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Classical text on Mathematical Problem Solving.
What does this book have to offer, and to whom? It is addressed to people with research interests in the nature of mathematical thinking at any level, to people 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 road-blocks 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.
Polya's "How To Solve It" is among the classical books on Mathematical Problem Solving, and Schoenfeld manages to build on that work in a meaningful way. In particular, the experiments that are detailed throughout the book are very compelling and offer strong supporting evidence for Schoenfeld's theories, which can be applied to great effect.
The book thouroghly speaks of problem solving and about what is needed to become an expert problem solver.
Introduction and Overview
Aspects of Mathematical Thinking: A Theoretical Overview
A Framework for the Analysis of Mathematical Behavior
Resources
Heuristics
Control
Belief Systems
Experimental and Observational Studies, Issues of Methodology, and Questions of Where We Go Next
Explicit Heuristic Training as a Variable in Problem- Solving Performance
Measures of Problem-Solving Performance and Problem-Solving Instruction
Problem Perception, Knowledge Structure, and Problem-Solving Performance
Verbal Data, Protocol Analysis, and the Issue of Control
The Roots of Belief
References
Author Index
Subject Index