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.     

The influence of self-efficacy and working memory capacity on problem-solving efficiency

We investigated the influence of self-efficacy beliefs and working memory capacity on mathematical problem-solving performance, response time, and efficiency (i.e., the ratio of problems solved correctly to time). Students completed a letter-recoding task (Experiment 1) or an operation span task (Experiment 2), rated their self-efficacy for solving mental multiplication problems, and then solved similar problems of varying complexity. We tested the motivational efficiency hypothesis, which predicted that motivational beliefs, such as self-efficacy, increase problem-solving efficiency through focused effort and strategy use. Experiments 1 and 2 reported a significant effect for self-efficacy on problem-solving performance and efficiency, but limited effects for time. A self-efficacy by working memory interaction occurred in Experiment 1, suggesting self-efficacy is beneficial as demands on working memory increase. These findings suggested that self-efficacy increased problem-solving efficiency through strategic performance rather than faster solution times, and were consistent with the motivational efficiency hypothesis.     

Improving children's working memory and classroom performance

Previous research has demonstrated close relationships between working memory and children's scholastic attainment. The aim of the present study was to explore a method of improving working memory, using memory strategy training. Two hundred and fifty‐four children aged five to eight years were tested on measures of the phonological loop, visuo‐spatial sketchpad and central executive components of the multiple component model of working memory. Subgroups of children also completed tasks of following instructions and mental arithmetic in the classroom, and standardised tests of reading, arithmetic and mathematics. Half of the children then used Memory Booster, a computer game that teaches memory strategies, over a period of six to eight weeks. All the children were then retested on the memory and ability measures. The standardised tests were also administered five months later. The results revealed that working memory strategy training resulted in significant improvements in tasks assessing the phonological loop and central executive components of working memory, and tasks assessing following instructions and mental arithmetic in the classroom. However, no improvements were observed on standardised tests of reading, arithmetic or mathematics, either immediately following training or five months later. The results are discussed in terms of implications for educational practice.     

谈工作记忆与类比推理

类比推理是一种特殊的推理形式,也是个体抽象思维的一种主要形式。有些心理学家着重研究与类比推理有关的问题,以期可以找到发展类比推理能力的有效策略,以提高学生学习新知识、新技能等的效率。经过大量研究与调查数据显示,工作记忆的各个子成分与类比推理之间有着密切的联系,工作记忆直接影响着个体类比推理能力的发展。因此,文章主要探讨工作记忆与类比推理的含义,以及工作记忆对类比推理的影响,希望能够为相关研究者提供参考。     

Additive and multiplicative effects of working memory and test anxiety on mathematics performance in grade 3 students

The aim of this study was to investigate the interplay between test anxiety and working memory (WM) on mathematics performance in younger children. A sample of 624 grade 3 students completed a test battery consisting of a test anxiety scale, WM tasks and the Swedish national examination in mathematics for grade 3. The main effects of test anxiety and WM, and the two-way interaction between test anxiety and WM on mathematics performance, were modelled with structural equation modelling techniques. Additionally, the effects were also tested separately on tasks with high WM demands (mathematical problem-solving) versus low WM demands (basic arithmetic). As expected, WM positively predicted mathematics performance in all three models (overall mathematics performance, problem-solving tasks, and basic arithmetic). Test anxiety had a negative effect on problem-solving on the whole sample level but concerning basic arithmetic only students with lower WM were affected by the negative effects of test anxiety on performance. Thus, students with low WM are more vulnerable to the negative effects of test anxiety in low WM tasks like basic arithmetic. The results are discussed in relation to the early identification of test anxiety.     

言语工作记忆在双语加工中的作用

本文探讨了言语工作记忆在双语者第一语言和第二语言学习中发挥的不同作用,尤其在两种语言的阅读理解、语法理解、口语水平、词汇密度和语言成绩等方面,并推论它在形式不同的语言加工中具有特殊性和普适性。     

中学数学解题思维错误的成因及对策

正确处理学生的解题错误,不仅能提高学生解题能力,同时也从某种程度上反映教师的数学素养.这里从错误的特性,生成因素,以及对应策略三方面,并结合实际题例,进行简单的定性分析.     

Visuospatial working memory in adolescents with poor performance in mathematics: variation depending on reading skills

The first purpose of this study was to investigate whether the visuospatial working memory (VSWM) skills of 15–16‐year‐old pupils with difficulties in mathematics differ from those of their normally achieving peers. The goal was to broaden the view of the complex system of VSWM. A set of passive and active VSWM tasks was used. The study’s second purpose was to investigate whether pupils with mathematical difficulties differed in their VSWM skills based on whether they had signs of reading deficits or not. Results indicate that the pupils with poor performance in maths showed poorer performance on certain VSWM tasks. The group with deficits only in maths had less capacity for storing passive visual simultaneous information, while the group with difficulties both in maths and reading had deficits in both storing (passive visual and visuospatial information) and processing, and had less ability to control irrelevant visuospatial information compared to their peers of the same age. The results indicate a general VSWM deficit in pupils with both mathematics and reading problems and a specific VSWM deficit in pupils with only mathematics problems.     

