Teaching Secondary and Middle School Mathematics PDF
Preview Teaching Secondary and Middle School Mathematics
TEACHING SECONDARY AND MIDDLE SCHOOL MATHEMATICS Teaching Secondary and Middle School Mathematics combines the latest developments in research, standards, and technology with a vibrant writing style to help teachers prepare for the excitement and challenges of teaching secondary and middle school mathematics today. In the fully revised fifth edition, scholar and mathematics educator Daniel J. Brahier invites teachers to investigate the nature of the mathematics curriculum and reflect on research-based “best practices” as they define and sharpen their own personal teaching styles. The fifth edition has been updated and expanded with a particular emphasis on the continued impact of the Common Core State Standards for Mathematics and NCTM’s 2014 document Principles to Actions, as well as increased attention to teaching with technology, classroom management, and differentiated instruction. Features: • A new Chapter 7 on selection and use of specific tools and technology combined with “Spotlight on Technology” features throughout clearly illustrate the practical aspects of how technology can be used for teaching or professional development. • Foundational Chapters 1 and 2 on the practices and principles of mathematics education have been revised to build directly on Common Core State Standards for Mathematics and Principles to Actions, with additional references to both documents throughout all chapters. • A new Chapter 4 focuses on the use of standards in writing objectives and organizing lesson plan resources, while an updated Chapter 5 details each step of the lesson-planning process. • A fully revised Chapter 12 provides new information on teaching diverse populations and outlines specific details and suggestions for classroom management for mathematics teachers. • A “Classroom Dialogues” feature draws on the author’s 35-year experience as an educator to present real-world teacher-student conversations about specific mathematical problems or ideas. • A “How Would You React?” feature prepares future teachers for real-life scenarios by engaging them in common classroom situations and offering tried-and-true solutions. With more than 60 practical, classroom-tested teaching ideas, sample lessons, and activities, Teaching Secondary and Middle School Mathematics combines the best of theory and practice to pro- vide clear descriptions of what it takes to be an effective teacher of mathematics. Daniel J. Brahier is a professor of mathematics education at Bowling Green State University in Ohio and Director of the Science and Math Education in ACTION scholarship program at BGSU. He also teaches junior high (grades 7 and 8) mathematics at St. Rose School in Perrysburg, Ohio. This Page is Intentionally Left Blank TEACHING SECONDARY AND MIDDLE SCHOOL MATHEMATICS Fifth Edition Daniel J. Brahier First published 2016 by Routledge 711 Third Avenue, New York, NY 10017 and by Routledge 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN Routledge is an imprint of the Taylor & Francis Group, an informa business © 2016 Taylor & Francis The right of Daniel J. Brahier to be identified as author of this work has been asserted by him in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging in Publication Data Names: Brahier, Daniel J. Title: Teaching secondary and middle school mathematics / Daniel Brahier. Description: Fifth edition. | New York : Routledge, 2016. | Includes bibliographical references and index. Identifiers: LCCN 2015032124| ISBN 9781138922778 (Hardback) | ISBN 9781138922785 (Paperback) | ISBN 9781315685502 (ebook) Subjects: LCSH: Mathematics—Study and teaching (Middle school) | Mathematics—Study and teaching (Secondary) Classification: LCC QA11.2 .B73 2016 | DDC 510.71/2—dc23 LC record available at http://lccn.loc.gov/2015032124 ISBN: 978-1-138-92277-8 (hbk) ISBN: 978-1-138-92278-5 (pbk) ISBN: 978-1-315-68550-2 (ebk) Typeset in StoneSans and StoneSerif by Swales & Willis Ltd, Exeter, Devon, UK This book is dedicated to my father, Frank, who has taught me the values of patience and persistence, and to the memory of my mother, Joyce, who was the author of the first book in the Brahier family and a source of support and love beyond imagination . . .“the finest of teachers I’ve ever known.” This Page is Intentionally Left Blank CONTENTS Preface xiv Acknowledgments xviii About the Author xix UNIT 1 What Does It Mean to “Do,” “Teach,” and “Learn” Mathematics? 1 CHAPTER 1 Mathematics as a Process 3 National and International Assessment Data 5 How Would You React? 