Learn Methods for Teaching Mathematics Through Visual Pedagogy Visuals help illustrate mathematical concepts and procedures for teaching mathematics to children. Through visuals with guided instruction, you learn to organize and prioritize information, select and use appropriate representations, and integrate visuals with other pedagogical tools. Mathematical patterns are abundant in the natural world. Here the seedhead of a sunfl ower demonstrates the Fibonacci sequence, the chambered nautilus illustrates the Golden Ratio, and the beehive is constructed from regular hexagons. Connect Mathematics to Our Everyday Lives Children will be motivated to learn mathematics more successfully if they understand how it is a part of their lives inside and outside of school. Throughout this text, mathematics in familiar contexts is illustrated in chapter openers, discussion of children’s literature, lesson plans, examples, and activities. Infuse your lessons with these examples to motivate student interest and notice the difference in how students respond. Classifi cation is an important process linked to the acquisition of counting skills and is also a part of our everyday lives, as illustrated by these students sorting recyclables. Teach Mathematics as a Social Activity Mathematics concepts are addressed in the text through collaborative activities as well as techniques that encourage communication and discourse. Mathematics is foremost a social activity that involves working with others to solve problems and generate new ideas. Vignettes and research projects from real classrooms appear throughout Visualizing Elementary and Middle School Mathematics with questions on how to apply the results of these situations in the These kindergarteners are learning how to ask statistical questions and classroom. collect and interpret data in a collaborative setting. There are four principles to consider when Make Mathematics Accessible to All Populations differentiating instruction for English-language Incorporate how diverse cultures have used and contributed learners: to mathematics, how these contributions can be integrated 1. Comprehensible input into the mathematics curriculum, and how mathematics can 2. Contextualized instruction 3. A safe learning environment be made accessible to all populations. Use real-world and 4. Meaningful learning activities cultural perspectives of mathematics to teach the strong connection between mathematics, culture, and learning. Finger counting has been used by many cultures around the world. Children who learn fi nger-counting techniques can enhance their number sense. (From Count on Your Fingers African Style, written by Claudia Zaslavsky, illustrated by Wangechi Mutu) V I S U A L I Z I N G ELEMENTARY AND MIDDLE SCHOOL MATHEMATICS METHODS JOAN COHEN JONES, PhD Eastern Michigan University Visualizing Elementary & Middle School Mathematics Methods offers future teachers the opportunity to learn about teaching mathematics with real-life examples, multicultural perspectives, and powerful visuals. This dynamic approach enables students to set aside their previous beliefs about mathematics and to learn concepts and pedagogy from a new perspective. For example, using a real-life visual like a lighthouse can help teach math in a meaningful way. Many lighthouses, like the one pictured above (an interior and an exterior photo) and on the front cover, were built with spiral staircases because they take up less floor space than traditional staircases. In addition to being used for decorative and architectural purposes, spiral curves have been studied by mathematicians since the time of the ancient Greeks. They appear in many forms—including the shell of a snail, the structure of a chambered nautilus, and the shape of a whirlpool—a reminder that math is everywhere. Credits VICE PRESIDENT AND EXECUTIVE PUBLISHER Jay O’Callaghan EXECUTIVE EDITOR Christopher Johnson ACQUISITIONS EDITOR Robert Johnston DIRECTOR OF DEVELOPMENT Barbara Heaney MANAGER, PRODUCT DEVELOPMENT Nancy Perry WILEY VISUALIZING PROJECT EDITOR Beth Tripmacher WILEY VISUALIZING SENIOR EDITORIAL ASSISTANT Tiara Kelly PROGRAM ASSISTANT Brittany Cheetham AAA EDITORIAL ASSISTANT Mariah Maguire-Fong AAA BBB ASSOCIATE DIRECTOR, MARKETING Jeffrey Rucker AAA AAA SENIOR MARKETING MANAGER Danielle Torio Hagey BBB CONTENT MANAGER Micheline Frederick SENIOR MEDIA EDITOR Lynn Pearlman CREATIVE DIRECTOR Harry Nolan COVER DESIGN Harry Nolan INTERIOR DESIGN Jim O’Shea PHOTO MANAGER Elle Wagner PHOTO RESEARCHER Teri Stratford SENIOR ILLUSTRATION EDITOR Sandra Rigby PRODUCTION SERVICES Camelot Editorial Services, LLC COVER CREDITS: Main Image: Geri Lynn Smith/iStockphoto Filmstrip (from left to right): Myrleen Ferguson Cate/PhotoEdit; GEORGE GRALL/ NG Image Collection; Clare Hooper/Alamy; Wealan Pollard/OJO Images/Getty Images, Inc.; RAYMOND GEHMAN/NG Image Collection Back Main Image: Kenneth C. Zirkel/iStock Exclusive/Getty Images, Inc. Back Inset: Myrleen Ferguson Cate/PhotoEdit This book was set in New Baskerville by Silver Editions, Inc. and printed and bound by Quad/ Graphics, Inc. The cover was printed by Quad/Graphics, Inc. Copyright © 2012 John Wiley & Sons, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc. 222 Rosewood Drive, Danvers, MA 01923, Web site www. copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, (201) 748-6011, fax (201) 748-6008, Web site http://www.wiley.com/go/permissions. Evaluation copies are provided to qualifi ed academics and professionals for review purposes only, for use in their courses during the next academic year. These copies are licensed and may not be sold or transferred to a third party. Upon completion of the review period, please