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Deep Thinking

Author: William Byers
Publisher: World Scientific Publishing Company Incorporated
ISBN: 9789814618038
Size: 29.16 MB
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There is more than one way to think. Most people are familiar with the systematic, rule-based thinking that one finds in a mathematical proof or a computer program. But such thinking does not produce breakthroughs in mathematics and science nor is it the kind of thinking that results in significant learning. Deep thinking is a different and more basic way of using the mind. It results in the discontinuous "aha!" experience, which is the essence of creativity. It is at the heart of every paradigm shift or reframing of a problematic situation. The identification of deep thinking as the default state of the mind has the potential to reframe our current approach to technological change, education, and the nature of mathematics and science. For example, there is an unbridgeable gap between deep thinking and computer simulations of thinking. Many people suspect that such a gap exists, but find it difficult to make this intuition precise. This book identifies the way in which the authentic intelligence of deep thinking differs from the artificial intelligence of "big data" and "analytics." Deep thinking is the essential ingredient in every significant learning experience, which leads to a new way to think about education. It is also essential to the construction of conceptual systems that are at the heart of mathematics and science, and of the technologies that shape the modern world. Deep thinking can be found whenever one conceptual system morphs into another. The sources of this study include the cognitive development of numbers in children, neuropsychology, the study of creativity, and the historical development of mathematics and science. The approach is unusual and original. It comes out of the author's lengthy experience as a mathematician, teacher, and writer of books about mathematics and science, such as How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics and The Blind Spot: Science and the Crisis of Uncertainty.

Deep Thinking

Author: William Byers
Publisher: World Scientific
ISBN: 9814618055
Size: 34.64 MB
Format: PDF, Kindle
View: 4219
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There is more than one way to think. Most people are familiar with the systematic, rule-based thinking that one finds in a mathematical proof or a computer program. But such thinking does not produce breakthroughs in mathematics and science nor is it the kind of thinking that results in significant learning. Deep thinking is a different and more basic way of using the mind. It results in the discontinuous “aha!” experience, which is the essence of creativity. It is at the heart of every paradigm shift or reframing of a problematic situation. The identification of deep thinking as the default state of the mind has the potential to reframe our current approach to technological change, education, and the nature of mathematics and science. For example, there is an unbridgeable gap between deep thinking and computer simulations of thinking. Many people suspect that such a gap exists, but find it difficult to make this intuition precise. This book identifies the way in which the authentic intelligence of deep thinking differs from the artificial intelligence of “big data” and “analytics”. Deep thinking is the essential ingredient in every significant learning experience, which leads to a new way to think about education. It is also essential to the construction of conceptual systems that are at the heart of mathematics and science, and of the technologies that shape the modern world. Deep thinking can be found whenever one conceptual system morphs into another. The sources of this study include the cognitive development of numbers in children, neuropsychology, the study of creativity, and the historical development of mathematics and science. The approach is unusual and original. It comes out of the author's lengthy experience as a mathematician, teacher, and writer of books about mathematics and science, such as How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics and The Blind Spot: Science and the Crisis of Uncertainty. Contents:What is Deep Thinking?Conceptual SystemsDeep Thinking in Mathematics and ScienceDeep Thinking in the Mind and the BrainDeep Thinking and CreativityDeep LearningGood TeachingUndergraduate MathematicsWhat the Mind Can Teach Us About MathematicsWhat Mathematics Can Teach Us About the MindReferences Readership: Students, graduate students and researchers with an interest in mathematics, mathematicians, scientists, philosophers, psychologists, and readers who use mathematics in their work. Key Features:In this book, the author, a mathematician, demonstrates the specific difference between creative mathematical thought and the analytic thought of logic and the artificial intelligence of computersThis book shows why learning is a creative activity and demonstrates how teaching and learning must undergo radical changes in this age of rapid technological changeMathematics is a model for how people think and it reveals the essence of intelligenceThe author, a mathematician, demonstrates that creativity is a basic feature of the world. The same phenomenon of creative intelligence underlies the theory of evolution, child development, learning, and scientific and mathematical research. Creativity is so natural that even babies are capable of it but so difficult that adults have great trouble with it. This book explains the reasons behind this apparent paradoxKeywords:Mathematics;Cognitive Development;Thinking;Education;Mathematics Education;Conceptual Systems

The Best Writing On Mathematics 2015

Author: Mircea Pitici
Publisher: Princeton University Press
ISBN: 1400873371
Size: 15.54 MB
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This annual anthology brings together the year's finest mathematics writing from around the world. Featuring promising new voices alongside some of the foremost names in the field, The Best Writing on Mathematics 2015 makes available to a wide audience many articles not easily found anywhere else—and you don’t need to be a mathematician to enjoy them. These writings offer surprising insights into the nature, meaning, and practice of mathematics today. They delve into the history, philosophy, teaching, and everyday occurrences of math, and take readers behind the scenes of today’s hottest mathematical debates. Here David Hand explains why we should actually expect unlikely coincidences to happen; Arthur Benjamin and Ethan Brown unveil techniques for improvising custom-made magic number squares; Dana Mackenzie describes how mathematicians are making essential contributions to the development of synthetic biology; Steven Strogatz tells us why it’s worth writing about math for people who are alienated from it; Lisa Rougetet traces the earliest written descriptions of Nim, a popular game of mathematical strategy; Scott Aaronson looks at the unexpected implications of testing numbers for randomness; and much, much more. In addition to presenting the year’s most memorable writings on mathematics, this must-have anthology includes a bibliography of other notable writings and an introduction by the editor, Mircea Pitici. This book belongs on the shelf of anyone interested in where math has taken us—and where it is headed.

