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How Mathematicians Think

Author : William Byers
Publisher : Princeton University Press
Release Date : 2010-04-12
ISBN : 1400833957
Pages : 424 pages
Rating : 4.0/5 (833 Download)

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Download Free How Mathematicians Think PDF by William Byers Full Book and published by Princeton University Press. This book was released on 2010-04-12 with total page 424 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Thinking Like a Mathematician

Author : Eugene F. Provenzo
Publisher :
Release Date : 1998
ISBN : 9780760902622
Pages : pages
Rating : 4.2/5 (623 Download)

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Download Free Thinking Like a Mathematician PDF by Eugene F. Provenzo Full Book and published by . This book was released on 1998 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Deep Thinking

Author : William Byers
Publisher : World Scientific Publishing Company Incorporated
Release Date : 2015
ISBN : 9789814618038
Pages : 246 pages
Rating : 4.8/5 (39 Download)

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Download Free Deep Thinking PDF by William Byers Full Book and published by World Scientific Publishing Company Incorporated. This book was released on 2015 with total page 246 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

The Foundations of Mathematics

Author : Ian Stewart
Publisher : Oxford University Press
Release Date : 2015
ISBN : 0198706448
Pages : 408 pages
Rating : 4.8/5 (76 Download)

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Download Free The Foundations of Mathematics PDF by Ian Stewart Full Book and published by Oxford University Press. This book was released on 2015 with total page 408 pages. Available in PDF, EPUB and Kindle. Book excerpt: The transition from school mathematics to university mathematics is seldom straightforward. Students are faced with a disconnect between the algorithmic and informal attitude to mathematics at school, versus a new emphasis on proof, based on logic, and a more abstract development of general concepts, based on set theory. The authors have many years' experience of the potential difficulties involved, through teaching first-year undergraduates and researching the ways in which students and mathematicians think. The book explains the motivation behind abstract foundational material based on students' experiences of school mathematics, and explicitly suggests ways students can make sense of formal ideas. This second edition takes a significant step forward by not only making the transition from intuitive to formal methods, but also by reversing the process- using structure theorems to prove that formal systems have visual and symbolic interpretations that enhance mathematical thinking. This is exemplified by a new chapter on the theory of groups. While the first edition extended counting to infinite cardinal numbers, the second also extends the real numbers rigorously to larger ordered fields. This links intuitive ideas in calculus to the formal epsilon-delta methods of analysis. The approach here is not the conventional one of 'nonstandard analysis', but a simpler, graphically based treatment which makes the notion of an infinitesimal natural and straightforward. This allows a further vision of the wider world of mathematical thinking in which formal definitions and proof lead to amazing new ways of defining, proving, visualising and symbolising mathematics beyond previous expectations.

How Mathematicians Think

Author : British Broadcasting Corporation
Publisher :
Release Date : 1950
ISBN :
Pages : pages
Rating : 4.1/5 (11 Download)

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Download Free How Mathematicians Think PDF by British Broadcasting Corporation Full Book and published by . This book was released on 1950 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

How Not to Be Wrong

Author : Jordan Ellenberg
Publisher : Penguin
Release Date : 2015-05-26
ISBN : 0143127535
Pages : 482 pages
Rating : 4.3/5 (127 Download)

