Proofs Without Words

Exercises in Visual Thinking


Author: Malcolm Scott MacKenzie,Roger B. Nelsen

Publisher: MAA

ISBN: 9780883857007

Category: Mathematics

Page: 140

View: 1873

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.

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: 2859

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.

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: 856

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.

Cameos for Calculus

Visualization in the First-Year Course


Author: Roger B. Nelsen

Publisher: The Mathematical Association of America

ISBN: 088385788X

Category: Mathematics

Page: 186

View: 3959

A thespian or cinematographer might define a cameo as “a brief appearance of a known figure,” while a gemologist or lapidary might define it as “a precious or semiprecious stone.” This book presents fifty short enhancements or supplements (the Cameos) for the first-year calculus course in which a geometric figure briefly appears. Some of the Cameos illustrate mainstream topics such as the derivative, combinatorial formulas used to compute Riemann sums, or the geometry behind many geometric series. Other Cameos present topics accessible to students at the calculus level but not usually encountered in the course, such as the Cauchy-Schwarz inequality, the arithmetic mean-geometric mean inequality, and the Euler-Mascheroni constant. There are fifty Cameos in the book, grouped into five sections: Part I Limits and Differentiation; Part II Integration; Part III Infinite Series; Part IV Additional Topics, and Part V Appendix: Some Precalculus Topics. Many of the Cameos include exercises, so Solutions to all the Exercises follows Part V. The book concludes with References and an Index. Many of the Cameos are adapted from articles published in journals of the MAA, such as The American Mathematical Monthly, Mathematics Magazine, and The College Mathematics Journal. Some come from other mathematical journals, and some were created for this book. By gathering the Cameos into a book we hope that they will be more accessible to teachers of calculus, both for use in the classroom and as supplementary explorations for students.

Nuggets of Number Theory

A Visual Approach


Author: Roger B. Nelsen

Publisher: MAA Press

ISBN: 9781470443986

Category: Mathematics

Page: 153

View: 5670

Nuggets of Number Theory will attract fans of visual thinking, number theory, and surprising connections. This book contains hundreds of visual explanations of results from elementary number theory. Figurate numbers and Pythagorean triples feature prominently, of course, but there are also proofs of Fermat's Little and Wilson's Theorems. Fibonacci and perfect numbers, Pell's equation, and continued fractions all find visual representation in this charming collection. It will be a rich source of visual inspiration for anyone teaching, or learning, number theory and will provide endless pleasure to those interested in looking at number theory with new eyes. Author Roger Nelsen is a long-time contributor of ``Proofs Without Words'' in the MAA's Mathematics Magazine and College Mathematics Journal. This is his twelfth book with MAA Press.

The Calculus Collection

A Resource for AP and Beyond


Author: Caren L. Diefenderfer,Roger B. Nelsen

Publisher: MAA

ISBN: 9780883857618

Category: Juvenile Nonfiction

Page: 507

View: 4111

The Calculus Collection is a useful resource for everyone who teaches calculus, in high school or in a 2- or 4-year college or university. It consists of 123 articles, selected by a panel of six veteran high school teachers, each of which was originally published in Math Horizons, MAA Focus, The American Mathematical Monthly, The College Mathematics Journal, or Mathematics Magazine. The articles focus on engaging students who are meeting the core ideas of calculus for the first time. The Calculus Collection is filled with insights, alternate explanations of difficult ideas, and suggestions for how to take a standard problem and open it up to the rich mathematical explorations available when you encourage students to dig a little deeper. Some of the articles reflect an enthusiasm for bringing calculators and computers into the classroom, while others consciously address themes from the calculus reform movement. But most of the articles are simply interesting and timeless explorations of the mathematics encountered in a first course in calculus.The MAA has twice previously issued a calculus reader, collecting articles on calculus from its journals: Selected Papers in Calculus, published in 1969 and reprinted as Part I of A Century of Calculus, and Part II published in 1992. In a sense The Calculus Collection is the third volume in that series, but different in that it is a collection chosen for its usefulness to those who teach first-year calculus in high schools as well as colleges and universities.

Making Thinking Visible

How to Promote Engagement, Understanding, and Independence for All Learners


Author: Ron Ritchhart,Mark Church,Karin Morrison

Publisher: John Wiley & Sons

ISBN: 047091551X

Category: Education

Page: 294

View: 5900

"Visible Thinking is a research-based approach to teaching thinking that develops students' thinking dispositions, while at the same time deepening their understanding of the topics they study. Rather than a set of fixed lessons, Visible Thinking is an extensive and adaptable collection of practices that include thinking routines and the documentation of student thinking. The routines are a central element of the practical, functional and accessible nature of Visible Thinking. Thinking routines are easy to use mini-strategies that are repeatedly used in the classroom. They are a small set of questions or a short sequence of steps that can be used across various grade levels and content. Each routine targets a different type of thinking and by bringing their own content, teachers can integrate the routines into the fabric of their classrooms. Thinking Routines help direct student thinking and structure classroom discussion. Thinking becomes visible as the students' different viewpoints are expressed, documented, discussed and reflected upon"--

Proofs without Words II


Author: Roger B. Nelsen

Publisher: Mathematical Association of America

ISBN: 9780883857212

Category: Mathematics

Page: 142

View: 8359

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.

