This survey book reviews four interrelated areas: (i) the relevance of heuristics in problem-solving approaches – why they are important and what research tells us about their use; (ii) the need to characterize and foster creative problem-solving approaches – what type of heuristics helps learners devise and practice creative solutions; (iii) the importance that learners formulate and pursue their own problems; and iv) the role played by the use of both multiple-purpose and ad hoc mathematical action types of technologies in problem-solving contexts – what ways of reasoning learners construct when they rely on the use of digital technologies, and how technology and technology approaches can be reconciled.

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.

This volume presents a state-of-the-science review of the most promising current European research -- and its historic roots of research -- on complex problem solving (CPS) in Europe. It is an attempt to close the knowledge gap among American scholars regarding the European approach to understanding CPS. Although most of the American researchers are well aware of the fact that CPS has been a very active research area in Europe for quite some time, they do not know any specifics about even the most important research. Part of the reason for this lack of knowledge is undoubtedly the fact that European researchers -- for the most part -- have been rather reluctant to publish their work in English-language journals. The book concentrates on European research because the basic approach European scholars have taken to studying CPS is very different from one taken by North American researchers. Traditionally, American scholars have been studying CPS in "natural" domains -- physics, reading, writing, and chess playing -- concentrating primarily on exploring novice-expert differences and the acquisition of a complex skill. European scholars, in contrast, have been primarily concerned with problem solving behavior in artificially generated, mostly computerized, complex systems. While the American approach has the advantage of high external validity, the European approach has the advantage of system variables that can be systematically manipulated to reveal the effects of system parameters on CPS behavior. The two approaches are thus best viewed as complementing each other. This volume contains contributions from four European countries -- Sweden, Switzerland, Great Britain, and Germany. As such, it accurately represents the bulk of empirical research on CPS which has been conducted in Europe. An international cooperation started two years ago with the goal of bringing the European research on complex problem solving to the awareness of American scholars. A direct result of that effort, the contributions to this book are both informative and comprehensive.

"Polya reveals how the mathematical method of demonstrating a proof or finding an unknown can be of help in attacking any problem that can be "reasoned" out--from building a bridge to winning a game of anagrams."--Back cover.