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.     

“I think I can,but I'm afraid to try”: The role of self-efficacy beliefs and mathematics anxiety in mathematics problem-solving efficiency

This study investigated the role of self-efficacy beliefs, mathematics anxiety, and working memory capacity in problem-solving accuracy, response time, and efficiency (the ratio of problem-solving accuracy to response time). Pre-service teachers completed a mathematics anxiety inventory measuring cognitive and affective dispositions for mathematics, before completing an operation span task to measure working memory capacity, rating self-efficacy for mental multiplication, and then solved computer-based multiplication problems at two complexity levels. A simultaneous regression design was used to assess the unique variance associated with each variable. There were two new findings; the differential role of self-efficacy on response time and efficiency, and the potential compensatory relationship between self-efficacy and mathematics anxiety related to efficiency outcomes. Educational implications and suggestions for future research were proposed.     

An investigation of undergraduates' transformational problem solving strategies: cognitive/metacognitive processes as predictors of holistic/analytic strategies

Recent years have seen increasing interest in the role of metacognition in mathematical problem solving. The study described in this paper explored problem solving strategies used by undergraduates. Furthermore, cognitive/metacognitive processes are predicted each of holistic and analytic strategies. Educational sciences students (n=178) were asked to talk/think aloud while solving two constructed response transformational problems. The protocols were transcribed and analysed, revealing that the cohort used nine strategies. The results showed that a holistic strategy was predicted by five cognitive/metacognitive processes, two of which were suppressors; whilst an analytic strategy was predicted by four cognitive/metacognitive processes, three of which were suppressors.     

问题解决策略的认知和元认知研究

介绍了国内外学者近年来对问题解决策略的认知和元认知研究的最新成果:Alessandro Antonietti和梁宁建等人所做的大学生对一般问题解决策略的元认知的调查,Daniela Lueangeli等人所做的算术应用题解决中的认知和元认知能力的研究.     

The effects of cognitive and metacognitive strategy instruction on the mathematical problem solving of middle school students with learning disabilities.

This study investigated the effects of cognitive and metacognitive strategy instruction on the mathematical problem solving of six middle school students with learning disabilities. Conditions of the multiple baseline, across-subjects design included baseline, two levels of treatment, setting and temporal generalization, and retraining. For Treatment 1, subjects received either cognitive or metacognitive strategy instruction. Treatment 2 consisted of instruction in the complementary component of the instructional program so that all subjects eventually received both cognitive and metacognitive strategy instruction. This design allowed a componential analysis of the content as well as sequence of instruction. Generally, subjects improved their mathematical problem solving as measured by performance on one-, two-, and three-step word problems. Discussion focused on effectiveness of treatment, acquisition and application of strategic knowledge, error pattern analysis, and the need to tailor instruction to the learner's individual characteristics.     

Cognitive,metacognitive, and motivational aspects of problem solving

This article examines the role of cognitive, metacognitive, and motivational skills in problem solving. Cognitive skills include instructional objectives, components in a learning hierarchy, and components in information processing. Metacognitive skills include strategies for reading comprehension, writing, and mathematics. Motivational skills include motivation based on interest, self-efficacy, and attributions. All three kinds of skills are required for successful problem solving in academic settings.     

Do it this way! Metacognitive strategies in collaborative mathematical problem solving

Recent years have seen increasing interest in the role of metacognition in mathematical problem solving, and in the use of small group work in classroom settings. However, little is known about the nature of secondary students' metacognitive strategy use, and how these strategies are applied when students work together on problems. The study described in this paper investigated the monitoring behaviour of a pair of senior secondary school students as they worked collaboratively on problems in applied mathematics. Analysis of verbal protocols from think aloud problem solving sessions showed that, although the students generally benefited from adopting complementary metacognitive roles, unhelpful social interactions sometimes impeded progress. The findings shed some light on the nature of individual and interactive metacognitive strategy use during collaborative activity.     

