Understanding Mathematical Proof

Author: John Taylor

Publisher: CRC Press

ISBN: 9781138466852


Page: N.A

View: 7562

The notion of proof is central to mathematics yet it is one of the most difficult aspects of the subject to teach and master. In particular, undergraduate mathematics students often experience difficulties in understanding and constructing proofs. Understanding Mathematical Proof describes the nature of mathematical proof, explores the various techniques that mathematicians adopt to prove their results, and offers advice and strategies for constructing proofs. It will improve students� ability to understand proofs and construct correct proofs of their own. The first chapter of the text introduces the kind of reasoning that mathematicians use when writing their proofs and gives some example proofs to set the scene. The book then describes basic logic to enable an understanding of the structure of both individual mathematical statements and whole mathematical proofs. It also explains the notions of sets and functions and dissects several proofs with a view to exposing some of the underlying features common to most mathematical proofs. The remainder of the book delves further into different types of proof, including direct proof, proof using contrapositive, proof by contradiction, and mathematical induction. The authors also discuss existence and uniqueness proofs and the role of counter examples.

Introduction to Advanced Mathematics: A Guide to Understanding Proofs

Author: Connie M. Campbell

Publisher: Cengage Learning

ISBN: 1133168787

Category: Mathematics

Page: 144

View: 7873

This text offers a crucial primer on proofs and the language of mathematics. Brief and to the point, it lays out the fundamental ideas of abstract mathematics and proof techniques that students will need to master for other math courses. Campbell presents these concepts in plain English, with a focus on basic terminology and a conversational tone that draws natural parallels between the language of mathematics and the language students communicate in every day. The discussion highlights how symbols and expressions are the building blocks of statements and arguments, the meanings they convey, and why they are meaningful to mathematicians. In-class activities provide opportunities to practice mathematical reasoning in a live setting, and an ample number of homework exercises are included for self-study. This text is appropriate for a course in Foundations of Advanced Mathematics taken by students who've had a semester of calculus, and is designed to be accessible to students with a wide range of mathematical proficiency. It can also be used as a self-study reference, or as a supplement in other math courses where additional proofs practice is needed. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.

Making the Connection

Research and Teaching in Undergraduate Mathematics Education

Author: Marilyn Paula Carlson,Chris Rasmussen

Publisher: MAA

ISBN: 9780883851838

Category: Mathematics

Page: 319

View: 6505

The chapters in this volume convey insights from mathematics education research that have direct implications for anyone interested in improving teaching and learning in undergraduate mathematics. This synthesis of research on learning and teaching mathematics provides relevant information for any math department or individual faculty member who is working to improve introductory proof courses, the longitudinal coherence of precalculus through differential equations, students' mathematical thinking and problem-solving abilities, and students' understanding of fundamental ideas such as variable and rate of change. Other chapters include information about programs that have been successful in supporting students' continued study of mathematics. The authors provide many examples and ideas to help the reader infuse the knowledge from mathematics education research into mathematics teaching practice. University mathematicians and community college faculty spend much of their time engaged in work to improve their teaching. Frequently, they are left to their own experiences and informal conversations with colleagues to develop new approaches to support student learning and their continuation in mathematics. Over the past 30 years, research in undergraduate mathematics education has produced knowledge about the development of mathematical understandings and models for supporting students' mathematical learning. Currently, very little of this knowledge is affecting teaching practice. We hope that this volume will open a meaningful dialogue between researchers and practitioners toward the goal of realizing improvements in undergraduate mathematics curriculum and instruction.

Das BUCH der Beweise

Author: Martin Aigner,Günter M. Ziegler

Publisher: Springer-Verlag

ISBN: 3662064545

Category: Mathematics

Page: 247

View: 6386

Die elegantesten mathematischen Beweise, spannend und für jeden Interessierten verständlich. "Der Beweis selbst, seine Ästhetik, seine Pointe geht ins Geschichtsbuch der Königin der Wissenschaften ein. Ihre Anmut offenbart sich in dem gelungenen und geschickt illustrierten Buch." Die Zeit

Explanation and Proof in Mathematics

Philosophical and Educational Perspectives

Author: Gila Hanna,Hans Niels Jahnke,Helmut Pulte

Publisher: Springer Science & Business Media

ISBN: 9781441905765

Category: Education

Page: 294

View: 8750

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.

