Diagrams and illustrations as aids to problem solving

The psychological literature relating to factors which should be taken into account in the design of illustrations and pictorial instructions as aids to problem solving is reviewed. The problem solver can invoke a wide range of strategies for using pictorial materials to achieve his goal, although pictorial instructions which indicate how to perform actions place a tighter constraint on the choice of strategy than illustrations showing only states to be attained. The designer must make his graphics compatible with as wide a range of strategies for using them as possible while building on the knowledge the problem solver already possesses.The author now works for the Post Office Telecommunications Systems Strategy Department, 88 Hills Road, Cambridge.     

The role of epistemic emotions in mathematics problem solving

The purpose of this research was to examine the antecedents and consequences of epistemic and activity emotions in the context of complex mathematics problem solving. Seventy-nine elementary students from the fifth grade participated. Students self-reported their perceptions of control and value specific to mathematics problem solving, and were given a complex mathematics problem to solve over a period of several days. At specific time intervals during problem solving, students reported their epistemic and activity emotions. To capture self-regulatory processes, students thought out loud as they solved the problem. Path analyses revealed that both perceived control and value served as important antecedents to the epistemic and activity emotions students experienced during problem solving. Epistemic and activity emotions also predicted the types of processing strategies students used across three phases of self-regulated learning during problem solving. Finally, shallow and deep processing cognitive and metacognitive strategies positively predicted problem-solving performance. Theoretical and educational implications are discussed.     

The role of model building in problem solving and conceptual change

This study examines the effects of the activity of building systems models for school-based problems on problem solving and on conceptual change in elementary science classes. During a unit on the water cycle in an Asian elementary school, students constructed systems models of the water cycle. We found that representing ill-structured problems as system models improves problem solving as well as creating beneficial conceptual change related to the topic.     

Common sense clarified: The role of intuitive knowledge in physics problem solving

Over the last two decades, a significant body of research has documented the nature of intuitive physics knowledge—the knowledge of the world that students bring to the learning of formal physics. However, this research has yet to document the roles played by intuitive physics knowledge in expert physics practice. In this article, I discuss three related questions: (1) What role, if any, does intuitive knowledge play in physics problem solving? (2) How does intuitive physics knowledge change in order to play that role, if at all? (3) When and how do these changes typically occur? In answer to these questions, I attempt to show that intuitive physics knowledge can play a variety of roles in expert problem solving, including some roles that are central and directly connected to equations. This research draws on observations of college students working in pairs to solve physics problems. © 2006 Wiley Periodicals, Inc. J Res Sci Teach 43: 535–555, 2006     

Cognitive restructuring as an early stage in problem solving

This article examines the hypothesis that there are preliminary stages in problem solving which most chemists neglect when trying to teach their students how to solve problems in introductory chemistry courses. It is during these early stages that relevant information is disembedded from the question and the problem is restructured. Unless students can successfully complete these cognitive restructuring stages, they cannot proceed on to the more analytic stages in problem solving that have received more attention from chemists. Preliminary evidence for this hypothesis consists of linear correlations between student ability to handle disembedding and cognitive restructuring tasks in the spatial domain and their ability to solve chemistry problems.     

The role of strategy-based motivation in mathematical problem solving: The case of learner-generated drawings

This study investigated the role of strategy-based motivation (SBM) in solving real-world geometry problems. Students from 19 classes (N = 437) were assigned to the strategy training or control condition. Before the treatment, students were asked about their SBM (i.e., self-efficacy, cost, and value). After the treatment, they were instructed to generate drawings and solve problems. The results revealed that self-efficacy expectations and cost (but not value) regarding the drawing strategy predicted drawing quality and performance. The relation between SBM and performance was mediated by drawing quality. SBM did not moderate the effects of strategy training on drawing quality or performance. The results emphasize the importance of SBM in the context of strategy training for predicting the quality of strategy use and problem-solving performance. The results are consistent with assumptions of expectancy-value and learner-generated drawing theories. One practical implication is that attention should be given to SBM in mathematics lessons.     

