Understanding Mathematical Proof

  • Filename: understanding-mathematical-proof.
  • ISBN: 9781466514911
  • Release Date: 2016-04-19
  • Number of pages: 414
  • Author: John Taylor
  • Publisher: CRC Press



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.

100 Mathematical Proof

  • Filename: 100-mathematical-proof.
  • ISBN: 0471961981
  • Release Date: 1996-08-01
  • Number of pages: 326
  • Author: Rowan Garnier
  • Publisher: Wiley



"Proof" has been and remains one of the concepts which characterises mathematics. Covering basic propositional and predicate logic as well as discussing axiom systems and formal proofs, the book seeks to explain what mathematicians understand by proofs and how they are communicated. The authors explore the principle techniques of direct and indirect proof including induction, existence and uniqueness proofs, proof by contradiction, constructive and non-constructive proofs, etc. Many examples from analysis and modern algebra are included. The exceptionally clear style and presentation ensures that the book will be useful and enjoyable to those studying and interested in the notion of mathematical "proof."

Fundamentals of Mathematics

  • Filename: fundamentals-of-mathematics.
  • ISBN: 0470551380
  • Release Date: 2010-08-16
  • Number of pages: 338
  • Author: Bernd S. W. Schröder
  • Publisher: Wiley



An accessible introduction to abstract mathematics with an emphasis on proof writing Addressing the importance of constructing and understanding mathematical proofs, Fundamentals of Mathematics: An Introduction to Proofs, Logic, Sets, and Numbers introduces key concepts from logic and set theory as well as the fundamental definitions of algebra to prepare readers for further study in the field of mathematics. The author supplies a seamless, hands-on presentation of number systems, utilizing key elements of logic and set theory and encouraging readers to abide by the fundamental rule that you are not allowed to use any results that you have not proved yet. The book begins with a focus on the elements of logic used in everyday mathematical language, exposing readers to standard proof methods and Russell's Paradox. Once this foundation is established, subsequent chapters explore more rigorous mathematical exposition that outlines the requisite elements of Zermelo-Fraenkel set theory and constructs the natural numbers and integers as well as rational, real, and complex numbers in a rigorous, yet accessible manner. Abstraction is introduced as a tool, and special focus is dedicated to concrete, accessible applications, such as public key encryption, that are made possible by abstract ideas. The book concludes with a self-contained proof of Abel's Theorem and an investigation of deeper set theory by introducing the Axiom of Choice, ordinal numbers, and cardinal numbers. Throughout each chapter, proofs are written in much detail with explicit indications that emphasize the main ideas and techniques of proof writing. Exercises at varied levels of mathematical development allow readers to test their understanding of the material, and a related Web site features video presentations for each topic, which can be used along with the book or independently for self-study. Classroom-tested to ensure a fluid and accessible presentation, Fundamentals of Mathematics is an excellent book for mathematics courses on proofs, logic, and set theory at the upper-undergraduate level as well as a supplement for transition courses that prepare students for the rigorous mathematical reasoning of advanced calculus, real analysis, and modern algebra. The book is also a suitable reference for professionals in all areas of mathematics education who are interested in mathematical proofs and the foundation upon which all mathematics is built.

Book of Proof

  • Filename: book-of-proof.
  • ISBN: 0989472116
  • Release Date: 2016-01-01
  • Number of pages: 314
  • Author: Richard H. Hammack
  • Publisher:



This book is an introduction to the language and standard proof methods of mathematics. It is a bridge from the computational courses (such as calculus or differential equations) that students typically encounter in their first year of college to a more abstract outlook. It lays a foundation for more theoretical courses such as topology, analysis and abstract algebra. Although it may be more meaningful to the student who has had some calculus, there is really no prerequisite other than a measure of mathematical maturity.

Proof and Proving in Mathematics Education

  • Filename: proof-and-proving-in-mathematics-education.
  • ISBN: 9789400721296
  • Release Date: 2012-06-14
  • Number of pages: 476
  • Author: Gila Hanna
  • Publisher: Springer Science & Business Media



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.

Making the Connection

  • Filename: making-the-connection.
  • ISBN: 0883851830
  • Release Date: 2008
  • Number of pages: 319
  • Author: Marilyn Paula Carlson
  • Publisher: MAA



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.

Introduction to Advanced Mathematics A Guide to Understanding Proofs

  • Filename: introduction-to-advanced-mathematics-a-guide-to-understanding-proofs.
  • ISBN: 9781133168782
  • Release Date: 2011-01-01
  • Number of pages: 144
  • Author: Connie M. Campbell
  • Publisher: Cengage Learning



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.

