A Transition to Abstract Mathematics

Learning Mathematical Thinking and Writing

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Author: Randall Maddox

Publisher: Academic Press

ISBN: 0080922716

Category: Mathematics

Page: 384

View: 4573

Constructing concise and correct proofs is one of the most challenging aspects of learning to work with advanced mathematics. Meeting this challenge is a defining moment for those considering a career in mathematics or related fields. A Transition to Abstract Mathematics teaches readers to construct proofs and communicate with the precision necessary for working with abstraction. It is based on two premises: composing clear and accurate mathematical arguments is critical in abstract mathematics, and that this skill requires development and support. Abstraction is the destination, not the starting point. Maddox methodically builds toward a thorough understanding of the proof process, demonstrating and encouraging mathematical thinking along the way. Skillful use of analogy clarifies abstract ideas. Clearly presented methods of mathematical precision provide an understanding of the nature of mathematics and its defining structure. After mastering the art of the proof process, the reader may pursue two independent paths. The latter parts are purposefully designed to rest on the foundation of the first, and climb quickly into analysis or algebra. Maddox addresses fundamental principles in these two areas, so that readers can apply their mathematical thinking and writing skills to these new concepts. From this exposure, readers experience the beauty of the mathematical landscape and further develop their ability to work with abstract ideas. Covers the full range of techniques used in proofs, including contrapositive, induction, and proof by contradiction Explains identification of techniques and how they are applied in the specific problem Illustrates how to read written proofs with many step by step examples Includes 20% more exercises than the first edition that are integrated into the material instead of end of chapter

Mathematical Thinking and Writing

A Transition to Abstract Mathematics

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Author: Randall B. Maddox

Publisher: Academic Press

ISBN: 0124649769

Category: Mathematics

Page: 304

View: 7957

The ability to construct proofs is one of the most challenging aspects of the world of mathematics. It is, essentially, the defining moment for those testing the waters in a mathematical career. Instead of being submerged to the point of drowning, readers of Mathematical Thinking and Writing are given guidance and support while learning the language of proof construction and critical analysis. Randall Maddox guides the reader with a warm, conversational style, through the task of gaining a thorough understanding of the proof process, and encourages inexperienced mathematicians to step up and learn how to think like a mathematician. A student's skills in critical analysis will develop and become more polished than previously conceived. Most significantly, Dr. Maddox has the unique approach of using analogy within his book to clarify abstract ideas and clearly demonstrate methods of mathematical precision.

A Transition to Abstract Mathematics, Learning Mathematical Thinking and Writing

Mathematics, Mathematics

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Author: CTI Reviews

Publisher: Cram101 Textbook Reviews

ISBN: 1478422505

Category: Education

Page: 73

View: 3057

Facts101 is your complete guide to A Transition to Abstract Mathematics, Learning Mathematical Thinking and Writing. In this book, you will learn topics such as Sets and Their Properties, Functions, The Real Numbers, and Sequences of Real Numbers plus much more. With key features such as key terms, people and places, Facts101 gives you all the information you need to prepare for your next exam. Our practice tests are specific to the textbook and we have designed tools to make the most of your limited study time.

Mathematik für Software Engineering

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Author: Stephan Dreiseitl

Publisher: Springer-Verlag

ISBN: 3662567334

Category: Computers

Page: 467

View: 5270

Warum müssen InformatikerInnen und SoftwareentwicklerInnen im Studium eigentlich Mathe hören? Wie kann ihnen die Mathematik beim Programmieren helfen? Dieses Lehrbuch vermittelt StudienanfängerInnen die Sprache und Methode der Mathematik als Grundlage strukturierten Problemlösens, welches essenziell für das Entwickeln von Softwaresystemen ist. Deshalb liegt der didaktische Fokus hier darauf aufzuzeigen, wie mathematische Konzepte aufeinander aufbauen, welche Muster sich daraus ergeben, und welche klar strukturierten Regeln es in der mathematischen Argumentation (dem Beweisen) gibt. Dieses Buch richtet den inhaltlichen Fokus auf Logik, Mengenlehre, diskrete Strukturen und Wahrscheinlichkeitsrechnung und orientiert sich damit an den Empfehlungen von ACM und IEEE zur Mathematikausbildung im Software-Engineering-Studium. Da man Mathematik - ebenso wie die Softwareentwicklung - nicht durch Lesen, sondern nur durch Tun erlernt, schließt jeder Abschnitt mit einer Reihe von Verständnisfragen und Übungsaufgaben. Es eignet sich daher bestens zum Nacharbeiten einer Vorlesung und zur Prüfungsvorbereitung. Durch den verständlichen Schreibstil und die Lösungen auf der Webseite des Autors kann dieses Buch aber auch gut zum Selbststudium genutzt werden.