Academic essay writing as imitative problem solving: examples from distance learning

Students in tertiary education are often faced with the prospect of writing an essay on a topic they know nothing about in advance. In distance learning institutions, essays are a common method of assessment in the UK, and specified course texts remain the main sources of information the students have. How do students use a source text to construct an essay? The present paper presents a methodology for mapping the source text on to the finished student essay. The underlying assumption is that students are using a form of imitative problem solving when faced with the complex task of writing an essay. Twenty-two essays written by Open University students in the UK, based on three different questions, were analysed on the basis of the order in which novel concepts were introduced and the extent to which this order mirrored that of the source textbook. Correlations were then carried out between the structure of the essay, the structure of the source text and the eventual grade awarded. The average correlation for all three essays and source texts was 0.8, with some individual essays having a correlation of 0.98, demonstrating that the students were closely imitating the argument structure of the source text.     

Mathematical problem posing: Implications for facilitating student inquiry in classrooms

From 1984 through 1988, the authors worked with teachers using an inquiry approach to teach high school geometry courses with the aid of the GEOMETRIC SUPPOSERS. Problems are a critical component of the approach, as they are of any instructional process, because they focus attention and energy and guide students in the application, integration, and extension of knowledge. Inquiry problems differ from traditional, single-answer textbook exercises in that they must leave room for student initiative and creativity. The observations presented in this paper about the delicate balance between specifying too much instruction and too little, which is part of creating and posing inquiry problems, are based on careful examination of students' papers and classroom observations. The paper closes with speculations on whether these observations suggest general lessons for those seeking practical and successful strategies to introduce student inquiry into classrooms, with the hope of stimulating interest in and discussion of such strategies.Preliminary versions of this paper were presented at PME-XI Montreal, 1987, and the 1988 AERA conference in New Orleans.     

Assessing problem solving in mathematics: Some variables related to student performance

A mathematical problem is defined here as a question not dependent on specific syllabus content, and one sufficiently new to the student such that it cannot be solved by a previously known method. With increased attention being paid to this type of mathematical problem solving at the primary school level, the need for reliable and valid methods of assessment has become more apparent. This paper reports the results of using a new problem solving test, developed for use in the upper primary school, with 371 students in Years 4,5 and 6 at government schools in Melbourne. Particular attention is given to the effects of year level, sex and the method of test administration on student performance for different types of items and different problem solving processes. The performance of Year 4 students was generally lower than that of other students, but differences were small for most items and processes between Years 5 and 6. Although most of the differences in performance between the sexes were not significant, the girls had higher scores than the boys for the total score, for all processes and for all items except the spatial item. The method of administration was important for performance, especially for the girls. The marking schedule developed enabled high intra- and inter-marker reliabilities to be obtained.     

Case-based reasoning and instructional design: Using stories to support problem solving

With an increased emphasis on problem solving and problem-based learning in the instructional design field, new methods for task analysis and models for designing instruction are needed. An important methodology for both entails the elicitation, analysis, and inclusion of stories as a primary form of instructional support while learning to solve problems. Stories are the most natural and powerful formalism for storing and describing experiential knowledge that is essential to problem solving. The rationale and means for analyzing, organizing, and presenting stories to support problem solving are defined by case-based reasoning. Problems are solved by retrieving similar past experiences in the form of stories and applying the lessons learned from those stories to the new problems. In this paper, after justifying the use of stories as instructional supports, we describe methods for eliciting, indexing, and making stories available as instructional support for learning to solve problems.     

再探讨数学“问题解决”

本回眸数学问题解决的历史，对一些关键性的论题作了探讨，并提出的展望，期待为我国数学创新教育提供参考。     

Exploring problem conceptualization and performance in STEM problem solving contexts

