Visual Thinking in Mathematics

Author: Marcus Giaquinto

Publisher: Oxford University Press

ISBN: 0199285942

Category: Philosophy

Page: 287

View: 8804

Drawing from philosophical work on the nature of concepts and from empirical studies of visual perception, mental imagery, and numerical cognition, Giaquinto explores a major source of our grasp of mathematics, using examples from basic geometry, arithmetic, algebra, and real analysis.

Proofs Without Words

Exercises in Visual Thinking

Author: Malcolm Scott MacKenzie,Roger B. Nelsen

Publisher: MAA

ISBN: 9780883857007

Category: Mathematics

Page: 140

View: 6059

Proofs without words are generally pictures or diagrams that help the reader see why a particular mathematical statement may be true, and how one could begin to go about proving it. While in some proofs without words an equation or two may appear to help guide that process, the emphasis is clearly on providing visual clues to stimulate mathematical thought. The proofs in this collection are arranged by topic into five chapters: Geometry and algebra; Trigonometry, calculus and analytic geometry; Inequalities; Integer sums; and Sequences and series. Teachers will find that many of the proofs in this collection are well suited for classroom discussion and for helping students to think visually in mathematics.

Visualization in Teaching and Learning Mathematics

A Project

Author: Walter Zimmermann,Steve Cunningham,Mathematical Association of America. Committee on Computers in Mathematics Education

Publisher: MAA Press

ISBN: N.A

Category: Mathematics

Page: 224

View: 7641

The twenty papers in the book give an overview of research analysis, practical experience, and informed opinion about the role of visualization in teaching and learning mathematics, especially at the undergraduate level. Visualization, in its broadest level. Visualization, in its broadest sense, is as old as mathematics, but progress in computer graphics has generated a renaissance of interest in visual representations and visual thinking in mathematics.

Visible Thinking in the K–8 Mathematics Classroom

Author: Ted H. Hull,Don S. Balka,Ruth Harbin Miles

Publisher: Corwin Press

ISBN: 1452269408

Category: Education

Page: 184

View: 4358

Seeing is believing with this interactive approach to math instruction Do you ever wish your students could read each other’s thoughts? Now they can—and so can you! This newest book by veteran mathematics educators provides instructional strategies for maximizing students’ mathematics comprehension by integrating visual thinking into the classroom. Included are numerous grade-specific sample problems for teaching essential concepts such as number sense, fractions, and estimation. Among the many benefits of visible thinking are: Interactive student-to-student learning Increased class participation Development of metacognitive thinking and problem-solving skills

Toward a Visually-Oriented School Mathematics Curriculum

Research, Theory, Practice, and Issues

Author: Ferdinand Rivera

Publisher: Springer Science & Business Media

ISBN: 9789400700147

Category: Education

Page: 316

View: 1450

What does it mean to have a visual representation of a mathematical object, concept, or process? What visualization strategies support growth in mathematical thinking, reasoning, generalization, and knowledge? Is mathematical seeing culture-free? How can information drawn from studies in blind subjects help us understand the significance of a multimodal approach to learning mathematics? Toward a Visually-Oriented School Mathematics Curriculum explores a unified theory of visualization in school mathematical learning via the notion of progressive modeling. Based on the author’s longitudinal research investigations in elementary and middle school classrooms, the book provides a compelling empirical account of ways in which instruction can effectively orchestrate the transition from personally-constructed visuals, both externally-drawn and internally-derived, into more structured visual representations within the context of a socioculturally grounded mathematical activity. Both for teachers and researchers, a discussion of this topic is relevant in the history of the present. The ubiquity of technological tools and virtual spaces for learning and doing mathematics has aroused interest among concerned stakeholders about the role of mathematics in these contexts. The book begins with a prolegomenon on the author’s reflections on past and present visual studies in mathematics education. In the remaining seven chapters, visualization is pursued in terms of its role in bringing about progressions in mathematical symbolization, abduction, pattern generalization, and diagrammatization. Toward a Visually-Oriented School Mathematics Curriculum views issues surrounding visualization through the eyes of a classroom teacher-researcher; it draws on findings within and outside of mathematics education that help practitioners and scholars gain a better understanding of what it means to pleasurably experience the symmetric visual/symbolic reversal phenomenon – that is, seeing the visual in the symbolic and the symbolic in the visual."

