furthermore, we found that higher levels of mathematics anxiety in second grade predicted lower gains in children’s mathematical applications between second and third grade, but only for children with higher levels of working memory. Overall, our results indicate that mathematics anxiety is an important construct to consider when examining sources of individual differences in young children’s mathematical performance. Furthermore, our findings suggest that mathematics anxiety may affect how some children use working memory resources to learn mathematical applications.     

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.     

The consistency effect in word problem solving is effectively reduced through verbal instruction

In mathematical word problem solving, a relatively well-established finding is that more errors are made on word problems in which the relational keyword is inconsistent instead of consistent with the required arithmetic operation. This study aimed at reducing this consistency effect. Children solved a set of compare word problems before and after receiving a verbal instruction focusing on the consistency effect (or a control verbal instruction). Additionally, we explored potential transfer of the verbal instruction to word problems containing other relational keywords (e.g., larger/smaller than) than those in the verbal instruction (e.g., more/less than). Results showed a significant pretest-to posttest reduction of the consistency effect (but also an unexpected decrement on marked consistent problems) after the experimental verbal instruction but not after the control verbal instruction. No significant effects were found regarding transfer. It is concluded that our verbal instruction was useful for reducing the consistency effect, but future research should address how this benefit can be maintained without hampering performance on marked consistent problems.     

Attentional and executive function behaviours in children with poor working memory

The purpose of this study was to explore the profiles of classroom behaviour relating to attention and executive functions in children with very poor working memory, and to test the hypothesis that inattentive behaviour and working memory problems co-occur. Teachers rated problem behaviours of 52 children with low working memory scores aged 5/6 and 9/10 years on teacher rating measures of attention and executive function behaviours. The majority of children with low working memory scores 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 results extend previous findings that working memory problems and inattentive behaviour co-occur to a non-clinical sample. It is suggested that reduced working memory capacity may play a causal role in the problem behaviours of these children.     

Non‐mathematical problem solving in organic chemistry

Differences in problem‐solving ability among organic chemistry graduate students and faculty were studied within the domain of problems that involved the determination of the structure of a molecule from the molecular formula of the compound and a combination of IR and ¹H NMR spectra. The participants' performance on these tasks was compared across variables that included amount of research experience, year of graduate study, and level of problem‐solving confidence. Thirteen of the 15 participants could be classified as either “more successful” or “less successful.” The participants in this study who were “more successful” adopted consistent approaches to solving the problems; were more likely to draw molecular fragments obtained during intermediate stages in the problem‐solving process; were better at mining the spectral data; and were more likely to check their final answer against the spectra upon which the answer was based. Experience from research, teaching, and course work were found to be important factors influencing the level of participants' success. © 2009 Wiley Periodicals, Inc. J Res Sci Teach 47:643–660, 2010     

The role of the working memory and language skills in the prediction of word problem solving in 4- to 7-year-old children

The aim of this study was to analyse the role of verbal and visuo-spatial working memory (WM) and language skills (vocabulary, listening comprehension) in predicting preschool and kindergarten-aged children’s ability to solve mathematical word problems presented orally. The participants were 116 Finnish-speaking children aged 4–7?years. The results showed that verbal WM (VWM) did not have a direct effect on word problems in young children but was indirectly related to word problems through vocabulary and listening comprehension. These results suggest that in young children, VWM resources support language skills which, furthermore, contribute to variation in solving orally presented word problems. The results also showed that visuo-spatial WM had a direct effect on performance in word problems, suggesting that it plays an important role in word problem solving among this age group.     

Problem solving in mathematics: the significance of visualisation and related working memory

The importance of educational psychologists’ (EPs’) skills to the formulation of evidence-based educational strategies, as well as in response to learning difficulties, is demonstrated here in relation to mathematical problem solving. Initiatives to improve the link between mathematical skills from school to everyday life have drawn significantly on problem solving tasks. Through critical evaluation of research, the relevance of visualisation methods and working memory to problem solving is considered within this article. Studies suggest that differences exist in the effectiveness of particular visualisation methods, but that training can improve their utility and thereby problem solving performance. Additionally, differences between individuals and contexts can influence visualisation use, and some pupils experience specific difficulties in this area. Recommendations are made to enhance the teaching of visualisation as a strategy for problem solving, and to support those pupils with specific difficulties.     