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Download Free How Not to Be Wrong PDF by Jordan Ellenberg Full Book and published by Penguin. This book was released on 2015-05-26 with total page 482 pages. Available in PDF, EPUB and Kindle. Book excerpt: “Witty, compelling, and just plain fun to read . . ." —Evelyn Lamb, Scientific American The Freakonomics of math—a math-world superstar unveils the hidden beauty and logic of the world and puts its power in our hands The math we learn in school can seem like a dull set of rules, laid down by the ancients and not to be questioned. In How Not to Be Wrong, Jordan Ellenberg shows us how terribly limiting this view is: Math isn’t confined to abstract incidents that never occur in real life, but rather touches everything we do—the whole world is shot through with it. Math allows us to see the hidden structures underneath the messy and chaotic surface of our world. It’s a science of not being wrong, hammered out by centuries of hard work and argument. Armed with the tools of mathematics, we can see through to the true meaning of information we take for granted: How early should you get to the airport? What does “public opinion” really represent? Why do tall parents have shorter children? Who really won Florida in 2000? And how likely are you, really, to develop cancer? How Not to Be Wrong presents the surprising revelations behind all of these questions and many more, using the mathematician’s method of analyzing life and exposing the hard-won insights of the academic community to the layman—minus the jargon. Ellenberg chases mathematical threads through a vast range of time and space, from the everyday to the cosmic, encountering, among other things, baseball, Reaganomics, daring lottery schemes, Voltaire, the replicability crisis in psychology, Italian Renaissance painting, artificial languages, the development of non-Euclidean geometry, the coming obesity apocalypse, Antonin Scalia’s views on crime and punishment, the psychology of slime molds, what Facebook can and can’t figure out about you, and the existence of God. Ellenberg pulls from history as well as from the latest theoretical developments to provide those not trained in math with the knowledge they need. Math, as Ellenberg says, is “an atomic-powered prosthesis that you attach to your common sense, vastly multiplying its reach and strength.” With the tools of mathematics in hand, you can understand the world in a deeper, more meaningful way. How Not to Be Wrong will show you how.

Thinking Mathematics: Foundations

Author :
Publisher :
Release Date : 1993
ISBN :
Pages : pages
Rating : 4.L/5 (31 Download)

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Download Free Thinking Mathematics: Foundations PDF by Full Book and published by . This book was released on 1993 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Algebra and Algebraic Thinking in School Mathematics

Author : Carole E. Greenes
Publisher :
Release Date : 2008
ISBN :
Pages : 352 pages
Rating : 4.R/5 (3 Download)

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Download Free Algebra and Algebraic Thinking in School Mathematics PDF by Carole E. Greenes Full Book and published by . This book was released on 2008 with total page 352 pages. Available in PDF, EPUB and Kindle. Book excerpt: Examines the status of algebra in our schools and the changes that the curriculum has undergone over the past several years. Includes successful classroom practises for developing algebraic reasoning abilities and improving overall understanding.

Fostering and Sustaining Mathematics Thinking Through Problem Solving

Author : John Mason
Publisher : Deakin University Press
Release Date : 1991
ISBN :
Pages : 120 pages
Rating : 4.0/5 (15 Download)

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Download Free Fostering and Sustaining Mathematics Thinking Through Problem Solving PDF by John Mason Full Book and published by Deakin University Press. This book was released on 1991 with total page 120 pages. Available in PDF, EPUB and Kindle. Book excerpt: Prepared for courses ECT405, ECT705 offered by the Faculty of Education in Deakin University's Open Campus Program.

Journal of the Transactions of the Victoria Institute, Or Philosophical Society of Great Britain

Author : Victoria Institute (Great Britain)
Publisher :
Release Date : 1871
ISBN :
Pages : 566 pages
Rating : 4.1/5 (951 Download)

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Download Free Journal of the Transactions of the Victoria Institute, Or Philosophical Society of Great Britain PDF by Victoria Institute (Great Britain) Full Book and published by . This book was released on 1871 with total page 566 pages. Available in PDF, EPUB and Kindle. Book excerpt: Each volume includes list of members, and "objects of the institute" (except v. 31, which has no list of members). Beginning with v. 12, a list of the papers contained in preceding volumes is issued regularly with each volume.

Thinking Globally about Mathematics and Science Education

Author : University of British Columbia. Centre for the Study of Curriculum and Instruction
Publisher :
Release Date : 1995
ISBN :
Pages : 178 pages
Rating : 4.0/5 (25 Download)

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Download Free Thinking Globally about Mathematics and Science Education PDF by University of British Columbia. Centre for the Study of Curriculum and Instruction Full Book and published by . This book was released on 1995 with total page 178 pages. Available in PDF, EPUB and Kindle. Book excerpt:

First, supplementary, and second reports, with minutes of evidence and appendices. 1872 (c.536)

Author : Great Britain. Royal Commission on Scientific Instruction and the Advancement of Science
Publisher :
Release Date : 1872
ISBN :
Pages : pages
Rating : 4.0/5 (15 Download)

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Download Free First, supplementary, and second reports, with minutes of evidence and appendices. 1872 (c.536) PDF by Great Britain. Royal Commission on Scientific Instruction and the Advancement of Science Full Book and published by . This book was released on 1872 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Thinking Through Mathematics

