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Mathematical Problem Solving - 1st Edition - ISBN: 9780126288704, 9781483295480

Mathematical Problem Solving

1st Edition

eBook ISBN: 9781483295480
Imprint: Academic Press
Published Date: 28th October 1985
Page Count: 409
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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



Introduction and Overview

Part One Aspects of Mathematical Thinking: A Theoretical Overview

1. A Framework for the Analysis of Mathematical Behavior


Typical Problems, Typical Behavior: The Four Categories Illustrated




Belief Systems


2. Resources

Routine Access to Relevant Knowledge

The Broad Spectrum of Resources

Flawed Resources and Consistent Error Patterns


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


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


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


Part Two Experimental and Observational Studies, Issues of Methodology, and Questions of where we Go Next


6. Explicit Heuristic Training as a Variable in Problem-Solving Performance

A Brief Discussion of Relevant Literature

Experimental Design


Two Methodological Questions


Implications and Directions for Extension


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


8. Problem Perception, Knowledge Structure, and Problem-Solving Performance



Results of the Sortings



Appendix: Problems Used in the Card Sort

9. Verbal Data, Protocol Analysis, and the Issue of Control


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


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



Appendix 10.1: Protocol 10.1

Appendix 10.2: Protocol 10.2


Author Index

Subject Index


No. of pages:
© Academic Press 1985
28th October 1985
Academic Press
eBook ISBN:

About the Author


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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