The Foundations of Mathematics

Author: Ian Stewart,David Tall

Publisher: OUP Oxford

ISBN: 0191016489

Category: Mathematics

Page: 432

View: 8888

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.

Visualization, Explanation and Reasoning Styles in Mathematics

Author: P. Mancosu,Klaus Frovin Jørgensen,S.A. Pedersen

Publisher: Springer Science & Business Media

ISBN: 1402033354

Category: Mathematics

Page: 300

View: 8616

In the 20th century philosophy of mathematics has to a great extent been dominated by views developed during the so-called foundational crisis in the beginning of that century. These views have primarily focused on questions pertaining to the logical structure of mathematics and questions regarding the justi?cation and consistency of mathematics. Paradigmatic in this - spect is Hilbert’s program which inherits from Frege and Russell the project to formalize all areas of ordinary mathematics and then adds the requi- ment of a proof, by epistemically privileged means (?nitistic reasoning), of the consistency of such formalized theories. While interest in modi?ed v- sions of the original foundational programs is still thriving, in the second part of the twentieth century several philosophers and historians of mat- matics have questioned whether such foundational programs could exhaust the realm of important philosophical problems to be raised about the nature of mathematics. Some have done so in open confrontation (and hostility) to the logically based analysis of mathematics which characterized the cl- sical foundational programs, while others (and many of the contributors to this book belong to this tradition) have only called for an extension of the range of questions and problems that should be raised in connection with an understanding of mathematics. The focus has turned thus to a consideration of what mathematicians are actually doing when they produce mathematics. Questions concerning concept-formation, understanding, heuristics, changes instyle of reasoning, the role of analogies and diagrams etc.

How to Think About Analysis

Author: Lara Alcock

Publisher: OUP Oxford

ISBN: 0191035386

Category: Mathematics

Page: 272

View: 7946

Analysis (sometimes called Real Analysis or Advanced Calculus) is a core subject in most undergraduate mathematics degrees. It is elegant, clever and rewarding to learn, but it is hard. Even the best students find it challenging, and those who are unprepared often find it incomprehensible at first. This book aims to ensure that no student need be unprepared. It is not like other Analysis books. It is not a textbook containing standard content. Rather, it is designed to be read before arriving at university and/or before starting an Analysis course, or as a companion text once a course is begun. It provides a friendly and readable introduction to the subject by building on the student's existing understanding of six key topics: sequences, series, continuity, differentiability, integrability and the real numbers. It explains how mathematicians develop and use sophisticated formal versions of these ideas, and provides a detailed introduction to the central definitions, theorems and proofs, pointing out typical areas of difficulty and confusion and explaining how to overcome these. The book also provides study advice focused on the skills that students need if they are to build on this introduction and learn successfully in their own Analysis courses: it explains how to understand definitions, theorems and proofs by relating them to examples and diagrams, how to think productively about proofs, and how theories are taught in lectures and books on advanced mathematics. It also offers practical guidance on strategies for effective study planning. The advice throughout is research based and is presented in an engaging style that will be accessible to students who are new to advanced abstract mathematics.

Aspects of Teaching Secondary Mathematics

Perspectives on Practice

Author: Linda Haggarty

Publisher: Routledge

ISBN: 1134500963

Category: Education

Page: 312

View: 8577

If learners in the classroom are to be excited by mathematics, teachers need to be both well informed about current initiatives and able to see how what is expected of them can be translated into rich and stimulating classroom strategies. The book examines current initiatives that affect teaching mathematics and identifies pointers for action in the classroom. Divided into three major sections, it looks at: the changing mathematics classroom at primary, secondary and tertiary level major components of the secondary curriculum practical pedagogical issues of particular concern to mathematics teachers. Each issue is explores in terms of major underpinnings and research in that area, and practical ideas can be drawn from the text and implemented in the reader's classroom practice. Each chapter has been written by a well-respected writer, researcher and practitioner in their field and all share a common goal: to look thoughtfully and intelligently at some of the practical issues facing mathematics teachers and offer their perspectives on those issues.