Anschauliche Funktionentheorie

Author: Tristan Needham

Publisher: Oldenbourg Verlag

ISBN: 348670902X

Category: Mathematics

Page: 685

View: 6969

Needhams neuartiger Zugang zur Funktionentheorie wurde von der Fachpresse begeistert aufgenommen. Mit über 500 zum großen Teil perspektivischen Grafiken vermittelt er im wahrsten Sinne des Wortes eine Anschauung von der sonst oft als trocken empfundenen Funktionentheorie. "Anschauliche Funktionentheorie ist eine wahre Freude und ein Buch so recht nach meinem Herzen. Indem er ausschließlich seine neuartige geometrische Perspektive verwendet, enthüllt Tristan Needham viele überraschende und bisher weitgehend unbeachtete Facetten der Schönheit der Funktionentheorie." (Sir Roger Penrose)

Proofs Without Words III

Further Exercises in Visual Thinking

Author: Roger B. Nelsen

Publisher: The Mathematical Association of America

ISBN: 0883857901

Category: Mathematics

Page: 187

View: 7803

Proofs without words (PWWs) are figures or diagrams that help the reader see why a particular mathematical statement is true, and how one might begin to formally prove it true. PWWs are not new, many date back to classical Greece, ancient China, and medieval Europe and the Middle East. PWWs have been regular features of the MAA journals Mathematics Magazine and The College Mathematics Journal for many years, and the MAA published the collections of PWWs Proofs Without Words: Exercises in Visual Thinking in 1993 and Proofs Without Words II: More Exercises in Visual Thinking in 2000. This book is the third such collection of PWWs. The proofs in the book are divided by topic into five chapters: Geometry & Algebra; Trigonometry, Calculus & Analytic Geometry; Inequalities; Integers & Integer Sums; and Infinite Series & Other Topics. The proofs in the book are intended primarily for the enjoyment of the reader, however, teachers will want to use them with students at many levels: high school courses from algebra through precalculus and calculus; college level courses in number theory, combinatorics, and discrete mathematics; and pre-service and in-service courses for teachers.

Proofs without Words II

Author: Roger B. Nelsen

Publisher: Mathematical Association of America

ISBN: 9780883857212

Category: Mathematics

Page: 142

View: 9786

Like its predecessor, Proofs without Words, this book is a collection of pictures or diagrams that help the reader see why a particular mathematical statement may be true, and how one could begin to go about proving it. While in some proofs without words an equation or two may appear to help guide that process, the emphasis is clearly on providing visual clues to stimulate mathematical thought. The proofs in this collection are arranged by topic into five chapters: geometry and algebra; trigonometry, calculus and analytic geometry; inequalities; integer sums; and sequences and series. Teachers will find that many of the proofs in this collection are well suited for classroom discussion and for helping students to think visually in mathematics.

An Introduction to Copulas

Author: Roger B. Nelsen

Publisher: Springer Science & Business Media

ISBN: 0387286780

Category: Mathematics

Page: 272

View: 4021

The study of copulas and their role in statistics is a new but vigorously growing field. In this book the student or practitioner of statistics and probability will find discussions of the fundamental properties of copulas and some of their primary applications. The applications include the study of dependence and measures of association, and the construction of families of bivariate distributions. This book is suitable as a text or for self-study.

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: 9892

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.

Math Made Visual

Creating Images for Understanding Mathematics

Author: Roger Nelsen,Claudi Alsina,Roger B. Nelsen

Publisher: MAA

ISBN: 9780883857465

Category: Mathematics

Page: 173

View: 4895

The object of this book is to show how visualization techniques may be employed to produce pictures that have interest for the creation, communication and teaching of mathematics. Mathematical drawings related to proofs have been produced since antiquity in China, Arabia, Greece and India but only in the last thirty years has there been a growing interest in so-called 'proofs without words.' In this book the authors show that behind most of the pictures 'proving' mathematical relations are some well-understood methods. The first part of the book consists of twenty short chapters, each one describing a method to visualize some mathematical idea (a proof, a concept, an operation,...) and several applications to concrete cases. Following this the book examines general pedagogical considerations concerning the development of visual thinking, practical approaches for making visualizations in the classroom and a discussion of the role that hands-on material plays in this process.