The Conditional and Interaction Effects of Epistemological Beliefs on the Self-Regulated Learning of College Students: Motivational Strategies

This study examines the conditional and interaction effects of each of four dimensions of the epistemological beliefs of college students regarding the ability to learn, the speed of learning, the structure of knowledge, and the stability of knowledge on six measures of the motivational components of self-regulated learning strategies (intrinsic goal orientations, extrinisic goal orientation, task value, self-efficacy, control of learning and test anxiety). Students with more sophisticated beliefs about the nature of knowledge and learning were more likely than their peers to use educationally productive motivational strategies in their learning. Beliefs about one’s ability to learn and the structure of knowledge had the most significant and substantial effects on students’ use of self-regulated motivational strategies. Although a student’s belief about the stability of knowledge by itself had a statistically significant effect on only one motivational strategy, this belief did have four statistically significant interaction effects with beliefs about ability to learn and the structure of knowledge. Implications of these findings for theory, research, policy and practice are examined.     

he Challenges of Teaching Good Information Processing to Learning Disabled Students

A MODEL of good information processing is sketched, describing how metacognitive knowledge influences strategy selection and use. Three factors pose particular problems for learning disabled students as they attempt to acquire metacognitive knowledge and to use study strategies productively: neurological impairments; deficiencies in general world knowledge; and negative beliefs, attitudes, and styles that limit self‐efficacy. Creating an educational atmosphere that explicitly builds conceptual (domain‐specific) knowledge and teaches positive beliefs about learning potential is essential in promoting metacognitively‐oriented instruction.     

Metacognitive skills in Belgian third grade children (age 8 to 9) with and without mathematical learning disabilities

This paper presents a study on mathematical problem solving in third-grade pupils. The relationship between mathematics, metacognition and intelligence was investigated in children with (n = 191) and without mathematical learning disabilities (n = 268). A significant relationship was found between prediction, evaluation, intelligence, procedural and mathematical fact retrieval skills in children without mathematical learning disabilities. In the children with mathematical learning disabilities a relationship was found between metacognitive and procedural skills. No such relationship was found between intelligence and metacognition or between metacognition and mathematical fact retrieval skills. In addition it was investigated if children with mathematical learning disabilities had less adequate metacognitive skills than peers without learning problems. At group level significant differences were found between both groups. However on analyzing these results further, it was found that four out of five children with combined mathematical learning disabilities, half of the children with procedural disabilities and only 5% of the children with a retrieval deficiency had low metacognitive skills. Furthermore, metacognitive problems were found in one out of five children without learning disabilities. Moreover, a majority of the children with mathematical learning disabilities and inadequate metacognitive skills had problems with prediction and evaluation skills. Most third graders with low metacognitive skills only appeared to have problems predicting the level of difficulty of tasks. Inaccurate evaluations were found on a more regular basis in children with mathematical learning disabilities and inadequate metacognitive skills as opposed to the sample of children with inadequate metacognitive skills but without learning difficulties, where their occurrence was rather a one off. The implications of this study for diagnosis and treatment will be discussed later in this paper.     

Accuracy of metacognitive judgments as a moderator of learner control effectiveness in problem-solving tasks

A possible explanation for why students do not benefit from learner-controlled instruction is that they are not able to accurately monitor their own performance. The purpose of this study was to investigate whether and how the accuracy of metacognitive judgments made during training moderates the effect of learner control on performance when solving genetics tasks. Eighty-six undergraduate students solved self-selected genetics tasks using either a full learner control or a restricted learner control. Results indicated that learner control effectiveness was moderated by the absolute accuracy (i.e., absolute bias) of metacognitive judgments, and this accuracy was a better predictor of learning performance for full learner control than for restricted learner control. Furthermore, students’ prior knowledge predicted absolute accuracy of both ease-of-learning judgments (EOLs) and retrospective confidence judgments (RCJs) during training, with higher prior knowledge resulting in a better absolute accuracy. Overall, monitoring guided control, that is, EOLs predicted time-on-task and invested mental effort regardless of the degree of learner control, whereas RCJs predicted the total training time, but not the number of tasks selected during training. These results suggest that monitoring accuracy plays an important role in effective regulation of learning from problem-solving tasks, and provide further evidence that metacognitive judgments affect study time allocation in problem solving context.     