return the evaluation copy to Wiley. Return instructions and a free of charge return shipping label are available at www.wiley.com/go/returnlabel. Outside of the United States, please contact your local representative. ISBN 13: 978-0470-450314 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 Preface How Is Wiley Visualizing Different? Wiley Visualizing differs from competing textbooks by with an abundance of class-tested hands-on activities and uniquely combining several powerful elements: a visual full Lesson Plans based on NCTM and Common Core State pedagogy, integrated with comprehensive text; the use Standards. In the Classroom features present images and of authentic classroom situations and activities, actual research-based classroom practices, and Multicultural materials from children’s literature and publications such as Perspectives in Mathematics features provide content-rich, Mathematics Teaching Today, Teaching Children Mathematics, culturally relevant examples of mathematics and its place in the and Mathematics Teaching in the Middle School, and the world. Each chapter presents illustrations from children’s books integration of Teachscape videos. that contain exciting connections to mathematics content and offers detailed teaching strategies. These authentic 1. Visual Pedagogy. Wiley Visualizing is based on decades of situations and materials immerse the student in real-life issues research on the use of visuals in learning (Mayer, 2005).1 Using in mathematics education, thereby enhancing motivation, the Cognitive Theory of Multimedia Learning, which is backed up learning, and retention (Donovan & Bransford, 2005).2 by hundreds of empirical research studies, Wiley’s authors select visualizations for their texts that specifi cally support students’ 3. Teachscape Videos. Through a partnership with Teachscape thinking and learning. Visuals and text are conceived and planned professional development series, Wiley Visualizing provides together in ways that clarify and reinforce major concepts while a collection of online videocases featuring rich, authentic allowing students to understand the details. This commitment to classroom situations, teacher refl ection, and interviews. Each distinctive and consistent visual pedagogy sets Wiley Visualizing of the videocases is referenced within the chapters, supporting apart from other textbooks. the relevant content. The combination of textbook and video 2. Authentic Classroom Situations, Activities, and provides learners with multiple entry points to the content, giving Materials. Wiley Visualizing provides the pre-service teacher them greater opportunity to explore and apply concepts. Wiley Visualizing is designed as a natural extension of how we learn To understand why the visualizing approach is effective, it is engaged in learning or exert positive effort to create mental fi rst helpful to understand how we learn. models; and unproductive cognitive load, which occurs when the brain is trying to make sense of needlessly complex content 1. Our brain processes information using two main channels: visual or when information is not presented well. The learning and verbal. Our working memory holds information that our process can be impaired when the information to be processed minds process as we learn. This “mental workbench” helps us with decisions, problem solving, and making sense of words and exceeds the capacity of working memory. Well-designed visuals pictures by building verbal and visual models of the information. and text with effective pedagogical guidance can reduce the unproductive cognitive load in our working memory. 2. When the verbal and visual models of corresponding information are integrated in working memory, we form more comprehensive, Research shows that well-designed visuals, integrated with lasting mental models. comprehensive text, can improve the effi ciency with which a 3. When we link these integrated mental models to our prior learner processes information. In this regard, SEG Research, an knowledge, which is stored in our long-term memory, we build independent research fi rm, conducted a national, multisite study even stronger mental models. When an integrated (visual plus evaluating the effectiveness of Wiley Visualizing. Its fi ndings verbal) mental model is formed and stored in long-term memory, indicate that students using Wiley Visualizing products (both print real learning begins. and multimedia) were more engaged in the course, exhibited The effort our brains put forth to make sense of instructional greater retention throughout the course, and made signifi cantly information is called cognitive load. There are two kinds of greater gains in content area knowledge and skills, as compared cognitive load: productive cognitive load, such as when we’re to students in similar classes that did not use Wiley Visualizing.3 1Mayer, R. E. (Ed.) (2005). The Cambridge Handbook of Multimedia Learning. New York: Cambridge University Press. 2Donovan, M. S., & Bransford, J. (Eds.) (2005). How Students Learn: Science in the Classroom. The National Academy Press. Available online at http://www.nap.edu/openbook.php?record_id=11102&page=1. 