Mathematical Problem Solving

Mathematical Problem Solving provides information pertinent to the nature of mathematical thinking at any level. This book provides a framework for the analysis of complex problem-solving behavior. Organized into two parts encompassing 10 chapters, this book begins with an overview of the four qualitatively different aspects of complex intellectual activity, namely, cognitive resources, heuristics, control, and belief systems. This text then presents a series of empirical investigations that flesh out the analytical framework. Other chapters consider the ways that competent problem solvers make the most of the knowledge at their disposal. This book discusses as well the kind s of learning that can result from carefully designed instruction. The final chapter deals with observations made in typical high school classrooms, which serve to indicate some of the sources of students' mathematical behavior. This book is a valuable resource for mathematicians and mathematics teachers. Readers who are interested in higher-order thinking skills in any domain will also find this book useful.

Table of Contents:
Preface Acknowledgments Introduction and Overview Part One Aspects of Mathematical Thinking: A Theoretical Overview 1. A Framework for the Analysis of Mathematical Behavior Overview Typical Problems, Typical Behavior: The Four Categories Illustrated Resources Heuristics Control Belief Systems Summary 2. Resources Routine Access to Relevant Knowledge The Broad Spectrum of Resources Flawed Resources and Consistent Error Patterns Summary 3. Heuristics Introduction and Overview What a Problem Is and Who the Students Are Toward More Precise and Usable Descriptions of Heuristic Strategies The Complexity of Implementing a "Straightforward" Heuristic Solution Heuristics and Resources Deeply Intertwined Summary 4. Control Introduction and Overview On the Importance of Control: A Look at a Microcosm Modeling a Control Strategy for Heuristic Problem Solving Toward a Broader View of Control Literature Related to Control Summary 5. Belief Systems Selections from the Relevant Literature A Mathematician Works a Construction Problem The Student as Pure Empiricist: A Model of Empirical Behavior How the Model Corresponds to Performance A Deeper Look at Empiricism: CS and AM Work Problem 1.1 Further Evidence Regarding Naive Empiricism: DW and SP Work Four Related Problems Summary Part Two Experimental and Observational Studies, Issues of Methodology, and Questions of where we Go Next Overview 6. Explicit Heuristic Training as a Variable in Problem-Solving Performance A Brief Discussion of Relevant Literature Experimental Design Results Two Methodological Questions Discussion Implications and Directions for Extension Summary 7. Measures of Problem-Solving Performance and Problem-Solving Instruction A Brief Discussion of Relevant Work The Experimental and Control Treatments Measure 1 : A Plausible-Approach Analysis of Fully Solved Questions Discussion of Testing Results Measure 2: Students' Qualitative Assessments of Their Problem Solving Measure 3: Heuristic Fluency and Transfer A Brief Discussion of Control Issues Summary 8. Problem Perception, Knowledge Structure, and Problem-Solving Performance Background Method Results of the Sortings Discussion Summary Appendix: Problems Used in the Card Sort 9. Verbal Data, Protocol Analysis, and the Issue of Control Overview Background, Part 1: Verbal Methods Through a Glass Darkly: A Close Look at Verbal Data Background, Part 2: Other Protocol Coding Schemes and Issues of Control The Major Issues for Analysis: A Brief Discussion of Two Protocols A Framework for the Macroscopic Analysis of Problem-Solving Protocols Episodes and the Associated Questions A Full Analysis of a Protocol A Further Discussion of Control: More Data from Students, and the Analysis of an Expert Problem Solver's Protocol Brief Discussion: Limitations and Needed Work Summary Appendix 9.1 : A Single-Person Protocol of the Cells Problem Appendix 9.2: Protocol 9.2 Appendix 9.3: Protocol 9.3 Appendix 9.4: Protocol 9.4 Appendix 9.5: Protocol 9.5 10. The Roots of Belief A Discussion of Two Geometry Protocols A Brief Analysis of Protocol 10.1 A Brief Analysis of Protocol 10.2 A Brief Discussion The Strength of Empiricism: More Data The Origins of Empiricism Summary Postscript Appendix 10.1: Protocol 10.1 Appendix 10.2: Protocol 10.2 References Author Index Subject Index