Proofs Without Words

Exercises in Visual Thinking

Author: Malcolm Scott MacKenzie,Roger B. Nelsen

Publisher: MAA

ISBN: 9780883857007

Category: Mathematics

Page: 140

View: 7323

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: 1365

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: Claudi Alsina,Roger B. Nelsen

Publisher: MAA

ISBN: 9780883857465

Category: Mathematics

Page: 173

View: 6832

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

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: 2668

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.

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: 9479

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.

Nuggets of Number Theory

A Visual Approach

Author: Roger B. Nelsen

Publisher: MAA Press

ISBN: 9781470443986

Category: Mathematics

Page: 153

View: 1794

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.

Thinking Geometrically

A Survey of Geometries

Author: Thomas Q. Sibley

Publisher: The Mathematical Association of America

ISBN: 1939512085

Category: Mathematics

Page: 586

View: 2779

Thinking Geometrically: A Survey of Geometries is a well written and comprehensive survey of college geometry that would serve a wide variety of courses for both mathematics majors and mathematics education majors. Great care and attention is spent on developing visual insights and geometric intuition while stressing the logical structure, historical development, and deep interconnectedness of the ideas. Students with less mathematical preparation than upper-division mathematics majors can successfully study the topics needed for the preparation of high school teachers. There is a multitude of exercises and projects in those chapters developing all aspects of geometric thinking for these students as well as for more advanced students. These chapters include Euclidean Geometry, Axiomatic Systems and Models, Analytic Geometry, Transformational Geometry, and Symmetry. Topics in the other chapters, including Non-Euclidean Geometry, Projective Geometry, Finite Geometry, Differential Geometry, and Discrete Geometry, provide a broader view of geometry. The different chapters are as independent as possible, while the text still manages to highlight the many connections between topics. The text is self-contained, including appendices with the material in Euclid’s first book and a high school axiomatic system as well as Hilbert’s axioms. Appendices give brief summaries of the parts of linear algebra and multivariable calculus needed for certain chapters. While some chapters use the language of groups, no prior experience with abstract algebra is presumed. The text will support an approach emphasizing dynamical geometry software without being tied to any particular software.

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: 6613

"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"--

How People Learn

Brain, Mind, Experience, and School: Expanded Edition

Author: National Research Council,Division of Behavioral and Social Sciences and Education,Board on Behavioral, Cognitive, and Sensory Sciences,Committee on Developments in the Science of Learning with additional material from the Committee on Learning Research and Educational Practice

Publisher: National Academies Press

ISBN: 0309131979

Category: Education

Page: 384

View: 6893

First released in the Spring of 1999, How People Learn has been expanded to show how the theories and insights from the original book can translate into actions and practice, now making a real connection between classroom activities and learning behavior. This edition includes far-reaching suggestions for research that could increase the impact that classroom teaching has on actual learning. Like the original edition, this book offers exciting new research about the mind and the brain that provides answers to a number of compelling questions. When do infants begin to learn? How do experts learn and how is this different from non-experts? What can teachers and schools do-with curricula, classroom settings, and teaching methods--to help children learn most effectively? New evidence from many branches of science has significantly added to our understanding of what it means to know, from the neural processes that occur during learning to the influence of culture on what people see and absorb. How People Learn examines these findings and their implications for what we teach, how we teach it, and how we assess what our children learn. The book uses exemplary teaching to illustrate how approaches based on what we now know result in in-depth learning. This new knowledge calls into question concepts and practices firmly entrenched in our current education system. Topics include: How learning actually changes the physical structure of the brain. How existing knowledge affects what people notice and how they learn. What the thought processes of experts tell us about how to teach. The amazing learning potential of infants. The relationship of classroom learning and everyday settings of community and workplace. Learning needs and opportunities for teachers. A realistic look at the role of technology in education.

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: 3193

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.

Proofs without Words II

Author: Roger B. Nelsen

Publisher: Mathematical Association of America

ISBN: 9780883857212

Category: Mathematics

Page: 142

View: 3782

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.

Geometry Illuminated

An Illustrated Introduction to Euclidean and Hyperbolic Plane Geometry

Author: Matthew Harvey

Publisher: The Mathematical Association of America

ISBN: 1939512115

Category: Mathematics

Page: 560

View: 3465

Geometry Illuminated is an introduction to geometry in the plane, both Euclidean and hyperbolic. It is designed to be used in an undergraduate course on geometry, and as such, its target audience is undergraduate math majors. However, much of it should be readable by anyone who is comfortable with the language of mathematical proof. Throughout, the goal is to develop the material patiently. One of the more appealing aspects of geometry is that it is a very "visual" subject. This book hopes to takes full advantage of that, with an extensive use of illustrations as guides. Geometry Illuminated is divided into four principal parts. Part 1 develops neutral geometry in the style of Hilbert, including a discussion of the construction of measure in that system, ultimately building up to the Saccheri-Legendre Theorem. Part 2 provides a glimpse of classical Euclidean geometry, with an emphasis on concurrence results, such as the nine-point circle. Part 3 studies transformations of the Euclidean plane, beginning with isometries and ending with inversion, with applications and a discussion of area in between. Part 4 is dedicated to the development of the Poincaré disk model, and the study of geometry within that model. While this material is traditional, Geometry Illuminated does bring together topics that are generally not found in a book at this level. Most notably, it explicitly computes parametric equations for the pseudosphere and its geodesics. It focuses less on the nature of axiomatic systems for geometry, but emphasizes rather the logical development of geometry within such a system. It also includes sections dealing with trilinear and barycentric coordinates, theorems that can be proved using inversion, and Euclidean and hyperbolic tilings.

