Proof And Knowledge In Mathematics

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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.

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 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.

Teaching And Learning Proof Across The Grades

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Author : Despina A. Stylianou
language : en
Publisher: Routledge
Release Date : 2010-09-23

Teaching And Learning Proof Across The Grades written by Despina A. Stylianou and has been published by Routledge this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-09-23 with Education categories.

A Co-Publication of Routledge for the National Council of Teachers of Mathematics (NCTM) In recent years there has been increased interest in the nature and role of proof in mathematics education; with many mathematics educators advocating that proof should be a central part of the mathematics education of students at all grade levels. This important new collection provides that much-needed forum for mathematics educators to articulate a connected K-16 "story" of proof. Such a story includes understanding how the forms of proof, including the nature of argumentation and justification as well as what counts as proof, evolve chronologically and cognitively and how curricula and instruction can support the development of students’ understanding of proof. Collectively these essays inform educators and researchers at different grade levels about the teaching and learning of proof at each level and, thus, help advance the design of further empirical and theoretical work in this area. By building and extending on existing research and by allowing a variety of voices from the field to be heard, Teaching and Learning Proof Across the Grades not only highlights the main ideas that have recently emerged on proof research, but also defines an agenda for future study.

Problems And Proofs In Numbers And Algebra

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Author : Richard S. Millman
language : en
Publisher: Springer
Release Date : 2015-02-09

Problems And Proofs In Numbers And Algebra written by Richard S. Millman and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-02-09 with Mathematics categories.

Focusing on an approach of solving rigorous problems and learning how to prove, this volume is concentrated on two specific content themes, elementary number theory and algebraic polynomials. The benefit to readers who are moving from calculus to more abstract mathematics is to acquire the ability to understand proofs through use of the book and the multitude of proofs and problems that will be covered throughout. This book is meant to be a transitional precursor to more complex topics in analysis, advanced number theory, and abstract algebra. To achieve the goal of conceptual understanding, a large number of problems and examples will be interspersed through every chapter. The problems are always presented in a multi-step and often very challenging, requiring the reader to think about proofs, counter-examples, and conjectures. Beyond the undergraduate mathematics student audience, the text can also offer a rigorous treatment of mathematics content (numbers and algebra) for high-achieving high school students. Furthermore, prospective teachers will add to the breadth of the audience as math education majors, will understand more thoroughly methods of proof, and will add to the depth of their mathematical knowledge. In the past, PNA has been taught in a "problem solving in middle school” course (twice), to a quite advanced high school students course (three semesters), and three times as a secondary resource for a course for future high school teachers. PNA is suitable for secondary math teachers who look for material to encourage and motivate more high achieving students.

Understanding Proof

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Author : Tom Bennison
language : en
Publisher: Tarquin Group
Release Date : 2020-12

Understanding Proof written by Tom Bennison and has been published by Tarquin Group this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-12 with Logic, Symbolic and mathematical categories.

"Proof is central to all mathematical thinking. This book provides students with an excellent all round guide to proof including clear explanation and examples. Problems presented have full worked solutions within the book. ... [T]his book is an essential teaching, learning and revision guide."--Back cover.

Investigating Notions Of Proof

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Author : Keir Finlow-Bates
language : en
Publisher: Lulu.com
Release Date : 2009-10

Investigating Notions Of Proof written by Keir Finlow-Bates and has been published by Lulu.com this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-10 with Education categories.

Although proof is seen by most mathematicians as lying at the heart of mathematics, it is rarely explicitly taught at any point in the mathematics curriculum. This is compounded by the fact that within the mathematics and education communities there is no clear definition of or consensus on what actually constitutes proof. In this book a fallibilist approach based on the work of Imre Lakatos is adopted, and proof and proving are set within the context of a form of social knowledge in order to gain insight into the proof-activities of degree level mathematics students.

Forms Of Mathematical Knowledge

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Author : Dina Tirosh
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-14

Forms Of Mathematical Knowledge written by Dina Tirosh 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 2013-03-14 with Education categories.

What mathematics is entailed in knowing to act in a moment? Is tacit, rhetorical knowledge significant in mathematics education? What is the role of intuitive models in understanding, learning and teaching mathematics? Are there differences between elementary and advanced mathematical thinking? Why can't students prove? What are the characteristics of teachers' ways of knowing? This book focuses on various types of knowledge that are significant for learning and teaching mathematics. The first part defines, discusses and contrasts psychological, philosophical and didactical issues related to various types of knowledge involved in the learning of mathematics. The second part describes ideas about forms of mathematical knowledge that are important for teachers to know and ways of implementing such ideas in preservice and in-service education. The chapters provide a wide overview of current thinking about mathematics learning and teaching which is of interest for researchers in mathematics education and mathematics educators. Topics covered include the role of intuition in mathematics learning and teaching, the growth from elementary to advanced mathematical thinking, the significance of genres and rhetoric for the learning of mathematics and the characterization of teachers' ways of knowing.

The Nature Of Mathematical Knowledge

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Author : Philip Kitcher
language : en
Publisher: Oxford University Press, USA
Release Date : 1984

The Nature Of Mathematical Knowledge written by Philip Kitcher and has been published by Oxford University Press, USA this book supported file pdf, txt, epub, kindle and other format this book has been release on 1984 with Electronic books categories.

This book argues against the view that mathematical knowledge is a priori, contending that mathematics is an empirical science and develops historically, just as natural sciences do. Kitcher presents a complete, systematic, and richly detailed account of the nature of mathematical knowledge and its historical development, focusing on such neglected issues as how and why mathematical language changes, why certain questions assume overriding importance, and how standards of proof are modified.

Proof Technology In Mathematics Research And Teaching

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Author : Gila Hanna
language : en
Publisher: Springer Nature
Release Date : 2019-10-02

Proof Technology In Mathematics Research And Teaching written by Gila Hanna and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-10-02 with Education categories.

This book presents chapters exploring the most recent developments in the role of technology in proving. The full range of topics related to this theme are explored, including computer proving, digital collaboration among mathematicians, mathematics teaching in schools and universities, and the use of the internet as a site of proof learning. Proving is sometimes thought to be the aspect of mathematical activity most resistant to the influence of technological change. While computational methods are well known to have a huge importance in applied mathematics, there is a perception that mathematicians seeking to derive new mathematical results are unaffected by the digital era. The reality is quite different. Digital technologies have transformed how mathematicians work together, how proof is taught in schools and universities, and even the nature of proof itself. Checking billions of cases in extremely large but finite sets, impossible a few decades ago, has now become a standard method of proof. Distributed proving, by teams of mathematicians working independently on sections of a problem, has become very much easier as digital communication facilitates the sharing and comparison of results. Proof assistants and dynamic proof environments have influenced the verification or refutation of conjectures, and ultimately how and why proof is taught in schools. And techniques from computer science for checking the validity of programs are being used to verify mathematical proofs. Chapters in this book include not only research reports and case studies, but also theoretical essays, reviews of the state of the art in selected areas, and historical studies. The authors are experts in the field.