Taking working memory training from the laboratory into schools

Working memory skills have been shown to be enhanced by adaptive training in several randomised controlled trials. Here, two field trials were conducted in which teachers administered working memory training to their own pupils in school. Twenty-two children aged 8–9?years participated in Trial 1. In Trial 2, 50 children aged 9–11?years with the lowest academic performance completed training. They were matched with a group of 50 children who were not trained. Following training, children in Trial 1 improved significantly in both trained and untrained working memory tasks, with effect sizes comparable to those reported in research studies. Improvements on the trained tasks in Trial 2 were comparable, and training was associated with significantly greater progress at school across the academic year in maths and English. These findings indicate that teacher-administered training leads to generalised and robust gains in working memory and educationally significant gains in academic performance.     

Phonological awareness,verbal working memory and the acquisition of literacy

This paper describes a 2-year longitudinal study of 76 initially prereading children. The study examined the relationships between phonological awareness (measured by tests of onset and rime, phonemic segmentation and phoneme deletion), verbal working memory and the development of reading and spelling. Factor analyses showed that the verbal working memory tests which were administered loaded on two distinct but highly related factors, the first of which,simple repetition, involved the repetition of verbal items exactly as spoken by the experimenter, whereas the second,backwards repetition, involved repetition of items in reverse order. Factor analyses also showed that, whist the phonological awareness variables consistently loaded on the backwards repetition factor at the beginning and end of Grade 1, by Grade 2 the phonological awareness variables loaded on a separate factor which also included sentence repetition. Results of multiple regression analyses, with reading and spelling as a compound criterion variable, indicated that phonological awareness consistently predicted later reading and spelling even when both simple and backwards repetition were controlled. In contrast, verbal working memory did not consistently predict reading and spelling across testing times. Whilst there was some indication that verbal working memory, especially backwards repetition, measured during Grade 1 did predict reading and spelling in Grade 2, these effects were no longer evident when all three phonological variables were controlled. Nevertheless, with 4 individual reading and 2 individual spelling measures as the criterion variables, it was shown that phonological awareness was not quite such a consistent predictor of reading and spelling: it was most highly related to reading pseudowords and spelling real words; but it was not so highly related to spelling pseudowords, apparently because the processing demands of the task for the young children in the study were extremely high. Given the importance of verbal working memory for the completion of phonological awareness, reading and spelling tasks, in particular for spelling pseudowords, the findings are interpreted as providing some support for a theoretical position which posits that both phonological awareness and verbal working memory contribute to the early stages of literacy acquisition. Whilst the findings suggest some support for a general underlying phonological ability, there is also evidence that, as children learn to read and write, verbal working memory and phonological awareness become more differentiated.     

加拿大的问题解决教学

本文介绍了问题解决教学的指导思想,及加拿大问题解决教学的有关情况,如师资培训、教材、问题解决的原则,及问题解决课堂教学过程,可供参考.     

二维客体在视觉工作记忆中的存储单位

本实验对由不同维度(颜色和形状)的不同特征构成的客体进行探测,以正确率和反应时为指标,探索视觉工作记忆存储的基本单位是客体还是特征。实验1结果显示特征数显著影响了被试的记忆成绩。实验2结果显示客体数对被试的记忆成绩没有显著的影响。最终实验得出记忆由不同维度的特征组成的客体时,视觉工作记忆存储的基本单位是特征。     

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.     

The effects of instructional context and reasoning complexity on mathematics problem-solving

The focus of this study was on the effects of relevance in instructional context and reasoning complexity on mathematics problem-solving achievement, transfer, and attitude. Forty-six fifth graders participated in the five-day study. Students received either contextualized or decontextualized instruction involving either simple or complex reasoning. Two types of achievement questions were used, context-rich and context-poor, which required either single or multiple computational steps to solve. An interaction was found between complexity of treatment and complexity of questions. Students who studied simple problems in decontextualized contexts performed best on one-step questions, while students who studied complex problems in contextualized contexts performed best on multi-step questions (p = .003). A complexity effect was also found for student attitudes toward perceived lesson difficulty (p = .0001) as well as for perceived relevance of mathematics (p = .015). Students who studied simple problems perceived lesson difficulty and mathematics relevance more favorably than those who studied complex problems. These findings suggest that rich mathematical instructional contexts best support mathematics problem-solving, but simple, decontextualized instruction yields the most favorable attitudes.     

The contribution of domain-specific knowledge in predicting students' proportional word problem-solving performance

This study explored the extent to which domain-specific knowledge predicted proportional word problem-solving performance. We tested 411 seventh-grade students on conceptual and procedural fraction knowledge, conceptual and procedural proportion knowledge, and proportional word problem solving. Multiple regression analyses indicated that all four domain-specific knowledge variables (i.e., conceptual and procedural fraction knowledge, conceptual and procedural proportion knowledge) significantly predicted proportional word problem-solving performance. Conceptual fraction and procedural proportion knowledge contributed the most unique variance (10.0 and 6.7%, respectively, of the total variance) to proportional word problem solving. Procedural fraction and conceptual proportion knowledge each also contributed significant unique variance to proportional word problem solving explaining 5.6 and 2.8%, respectively. The results support the notion that both conceptual fraction and proportion knowledge and procedural fraction and proportion knowledge play a major role in understanding individual differences in proportional word problem-solving performance to inform interventions.     