Discrete Mathematics: Introduction to Mathematical Reasoning

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Author: Susanna S. Epp

Publisher: Cengage Learning

ISBN: 0495826170

Category: Mathematics

Page: 648

View: 8896

Susanna Epp's DISCRETE MATHEMATICS: AN INTRODUCTION TO MATHEMATICAL REASONING, provides the same clear introduction to discrete mathematics and mathematical reasoning as her highly acclaimed DISCRETE MATHEMATICS WITH APPLICATIONS, but in a compact form that focuses on core topics and omits certain applications usually taught in other courses. The book is appropriate for use in a discrete mathematics course that emphasizes essential topics or in a mathematics major or minor course that serves as a transition to abstract mathematical thinking. The ideas of discrete mathematics underlie and are essential to the science and technology of the computer age. This book offers a synergistic union of the major themes of discrete mathematics together with the reasoning that underlies mathematical thought. Renowned for her lucid, accessible prose, Epp explains complex, abstract concepts with clarity and precision, helping students develop the ability to think abstractly as they study each topic. In doing so, the book provides students with a strong foundation both for computer science and for other upper-level mathematics courses. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.

Discovering Group Theory

A Transition to Advanced Mathematics

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Author: Tony Barnard,Hugh Neill

Publisher: CRC Press

ISBN: 1315405776

Category: Mathematics

Page: 231

View: 499

Discovering Group Theory: A Transition to Advanced Mathematics presents the usual material that is found in a first course on groups and then does a bit more. The book is intended for students who find the kind of reasoning in abstract mathematics courses unfamiliar and need extra support in this transition to advanced mathematics. The book gives a number of examples of groups and subgroups, including permutation groups, dihedral groups, and groups of integer residue classes. The book goes on to study cosets and finishes with the first isomorphism theorem. Very little is assumed as background knowledge on the part of the reader. Some facility in algebraic manipulation is required, and a working knowledge of some of the properties of integers, such as knowing how to factorize integers into prime factors. The book aims to help students with the transition from concrete to abstract mathematical thinking. ? Features Full proofs with all details clearly laid out and explained Reader-friendly conversational style Complete solutions to all exercises Focus on deduction, helping students learn how to construct proofs "Asides" to the reader, providing overviews and connections "What you should know" reviews at the end of each chapter

The Nuts and Bolts of Proofs

An Introduction to Mathematical Proofs

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Author: Antonella Cupillari

Publisher: Academic Press

ISBN: 0123822181

Category: Mathematics

Page: 296

View: 3765

The Nuts and Bolts of Proofs: An Introduction to Mathematical Proofs provides basic logic of mathematical proofs and shows how mathematical proofs work. It offers techniques for both reading and writing proofs. The second chapter of the book discusses the techniques in proving if/then statements by contrapositive and proofing by contradiction. It also includes the negation statement, and/or. It examines various theorems, such as the if and only-if, or equivalence theorems, the existence theorems, and the uniqueness theorems. In addition, use of counter examples, mathematical induction, composite statements including multiple hypothesis and multiple conclusions, and equality of numbers are covered in this chapter. The book also provides mathematical topics for practicing proof techniques. Included here are the Cartesian products, indexed families, functions, and relations. The last chapter of the book provides review exercises on various topics. Undergraduate students in engineering and physical science will find this book invaluable. Jumps right in with the needed vocabulary—gets students thinking like mathematicians from the beginning Offers a large variety of examples and problems with solutions for students to work through on their own Includes a collection of exercises without solutions to help instructors prepare assignments Contains an extensive list of basic mathematical definitions and concepts needed in abstract mathematics

Brain Literacy for Educators and Psychologists

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Author: Virginia Wise Berninger,Todd L. Richards