How to Prove It

  • Filename: how-to-prove-it.
  • ISBN: 0521675995
  • Release Date: 2006-01-16
  • Number of pages: 384
  • Author: Daniel J. Velleman
  • Publisher: Cambridge University Press



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.

Introduction to Advanced Mathematics A Guide to Understanding Proofs

  • Filename: introduction-to-advanced-mathematics-a-guide-to-understanding-proofs.
  • ISBN: 9781133168782
  • Release Date: 2011-01-01
  • Number of pages: 144
  • Author: Connie M. Campbell
  • Publisher: Cengage Learning



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.

What Does Understanding Mathematics Mean for Teachers

  • Filename: what-does-understanding-mathematics-mean-for-teachers.
  • ISBN: 9781136870538
  • Release Date: 2013-05-13
  • Number of pages: 168
  • Author: Yuichi Handa
  • Publisher: Taylor & Francis



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.

Understanding Analysis

  • Filename: understanding-analysis.
  • ISBN: 9781493927128
  • Release Date: 2015-05-19
  • Number of pages: 312
  • Author: Stephen Abbott
  • Publisher: Springer



This lively introductory text exposes the student to the rewards of a rigorous study of functions of a real variable. In each chapter, informal discussions of questions that give analysis its inherent fascination are followed by precise, but not overly formal, developments of the techniques needed to make sense of them. By focusing on the unifying themes of approximation and the resolution of paradoxes that arise in the transition from the finite to the infinite, the text turns what could be a daunting cascade of definitions and theorems into a coherent and engaging progression of ideas. Acutely aware of the need for rigor, the student is much better prepared to understand what constitutes a proper mathematical proof and how to write one. Fifteen years of classroom experience with the first edition of Understanding Analysis have solidified and refined the central narrative of the second edition. Roughly 150 new exercises join a selection of the best exercises from the first edition, and three more project-style sections have been added. Investigations of Euler’s computation of ζ(2), the Weierstrass Approximation Theorem, and the gamma function are now among the book’s cohort of seminal results serving as motivation and payoff for the beginning student to master the methods of analysis. Review of the first edition: “This is a dangerous book. Understanding Analysis is so well-written and the development of the theory so w ell-motiv ated that exposing students to it could well lead them to expect such excellence in all their textbooks. ... Understanding Analysis is perfectly titled; if your students read it, that’s what’s going to happen. ... This terrific book will become the text of choice for the single-variable introductory analysis course ... ” — Steve Kennedy, MAA Reviews

Roads to Infinity

  • Filename: roads-to-infinity.
  • ISBN: 9781439865507
  • Release Date: 2010-07-13
  • Number of pages: 250
  • Author: John C. Stillwell
  • Publisher: CRC Press



Winner of a CHOICE Outstanding Academic Title Award for 2011! This book offers an introduction to modern ideas about infinity and their implications for mathematics. It unifies ideas from set theory and mathematical logic, and traces their effects on mainstream mathematical topics of today, such as number theory and combinatorics. The treatment is historical and partly informal, but with due attention to the subtleties of the subject. Ideas are shown to evolve from natural mathematical questions about the nature of infinity and the nature of proof, set against a background of broader questions and developments in mathematics. A particular aim of the book is to acknowledge some important but neglected figures in the history of infinity, such as Post and Gentzen, alongside the recognized giants Cantor and Gödel.

Math Made Visual

  • Filename: math-made-visual.
  • ISBN: 0883857464
  • Release Date: 2006-07-06
  • Number of pages: 173
  • Author: Claudi Alsina
  • Publisher: MAA



A book describing how visualization techniques can be used in the teaching of mathematics.

Deepening Our Understanding of Wittgenstein

  • Filename: deepening-our-understanding-of-wittgenstein.
  • ISBN: 9789042020108
  • Release Date: 2006
  • Number of pages: 283
  • Author: Michael Kober
  • Publisher: Rodopi



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).

Themes of Times on Quantitative Literacy for Using and Understanding Mathematics

  • Filename: themes-of-times-on-quantitative-literacy-for-using-and-understanding-mathematics.
  • ISBN: 0321487257
  • Release Date: 2007-07
  • Number of pages: 128
  • Author: Jeffrey O. Bennett
  • Publisher: Addison-Wesley



This collection of engaging articles fromThe New York Timesexplores a wide range of quantitative topics appearing in recent news items and stories. Articles are correlated to chapters inUsing and Understanding Mathematics: A Quantitative Reasoning Approach,and chapter-specific references make it easy for you to relate each article to the textbook. A series of follow-up questions are also included for homework or class discussion.

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