Proofs Without Words

Exercises in Visual Thinking

Author: Malcolm Scott MacKenzie,Roger B. Nelsen

Publisher: MAA

ISBN: 9780883857007

Category: Mathematics

Page: 140

View: 5296

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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: 8025

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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

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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: 6026

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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: 5597

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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: 9781118015032

Category: Education

Page: 320

View: 9057

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A proven program for enhancing students' thinking and comprehension abilities Visible Thinking is a research-based approach to teaching thinking, begun at Harvard's Project Zero, 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 a varied collection of practices, including thinking routines?small sets of questions or a short sequence of steps?as well as the documentation of student thinking. Using this process thinking becomes visible as the students' different viewpoints are expressed, documented, discussed and reflected upon. Helps direct student thinking and structure classroom discussion Can be applied with students at all grade levels and in all content areas Includes easy-to-implement classroom strategies The book also comes with a DVD of video clips featuring Visible Thinking in practice in different classrooms.

The Differentiated Classroom

Responding to the Needs of All Learners, 2nd Edition

Author: Carol Ann Tomlinson

Publisher: ASCD

ISBN: 1416618635

Category: Education

Page: 197

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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.

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: 8728

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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.

A Handbook for Teaching and Learning in Higher Education

Enhancing academic practice

Author: Heather Fry,Steve Ketteridge,Stephanie Marshall

Publisher: Routledge

ISBN: 1317650220

Category: Education

Page: 452

View: 3024

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This entirely new edition of a very successful book focuses on developing professional academic skills for supporting and supervising student learning and effective teaching. It is built on the premise that the roles of those who teach in higher education are complex and multi-faceted. A Handbook for Teaching and Learning in Higher Education is sensitive to the competing demands of teaching, research, scholarship, and academic management. The new edition reflects and responds to the rapidly changing context of higher education and to current understanding of how to best support student learning. Drawing together a large number of expert authors, it continues to feature extensive use of case studies that show how successful teachers have implemented these ideas. It includes key topics such as student engagement and motivation, internationalisation, employability, inclusive strategies for teaching, effective use of technology and issues relating to postgraduate students and student retention. Part 1 explores a number of aspects of the context of UK higher education that affect the education of students, looking at the drivers of institutional behaviours and how to achieve success as a university teacher. Part 2 examines learning, teaching and supervising in higher education and includes chapters on working with diversity, encouraging independent learning and learning gain. Part 3 considers approaches to teaching and learning in different disciplines, covering a full range including arts and humanities, social sciences, experimental sciences through to medicine and dentistry. Written to support the excellence in teaching and learning design required to bring about student learning of the highest quality, this will be essential reading for all new lecturers, particularly anyone taking an accredited course in teaching and learning in higher education, as well as those experienced lecturers who wish to improve their teaching practice. Those working in adult learning and educational development will also find the book to be a particularly useful resource. In addition it will appeal to staff who support learning and teaching in various other roles.

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: 5082

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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.

Visual Group Theory

Author: Nathan Carter

Publisher: MAA

ISBN: 9780883857571

Category: Mathematics

Page: 297

View: 8700

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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: 6294

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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.

Biscuits of Number Theory

Author: Arthur T. Benjamin,Ezra Brown

Publisher: MAA

ISBN: 9780883853405

Category: Mathematics

Page: 311

View: 8351

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

The Chicago Handbook for Teachers, Second Edition

A Practical Guide to the College Classroom

Author: Alan Brinkley,Esam E. El-Fakahany,Betty Dessants

Publisher: University of Chicago Press

ISBN: 0226075281

Category: Education

Page: 213

View: 4416

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This new edition is thoroughly revised for contemporary concerns, with updated coverage on the use of electronic resources and on the challenge of creating and sustaining an inclusive classroom. A new chapter on science education and new coverage of the distinctive issues faced by adjunct faculty broaden the book's audience considerably.

Proofs without Words II

Author: Roger B. Nelsen

Publisher: Mathematical Association of America

ISBN: 9780883857212

Category: Mathematics

Page: 142

View: 1707

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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.

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: 1931

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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.

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: 9139

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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.

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: 8947

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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.

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: 3964

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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.

Thinking in Pictures, Expanded Edition

My Life with Autism

Author: Temple Grandin

Publisher: Vintage

ISBN: 9780307548030

Category: Psychology

Page: 304

View: 5083

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Temple Grandin, Ph.D., is a gifted animal scientist who has designed one third of all the livestock-handling facilities in the United States. She also lectures widely on autism—because Temple Grandin is autistic, a woman who thinks, feels, and experiences the world in ways that are incomprehensible to the rest of us. In this unprecedented book, Grandin delivers a report from the country of autism. Writing from the dual perspectives of a scientist and an autistic person, she tells us how that country is experienced by its inhabitants and how she managed to breach its boundaries to function in the outside world. What emerges in Thinking in Pictures is the document of an extraordinary human being, one who, in gracefully and lucidly bridging the gulf between her condition and our own, sheds light on the riddle of our common identity. From the Trade Paperback edition.