12 The Need for Reform 13 “Doing” Mathematics 15 Problem Solving 15 Mathematical Reasoning 19 Spotlight on Technology 20 Communication 22 Connections and Structure 23 Representation 25 Classroom Dialogues 26 Conclusion 29 Glossary 30 Discussion Questions and Activities 31 CHAPTER 2 Principles of Mathematics Education 33 The Six Principles 34 Teaching and Learning 34 How Would You React? 37 Access and Equity 38 Curriculum 39 Tools and Technology 41 Spotlight on Technology 42 viii Contents Assessment 43 Professionalism 44 Classroom Dialogues 45 Connecting the Principles 47 Conclusion 48 Glossary 49 Discussion Questions and Activities 49 CHAPTER 3 Learning Theories and Psychology in Mathematics Education 51 Research in Mathematics Education 52 Quantitative and Qualitative Methods 53 How Would You React? 54 Experimental and Descriptive Research 55 Learning Theories in Mathematics 57 Development of Learning Theory 57 Bruner’s Stages of Representation 59 The van Hiele Model 63 The Inquiry Approach and Constructivist Model 65 Spotlight on Technology 69 Inductive versus Deductive Teaching 70 Classroom Dialogues 73 Motivation 75 Conclusion 78 Glossary 79 Discussion Questions and Activities 83 UNIT 2 Mathematics Curriculum and Planning 85 CHAPTER 4 Standards, Objectives, and Resources 87 Developing Curriculum Standards—Historical Perspective 88 How Would You React? 90 The Common Core State Standards for Mathematics 92 The Core Curriculum 95 Traditional versus Integrated Sequences 101 Spotlight on Technology 108 Goals and Objectives in the Mathematics Classroom 109 Classifying Objectives 111 Affective Objectives 112 Cognitive Objectives 112 Selection and Organization of Resources 118 Textbooks 118 Contents ix Print Resources 121 Classroom Dialogues 123 The Internet 125 Organizing a Resource File 125 Conclusion 127 Glossary 128 Discussion Questions and Activities 129 CHAPTER 5 Planning for Instruction 131 Unit Planning 132 How Would You React? 137 Lesson Planning 138 General Information 142 Standards and Objectives 142 Materials and Resources 143 Motivation 143 Lesson Procedure 144 Closure 146 Extension 146 Assessment 147 Spotlight on Technology 147 Sample Lesson Plans 149 Analysis of Sample Lesson Plan 1 154 Analysis of Sample Lesson Plan 2 155 Classroom Dialogues 158 Lesson Imaging versus Lesson Planning 160 Reflecting on a Lesson 161 Conclusion 162 Glossary 163 Discussion Questions and Activities 164 UNIT 3 Mathematics Teaching 167 CHAPTER 6 Teaching and Learning: Tasks, Learning Environment, and Discourse 169 Teaching Standards and Related Principles 171 The NCTM Standards for Teaching and Learning Mathematics 171 Knowledge Standards 173 The Teaching and Learning Principle 174 Selecting Activities and Problems 174 How Would You React? 179 Learning Environment 180 x Contents Spotlight on Technology 184 Classroom Discourse 186 The Role of Discourse 186 Questioning Skills 188 Reading and Literacy 191 Classroom Dialogues 193 Cooperative Learning 194 Reflective Practices 197 Conclusion 199 Glossary 200 Discussion Questions and Activities 200 CHAPTER 7 Teaching and Learning: Tools and Technology 202 How Would You React? 205 Classroom Tools 206 Hands-on Manipulative Materials 206 Spotlight on Technology 210 Diagrams, Graphs, and Other Discussion Starters 212 Classroom Technology 214 Calculators 215 CAS Technology 216 Interactive Whiteboards 217 Mobile Devices 219 Classroom Dialogues 221 Dynamic Geometry Software 223 Document Cameras 225 Blended Classes and Flipped Classrooms 226 Conclusion 227 Glossary 228 Discussion Questions and Activities 228 CHAPTER 8 Teaching Number Sense and Algebra 230 Reasoning and Sense Making 231 How Would You React? 233 The Teaching of Number Sense 235 Sample Lesson Plan: Number Sense 236 Number Sense—Activities Sampler 239 Spotlight on Technology 243 The Teaching of Algebra 244 Sample Lesson Plan: Algebra 247 Classroom Dialogues 249 Algebra—Activities Sampler 251 Contents xi Conclusion 255 Glossary 256 Discussion Questions and Activities 256 CHAPTER 9 Teaching Geometry, Statistics/Probability, and Discrete Mathematics 258 The Teaching of Geometry 259 Sample Lesson Plan: Geometry 260 How Would You React? 263 Geometry—Activities Sampler 265 Spotlight on Technology 269 The Teaching of Statistics and Probability 271 Sample Lesson Plan: Statistics and Probability 273 Statistics and Probability—Activities Sampler 276 The Teaching of Discrete Mathematics 281 Sample Lesson Plan: Discrete Mathematics 283 Classroom Dialogues 287 Discrete Mathematics—Activities Sampler 288 A Word on Pre-Calculus and Calculus 294 Conclusion 295 Glossary 295 Discussion Questions and Activities 296 UNIT 4 Assessment in Mathematics 297 CHAPTER 10 Assessment: Purposes and Strategies 299 What Is Assessment? 300 Purposes for Assessment 302 Test Construction 304 Preparing Items 305 Ensuring Validity and Reliability 307 Scoring Student Work 308 How Would You React? 