Problem solving abilities are critical components of contemporary Science, Technology, Engineering and Mathematics (STEM) education. Research in the area of problem solving has uncovered much about the representation, processes and heuristic approaches to problem solving. However, critics claim this overemphasis on the process of solving problems has led to a dearth in understanding of the earlier stages such as problem conceptualization. This paper aims to address some of these concerns by exploring the area of problem conceptualization and the underlying cognitive mechanisms that may play a supporting role in reasoning success. Participants (N?=?12) were prescribed a series of convergent problem-solving tasks representative of those used for developmental purposes in STEM education. During the problem-solving episodes, cognitive data were gathered by means of an electroencephalographic headset and used to investigate students’ cognitive approaches to conceptualizing the tasks. In addition, interpretive qualitative data in the form of post-task interviews and problem solutions were collected and analyzed. Overall findings indicated a significant reliance on memory during the conceptualization of the convergent problem-solving tasks. In addition, visuospatial cognitive processes were found to support the conceptualization of convergent problem-solving tasks. Visuospatial cognitive processes facilitated students during the conceptualization of convergent problems by allowing access to differential semantic content in long-term memory.

     

The Instructional Design Environment: technology to support design problem solving

The problems of efficiently producing effective instruction in areas such as industry, military, and vocational education are exacerbated by several complex factors: increasingly rapid change in technology; substantial variation in training needs and target populations within large organizations; inefficient mechanisms for performing formative evaluations; and a lack of means for efficiently incorporating new models of design into practice. In the context of these increasingly difficult problems, we present the Instructional Design Environment (IDE), a hypermedia system for designing and developing instructional material, including texts, interactive video disk, and intelligent tutoring systems. The representation of design analyses and specifications, and the design activities of IDE users are not constrained by any particular model of instructional design, but can be tailored to suit a wide range of such models. Although the system is continually evolving and exists in several forms, (Swartz and Russell, 1989) we discuss how the features of IDE suggest how computer-based design environments may provide ways simplifying the design problems for technical training in rapidly changing areas.     

Problem solving strategies and mathematical resources: A longitudinal view on problem solving in a function based approach to algebra

This study is an attempt to analyze students' construction of function based problem solving methods in introductory algebra. It claims that for functions to be a main concept for learning school algebra, a complex process that has to be developed during a long period of learning must take place. The article describes a longitudinal observation of a pair of students that studied algebra for 3 years using a function approach, including intensive use of graphing technology. Such a long observation is difficult to carry out and even more difficult to report. We watched for three years classrooms using the ‘Visual-Math’ sequence, and sampled students that exhibited various levels of mathematics achievement. The analysis method presented here is a non-standard case study of a pair of lower achievers students and their work is often juxtaposed to the work of other pairs participating in the study. The students' attempts to solve a linear break-even problem is analyzed along three interviews which present the development of the use of mathematical resources and the patterns of problem solving at different learning phases. Beyond describing solving attempts, the article offers terms for describing and explaining what and how do learners appreciate and make out of solving introductory school algebra problems over a three years course. This revised version was published online in July 2006 with corrections to the Cover Date.     

Relationship of children's conceptual tempo to problem solving and creativity

The present study examines the role of conceptual tempo in creativity and problem solving. It was hypothesized that reflective children would do well on tasks involving an evaluation component, while impulsive children would do well on typical creativity tasks. Measures of creativity, assessing fluency, flexibility, and originality, as well as measures of problem solving, stressing an evaluation component, were administered to 101 white suburban fifth-grade children. Using MFF scores, four groups were identified: reflectives, impulsives, fast-accurates, and slow-inaccurates. Results indicated no significant differences among the conceptual tempo groups on any of the creativity or problem-solving tasks. It is suggested that conceptual tempo may be less effective in predicting differences in some aspects of cognitive functioning at this age. Examination of this hypothesis with younger children is recommended.     

Student perceptions of student perception of module questionnaires: questionnaire completion as problem solving

A sample of 202 students filling in a student evaluation of teaching (SET) questionnaire were asked to complete another questionnaire asking about the specific reasons for awarding a score to the specific SET questionnaire items. The aim was to find out what influenced students' judgements on those items. It was found that students' interpretation of some questions differed from the ‘expected’ interpretation. Several factors, such as the placing of questions and the salience of items retrieved from memory, could influence a score. It was also found that asking for an explanation improved scores overall. The conclusions were that questionnaire completion could be understood as a form of problem solving and judgement under uncertainty. The specific heuristics used led to variability in students' interpretation of the task.