Church's Thesis After 70 Years

Author: Adam Olszewski,Jan Wolenski,Robert Janusz

Publisher: Walter de Gruyter

ISBN: 3110325462

Category: Philosophy

Page: 551

View: 3827

Church's Thesis (CT) was first published by Alonzo Church in 1935. CT is a proposition that identifies two notions: an intuitive notion of a effectively computable function defined in natural numbers with the notion of a recursive function. Despite of the many efforts of prominent scientists, Church's Thesis has never been falsified. There exists a vast literature concerning the thesis. The aim of the book is to provide one volume summary of the state of research on Church's Thesis. These include the following: different formulations of CT, CT and intuitionism, CT and intensional mathematics, CT and physics, the epistemic status of CT, CT and philosophy of mind, provability of CT and CT and functional programming.

What Does Understanding Mathematics Mean for Teachers?

Relationship as a Metaphor for Knowing

Author: Yuichi Handa

Publisher: Taylor & Francis

ISBN: 1136870539

Category: Education

Page: 168

View: 6394

This book opens up alternative ways of thinking and talking about ways in which a person can "know" a subject (in this case, mathematics), leading to a reconsideration of what it may mean to be a teacher of that subject. In a number of European languages, a distinction is made in ways of knowing that in the English language is collapsed into the singular word know. In French, for example, to know in the savoir sense is to know things, facts, names, how and why things work, and so on, whereas to know in the connaître sense is to know a person, a place, or even a thing—namely, an other— in such a way that one is familiar with, or in relationship with this other. Primarily through phenomenological reflection with a touch of empirical input, this book fleshes out an image for what a person’s connaître knowing of mathematics might mean, turning to mathematics teachers and teacher educators to help clarify this image.

Investigating Notions of Proof

Author: Keir Finlow-Bates

Publisher: Lulu.com

ISBN: 9529262922

Category: Education

Page: 200

View: 5137

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.

Understanding Real Analysis

Author: Paul Zorn

Publisher: CRC Press

ISBN: 1439894329

Category: Mathematics

Page: 362

View: 9175

This book is a one-semester text for an introduction to real analysis. The author's primary aims are to develop ideas already familiar from elementary calculus in a rigorous manner and to help students deeply understand some basic but crucial mathematical ideas, and to see how definitions, proofs, examples, and other forms of mathematical "apparatus" work together to create a unified theory. A key feature of the book is that it includes substantial treatment of some foundational material, including general theory of functions, sets, cardinality, and basic proof techniques.

An Introduction to Analysis

Second Edition

Author: James R. Kirkwood

Publisher: Waveland Press

ISBN: 1478609281

Category: Mathematics

Page: 278

View: 4692

An Introduction to Analysis, Second Edition provides a mathematically rigorous introduction to analysis of real-valued functions of one variable. The text is written to ease the transition from primarily computational to primarily theoretical mathematics. Numerous examples and exercises help students to understand mathematical proofs in an abstract setting, as well as to be able to formulate and write them. The material is as clear and intuitive as possible while still maintaining mathematical integrity. The author presents abstract mathematics in a way that makes the subject both understandable and exciting to students.

In Praise of the Cognitive Emotions (Routledge Revivals)

And Other Essays in the Philosophy of Education

Author: Israel Scheffler

Publisher: Routledge

ISBN: 1136955607

Category: Education

Page: 190

View: 7957

First published in 1991, In Praise of Cognitive Emotions comprises fourteen of Scheffler's most recent essays – all of which challenge contemporary notions of education and rationality. While defending the ideal of rationality, he insists that rationality not be identified with a mental faculty or a mechanism of inference but taken rather as the capactity to grasp principles and purposes and to evaluate them in the light of relevant reasons. Examining a broad range of issues – from computers in school to math education, from metaphor to morality – these essays are unified by Scheffler's conviction of the primacy of critical thought in education. Scheffler is especially concerned to promote a broad interpretation of rationality to counteract the narrowing of vision accompanying the technological revolution now sweeping education. Addressing three specific areas of curriculum, the work offers a critique of computer applications to education, develops a notion of strategic rationality in understanding mathematical reasoning, and, contrary to prevalent notions of moral education, connects reason with care, thus emphasizing the intimate connection between emotion and reason and challenging the dominant perception of the two as oppositional.