Involvement of working memory in mental multiplication in Chinese elementary students

The authors' aim was to examine the relation between two-digit mental multiplication and working memory. In Study 1, involving 30 fifth-grade students, we used digit span backward as an abbreviated measure of working memory. In Study 2, involving 41 fourth-grade students, working memory comprised measures of phonological loop, visuospatial sketchpad, and central executive (i.e., updating) based on A. Miyake et al. (2000 Miyake, A., Friedman, N. P., Emerson, M. J., Witzki, A. H., Howerter, A., &; Wager, T. D. (2000). The unity and diversity of executive functions and their contributions to complex “frontal lobe” tasks: A latent variable analysis. Cognitive Psychology, 41, 49–100.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). In Study 1, working memory measured through the use of digit span backward explained unique variance in response time. In Study 2, participants' accuracy and response time were most susceptible to phonological loop influence. The findings support the argument that multiplication facts are stored in a verbal form and the retrieval of multiplication facts relies more on verbal modality. Unique features of Chinese mathematics instruction of mental multiplication were discussed. The findings suggested that the involvement of different subcomponents of working memory in mental arithmetic may be subject to instructional, contextual, and linguistic factors.     

Deaf children and children with ADHD in the inclusive classroom: working memory matters

This study compared the working memory functioning of deaf children, children with ADHD and typically developing children. Working memory is involved in the storage and mental manipulation of information during classroom learning activities that are crucial for the acquisition of complex skills and knowledge. Thus, it is important to determine how these groups compare in this regard as this has implications for teaching them together in an inclusive classroom. Simple and complex visuo-spatial and verbal working memory were assessed in 24 children with ADHD and 20 control children with no diagnosed ADHD, to determine whether any differences existed between these groups. A second comparison occurred between the simple and complex visuo-spatial working memory of the latter two groups, as well as 24 deaf children, all matched on age, gender and home language. The control group scored significantly higher than the deaf children and the children with ADHD on all components of simple and complex working memory. The implications of this finding are that children with ADHD and deaf children may share similar working memory profiles, making it easier to accommodate both sets of children together in the inclusive classroom. Suggestions are given for how educators could assist these children by reducing the demands on working memory.     

A working memory model applied to mathematical word problem solving

The main objective of this study is (a) to explore the relationship among cognitive style (field dependence/independence), working memory, and mathematics anxiety and (b) to examine their effects on students’ mathematics problem solving. A sample of 161 school girls (13–14 years old) were tested on (1) the Witkin’s cognitive style (Group Embedded Figure Test) and (2) Digit Span Backwards Test, with two mathematics exams. Results obtained indicate that the effect of field dependency, working memory, and mathematics anxiety on students' mathematical word problem solving was significant. Moreover, the correlation among working memory capacity, cognitive style, and students’ mathematics anxiety was significant. Overall, these findings could help to provide some practical implications for adapting problem solving skills and effective teaching/learning.     

Common features of expertise in working life: implications for higher education

ABSTRACT

We need to know what it means to be an expert in working life today. Universities are often accused of neglecting the basic idea that higher education should be relevant to working life, and research on the subject of expertise in today’s workplace is lacking. Thirteen experts from different fields were interviewed and the obtained data were analysed using grounded theory as an analytical approach. The research questions were: (1) How do experts define expertise? (2) What kind of problems persist in their work? and (3) How are ‘routine’ and ‘adaptive’ expertise performed? The results revealed that expertise is more a social and collaborative phenomenon than an individual property. Experts develop new solutions and seek constant learning in their work. They excel at spontaneous problem solving. These findings suggest that, to prepare students to become experts, deliberate learning and practice should be provided within a framework of collaborative problem solving.     

Phonological working memory, phonological awareness and language in literacy difficulties in Brazilian children

One of the most usual flaws that lead to literacy disability regards cognitive difficulties and alterations some children present in the literacy process. Many studies have found alterations in phonological processing, more specifically in phonological working memory (PWM) and phonological awareness (PA). Therefore, our aim was to identify alterations in skills of phonological working memory, phonological awareness and language (semantic, syntactic and phonological aspects) of Brazilian children with literacy disabilities (age 7–8). They were divided into two groups: (1) Group of Normal Literacy (NL); and (2) Group With Literacy Disabilities (LD). The evaluation comprised tests that assessed general cognitive functions and the skills at stake. The LD presented a poorer performance than the NL in the skills of PWM, PA and language aspects. The factor analysis showed that these skills prevailed in differentiating the groups. Thus, children with literacy disabilities presented deficits in phonological processing and language. These deficits seem to be a result of alterations of the phonological representations and poor language skills that are prior to the literacy period. Consequently, we believe that the early identification of these alterations can be very useful for the prevention of future literacy problems.