Approach to mathematical problem solving and students’ belief systems: two case studies

The goal of the study reported here is to gain a better understanding of the role of belief systems in the approach phase to mathematical problem solving. Two students of high academic performance were selected based on a previous exploratory study of 61 students 12–13 years old. In this study we identified different types of approaches to problems that determine the behavior of students in the problem-solving process. The research found two aspects that explain the students’ approaches to problem solving: (1) the presence of a dualistic belief system originating in the student’s school experience; and (2) motivation linked to beliefs regarding the difficulty of the task. Our results indicate that there is a complex relationship between students’ belief systems and approaches to problem solving, if we consider a wide variety of beliefs about the nature of mathematics and problem solving and motivational beliefs, but that it is not possible to establish relationships of causality between specific beliefs and problem-solving activity (or vice versa).     

Applying mathematics with real world connections: metacognitive characteristics of secondary students

This paper reports an intensive study of problem-solving activity of female students at the senior secondary level. The study focussed both on the mathematical processing and the underlying cognitive and metacognitive activities that led to that processing. Response maps were used to analyse and categorize the written responses from individual students while videotaped problem solving sessions and structured and free response interviews probed the students' metacognitive knowledge, strategies, decision making, beliefs and affects. Metacognitive activities were involved in all phases of the solution process with key points in students' solutions identifiable in terms of the cognitive-metacognitive framework of Garofalo and Lester (1985). On average more time was spent on orientation and execution activities with little time being spent on organisation and verification activities, however, the successful groups spent less time on orientation than the other groups. All successful groups displayed a high number of key points where metacognitive decisions could influence cognitive action. Success was accompanied by a tendency to engage in a high number of organisational activities, regulation of execution activities and evaluation activities particularly evaluation of execution but fewer opportunities where metacognitive decisions could influence cognitive actions during orientation.     

Strategic development of multiplication problem solving: Patterns of students' strategy choices

Low mathematics achievement is a persistent problem in the United States, and multiplication is a fundamental area in which many students manifest learning difficulties. This study examined the strategic developmental levels of multiplication problem solving among 121 elementary school students in Grades 3 through 5. A latent class analysis modeling was used to identify three valid groups representing different patterns of strategy choices for each of three types of multiplication problems. Findings indicated intra-group variability for problem-solving accuracy, for frequency of using different strategies, and for accuracy of executing direct retrieval/algorithm (DR/AG) strategies. Students demonstrated relative consistency in their strategy choices for solving the three problem types. Students who used DR/AG strategies most frequently showed the highest problem-solving accuracy and the highest accuracy of executing the DR/AG strategies. Students who most frequently relied on incorrect operations or who indicated they did not know how to solve problems demonstrated the lowest problem-solving accuracy among the three groups; the number of students in this group increased with problem difficulty levels. Implications are discussed in terms of identifying students' strategic developmental levels and providing differentiated instruction based on the identified levels.     

Motives,beliefs, and perceptions among learners affect preparatory learning strategies

Preparation is an effective and necessary activity; however, most students do not prepare for future lessons. The present study addressed this problem and examined how learners' motives, beliefs, and perceptions affected their strategy use during preparation for future lessons. Participants were 219 Japanese junior high school students who responded to a questionnaire about mathematics learning. The result of a path analysis suggested that learners' intrinsic motives, extrinsic motives, and cognitive beliefs about learning positively related to their spontaneously obtaining prior knowledge and solving example problems. In addition, noncognitive beliefs positively affected perceived cost of preparation and decreased obtaining prior knowledge. Implications for educational practice, limitations of the present study, and suggestions for future research are discussed.     

The Relationship between Motivational Beliefs and Learning Strategy Use among Norwegian College Students

We studied how motivational beliefs were related to learning strategy use in 176 Norwegian college students who were in the second year of their teacher training. Students' implicit theories of intelligence, self-efficacy beliefs, and learning strategy use were assessed by self-report instruments. It was found that students who conceived of intelligence as a relatively modifiable quality reported using more strategies than students who had doubts about the modifiability of intelligence. However, the relation between students' theories of intelligence and their learning strategy use varied with the way their theories of intelligence were assessed, with only indirect questions about the modifiability of intelligence yielding a positive relation. Regression analysis and group comparisons suggested that beliefs in the modifiability of intelligence may override the contribution of self-efficacy to students' use of learning strategies. With this study, relations previously emphasized within American theory and research are extended to college students in a different cultural context.     