Mathematical Reasoning

Writing and Proof

Author: Theodore A. Sundstrom

Publisher: Prentice Hall

ISBN: 9780131877184

Category: Mathematics

Page: 514

View: 7842

This title is intended for one-semester courses in Transition to Advanced Mathematics that emphasize the construction and writing of mathematical proofs. Focusing on the formal development of mathematics, this text teaches students how to read and understand mathematical proofs and to construct and write mathematical proofs. Developed as a text for a writing course requirement, issues dealing with writing are addressed directly and practices of good writing are emphasized throughout the text. Active learning is emphasized with preview activities for each section and activities in each section that enable both teachers and students to test understanding and explore ideas in a traditional or non-lecture setting. Elementary number theory and congruence arithmetic are used throughout.

Understanding in Mathematics

Author: Anna Sierpinska

Publisher: Routledge

ISBN: 1135716323

Category: Education

Page: 248

View: 8442

The concept of understanding in mathematics with regard to mathematics education is considered in this volume. The main problem for mathematics teachers being how to facilitate their students' understanding of the mathematics being taught. In combining elements of maths, philosophy, logic, linguistics and the psychology of maths education from her own and European research, Dr Sierpinska considers the contributions of the social and cultural contexts to understanding. The outcome is an insight into both mathematics and understanding.

New Perspectives on Galileo

Papers Deriving from and Related to a Workshop on Galileo held at Virginia Polytechnic Institute and State University, 1975

Author: Robert E. Butts,Joseph C. Pitt

Publisher: Springer Science & Business Media

ISBN: 940099799X

Category: Science

Page: 264

View: 9469

The essays in this volume (except for the contribution of Dr. Le Grand) are extremely revised versions of papers originally delivered at a workshop on Galileo held in Blacksburg, Virginia in October, 1975. The meeting was organized by Professor Joseph Pitt and sponsored by the Department of Philosophy and Religion, The College of Arts and Sciences, and the Division of Research of Virginia Polytechnic Institute and State University. The papers that follow deal with problems OIf Galileo's philosophy of science, specific and general problems connected with his methodology, and with historical and conceptual questions concerning the relationship of his work to that of contemporaries and both earlier and later scientists. New perspectives take many forms. In this book the 'newness' has, for the most part, two forms. First, in the papers by Wisan, Shea, Le Grand and Wallace (the concerns will also appear in some of the other contributions), greatly enriched historical discoveries of how Galileo's science and its method ology developed are provided. It should be stressed that these papers are attempts to recapture a deep sense of the kind of science Galileo was creating. Other papers in the volume, for example, those by McMullin, Machamer, Butts and Pitt, underscore the importance of this historical venture by discussing various aspects of the philosophical background of Galileo's thought. The historical and philosophical evaluations and analyses compliment one another.

Learning to Teach Mathematics in the Secondary School

A Companion to School Experience

Author: Sue Johnston-Wilder,Peter Johnston-Wilder,David Pimm

Publisher: Psychology Press

ISBN: 9780415162807

Category: Education

Page: 267

View: 9963

This text covers a wide range of issues in the teaching of mathematics and importantly, provides supporting activities to the student to enable them to translate theory into practice.

Deepening Our Understanding of Wittgenstein

Author: Michael Kober

Publisher: Rodopi

ISBN: 9042020105

Category: Philosophy

Page: 283

View: 9488

This volume is of interest for anyone who aims at understanding the so-called 'later' or 'mature' Wittgenstein. Its contributions, written by leading German-speaking Wittgenstein-scholars like Hans Sluga, Hans-Johann Glock, Joachim Schulte, Eike von Savigny, and others, provide deeper insights to seemingly well discussed topics, such as family resemblance, Übersicht (perspicuous representation), religion, or grammar, or they explain in an eye-opening fashion hitherto enigmatic expressions of Wittgenstein, such as 'The pneumatic conception of thought' (PI §109), 'A mathematical proof must be surveyable' (RFM III §1), or 'On this a curious remark by H. Newman' (OC §1).