Mathematical Problem Solving (eBook)

Alan H. Schoenfeld (Autor)

eBook Download: PDF

2014 | 1. Auflage
409 Seiten
Elsevier Science (Verlag)
978-1-4832-9548-0 (ISBN)

Infos

This book is addressed to people with research interests in the nature of mathematical thinking at any level, topeople with an interest in 'higher-order thinking skills' in any domain, and to all mathematics teachers. The focal point of the book is a framework for the analysis of complex problem-solving behavior. That framework is presented in Part One, which consists of Chapters 1 through 5. It describes four qualitatively different aspects of complex intellectual activity: cognitive resources, the body of facts and procedures at one's disposal; heuristics, 'rules of thumb' for making progress in difficult situations; control, having to do with the efficiency with which individuals utilize the knowledge at their disposal; and belief systems, one's perspectives regarding the nature of a discipline and how one goes about working in it. Part Two of the book, consisting of Chapters 6 through 10, presents a series of empirical studies that flesh out the analytical framework. These studies document the ways that competent problem solvers make the most of the knowledge at their disposal. They include observations of students, indicating some typical roadblocks to success. Data taken from students before and after a series of intensive problem-solving courses document the kinds of learning that can result from carefully designed instruction. Finally, observations made in typical high school classrooms serve to indicate some of the sources of students' (often counterproductive) mathematical behavior.

Alan Schoenfeld is the Elizabeth and Edward Conner Professor of Education and Affiliated Professor of Mathematics at the University of California at Berkeley. He is a Fellow of the American Association for the Advancement of Science and of the American Educational Research Association (AERA), and a Laureate of the education honor society Kappa Delta Pi; he has served as President of AERA and vice President of the National Academy of Education. He holds the International Commission on Mathematics Instruction's Klein Medal, AERA's Distinguished Contributions to Research in Education award, and the Mathematical Association of America's Mary P. Dolciani award. Mathematical Problem Solving is laid the foundations for the field's work on mathematical thinking and problem solving. The ideas in the book have been referred to as the 'industry standard for research on mathematical problem solving."

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This book is addressed to people with research interests in the nature of mathematical thinking at any level, topeople with an interest in "e;higher-order thinking skills"e; in any domain, and to all mathematics teachers. The focal point of the book is a framework for the analysis of complex problem-solving behavior. That framework is presented in Part One, which consists of Chapters 1 through 5. It describes four qualitatively different aspects of complex intellectual activity: cognitive resources, the body of facts and procedures at one's disposal; heuristics, "e;rules of thumb"e; for making progress in difficult situations; control, having to do with the efficiency with which individuals utilize the knowledge at their disposal; and belief systems, one's perspectives regarding the nature of a discipline and how one goes about working in it. Part Two of the book, consisting of Chapters 6 through 10, presents a series of empirical studies that flesh out the analytical framework. These studies document the ways that competent problem solvers make the most of the knowledge at their disposal. They include observations of students, indicating some typical roadblocks to success. Data taken from students before and after a series of intensive problem-solving courses document the kinds of learning that can result from carefully designed instruction. Finally, observations made in typical high school classrooms serve to indicate some of the sources of students' (often counterproductive) mathematical behavior.

Inhalt

Front Cover 1
Mathematical Problem Solving 4
Copyright Page 5
Table of Contents 8
Dedication 6
Preface 12
Acknowledgments 16
Introduction and Overview 18
Part One: Aspects of Mathematical Thinking: A Theoretical Overview 26
Chapter 1. A Framework for the Analysis of Mathematical Behavior 28
Overview 28
Typical Problems, Typical Behavior:The Four Categories Illustrated 31
Resources 34
Heuristics 39
Control 44
Belief Systems 51
Summary 61
Chapter 2. Resources 63
Routine Access to Relevant Knowledge 63
The Broad Spectrum of Resources 71
Flawed Resources and Consistent Error Patterns 78
Summary 84
Chapter 3. Heuristics 86
Introduction and Overview 86
What a Problem Is and Who the Students Are 91
Toward More Precise and Usable Descriptionsof Heuristic Strategies 93
The Complexity of Implementing a "Straightforward" Heuristic Solution 101
Heuristics and Resources Deeply Intertwined 108
Summary 112
Chapter 4. Control 114
Introduction and Overview 114
On the Importance of Control: A Look at a Microcosm 117
Modeling a Control Strategy for Heuristic Problem Solving 123
Toward a Broader View of Control 131
Literature Related to Control 144
Summary 160
Chapter 5. Belief Systems 162
Selections from the Relevant Literature 163
A Mathematician Works a Construction Problem 174
The Student as Pure Empiricist: A Model of Empirical Behavior 177
How the Model Corresponds to Performance 181
A Deeper Look at Empiricism: CS and AM Work Problem 1.1 182
Further Evidence Regarding Naive Empiricism: DW and SP Work Four Related Problems 191
Summary 201
Part Two: Experimental and Observational Studies, Issues of Methodology, and Questions of Where We Go Next 204
Overview 204
Chapter 6. Explicit Heuristic Training as a Variable in Problem-Solving 
206 
A Brief Discussion of Relevant Literature 207
Experimental Design 210
Results 220
Two Methodological Questions 225
Discussion 226
Implications and Directions for Extension 229
Summary 231
Chapter 7. Measures of Problem-Solving Performance and Problem-Solving Instruction 233
A Brief Discussion of Relevant Work 234
The Experimental and Control Treatments 236
Measure 1 : A Plausible-Approach Analysis of Fully Solved Questions 240
Discussion of Testing Results 247
Measure 2: Students' Qualitative Assessments of Their Problem Solving 248
Measure 3: Heuristic Fluency and Transfer 250
A Brief Discussion of Control Issues 255
Summary 256
Chapter 8. Problem Perception, Knowledge Structure, and Problem-Solving Performance 259
Background 260
Method 264
Results of the Sortings 269
Discussion 276
Summary 280
Appendix: Problems Used in the Card Sort 282
Chapter 9. Verbal Data, Protocol Analysis,and the Issue of Control 287
Overview 287
Background, Part 1 : Verbal Methods 289
Through a Glass Darkly: A Close Look at Verbal Data 292
Background, Part 2: Other Protocol Coding Schemesand Issues of Control 300
The Major Issues for Analysis:A Brief Discussion of Two Protocols 305
A Framework for the Macroscopic Analysis ofProblem-Solving Protocols 309
Episodes and the Associated Questions 314
The Full Analysis of a Protocol 318
A Further Discussion of Control: More Data from Students, and the Analysis of an Expert Problem Solver's Protocol 323
Brief Discussion: Limitations and Needed Work 331
Summary 333
Appendix 9.1: A Single-Person Protocol of the Cells Problem 334
Appendix 9.2: Protocol 9.2 336
Appendix 9.3: Protocol 9.3 341
Appendix 9.4: Protocol 9.4 348
Appendix 9.5: Protocol 9.5 357
Chapter 10. The Roots of Belief 362
A Discussion of Two Geometry Protocols 363
A Brief Analysis of Protocol 10.1 364
A Brief Analysis of Protocol 10.2 366
A Brief Discussion 371
The Strength of Empiricism: More Data 372
The Origins of Empiricism 374
Summary 390
Postscript 392
Appendix 10.1: Protocol 10.1 392
Appendix 10.2: Protocol 10.2 399
References 405
Author Index 416
Subject Index 420

Produkt-Details

• Erscheint lt. Verlag: 28.6.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Technik
• ISBN-10: 1-4832-9548-6 / 1483295486
• ISBN-13: 978-1-4832-9548-0 / 9781483295480