The Differentiated Classroom

Responding to the Needs of All Learners, 2nd Edition


Author: Carol Ann Tomlinson

Publisher: ASCD

ISBN: 1416618635

Category: Education

Page: 197

View: 6391

Although much has changed in schools in recent years, the power of differentiated instruction remains the same--and the need for it has only increased. Today's classroom is more diverse, more inclusive, and more plugged into technology than ever before. And it's led by teachers under enormous pressure to help decidedly unstandardized students meet an expanding set of rigorous, standardized learning targets. In this updated second edition of her best-selling classic work, Carol Ann Tomlinson offers these teachers a powerful and practical way to meet a challenge that is both very modern and completely timeless: how to divide their time, resources, and efforts to effectively instruct so many students of various backgrounds, readiness and skill levels, and interests. With a perspective informed by advances in research and deepened by more than 15 years of implementation feedback in all types of schools, Tomlinson explains the theoretical basis of differentiated instruction, explores the variables of curriculum and learning environment, shares dozens of instructional strategies, and then goes inside elementary and secondary classrooms in nearly all subject areas to illustrate how real teachers are applying differentiation principles and strategies to respond to the needs of all learners. This book's insightful guidance on what to differentiate, how to differentiate, and why lays the groundwork for bringing differentiated instruction into your own classroom or refining the work you already do to help each of your wonderfully unique learners move toward greater knowledge, more advanced skills, and expanded understanding. Today more than ever, The Differentiated Classroom is a must-have staple for every teacher's shelf and every school's professional development collection.


Beauty in Mathematical Proof


Author: Burkard Polster

Publisher: Bloomsbury Publishing USA

ISBN: 0802714315

Category: Mathematics

Page: 58

View: 8939

Q.E.D. presents some of the most famous mathematical proofs in a charming book that will appeal to nonmathematicians and math experts alike. Grasp in an instant why Pythagoras's theorem must be correct. Follow the ancient Chinese proof of the volume formula for the frustrating frustum, and Archimedes' method for finding the volume of a sphere. Discover the secrets of pi and why, contrary to popular belief, squaring the circle really is possible. Study the subtle art of mathematical domino tumbling, and find out how slicing cones helped save a city and put a man on the moon.

Biscuits of Number Theory


Author: Arthur T. Benjamin,Ezra Brown

Publisher: MAA

ISBN: 9780883853405

Category: Mathematics

Page: 311

View: 5209

An anthology of articles designed to supplement a first course in number theory.

Mathematical Mindsets

Unleashing Students' Potential through Creative Math, Inspiring Messages and Innovative Teaching


Author: Jo Boaler

Publisher: John Wiley & Sons

ISBN: 1118418271

Category: Education

Page: 320

View: 3547

Banish math anxiety and give students of all ages a clear roadmap to success Mathematical Mindsets provides practical strategies and activities to help teachers and parents show all children, even those who are convinced that they are bad at math, that they can enjoy and succeed in math. Jo Boaler—Stanford researcher, professor of math education, and expert on math learning—has studied why students don't like math and often fail in math classes. She's followed thousands of students through middle and high schools to study how they learn and to find the most effective ways to unleash the math potential in all students. There is a clear gap between what research has shown to work in teaching math and what happens in schools and at home. This book bridges that gap by turning research findings into practical activities and advice. Boaler translates Carol Dweck's concept of 'mindset' into math teaching and parenting strategies, showing how students can go from self-doubt to strong self-confidence, which is so important to math learning. Boaler reveals the steps that must be taken by schools and parents to improve math education for all. Mathematical Mindsets: Explains how the brain processes mathematics learning Reveals how to turn mistakes and struggles into valuable learning experiences Provides examples of rich mathematical activities to replace rote learning Explains ways to give students a positive math mindset Gives examples of how assessment and grading policies need to change to support real understanding Scores of students hate and fear math, so they end up leaving school without an understanding of basic mathematical concepts. Their evasion and departure hinders math-related pathways and STEM career opportunities. Research has shown very clear methods to change this phenomena, but the information has been confined to research journals—until now. Mathematical Mindsets provides a proven, practical roadmap to mathematics success for any student at any age.

Kiselev's Geometry



Author: Andreĭ Petrovich Kiselev

Publisher: N.A


Category: Geometry

Page: 176

View: 3924

This volume completes the English adaptation of a classical Russian textbook in elementary Euclidean geometry. The 1st volume subtitled "Book I. Planimetry" was published in 2006 (ISBN 0977985202). This 2nd volume (Book II. Stereometry) covers solid geometry, and contains a chapter on vectors, foundations, and introduction in non-Euclidean geometry added by the translator. The book intended for high-school and college students, and their teachers. Includes 317 exercises, index, and bibliography.