Author : Edward A. Silver
Publisher :
Release Date : 1990
ISBN :
Pages : 68 pages
Rating : 4.R/5 (3 Download)

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Download Free Thinking Through Mathematics PDF by Edward A. Silver Full Book and published by . This book was released on 1990 with total page 68 pages. Available in PDF, EPUB and Kindle. Book excerpt: This document focuses on how mathematics teaching and learning can be improved by developing more powerful approaches to connect thinking and mathematics. It proposes changing perspectives on what it means to learn and do mathematics and explores how these perspectives can be incorporated into the teaching of secondary school mathematics. Chapter 1 offers a view of mathematics as emerging largely from individual and social activity rather than from textbooks, worksheets, and tradition. The learner is depicted as someone who actively constructs meaning instead of passively receiving it. Chapter 2 considers how a greater emphasis on communication (discussion, debate, recording, and writing) stimulates and uncovers students' learning and thinking and leads to a deeper understanding by both teachers and students. Chapter 3 explores how teachers might encourage greater inquiry and communication in a secondary school class by making minor, but thought-provoking changes in ordinary problems and situations. Finally, chapter 4 gives some practical advice on transforming the mathematics classroom into a place where students are expected not only to absorb and consume mathematics but also to produce and think about it. Contains 17 references and 17 figures. (MKR)

Great Currents of Mathematical Thought

Author : François Le Lionnais
Publisher :
Release Date : 1971
ISBN :
Pages : pages
Rating : 4.0/5 (3 Download)

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Download Free Great Currents of Mathematical Thought PDF by François Le Lionnais Full Book and published by . This book was released on 1971 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

The Nature of Mathematical Thinking

Author : Robert J. Sternberg
Publisher : Routledge
Release Date : 2012-10-12
ISBN : 1136487506
Pages : 352 pages
Rating : 4.6/5 (487 Download)

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Download Free The Nature of Mathematical Thinking PDF by Robert J. Sternberg Full Book and published by Routledge. This book was released on 2012-10-12 with total page 352 pages. Available in PDF, EPUB and Kindle. Book excerpt: Why do some children seem to learn mathematics easily and others slave away at it, learning it only with great effort and apparent pain? Why are some people good at algebra but terrible at geometry? How can people who successfully run a business as adults have been failures at math in school? How come some professional mathematicians suffer terribly when trying to balance a checkbook? And why do school children in the United States perform so dismally in international comparisons? These are the kinds of real questions the editors set out to answer, or at least address, in editing this book on mathematical thinking. Their goal was to seek a diversity of contributors representing multiple viewpoints whose expertise might converge on the answers to these and other pressing and interesting questions regarding this subject. The chapter authors were asked to focus on their own approach to mathematical thinking, but also to address a common core of issues such as the nature of mathematical thinking, how it is similar to and different from other kinds of thinking, what makes some people or some groups better than others in this subject area, and how mathematical thinking can be assessed and taught. Their work is directed to a diverse audience -- psychologists interested in the nature of mathematical thinking and abilities, computer scientists who want to simulate mathematical thinking, educators involved in teaching and testing mathematical thinking, philosophers who need to understand the qualitative aspects of logical thinking, anthropologists and others interested in how and why mathematical thinking seems to differ in quality across cultures, and laypeople and others who have to think mathematically and want to understand how they are going to accomplish that feat.

Journal for Research in Mathematics Education

Author :
Publisher :
Release Date : 2000
ISBN :
Pages : pages
Rating : 4.R/5 (3 Download)

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Download Free Journal for Research in Mathematics Education PDF by Full Book and published by . This book was released on 2000 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Connecting Mathematics for Elementary Teachers

Author : David Feikes
Publisher : Addison-Wesley Longman
Release Date : 2009
ISBN :
Pages : 358 pages
Rating : 4.1/5 (822 Download)

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Download Free Connecting Mathematics for Elementary Teachers PDF by David Feikes Full Book and published by Addison-Wesley Longman. This book was released on 2009 with total page 358 pages. Available in PDF, EPUB and Kindle. Book excerpt: --Book Jacket.