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Proof And Knowledge In Mathematics


Proof And Knowledge In Mathematics
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Explanation And Proof In Mathematics


Explanation And Proof In Mathematics
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Author : Gila Hanna
language : en
Publisher: Springer Science & Business Media
Release Date : 2009-12-04

Explanation And Proof In Mathematics written by Gila Hanna and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-12-04 with Education categories.


In the four decades since Imre Lakatos declared mathematics a "quasi-empirical science," increasing attention has been paid to the process of proof and argumentation in the field -- a development paralleled by the rise of computer technology and the mounting interest in the logical underpinnings of mathematics. Explanantion and Proof in Mathematics assembles perspectives from mathematics education and from the philosophy and history of mathematics to strengthen mutual awareness and share recent findings and advances in their interrelated fields. With examples ranging from the geometrists of the 17th century and ancient Chinese algorithms to cognitive psychology and current educational practice, contributors explore the role of refutation in generating proofs, the varied links between experiment and deduction, the use of diagrammatic thinking in addition to pure logic, and the uses of proof in mathematics education (including a critique of "authoritative" versus "authoritarian" teaching styles). A sampling of the coverage: The conjoint origins of proof and theoretical physics in ancient Greece. Proof as bearers of mathematical knowledge. Bridging knowing and proving in mathematical reasoning. The role of mathematics in long-term cognitive development of reasoning. Proof as experiment in the work of Wittgenstein. Relationships between mathematical proof, problem-solving, and explanation. Explanation and Proof in Mathematics is certain to attract a wide range of readers, including mathematicians, mathematics education professionals, researchers, students, and philosophers and historians of mathematics.



Proof And Knowledge In Mathematics


Proof And Knowledge In Mathematics
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Author : Michael Detlefsen
language : en
Publisher: Routledge
Release Date : 2005-08-18

Proof And Knowledge In Mathematics written by Michael Detlefsen and has been published by Routledge this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-08-18 with Education categories.


Distinguished contributors tackle the main problem that arizes when considering an epistemology for mathematics, the nature and sources of mathematical justification.



Proof And Knowledge In Mathematics


Proof And Knowledge In Mathematics
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Author : Michael Detlefsen
language : en
Publisher: Routledge
Release Date : 2005-08-18

Proof And Knowledge In Mathematics written by Michael Detlefsen and has been published by Routledge this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-08-18 with Philosophy categories.


These questions arise from any attempt to discover an epistemology for mathematics. This collection of essays considers various questions concerning the nature of justification in mathematics and possible sources of that justification. Among these are the question of whether mathematical justification is a priori or a posteriori in character, whether logical and mathematical differ, and if formalization plays a significant role in mathematical justification,



Proof And Proving In Mathematics Education


Proof And Proving In Mathematics Education
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Author : Gila Hanna
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-06-14

Proof And Proving In Mathematics Education written by Gila Hanna and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-06-14 with Education categories.


*THIS BOOK IS AVAILABLE AS OPEN ACCESS BOOK ON SPRINGERLINK* One of the most significant tasks facing mathematics educators is to understand the role of mathematical reasoning and proving in mathematics teaching, so that its presence in instruction can be enhanced. This challenge has been given even greater importance by the assignment to proof of a more prominent place in the mathematics curriculum at all levels. Along with this renewed emphasis, there has been an upsurge in research on the teaching and learning of proof at all grade levels, leading to a re-examination of the role of proof in the curriculum and of its relation to other forms of explanation, illustration and justification. This book, resulting from the 19th ICMI Study, brings together a variety of viewpoints on issues such as: The potential role of reasoning and proof in deepening mathematical understanding in the classroom as it does in mathematical practice. The developmental nature of mathematical reasoning and proof in teaching and learning from the earliest grades. The development of suitable curriculum materials and teacher education programs to support the teaching of proof and proving. The book considers proof and proving as complex but foundational in mathematics. Through the systematic examination of recent research this volume offers new ideas aimed at enhancing the place of proof and proving in our classrooms.



How To Prove It


How To Prove It
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Author : Daniel J. Velleman
language : en
Publisher: Cambridge University Press
Release Date : 2006-01-16

How To Prove It written by Daniel J. Velleman and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-01-16 with Mathematics categories.


Many students have trouble the first time they take a mathematics course in which proofs play a significant role. This new edition of Velleman's successful text will prepare students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. The author shows how complex proofs are built up from these smaller steps, using detailed 'scratch work' sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets. To give students the opportunity to construct their own proofs, this new edition contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software. No background beyond standard high school mathematics is assumed. This book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and of course mathematicians.



Connecting Mathematics And Mathematics Education


Connecting Mathematics And Mathematics Education
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Author : Erich Christian Wittmann
language : en
Publisher: Springer Nature
Release Date : 2020-12-09

Connecting Mathematics And Mathematics Education written by Erich Christian Wittmann and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-12-09 with Education categories.