309 Including Review Items 311 Limitations to Written Tests 311 Alternate Strategies for Assessing Student Progress 312 Journals 312 Open-Ended Questions and Rubrics 314 Spotlight on Technology 319 Individual and Team Projects 320 Observations and Checklists 323 Interviews 326 xii Contents Self-Assessment 328 Classroom Dialogues 329 Portfolios 330 Conclusion 333 Glossary 334 Discussion Questions and Activities 335 CHAPTER 11 Assessment: Standards, Homework, and Grading 337 NCTM Recommendations on Assessment 338 The Assessment Standards 338 How Would You React? 342 The Assessment Principle 343 Equity and Assessment 344 Homework Assignments 346 Homework Amount and Frequency 346 Checking Homework Assignments 350 Spotlight on Technology 351 Using Homework Assignments in Assessment 354 Classroom Dialogues 356 Evaluation: Maintaining and Determining Final Grades 358 Conclusion 364 Glossary 365 Discussion Questions and Activities 365 UNIT 5 Meeting Individual Student and Teacher Needs 367 CHAPTER 12 Managing a Mathematics Classroom 369 Defining and Achieving Equity 370 Students with Special Needs 373 How Would You React? 376 Gender 379 Spotlight on Technology 383 Ethnic and Cultural Issues 385 English Language Learners 388 Classroom Dialogues 389 Ideas for Meeting Diverse Student Needs 392 Differentiated Instruction 392 General Suggestions for Differentiating in the Classroom 394 Classroom Management Tips 397 Conclusion 399 Glossary 400 Discussion Questions and Activities 401 Contents xiii CHAPTER 13 The Teacher of Mathematics in the School Community 403 Working with Parents 404 How Would You React? 409 The Supervision and Evaluation of Teachers 410 Spotlight on Technology 416 Functioning in a Department 418 Ongoing Professional Development 419 Making the Case for Long-Term Development 419 Classroom Dialogues 421 Opportunities for Professional Development 423 The Professional Development Plan 425 Conclusion 426 Discussion Questions and Activities 427 Bibliographic References and Resources 429 Index 460 PREFACE Teaching secondary and middle school mathematics has never been more exciting than it is today. The Common Core State Standards for Mathematics, released in 2010, promotes a comprehensive mathematics curriculum for all students, rooted in significant research on how students learn mathematics and the role of technology in the teaching and learning process. More recently, the National Council of Teachers of Mathematics (NCTM) released a document entitled Principles to Actions: Ensuring Mathematical Success for All in which they describe a blueprint for reforming math- ematics education so that state and national Standards can be met. Also, in 2007, NCTM released the second edition of its teaching standards document, now entitled Mathematics Teaching Today. Meanwhile, the ongoing Trends in International Mathematics and Science Study has resulted in the most comprehensive international comparisons of mathematics education that have ever been conducted. In the midst of all of the reform efforts that have been put in motion, it is important that teacher candidates and inservice teachers investigate the nature of the mathematics curriculum and reflect on research-based “best practices” as they define and sharpen their own personal styles. The format of the fifth edition of Teaching Secondary and Middle School Mathematics is designed to provide the reader with the total picture of the multifaceted art of teaching. What Is New to the Fifth Edition? The fifth edition of Teaching Secondary and Middle School Mathematics features a number of significant changes and enhancements: • A brand new chapter describes the roles of tools and technology in the teaching and learning of mathematics. The former Chapter 7 on teaching strategies was expanded into two chapters (now Chapters 6 and 7) to provide a more in-depth look at how tools and technology can be creatively used in the teaching process. • Chapter 12 was completely revised to include more information on teaching diverse populations and outlines specific details on classroom management for mathematics teachers. The chapter was rewritten with over two dozen new citations with the latest research on meeting the diverse needs of students in the mathematics classroom. • Due to the release of the Common Core State Standards (CCSSM) in 2010 and Principles to Actions by NCTM in 2014, major revisions reflect quotes and philosophy from both of these documents. Specifically, Chapters 1 and 2 were rewritten with these documents as their cornerstone, and every chapter in the book now includes references to these landmark publications.
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