Working memory and mathematics in primary school children: A meta-analysis

Working memory, including central executive functions (inhibition, shifting and updating) are factors thought to play a central role in mathematical skill development. However, results reported with regard to the associations between mathematics and working memory components are inconsistent. The aim of this meta-analysis is twofold: to investigate the strength of this relation, and to establish whether the variation in the association is caused by tests, sample characteristics and study and other methodological characteristics. Results indicate that all working memory components are associated with mathematical performance, with the highest correlation between mathematics and verbal updating. Variation in the strength of the associations can consistently be explained by the type of mathematics measure used: general tests yield stronger correlations than more specific tests. Furthermore, characteristics of working memory measures, age and sample explain variance in correlations in some analyses. Interpretations of the contribution of moderator variables to various models are discussed.     

The relationship between math anxiety and math achievement in young children is mediated through working memory,not by number sense,and it is not direct

Math anxiety is considered a predictor of math achievement, although the cognitive mechanism whereby math anxiety impairs math achievement is unclear. The paper presents the results of cross-sectional (N = 241) and longitudinal (N = 369) studies conducted among early school-aged children on the cognitive mechanism whereby math anxiety impairs math achievement. The following hypotheses were tested: (1) math anxiety directly affects math achievement; (2) in accordance with processing efficiency and attentional cognitive theories, math anxiety indirectly affects math achievement through working memory; (3) in accordance with the cognitive deficit model, math anxiety indirectly affects math achievement through number sense. The results mostly confirm the mediating role of working memory and undermine the mediating role of number sense and the direct path in the relationship between math anxiety and math achievement. Because previous studies undertaken in adults show the direct path from math anxiety to math achievement and the role of symbolic number processing in explaining the relationship between the two, the methodological and developmental aspects of the obtained results are discussed in the paper.     

Inhibitory control and visuospatial working memory contribute to 5-year-old children's use of quantitative inversion

This study examined the hypothesis that general cognitive resources moderated 5-year-old children's performance differences between the Concrete Identical and the Pure Quantity conditions on inversion problems (a + b – b) but not on standard problems (a + b – c). Study 1 (N = 104) showed that children who experienced higher visuospatial working memory burden performed significantly poorer in solving the inversion problems in the Pure Quantity condition than in the Concrete Identical condition, whereas those who experienced lower working memory burden showed no such difference. Study 2 (N = 194) demonstrated that children with lower levels of inhibitory control solved significantly fewer inversion problems in the Pure Quantity condition than in the Concrete Identical condition, whereas no such difference was found in children with higher levels of inhibitory control. These findings suggest that inhibitory control and visuospatial working memory may support children's use of quantitative inversion.     

The role of phonological memory,word recognition,and comprehension skills in reading development: from preschool to grade 2

We examined the relationships among phonologicalawareness, phonological memory, and development ofreading skills in a longitudinal study, by following222 Finnish preschoolers through the grade 2.The main focus was on the role of phonological memoryin word recognition and comprehension. The skillsassessed were verbal abilities, phonological memory,phonological awareness, word recognition, listeningand reading comprehension, altogether comprising themost extensive set of variables so far used in thestudy of phonological memory and reading. We proposeda structural equation model for the developmentalrelationships among the variables. This model waslargely confirmed by the data. The most significantpredictor of word recognition was phonologicalawareness. Phonological memory had only a weak effecton phonological awareness at preschool age, andvia this connection, a weak indirect effect on grade 1 word recognition. Contrary toexpectations, phonological memory also had asignificant, albeit weak effect on grade 2word recognition. Phonological memory did notdirectly affect reading comprehension. However,it was strongly related to listeningcomprehension at preschool, and via the strongeffects of both listening comprehension and wordrecognition on reading comprehension, there weresignificant indirect effects of phonological memory onreading comprehension. The results also underline thestability of development of phonological memory, wordrecognition, and comprehension from preschool to theend of grade 2.     

Individual differences in children's writing: A function of working memory or reading or both processes?