This open access book features a selection of articles written by Erich Ch. Wittmann between 1984 to 2019, which shows how the “design science conception” has been continuously developed over a number of decades. The articles not only describe this conception in general terms, but also demonstrate various substantial learning environments that serve as typical examples. In terms of teacher education, the book provides clear information on how to combine (well-understood) mathematics and methods courses to benefit of teachers. The role of mathematics in mathematics education is often explicitly and implicitly reduced to the delivery of subject matter that then has to be selected and made palpable for students using methods imported from psychology, sociology, educational research and related disciplines. While these fields have made significant contributions to mathematics education in recent decades, it cannot be ignored that mathematics itself, if well understood, provides essential knowledge for teaching mathematics beyond the pure delivery of subject matter. For this purpose, mathematics has to be conceived of as an organism that is deeply rooted in elementary operations of the human mind, which can be seamlessly developed to higher and higher levels so that the full richness of problems of various degrees of difficulty, and different means of representation, problem-solving strategies, and forms of proof can be used in ways that are appropriate for the respective level. This view of mathematics is essential for designing learning environments and curricula, for conducting empirical studies on truly mathematical processes and also for implementing the findings of mathematics education in teacher education, where it is crucial to take systemic constraints into account.



Proof In Mathematics Education


Proof In Mathematics Education
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Author : David A. Reid
language : en
Publisher: BRILL
Release Date : 2010-01-01

Proof In Mathematics Education written by David A. Reid and has been published by BRILL this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-01-01 with Education categories.


Research on teaching and learning proof and proving has expanded in recent decades. This reflects the growth of mathematics education research in general, but also an increased emphasis on proof in mathematics education. This development is a welcome one for those interested in the topic, but also poses a challenge, especially to teachers and new scholars. It has become more and more difficult to get an overview of the field and to identify the key concepts used in research on proof and proving.



The Meaning Of Proofs


The Meaning Of Proofs
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Author : Gabriele Lolli
language : en
Publisher: MIT Press
Release Date : 2022-09-27

The Meaning Of Proofs written by Gabriele Lolli and has been published by MIT Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-09-27 with Mathematics categories.


Why mathematics is not merely formulaic: an argument that to write a mathematical proof is tantamount to inventing a story. In The Meaning of Proofs, mathematician Gabriele Lolli argues that to write a mathematical proof is tantamount to inventing a story. Lolli offers not instructions for how to write mathematical proofs, but a philosophical and poetic reflection on mathematical proofs as narrative. Mathematics, imprisoned within its symbols and images, Lolli writes, says nothing if its meaning is not narrated in a story. The minute mathematicians open their mouths to explain something—the meaning of x, how to find y—they are framing a narrative. Every proof is the story of an adventure, writes Lolli, a journey into an unknown land to open a new, connected route; once the road is open, we correct it, expand it. Just as fairy tales offer a narrative structure in which new characters can be inserted into recurring forms of the genre in original ways, in mathematics, each new abstract concept is the protagonist of a different theory supported by the general techniques of mathematical reasoning. In ancient Greece, there was more than an analogy between literature and mathematics, there was direct influence. Euclid’s proofs have roots in poetry and rhetoric. Mathematics, Lolli asserts, is not the mere manipulation of formulas.



Proof And The Art Of Mathematics


Proof And The Art Of Mathematics
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Author : Joel David Hamkins
language : en
Publisher: MIT Press
Release Date : 2021-02-23

Proof And The Art Of Mathematics written by Joel David Hamkins and has been published by MIT Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-02-23 with Mathematics categories.


How to write mathematical proofs, shown in fully-worked out examples. This is a companion volume Joel Hamkins's Proof and the Art of Mathematics, providing fully worked-out solutions to all of the odd-numbered exercises as well as a few of the even-numbered exercises. In many cases, the solutions go beyond the exercise question itself to the natural extensions of the ideas, helping readers learn how to approach a mathematical investigation. As Hamkins asks, "Once you have solved a problem, why not push the ideas harder to see what further you can prove with them?" These solutions offer readers examples of how to write a mathematical proofs. The mathematical development of this text follows the main book, with the same chapter topics in the same order, and all theorem and exercise numbers in this text refer to the corresponding statements of the main text.



Learning To Reason


Learning To Reason
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Author : Nancy Rodgers
language : en
Publisher: John Wiley & Sons
Release Date : 2011-09-15

Learning To Reason written by Nancy Rodgers and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-09-15 with Mathematics categories.


Learn how to develop your reasoning skills and how to writewell-reasoned proofs Learning to Reason shows you how to use the basic elements ofmathematical language to develop highly sophisticated, logicalreasoning skills. You'll get clear, concise, easy-to-followinstructions on the process of writing proofs, including thenecessary reasoning techniques and syntax for constructingwell-written arguments. Through in-depth coverage of logic, sets,and relations, Learning to Reason offers a meaningful, integratedview of modern mathematics, cuts through confusing terms and ideas,and provides a much-needed bridge to advanced work in mathematicsas well as computer science. Original, inspiring, and designed formaximum comprehension, this remarkable book: * Clearly explains how to write compound sentences in equivalentforms and use them in valid arguments * Presents simple techniques on how to structure your thinking andwriting to form well-reasoned proofs * Reinforces these techniques through a survey of sets--thebuilding blocks of mathematics * Examines the fundamental types of relations, which is "where theaction is" in mathematics * Provides relevant examples and class-tested exercises designed tomaximize the learning experience * Includes a mind-building game/exercise space atwww.wiley.com/products/subject/mathematics/