Humanizing Mathematics And Its Philosophy

Author: Bharath Sriraman
Publisher: Birkhäuser
ISBN: 331961231X
Size: 37.54 MB
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This Festschrift contains numerous colorful and eclectic essays from well-known mathematicians, philosophers, logicians, and linguists celebrating the 90th birthday of Reuben Hersh. The essays offer, in part, attempts to answer the following questions set forth by Reuben himself as a focus for this volume: Can practicing mathematicians, as such, contribute anything to the philosophy of math? Can or should philosophers of math, as such, say anything to practicing mathematicians? Twenty or fifty years from now, what will be similar, and what will, or could, or should be altogether different: About the philosophy of math? About math education? About math research institutions? About data processing and scientific computing? The essays also offer glimpses into Reuben’s fertile mind and his lasting influence on the mathematical community, as well as revealing the diverse roots, obstacles and philosophical dispositions that characterize the working lives of mathematicians. With contributions from a veritable “who’s who” list of 20th century luminaries from mathematics and philosophy, as well as from Reuben himself, this volume will appeal to a wide variety of readers from curious undergraduates to prominent mathematicians.

From Giftedness To Gifted Education

Author: Jonathan Plucker
Publisher: Sourcebooks, Inc.
ISBN: 1618217089
Size: 36.88 MB
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Gifted education and talent development have numerous theories and conceptions for how to identify and serve students. This book helps introduce and apply these ideas to help reflect theory in practice. Each chapter introduces readers to a different theory by providing definitions of key concepts, explaining the fundamental conceptual/theoretical approach, and concluding with advice on how the conception can be put into practice. Suggestions for further reading are also provided. Some chapters are based on theories that have been around for decades, and some have been developed more recently. But all chapters focus on helping empower readers to understand and take action without having to reinvent the wheel.

Qigong Through The Seasons

Author: Ronald H. Davis
Publisher: Singing Dragon
ISBN: 0857011855
Size: 26.72 MB
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Within the holistic framework of the Daoist way, Ronald H. Davis provides a comprehensive and straightforward guide to the principles and practice of actively living in harmony with the seasonal energetic changes throughout the year. The program focuses on the importance of Qigong, meditation, and diet to support good mental and physical health. The author includes clear explanations of the theory underlying this approach to a healthy life, including the Five Phases of Change (Five Elements) of Chinese Medicine, the body's Qi system of channels and reservoirs, and the concept of chronobiology. For each season he provides a Qigong specific for the season, meditations, herbal information, and dietary recommendations with simple recipes to support the functions of the organ in focus in that season. This accessible program of Qigong for the seasons will be of interest to anyone seeking a healthy life, students of Qigong at any level, and the seasonal emphasis makes this an excellent quick reference for Qigong teachers.

Proceedings Of The Fifth International Congress On Mathematical Education

Author: CARASS
Publisher: Springer Science & Business Media
ISBN: 147574238X
Size: 38.20 MB
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International Congresses on Mathematical Education (ICMEs), under the auspices of the International Commission on Mathematical Instruction, are held every four years. Previous Congresses have been held in France (Lyons), England (Exeter), the Federal Republic of Germany (Karlsruhe), and the United States of America (Berkeley). The Fifth International Congress on Mathematical Education (lCME 5) was held in Adelaide, Australia, from August 24-30, 1984. More than 1800 participants from over 70 countries participated in the Congress, while some additional 200 people attended social functions and excursions. The program for ICME 5 was planned and structured by an International Program Committee, and implemented by the National Program Committee in Australia. For the main body of the program, Chief Organisers, assisted by Australian Coordinators, were invited to plan and prepare the individual components of the program which addressed a wide range of topics and interest areas. Each of these teams involved many individuals from around the world in the detailed planning and preparation of the working sessions for their area of program responsibility. For the actual working sessions at the Congress, the smallest group had some 60 members, while the largest had well over 300. In addition to the working sessions, there were three major plenary addresses, several specially invited presentations, and over 420 individual papers in the form of short communications, either as posters or brief talks.

How Mathematicians Think

Author: William Byers
Publisher: Princeton University Press
ISBN: 9780691145990
Size: 60.72 MB
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To many outsiders, mathematicians appear to think like computers, grimly grinding away with a strict formal logic and moving methodically--even algorithmically--from one black-and-white deduction to another. Yet mathematicians often describe their most important breakthroughs as creative, intuitive responses to ambiguity, contradiction, and paradox. A unique examination of this less-familiar aspect of mathematics, How Mathematicians Think reveals that mathematics is a profoundly creative activity and not just a body of formalized rules and results. Nonlogical qualities, William Byers shows, play an essential role in mathematics. Ambiguities, contradictions, and paradoxes can arise when ideas developed in different contexts come into contact. Uncertainties and conflicts do not impede but rather spur the development of mathematics. Creativity often means bringing apparently incompatible perspectives together as complementary aspects of a new, more subtle theory. The secret of mathematics is not to be found only in its logical structure. The creative dimensions of mathematical work have great implications for our notions of mathematical and scientific truth, and How Mathematicians Think provides a novel approach to many fundamental questions. Is mathematics objectively true? Is it discovered or invented? And is there such a thing as a "final" scientific theory? Ultimately, How Mathematicians Think shows that the nature of mathematical thinking can teach us a great deal about the human condition itself.