Theory Of Problem Solving

Examples help explain the seven basic mathematical problem-solving methods, including inference, classification of action sequences, working backward, and contradiction

Psychological, educational, gerontological, marketing, and other literatures all report recent research in everyday problem solving, yet few sources have made these various types of results available in one state-of-the-art volume. Everyday Problem Solving makes accessible many of these points of view for all readers, coordinates them, and provides directions from which to formulate new studies. The wide but methodical scope of this work will interest researchers, clinicians, philosophers, marketing specialists, administrators, artificial intelligence scientists, educators, guidance counselors and psychologists. Undergraduate and graduate students in these fields will also find this an invaluable source. The collection of reports includes an examination of models from information processing theory and postformal cognitive developmental theory, and an overview of the tasks used in everyday problem solving research. Several leading theories, including Sinnott's, are applied to describe the thoughts and emotions of adults as they solve illstructured problems. Reports on applied research include: techniques of master teachers; the ways adults resolve conflicts; consumer behavior; academic intelligence; the connection of memory to problem solving; intervention strategies and the elderly.

One of the most active fields of educational research in recent years has been the investigation of problem-solving performance. Two opposing views of current research -- one suggesting that there are more differences than similarities within different domains, and the other stating that there is great similarity -- lead to a variety of questions: * Is problem solving a single construct? * Are there aspects of problem-solving performance that are similar across a variety of content domains? * What problem-solving skills learned within one context can be expected to transfer to other domains? The purpose of this book is to serve as the basis for the productive exchange of information that will help to answer these questions -- by drawing together preliminary theoretical understandings, sparking debate and disagreement, raising new questions and directions, and perhaps developing new world views.

We respect Herbert A. Simon as an established leader of empirical and logical analysis in the human sciences while we happily think of him as also the loner; of course he works with many colleagues but none can match him. He has been writing fruitfully and steadily for four decades in many fields, among them psychology, logic, decision theory, economics, computer science, management, production engineering, information and control theory, operations research, confirmation theory, and we must have omitted several. With all of them, he is at once the technical scientist and the philosophical critic and analyst. When writing of decisions and actions, he is at the interface of philosophy of science, decision theory, philosophy of the specific social sciences, and inventory theory (itself, for him, at the interface of economic theory, production engineering and information theory). When writing on causality, he is at the interface of methodology, metaphysics, logic and philosophy of physics, systems theory, and so on. Not that the interdisciplinary is his orthodoxy; we are delighted that he has chosen to include in this book both his early and little-appreciated treatment of straightforward philosophy of physics - the axioms of Newtonian mechanics, and also his fine papers on pure confirmation theory.

"We put together a book that would offer readers multiple perspectives, insights, and directions in understanding social problem solving as an important theory that has driven wide-ranging scientific research and as an important means of training to empower and elevate the lives of individuals. We believe that social problem solving can help individuals free themselves from the problems they face or the distress that these problems cause. We recognize that some problems may be difficult or impossible to solve, but we believe that considerable value remains in understanding and promoting effective social problem solving to foster the novel insights and methods in which problems that seem insurmountable ultimately may be conquered in incremental steps, across time and across individuals. Moreover, we believe that problems can be solved in different ways. When problematic situations or circumstances are manageable or controllable, a good problem solver tries to find ways to change them for the better. However, when such situations or circumstances are unchangeable or uncontrollable, one can still use problem solving to find ways to accept and tolerate with less distress that which cannot be changed or controlled"--Preface. (PsycINFO Database Record (c) 2005 APA, all rights reserved)

Through its engaging and unusual problems, this book demonstrates methods of reasoning necessary for learning number theory. Every technique is followed by problems (as well as detailed hints and solutions) that apply theorems immediately, so readers can solve a variety of abstract problems in a systematic, creative manner. New solutions often require the ingenious use of earlier mathematical concepts - not the memorization of formulas and facts. Questions also often permit experimental numeric validation or visual interpretation to encourage the combined use of deductive and intuitive thinking. The first chapter starts with simple topics like even and odd numbers, divisibility, and prime numbers and helps the reader to solve quite complex, Olympiad-type problems right away. It also covers properties of the perfect, amicable, and figurate numbers and introduces congruence. The next chapter begins with the Euclidean algorithm, explores the representations of integer numbers in different bases, and examines continued fractions, quadratic irrationalities, and the Lagrange Theorem. The last section of Chapter Two is an exploration of different methods of proofs. The third chapter is dedicated to solving Diophantine linear and nonlinear equations and includes different methods of solving Fermat’s (Pell’s) equations. It also covers Fermat’s factorization techniques and methods of solving challenging problems involving exponent and factorials. Chapter Four reviews the Pythagorean triple and quadruple and emphasizes their connection with geometry, trigonometry, algebraic geometry, and stereographic projection. A special case of Waring’s problem as a representation of a number by the sum of the squares or cubes of other numbers is covered, as well as quadratic residuals, Legendre and Jacobi symbols, and interesting word problems related to the properties of numbers. Appendices provide a historic overview of number theory and its main developments from the ancient cultures in Greece, Babylon, and Egypt to the modern day. Drawing from cases collected by an accomplished female mathematician, Methods in Solving Number Theory Problems is designed as a self-study guide or supplementary textbook for a one-semester course in introductory number theory. It can also be used to prepare for mathematical Olympiads. Elementary algebra, arithmetic and some calculus knowledge are the only prerequisites. Number theory gives precise proofs and theorems of an irreproachable rigor and sharpens analytical thinking, which makes this book perfect for anyone looking to build their mathematical confidence.