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: 2901

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: 8587

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.

Mathematik für Software Engineering

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Author: Stephan Dreiseitl

Publisher: Springer-Verlag

ISBN: 3662567334

Category: Computers

Page: 467

View: 2448

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.

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: 4729

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.

Research on Teaching and Learning Mathematics at the Tertiary Level

State-of-the-art and Looking Ahead

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Author: Irene Biza,Victor Giraldo,Reinhard Hochmuth,Azimehsadat Khakbaz,Chris Rasmussen

Publisher: Springer

ISBN: 3319418149

Category: Education

Page: 32

View: 9143

This topical survey focuses on research in tertiary mathematics education, a field that has experienced considerable growth over the last 10 years. Drawing on the most recent journal publications as well as the latest advances from recent high-quality conference proceedings, our review culls out the following five emergent areas of interest: mathematics teaching at the tertiary level; the role of mathematics in other disciplines; textbooks, assessment and students’ studying practices; transition to the tertiary level; and theoretical-methodological advances. We conclude the survey with a discussion of some potential directions for future research in this new and rapidly evolving domain of inquiry.

Discrete Mathematics: Introduction to Mathematical Reasoning

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Author: Susanna S. Epp

Publisher: Cengage Learning

ISBN: 1133417078

Category: Mathematics

Page: 648

View: 2291

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.

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: 8132

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.

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: 219

View: 9585

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.

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: 990

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.

Handbook of International Research in Mathematics Education

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Author: Lyn D. English,David Kirshner

Publisher: Routledge

ISBN: 1135192766

Category: Education

Page: 944

View: 9479

This book brings together mathematics education research that makes a difference in both theory and practice - research that anticipates problems and needed knowledge before they become impediments to progress.

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: 3179

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.

The Nuts and Bolts of Proofs

An Introduction to Mathematical Proofs

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Author: Antonella Cupillari

Publisher: Academic Press

ISBN: 0123822173

Category: Mathematics

Page: 283

View: 7766

The Nuts and Bolts of Proofs instructs students on the primary basic logic of mathematical proofs, showing how proofs of mathematical statements work. The text provides basic core techniques of how to read and write proofs through examples. The basic mechanics of proofs are provided for a methodical approach in gaining an understanding of the fundamentals to help students reach different results. A variety of fundamental proofs demonstrate the basic steps in the construction of a proof and numerous examples illustrate the method and detail necessary to prove various kinds of theorems. New chapter on proof by contradiction New updated proofs A full range of accessible proofs Symbols indicating level of difficulty help students understand whether a problem is based on calculus or linear algebra Basic terminology list with definitions at the beginning of the text

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: 7076

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.

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: 5837

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.

How to Prove It

A Structured Approach

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Author: Daniel J. Velleman

Publisher: Cambridge University Press

ISBN: 9780521446631

Category: Mathematics

Page: 309

View: 2738

Many mathematics students have trouble the first time they take a course, such as linear algebra, abstract algebra, introductory analysis, or discrete mathematics, in which they are asked to prove various theorems. This textbook 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 "scratchwork" sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets. Numerous exercises give students the opportunity to construct their own proofs. 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.