Publisher: Academic Press

ISBN: 9780120928712

Category: Education

Page: 373

View: 5754

Although educators are expected to bring about functional changes in the brain--the organ of human learning--they are given no formal training in the structure, function or development of the brain in formal or atypically developing children as part of their education. This book is organized around three conceptual themes: First, the interplay between nature (genetics) and nurture (experience and environment) is emphasized. Second, the functional systems of the brain are explained in terms of how they lead to reading, writing and mathematics and the design of instruction. Thirdly, research is presented, not as a finished product, but as a step forward within the field of educational neuropsychology. The book differs from neuropsychology and neuroscience books in that it is aimed at practitioners, focuses on high incidence neuropsychological conditions seen in the classroom, and is the only book that integrates both brain research with the practice of effective literacy, and mathematics instruction of the general and special education school-aged populations.

Transition to Higher Mathematics: Structure and Proof

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Author: Bob Dumas,John McCarthy

Publisher: McGraw-Hill Science/Engineering/Math

ISBN: 9780073533537

Category: Mathematics

Page: 304

View: 3167

This text is intended for the Foundations of Higher Math bridge course taken by prospective math majors following completion of the mainstream Calculus sequence, and is designed to help students develop the abstract mathematical thinking skills necessary for success in later upper-level majors math courses. As lower-level courses such as Calculus rely more exclusively on computational problems to service students in the sciences and engineering, math majors increasingly need clearer guidance and more rigorous practice in proof technique to adequately prepare themselves for the advanced math curriculum. With their friendly writing style Bob Dumas and John McCarthy teach students how to organize and structure their mathematical thoughts, how to read and manipulate abstract definitions, and how to prove or refute proofs by effectively evaluating them. Its wealth of exercises give students the practice they need, and its rich array of topics give instructors the flexibility they desire to cater coverage to the needs of their school’s majors curriculum. This text is part of the Walter Rudin Student Series in Advanced Mathematics.

Research in Collegiate Mathematics Education VII

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Author: Fernando Hitt,Derek Allan Holton,Patrick W. Thompson

Publisher: American Mathematical Soc.

ISBN: 0821849964

Category: Mathematics

Page: 261

View: 3813

The present volume of Research in Collegiate Mathematics Education, like previous volumes in this series, reflects the importance of research in mathematics education at the collegiate level. The editors in this series encourage communication between mathematicians and mathematics educators, and as pointed out by the International Commission of Mathematics Instruction (ICMI), much more work is needed in concert with these two groups. Indeed, editors of RCME are aware of this need and the articles published in this series are in line with that goal. Nine papers constitute this volume. The first two examine problems students experience when converting a representation from one particular system of representations to another. The next three papers investigate students learning about proofs. In the next two papers, the focus is instructor knowledge for teaching calculus. The final two papers in the volume address the nature of ``conception'' in mathematics. Whether they are specialists in education or mathematicians interested in finding out about the field, readers will obtain new insights about teaching and learning and will take away ideas that they can use.

Perspectives on Transitions in Schooling and Instructional Practice

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Author: Susan E. Elliott-Johns,Daniel H. Jarvis

Publisher: University of Toronto Press

ISBN: 1442667117

Category: Education

Page: 552

View: 3661

Perspectives on Transitions in Schooling and Instructional Practice examines student transitions between major levels of schooling, teacher transitions in instructional practice, and the intersection of these two significant themes in education research. Twenty-six leading international experts offer meaningful insights on current pedagogical practices, obstacles to effective transitions, and proven strategies for stakeholders involved in supporting students in transition. The book is divided into four sections, representing the four main transitions in formal schooling: Early Years (Home, Pre-school, and Kindergarten) to Early Elementary (Grades 1–3); Early Elementary to Late Elementary (Grades 4–8); Late Elementary to Secondary (Grades 9–12); and Secondary to Post-Secondary (College and University). A coda draws together over-arching themes from throughout the text to provide recommendations and a visual model that captures their interactions. Combining theoretical approaches with practical examples of school-based initiatives, this book will appeal to those involved in supporting either the student experience (both academically and emotionally) or teacher professional learning and growth.