The Philosophy of Mathematical Practice

Author: Paolo Mancosu

Publisher: Oxford University Press on Demand

ISBN: 0199296456

Category: Philosophy

Page: 447

View: 399

This book gives a coherent and unified presentation of a new direction of work in philosophy of mathematics. This new approach in philosophy of mathematics requires extensive attention to mathematical practice and provides philosophical analyses of important novel characteristics of contemporary (twentieth century) mathematics and of many aspects of mathematical activity-such as visualization, explanation, understanding etc.-- which escape purely formal logicaltreatment.The book consists of a lengthy introduction by the editor and of eight chapters written by some of the very best scholars in this area. Each chapter consists of a short introduction to the general topic of the chapter and of a longer research article in the very same area. Theeight topics selected represent a broad spectrum of the contemporary philosophical reflection on different aspects of mathematical practice: Diagrammatic reasoning and representational systems; Visualization; Mathematical Explanation; Purity of Methods; Mathematical Concepts; Philosophical relevance of category theory; Philosophical aspects of computer science in mathematics; Philosophical impact of recent developments in mathematical physics.

Key Ideas in Teaching Mathematics

Research-based guidance for ages 9-19

Author: Anne Watson,Keith Jones,Dave Pratt

Publisher: OUP Oxford

ISBN: 0191643424

Category: Mathematics

Page: 272

View: 5714

Big ideas in the mathematics curriculum for older school students, especially those that are hard to learn and hard to teach, are covered in this book. It will be a first port of call for research about teaching big ideas for students from 9-19 and also has implications for a wider range of students. These are the ideas that really matter, that students get stuck on, and that can be obstacles to future learning. It shows how students learn, why they sometimes get things wrong, and the strengths and pitfalls of various teaching approaches. Contemporary high-profile topics like modelling are included. The authors are experienced teachers, researchers and mathematics educators, and many teachers and researchers have been involved in the thinking behind this book, funded by the Nuffield Foundation. An associated website, hosted by the Nuffield Foundation, summarises the key messages in the book and connects them to examples of classroom tasks that address important learning issues about particular mathematical ideas.

Intelligent Computer Mathematics

9th International Conference, AISC 2008 15th Symposium, Calculemus 2008 7th International Conference, MKM 2008 Birmingham, UK, July 28 - August 1, 2008, Proceedings

Author: Serge Autexier,John Campbell,Julio Rubio,Volker Sorge,Masakazu Suzuki

Publisher: Springer Science & Business Media

ISBN: 3540851097

Category: Computers

Page: 600

View: 5730

This book constitutes the joint refereed proceedings of the 9th International Conference on Artificial Intelligence and Symbolic Computation, AISC 2008, the 15th Symposium on the Integration of Symbolic Computation and Mechanized Reasoning, Calculemus 2008, and the 7th International Conference on Mathematical Knowledge Management, MKM 2008, held in Birmingham, UK, in July/August as CICM 2008, the Conferences on Intelligent Computer Mathematics. The 14 revised full papers for AISC 2008, 10 revised full papers for Calculemus 2008, and 18 revised full papers for MKM 2008, plus 5 invited talks, were carefully reviewed and selected from a total of 81 submissions for a joint presentation in the book. The papers cover different aspects of traditional branches in CS such as computer algebra, theorem proving, and artificial intelligence in general, as well as newly emerging ones such as user interfaces, knowledge management, and theory exploration, thus facilitating the development of integrated mechanized mathematical assistants that will be routinely used by mathematicians, computer scientists, and engineers in their every-day business.

How to Study as a Mathematics Major

Author: Lara Alcock

Publisher: OUP Oxford

ISBN: 0191637351

Category: Mathematics

Page: 288

View: 4263

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.

When Less is More

Visualizing Basic Inequalities

Author: Claudi Alsina,Roger B. Nelsen

Publisher: MAA

ISBN: 9780883853429

Category: Mathematics

Page: 181

View: 1661

Introduces the richness and variety of inequalities in mathematics using illustration and visualisation.

Activities in Support of Two-Year College Science, Mathematics, Engineering, and Technology Educationn

Fiscal Year 1994 Highlights

Author: Robert F. Watson

Publisher: DIANE Publishing

ISBN: 9780788127748

Category:

Page: 79

View: 5252

Focuses on the need to meet the economic and social needs of today's society while looking at America's colleges and universities. Identifies colleges' goals focusing primarily on two-year college programs. Includes: leadership activities in education and human resources; leveraged program support (instrumentation and laboratory improvement, undergraduate faculty enhancement, young scholars, alliances for minority participation, rural systemic initiatives, teacher enhancement, and much more). Charts and tables.

Einführung in das mathematische Denken

die Begriffsbildung der modernen Mathematik

Author: Friedrich Waismann

Publisher: N.A

ISBN: 9783534243013

Category: Electronic books

Page: 184

View: 5463