Algebraic Problem Solving Research Articles

Research issues in the learning and teaching of algebra

Contents: C. Kieran, S. Wagner, The Research Agenda Conference on Algebra: Background and Issues. Part I:Past Research and Current Issues. J.A. Thorpe, Algebra: What Should We Teach and How Should We Teach It? J.R. Leitzel, Critical Considerations for the Future of Algebra Instruction. C. Kieran, The Early Learning of Algebra: A Structural Perspective. L.R. Booth, A Question of Structure. N. Herscovics, Cognitive Obstacles Encountered in the Learning of Algebra. D. Tall, Different Cognitive Obstacles in a Technological Paradigm. S. Chaiklin, Cognitive Studies of Algebra Problem Solving and Learning. R.B. Davis, Three Ways of Improving Cognitive Studies in Algebra. J.H. Larkin, Robust Performance in Algebra: The Role of the Problem Representation. P.W. Thompson, Artificial Intelligence, Advanced Technology, and Learning and Teaching Algebra. M.W. Lewis, Intelligent Tutoring Systems: First Steps and Future Directions. J.J. Kaput, Linking Representations in the Symbol Systems of Algebra. D. Kirshner, Critical Issues in Current Representation System Theory. J.T. Fey, School Algebra for the Year 2000. S.L. Senk, Toward School Algebra in the Year 2000. Part II:A Research Agenda. S. Wagner, C. Kieran, An Agenda for Research on the Learning and Teaching of Algebra. L.R. Booth, The Research Agenda in Algebra: A Mathematics Education Perspective. M.W. Lewis, The Research Agenda in Algebra: A Cognitive Science Perspective. S.L. Rachlin, The Research Agenda in Algebra: A Curriculum Development Perspective. Part III:Theoretical Considerations. R.B. Davis, Research Studies in How Humans Think About Algebra. J.H. Larkin, Eight Reasons for Explicit Theories in Mathematics Education. D. Wheeler, Contexts for Research on the Teaching and Learning of Algebra.

Exploring the Episodic Structure of Algebra Story Problem Solving

This article analyzes the quantitative and situational structure of algebra story problems, uses these materials to propose an interpretive framework for written problem-solving protocols, and then presents an exploratory study of the episodic structure of algebra story problem solving in a sizable group of mathematically competent undergraduates. Analyses of written protocols compare the strategic, tactical, and conceptual content of solution attempts, looking within these attempts at the interplay between problem comprehension and solution. Comprehension and solution of algebra story problems are complementary activities, giving rise to a succession of problem-solving episodes. Although direct algebraic problem solving is sometimes effective, results suggest that the algebraic formalism may be of little help in comprehending the quantitative constraints posed in a problem text. Instead, competent problem solvers often reason within the situational context presented by a story problem, using various forms of model-based reasoning to identify, pursue, and verify quantitative constraints required for solution. The article concludes by discussing the implications of these finding for acquiring mathematical concepts (e.g., related linear functions) and for supporting their acquisition through instruction.

Problem Solving in Ninth Grade Science and Algebra

Problem Solving in Ninth Grade Science and Algebra

STRATEGIES FOR TEACHING LEARNING DISABLED ADOLESCENTS ALGEBRAIC PROBLEMS

Learning disabled adolescents experience considerable difficulty learning to solve word problems in the high school algebra curriculum. This article describes instruction that enables learning disabled adolescents to master three types of algebraic word problems. A cognitive model of problem solving is presented. Components of effective cognitive strategy instruction are presented for each phase. These components include task analysis, modeling by thinking aloud, and the students' use of a self‐questioning strategy and a structured worksheet. Finally, there is a discussion of the prospects and problems faced by teachers using cognitive strategy instruction for algebraic problem solving.

Algebra slaves and agents in a Logo-based mathematics curriculum

For many students the ideas and methods of algebra appear obscure and mysterious, their sense and purpose unclear, and their applicability to anything genuinely real or interesting very remote. Students often fail to acquire an understanding of the key concepts, despite their inherent simplicity. Even when they gain the notion of variables, expressions and equations, students often lack the strategic knowledge required to motivate and direct the global planning and detailed execution of an attack on a problem. These conceptual and strategic difficulties are compounded by the needs for precise performance of the arithmetic and symbolic operations required in manipulating expressions. Extended operations like subtracting an expression from both sides of an equation or expanding a product of three terms, are very difficult for beginning students. Their buggy performance in carrying out the detailed manipulative work greatly confounds and frustrates their acquisition and assimilation of the most important and central ideas. In an effort to confront these difficulties and show how they can be overcome, we are developing a Logo-based introductory algebra course for sixth graders. Our approach has three major components: work on Logo programming projects in algebraically rich contexts whose content is meaning ful and compelling to students, the use of algebra microworlds with concrete iconic representations of formal objects and operations, and the introduction of the algebra workbench, an expert instructional system to aid students in performing extended algebraic operations. The algebra workbench will employ a set of powerful symbolic manipulation tools for performing the standard manipulations of high school algebra. It will have two main modes of use: demonstration mode, which uses an expert tutor program to solve algebra problems incrementally, explaining its strategy and its step by step operations in straightforward terms along the way; and practice mode, in which the student tries to solve a problem with the assistance of the tutor, which performs the operations requested by the student at each step and which can be called at any point to advise the student of the correctness of a step, to perform or explain any step, to evaluate the student's solution, or to perform a problem that she poses. These powerful aids make it possible to effectively separate out the difficulties in performing the formal and manipulative aspects of algebra work from those encountered in learning the central conceptual and strategic content. Distinctly different kinds of instructional tools and activities-Logo programming, expert tutors, or algebra microworlds-can thus be brought to bear where each is most appropriate and effective.

The Use of Worked Examples as a Substitute for Problem Solving in Learning Algebra

The knowledge required to solve algebra manipulation problems and procedures designed to hasten knowledge acquisition were studied in a series of five experiments. It was hypothesized that, as occurs in other domains, algebra problem-solving skill requires a large number of schemas and that schema acquisition is retarded by conventional problem-solving search techniques. Experiment 1, using Year 9, Year 11, and university mathematics students, found that the more experienced students had a better cognitive representation of algebraic equations than less experienced students as measured by their ability to (a) recall equations, and (b) distinguish between perceptually similar equations on the basis of solution mode. Experiments 2 through 5 studied the use of worked examples as a means of facilitating the acquisition of knowledge needed for effective problem solving. It was found that not only did worked examples, as expected, require considerably less time to process than conventional problems, but that su...

An evaluation of achievement in algebra of class X students in Delhi schools

The study was mainly aimed at: determining the structure of intellect abilities involved in problem solving ability in algebra; determining the selected structure of intellect abilities as predictors of problem solving ability in algebra at class X level; investigating into sex differences in the factorial nature and predicting capacity of selected abilities relevant in algebraic problem solving. On the basis of previous research findings, eight abilities (CSS, NSR, NST, CSR, CMU, NSS, CSI and CMS) were selected from Guilford's model [6] and tests were developed to measure them. Similarly, problem solving ability in algebra was analysed into eight components and eight tests were developed based on them. These sixteen tests were administered to a sample of 500 class X students: 241 boys and 259 girls. The data were analysed by using complex statistical techniques of factor analysis and regression analysis. The data for boys and girls were analysed separately. The major findings of the study were: The eight...

Problem Solving in Algebra

No apology is needed for re-introducing in the pages of The Mathematics Teacher a topic which may be time-worn. The subject of problem solving is a lways in place and ever calls for solution.