A Concise Approach to Mathematical Analysis

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Author: Mangatiana A. Robdera

Publisher: Springer Science & Business Media

ISBN: 9781852335526

Category: Mathematics

Page: 366

View: 9469

A Concise Approach to Mathematical Analysis introduces the undergraduate student to the more abstract concepts of advanced calculus. The main aim of the book is to smooth the transition from the problem-solving approach of standard calculus to the more rigorous approach of proof-writing and a deeper understanding of mathematical analysis. The first half of the textbook deals with the basic foundation of analysis on the real line; the second half introduces more abstract notions in mathematical analysis. Each topic begins with a brief introduction followed by detailed examples. A selection of exercises, ranging from the routine to the more challenging, then gives students the opportunity to practise writing proofs. The book is designed to be accessible to students with appropriate backgrounds from standard calculus courses but with limited or no previous experience in rigorous proofs. It is written primarily for advanced students of mathematics - in the 3rd or 4th year of their degree - who wish to specialise in pure and applied mathematics, but it will also prove useful to students of physics, engineering and computer science who also use advanced mathematical techniques.

How to Study as a Mathematics Major

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Author: Lara Alcock

Publisher: OUP Oxford

ISBN: 0191637351

Category: Mathematics

Page: 288

View: 1783

Every year, thousands of students in the USA declare mathematics as their major. Many are extremely intelligent and hardworking. However, even the best will encounter challenges, because upper-level mathematics involves not only independent study and learning from lectures, but also a fundamental shift from calculation to proof. This shift is demanding but it need not be mysterious — research has revealed many insights into the mathematical thinking required, and this book translates these into practical advice for a student audience. It covers every aspect of studying as a mathematics major, from tackling abstract intellectual challenges to interacting with professors and making good use of study time. Part 1 discusses the nature of upper-level mathematics, and explains how students can adapt and extend their existing skills in order to develop good understanding. Part 2 covers study skills as these relate to mathematics, and suggests practical approaches to learning effectively while enjoying undergraduate life. As the first mathematics-specific study guide, this friendly, practical text is essential reading for any mathematics major.

Logic for Programming, Artificial Intelligence, and Reasoning

8th International Conference, LPAR 2001, Havana, Cuba, December 3-7, 2001, Proceedings

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Author: Robert Nieuwenhuis,Andrei Voronkov

Publisher: Springer Science & Business Media

ISBN: 3540429573

Category: Computers

Page: 741

View: 3440

This volume contains the papers presented at the Eighth International C- ference on Logic for Programming, Arti?cial Intelligence and Reasoning (LPAR 2001), held on December 3-7, 2001, at the University of Havana (Cuba), together with the Second International Workshop on Implementation of Logics. There were 112 submissions, of which 19 belonged to the special subm- sion category of experimental papers, intended to describe implementations or comparisons of systems, or experiments with systems. Each submission was - viewed by at least three program committee members and an electronic program committee meeting was held via the Internet. The high number of submissions caused a large amount of work, and we are very grateful to the other 31 PC members for their e?ciency and for the quality of their reviews and discussions. Finally, the committee decided to accept 40papers in the theoretical ca- gory, and 9 experimental papers. In addition to the refereed papers, this volume contains an extended abstract of the invited talk by Frank Wolter. Two other invited lectures were given by Matthias Baaz and Manuel Hermenegildo. Apart from the program committee, we would also like to thank the other people who made LPAR 2001 possible: the additional referees; the Local Arran- ` gements Chair Luciano Garc ́?a; Andr ́es Navarro and Oscar Guell, ̈ who ran the internet-based submission software and the program committee discussion so- ware at the LSI Department lab in Barcelona; and Bill McCune, whose program committee management software was used.

Abstraction and Representation

Essays on the Cultural Evolution of Thinking

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Author: Peter Damerow

Publisher: Springer Science & Business Media

ISBN: 9780792338161

Category: History

Page: 418

View: 7652

This book deals with the development of thinking under different cultural conditions, focusing on the evolution of mathematical thinking in the history of science and education. Starting from Piaget's genetic epistemology, it provides a conceptual framework for describing and explaining the development of cognition by reflective abstractions from systems of actions.

The Foundations of Mathematics

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Author: Ian Stewart,David Tall

Publisher: Oxford University Press, USA

ISBN: 019870643X

Category: Mathematics

Page: 432

View: 7384

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.