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: 4680

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.

Bridge to Higher Mathematics

Author: Sam Vandervelde


ISBN: 055750337X

Category: Mathematics

Page: 244

View: 8694

This engaging math textbook is designed to equip students who have completed a standard high school math curriculum with the tools and techniques that they will need to succeed in upper level math courses. Topics covered include logic and set theory, proof techniques, number theory, counting, induction, relations, functions, and cardinality.

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: 9731

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.

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: 846

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.

Visual-Spatial Learners

Differentiation Strategies for Creating a Successful Classroom

Author: Alexandra Golon

Publisher: Sourcebooks, Inc.

ISBN: 1618216600

Category: Education

Page: 200

View: 5471

Are you looking for ways to differentiate your instruction to meet the needs of gifted visual-spatial learners? You've found it in Visual-Spatial Learners: Differentiation Strategies for Creating a Successful Classroom (2nd ed.). Visual-spatial learners are students who show advanced abilities with computers, maps, construction toys, and puzzles. The techniques outlined within these pages help all learners succeed—regardless of preferred learning style. Based on the most current understanding of the brain's hemispheric functions, the author provides a number of strategies and lesson plan ideas to help make your classroom a successful learning environment for all learners. These strategies address preparing students to succeed on timed tests; easing the pain of handwriting; teaching spelling using imagery; incorporating mnemonics, rhyme, and other tricks that engage the right hemisphere of the brain; helping students stay focused and on track; getting—and keeping—students organized; and much more!

Better Learning Through Structured Teaching

A Framework for the Gradual Release of Responsibility, 2nd Edition

Author: Douglas Fisher,Nancy Frey

Publisher: ASCD

ISBN: 1416618309

Category: Education

Page: 158

View: 9694

In this updated 2nd edition of the ASCD best-seller, Douglas Fisher and Nancy Frey dig deeper into the hows and whys of the gradual release of responsibility instructional framework. To gradually release responsibility is to equip students with what they need to be engaged and self-directed learners. On a day-to-day level, it means delivering lessons purposefully planned to incorporate four essential and interrelated instructional phases: 1. Focused Instruction: Preparing students for learning by establishing lesson purpose, modeling strategies and skills, thinking aloud, and noticing how students respond. 2. Guided Instruction: Strategically using prompts, cues, and questions to lead students to new understanding. 3. Collaborative Learning: Allowing students to consolidate their understanding through exploration, problem-solving, discussion, and thinking with their peers. 4. Independent Learning: Requiring students to use the skills and knowledge they've acquired to create authentic products and ask new questions. The authors explore each phase, using real-life examples from a variety of disciplines. You'll find tips and tools for classroom implementation, including checklists for planning and assessment; advice on feedback, homework, group work, differentiated instruction, and blended learning; answers to frequently asked questions; and examples that align to Common Core State Standards. No matter what grade level or subject you teach, Better Learning Through Structured Teaching is your essential guide to helping students expand their capacity for successful and long-lasting learning.

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: 2723

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.

Never Work Harder Than Your Students and Other Principles of Great Teaching

Author: Robyn R. Jackson

Publisher: ASCD

ISBN: 1416612262

Category: Education

Page: 249

View: 5032

Is great teaching a gift that only a few of us are born with, or is it a skill that can be learned? In Never Work Harder Than Your Students, Robyn Jackson makes a radical assertion: Any teacher can become a master teacher by developing a master teacher mindset. The master teacher mindset can be achieved by rigorously applying seven principles to your teaching until they become your automatic response to students in the classroom. The more you practice these seven principles, the more you begin to think like a master teacher: 1. Start where your students are. 2. Know where your students are going. 3. Expect to get your students to their goal. 4. Support your students along the way. 5. Use feedback to help you and your students get better. 6. Focus on quality rather than quantity. 7. Never work harder than your students. Using these principles, Jackson shows you how to become a master teacher no matter where you are in your practice. Each chapter provides a detailed explanation of one of the mastery principles, the steps you need to take to apply them to your own practice, and suggestions for how you can begin practicing the principle in your classroom right away. Jackson offers stories from her own teaching practice, as well as from other teachers she has helped, to show you how each principle works. Teaching is a hard job, but using Jackson’s principles will help you and your students reap the rich rewards of that hard work.