Discovering Discrete Dynamical Systems


Author: Aimee S. A. Johnson,Kathleen M. Madden,Ayşe A. Şahin

Publisher: The Mathematical Association of America

ISBN: 0883857936

Category: Mathematics

Page: 130

View: 9678

A dynamical system is a collection of possible states and a rule (or rules) that describes evolution of these states over time. The main purpose of this book is to introduce important concepts in dynamical systems - including fixed and periodic points, attractors and repellers, chaos and fractals - in a way that encourages readers to explore, discover , and learn independently. The book differs from other dynamical system textbooks in that topics have been carefully chosen both to give a coherent introduction to dynamical systems and to support inquiry-based learning.

Science Teaching Reconsidered

A Handbook


Author: National Research Council,Division of Behavioral and Social Sciences and Education,Board on Science Education,Committee on Undergraduate Science Education

Publisher: National Academies Press

ISBN: 9780309175449

Category: Education

Page: 104

View: 3539

Effective science teaching requires creativity, imagination, and innovation. In light of concerns about American science literacy, scientists and educators have struggled to teach this discipline more effectively. Science Teaching Reconsidered provides undergraduate science educators with a path to understanding students, accommodating their individual differences, and helping them grasp the methods--and the wonder--of science. What impact does teaching style have? How do I plan a course curriculum? How do I make lectures, classes, and laboratories more effective? How can I tell what students are thinking? Why don't they understand? This handbook provides productive approaches to these and other questions. Written by scientists who are also educators, the handbook offers suggestions for having a greater impact in the classroom and provides resources for further research.

Connecting Teachers, Students, and Standards

Strategies for Success in Diverse and Inclusive Classrooms


Author: Deborah L. Voltz,Michele Jean Sims,Betty Palmer Nelson

Publisher: ASCD

ISBN: 1416610243

Category: Education

Page: 159

View: 3344

Creating and sustaining a classroom where every learner succeeds is a challenge for any teacher--especially when the elements of diversity and inclusion are added to the mix. How can teachers differentiate instruction in ways that help all students meet standards and develop lifelong learning skills? The authors of Connecting Teachers, Students, and Standards provide a comprehensive framework for reaching and teaching English language learners, students from culturally diverse backgrounds, and students with disabilities. In this book, you'll learn how to * Select the best instructional methods and materials for diverse learners * Create classrooms that are welcoming, practical, and conducive to learning * Develop classroom content that allows every student to achieve standards while meeting the individual needs of diverse learners * Collaborate effectively with fellow teachers and education specialists * Administer assessments that challenge and accommodate diverse learners The book includes helpful, real-world scenarios that provide tips for connecting with diverse students in the classroom, communicating with their families, and coordinating efforts with colleagues. Packed with practical strategies for handling difficult issues, this is a go-to guide for any teacher facing the complexities of helping diverse learners flourish at school and beyond.

Visual Group Theory


Author: Nathan Carter

Publisher: MAA

ISBN: 9780883857571

Category: Mathematics

Page: 297

View: 6308

Group theory is the branch of mathematics that studies symmetry, found in crystals, art, architecture, music and many other contexts. But its beauty is lost on students when it is taught in a technical style that is difficult to understand. Visual Group Theory assumes only a high school mathematics background and covers a typical undergraduate course in group theory from a thoroughly visual perspective. The more than 300 illustrations in Visual Group Theory bring groups, subgroups, homomorphisms, products, and quotients into clear view. Every topic and theorem is accompanied with a visual demonstration of its meaning and import, from the basics of groups and subgroups through advanced structural concepts such as semidirect products and Sylow theory.

Concrete Mathematics

A Foundation for Computer Science


Author: Ronald L. Graham,Donald Ervin Knuth,Oren Patashnik

Publisher: Addison-Wesley Professional

ISBN: 9780201558029

Category: Computers

Page: 657

View: 8902

This book, updated and improved, introduces the mathematics that supports advanced computer programming and the analysis of algorithms. The primary aim of its well-known authors is to provide a solid and relevant base of mathematical skills--the skills needed to solve complex problems, to evaluate horrendous-looking sums, to solve complex recurrence relations, and to discover subtle patterns in data. It is an indispensable text and reference, not only for computer scientists but for all technical professionals in virtually every discipline.

Teaching and Learning Proof Across the Grades

A K-16 Perspective


Author: Despina A. Stylianou,Maria L. Blanton,Eric J. Knuth

Publisher: Routledge

ISBN: 1135856745

Category: Education

Page: 408

View: 8692

A Co-Publication of Routledge for the National Council of Teachers of Mathematics (NCTM) In recent years there has been increased interest in the nature and role of proof in mathematics education; with many mathematics educators advocating that proof should be a central part of the mathematics education of students at all grade levels. This important new collection provides that much-needed forum for mathematics educators to articulate a connected K-16 "story" of proof. Such a story includes understanding how the forms of proof, including the nature of argumentation and justification as well as what counts as proof, evolve chronologically and cognitively and how curricula and instruction can support the development of students’ understanding of proof. Collectively these essays inform educators and researchers at different grade levels about the teaching and learning of proof at each level and, thus, help advance the design of further empirical and theoretical work in this area. By building and extending on existing research and by allowing a variety of voices from the field to be heard, Teaching and Learning Proof Across the Grades not only highlights the main ideas that have recently emerged on proof research, but also defines an agenda for future study.