The purpose of this study was to investigate whether a general or specific working memory (WM) system is related to writing and whether individual differences in reading and/or processing efficiency underlie the correlations between WM and writing. Two studies correlated WM with writing (Test of Written Language-TOWL) and reading measures. In Study 1, WM was correlated significantly with a number of writing measures, particularly to those measures related to text generation. Working-memory also contributed unique variance to writing, beyond what is predicted by reading comprehension. Study 2 compared the correlations of verbal and visual-spatial WM measures with the TOWL under initial and enhanced memory processing (dynamic assessment) conditions. The coefficients were statistically comparable between initial and enhanced processing conditions, suggesting that individual differences in processing efficiency do not account for the correlations between WM and writing. Overall, the results indicated that (a) WM measures contribute unique variance to writing, especially text generation, and (b) working memory performance improves under gain conditions, but this enhanced processing efficiency did not appear to mediate the links between WM and writing. Taken together, the two studies support a general capacity explanation for the relationship between working memory and text generation.     

Fostering analogical transfer: The multiple components approach to algebra word problem solving in a chemistry context

Holyoak and Koh (1987) and Holyoak (1984) propose four critical tasks for analogical transfer to occur in problem solving. A study was conducted to test this hypothesis by comparing a multiple components (MC) approach against worked examples (WE) in helping students to solve algebra word problems in chemistry classes. The MC approach incorporated multiple components (symbolic equations, symbols, categorization, hint) in the source, or target, or both, to address the four analogical tasks. Different combinations of the components were tested in a series of four experiments. Symbolic equations (main component) fostered a mental construction of the problem in its solution mode. Categorization enabled an identification of the problem category. A hint in the target directed the learners to the source problem. The interaction between these components facilitated the mapping of the symbolic equations in the source onto the target, resulting in the superiority of the MC approach in fostering analogical transfer. Neither the main component alone nor the main component plus one sub-component was sufficient for analogical transfer. Hence for analogical transfer to occur, at least the main component (symbolic equations) and two sub-components (categorization and hint) are required. However, symbols may not have additional effects for transfer to occur.     

Examining the unique contributions and developmental stability of individual forms of relational reasoning to mathematical problem solving

Relational reasoning, a higher-order cognitive ability that identifies meaningful patterns among information streams, has been suggested to underlie STEM development. This study attempted to explore the potentially unique contributions of four forms of relational reasoning (i.e., analogy, anomaly, antinomy, and antithesis) to mathematical problem solving. Two separate samples, fifth graders (n = 254) and ninth graders (n = 198), were assessed on their mathematical problem solving ability and the different forms of relational reasoning ability. Linear regression analysis was conducted, with participants’ age, working memory, and spatial skills as covariates. The results showed that analogical and antithetical reasoning abilities uniquely predicted mathematical problem solving. This pattern demonstrated developmental stability across a four-year time frame. The findings clarify the unique significance of individual forms of relational reasoning to mathematical problem solving and call for a shift of research direction to reasoning abilities when exploring dissimilarity-based relations (opposites in particular).     

Middle school children's strategic behavior: Classification and relation to academic achievement and mathematical problem solving

Theories of problem solving (e.g., Verschaffelet al., 2000) hold strategic behavior centralto processing mathematical word problems. Thepresent study explores 80 sixth- andseventh-grade students' self-reported use of 14categories of strategies (Zimmerman &Martinez-Pons, 1986) and the relationship ofstrategy use to academic achievement,problem-solving behaviors, and problem-solvingsuccess. High and low achievement groupsdiffered in the number of different strategiesand categories of strategies reported but notin overall number of strategies, confidence inusing strategies, or frequency of strategy use.Students whose behaviors evidenced elaborationof the word problem's text reported moreself-evaluation; organizing and transforming;and goal setting and monitoring behavior.Implications for instructional practices thatsupport active stances toward problem solvingare discussed.     

The mathematical discourse of 13-year-old partnered problem solving and its relation to the mathematics that emerges

This paper, written within a discursive perspective, explores the co-shaping of public and private discourse, and some of the circumstances under which one occasions the other, in the evolution of mathematical thinking by pairs of 13-year-olds. The discourse of six pairs of students, engaged in interpreting and graphing problem situations involving rational functions, was analyzed by means of recently developed methodological tools. The nature of the mathematics that emerged for each pair was found to be related to several factors that included the characteristics of the interpersonal object-level utterances both before and after the solution path had been generated, the degree of activity of the personal channels of the interlocutors, and the extent to which the thoughts of participants were made explicit in the public discourse. The analysis of the discursive interactions provided evidence that adolescents within novel problem situations can experience some difficulty in making their emergent thinking available to their partners in such a way that the interaction be highly mathematically productive for both of them. This revised version was published online in July 2006 with corrections to the Cover Date.