3SEG Research (2009). Improving Student-Learning with Graphically-Enhanced Textbooks: A Study of the Effectiveness of the Wiley Visualizing Series. Preface iii How Are the Wiley Visualizing Chapters Organized? Student engagement requires more than just providing visuals, text, and interactivity—it entails motivating students to learn. It is easy to get bored or lose focus when presented with large amounts of information, and it is easy to lose motivation when the relevance of the information is unclear. Wiley Visualizing organizes course content into manageable learning modules and relates it to everyday life. It transforms learning into an interactive, stimulating, and outcomes-oriented experience for students. Each learning module has a clear instructional objective, one or more examples, and an opportunity for assessment. These modules are the building blocks of Wiley Visualizing. Each Wiley Visualizing chapter engages students from the start Chapter opening text and visuals introduce the subject and connect the student with the material that follows. Chapter Introductions Alongside Chapter Outlines provide Key striking photographs, narratives Questions to guide students recount intriguing classroom experiences to evoke student interest through the chapter. in the chapter’s central mathematics concept. For each chapter, the NCTM Principles and Standards are highlighted for the relevant grade-level band, giving the reader an overview of the standards-based mathematics the chapter will present. The Chapter Planner gives students a path through the learning aids in the chapter. Throughout the chapter, The Planner icon prompts students to use the learning aids and to set priorities as they study. iv VISUALIZING ELEMENTARY AND MIDDLE SCHOOL MATHEMATICS METHODS Wiley Visualizing guides students through the chapter The content of Wiley Visualizing gives students a variety of approaches—visuals, words, interactions, video, and assessments—that work together to provide students with a guided path through the content. Learning Objectives at the start of each section indicate in behavioral terms the concepts that students are expected to Process Diagrams provide in-depth explanation master while reading the section. of how to use mathematics pedagogy. Clear, step-by-step narrative enables students to grasp important topics with less effort. Throughout the text, visuals provide prospective teachers with samples of tools to use in the classroom. Several visuals offer tools for differentiating instruction to meet the needs of all learners. Other visuals support the text by providing glimpses of students using the materials Education InSight features are multipart visual sections and learning the that focus on a key concept or topic in the chapter, concepts presented in exploring it in detail or in broader context using a the narrative. combination of visuals. Preface v In each chapter, the Strategies for the Children’s Literature Classroom offers detailed feature presents illustrations suggestions of how to use from children’s books that children’s books to motivate Multicultural Perspectives in Strategies for the Classroom contain exciting connections mathematics learning. Mathematics present content- guide prospective teachers to to mathematics content. rich, culturally relevant examples analyze the material, develop of mathematics and its place in insights into essential concepts, and use them in the classroom. the world. Fully-developed Lesson Plans Prospective teachers are given an model ways to make mathematics abundance of hands-on Activities, culturally relevant and refl ective of which include illustrations of materials students’ lives outside the classroom, and complete instructions. They can while fulfi lling standards-based be used as mini-lessons for children to mathematics objectives. practice using mathematics concepts. vi VISUALIZING ELEMENTARY AND MIDDLE SCHOOL MATHEMATICS METHODS In the Classroom features provide a real-life look into a classroom and give students access to a wide range of ideas and classroom research. Many are from the pages of Teaching Children Mathematics. Through a partnership with Teachscape’s professional development series, a collection of videocases featuring rich, authentic classroom situations supplements the textbook’s instruction. In the textbook, Virtual Classroom Observations highlight a videocase that corresponds to the content in the text and provides Teaching Tips provide applications of best focal points for the viewer. practices. Concept Check questions at the end of Tech Tools help prospective teachers learn each section allow students to test their how to integrate technology in the classroom. comprehension of the learning objectives. Preface vii Student understanding is assessed at different levels Critical and Creative Thinking Questions challenge students to think more broadly about chapter concepts. The level of these questions Wiley Visualizing offers students lots of practice ranges from simple to advanced; they encourage students to think material in several modalities for assessing their critically and develop an analytical understanding of the ideas discussed understanding of each study objective. in the chapter. The Summary revisits each major section, with informative images taken from the chapter. These visuals reinforce important concepts. In the fi eld provides opportunities for Using Visuals calls upon students prospective teachers to explore the to use the visuals in this textbook concepts developed in the chapter in as a springboard for creating their a variety of real-world situations, from own classroom materials or for analyzing textbooks to observing and understanding the concepts of the interviewing teachers and students. chapter. What is happening in this picture? presents a new uncaptioned photograph or illustration, such as children’s work, that is relevant to a chapter topic. Visual end-of chapter Self-Tests pose review questions that ask students to demonstrate their understanding of key concepts. Think Critically questions ask the students to describe and explain what they can observe in the image based on what they have learned. viii VISUALIZING ELEMENTARY AND MIDDLE SCHOOL MATHEMATICS METHODS