The role of metacognition in the context of integrated strategy intervention

In the present study, the role of metacognition in the context of integrated strategy intervention was examined. The integrated strategy training in reading comprehension, mathematics, and metacognition was carried out over a period of seven months with young 9–10-years-old, 3rd grade elementary school students with learning problems. A pretest-post test design with matched intervention and control groups was applied. Before the intervention, these students and all their classmates had been followed up through 1st to 3rd grade, and tested with multiple cognitive and metacognitive tasks each year. The results showed that early, 1st grade, cognitive-metacognitive differences were strongly associated with later problem solving and reading comprehension proficiency, thus confirming the importance of reading comprehension strategies and metacognition in mathematical problem solving. Further, marked training effects were found. The growth of metacognitive awareness, experiences and self-regulation were observed on the basis of behaviour analyses during the intervention and of post-intervention interviews. However, despite significant training effects, students who were resistant and those who were responsive to training were identified. The results showed that early metacognitive proficiency is closely associated with the responsiveness to training efforts. The nature of metacognitive experiences and the early teaching of metacognitive awareness and self-regulation are emphasised in the conclusions.     

数学问题解决的实证研究述评

数学问题解决的心理学实证研究主要集中在数学应用问题、平面几何问题、解题中的迁移、解题中的元认知等方面。就目前的研究状况来看，存在研究选题面窄、研究层面较低、研究起点单一等问题。因而，开展深层次的研究，是数学解题心理研究的发展方向。     

The Relations Between Number Property Strategies,Working Memory,and Multiplication in Elementary Students

This study aimed to examine the relations among property strategies, working memory, and multiplication tasks with 101 Chinese fourth-grade students. Two multiplication property strategies (associative and distributive) were compared with no strategy and demonstrated differentiated effects on students’ accuracy and reaction time. Associative property strategy outperformed the distributive strategy and no strategy on the measures of accuracy and reaction time. The findings support the problem difficulty effect. The findings indicate that if a strategy does not necessarily simplify a problem, it might not help to enhance accuracy, suggesting that multiplication property strategies should be used with respect to the structural features of the problems. Among more difficult problems, visuospatial sketchpad consistently explained unique variances in problem accuracy.     

Metacognition in Mathematical Problem Solving

Metacognition is considered by most educationists as an element necessary for many cognitive tasks. In problem solving, it has been said that possessing knowledge alone is insufficient and problem solvers need to exhibit high level cognitive skills like “self-regulation skills” (also known as metacognitive strategies) for successful problem solving.

A study on students' metacognitive strategies was carried out with over a thousand secondary and pre-university students from 12 schools. A questionnaire adapted from Biggs (1987) was administered to students at various levels (Secondary 2, Secondary 4, Pre-University 1), from different academic tracks (General, Science, Arts) and academic streams (Special, Express, and Normal). They were required to self-report on their metacognitive beliefs; their use of metacognitive strategies in mental tasks involving memory, problem solving and comprehension; and their attitudes towards the learning of various academic subjects. 20 items from the questionnaire which were related to problem solving were categorized into four stages, namely, orientation, organisation, execution and verification and data from these items were analysed.

Some findings that emerged were:

• (a) Normal stream students exhibited a lower usage of metacognitive strategies as compared to students from the Express and Special streams.

• (b) Metacognitive strategies used by Normal stream students tended to be of the “surface” type.

• (c) There was no significant difference in the frequency of usage of metacognitive strategies between students from different academic tracks.

• (d) During the problem solving process, students spent most time on evaluation of answers rather than on monitoring their understanding.

• (e) Students from different levels (Secondary 2, Secondary 4 and Pre-University) exhibited similar frequency of usage of metacognitive strategies in problem solving.

• The implications of these findings on future research and development projects as well as the teaching of metacognitive strategies are discussed in the paper.