Teaching Mathematics Meaningfully Solutions for Reaching Struggling Learners Second Edition by David H. Allsopp, Ph.D. University of South Florida Tampa LouAnn H. Lovin, Ph.D. James Madison University Harrisonburg, Virginia and Sarah van Ingen, Ph.D. University of South Florida Tampa Baltimore • London • Sydney Paul H. Brookes Publishing Co. Post Office Box 10624 Baltimore, Maryland 21285-0624 USA www.brookespublishing.com Copyright © 2018 by Paul H. Brookes Publishing Co., Inc. All rights reserved. Previous edition copyright © 2007. “Paul H. Brookes Publishing Co.” is a registered trademark of Paul H. Brookes Publishing Co., Inc. Typeset by Absolute Service, Inc., Towson, Maryland. Manufactured in the United States of America by Sheridan Books, Inc., Chelsea, Michigan. All examples in this book are composites. Any similarity to actual individuals or circumstances is coincidental, and no implications should be inferred. Purchasers of Teaching Mathematics Meaningfully: Solutions for Reaching Struggling Learners, Second Edition, are granted permission to download, photocopy, and print the forms and activities found in Appendices A–D for educational and professional purposes. This material may not be reproduced to generate revenue for any program or individual. Photocopies may only be made from an original book. Unauthorized use beyond this privilege may be prosecutable under federal law. You will see the copyright protection notice at the bottom of each photocopiable page. Library of Congress Cataloging-in-Publication Data Names: Allsopp, David H., author. | Lovin, LouAnn H., author. | van Ingen, Sarah, author. Title: Teaching mathematics meaningfully: solutions for reaching struggling learners / by David H. Allsopp, Ph.D., University of South Florida, Tampa, LouAnn H. Lovin, Ph.D., James Madison University, Harrisonburg, Virginia, and, Sarah van Ingen, Ph.D., University of South Florida, Tampa. Description: Second edition. | Baltimore: Paul H. Brookes Publishing Co., [2018] | Includes bibliographical references and index. Identifiers: LCCN 2017027635 (print) | LCCN 2017033234 (ebook) | ISBN 9781598575590 (epub) | ISBN 9781598575637 (pdf) | ISBN 9781598575583 (pbk.) Subjects: LCSH: Mathematics—Study and teaching (Elementary) | Mathematics—Study and teaching (Middle school) | Mathematics—Study and teaching (Secondary) | Attention-deficit- disordered youth—Education. | Learning disabled teenagers—Education. Classification: LCC QA13 (ebook) | LCC QA13 .A44 2018 (print) | DDC 371.9/0447—dc23 LC record available at https://lccn.loc.gov/2017027635 British Library Cataloguing in Publication data are available from the British Library. 2021 2020 2019 2018 2017 10  9  8  7  6  5  4  3  2  1 Contents About the Activities and Forms ...........................................v About the Authors .....................................................vii Preface ............................................................... ix Acknowledgments .....................................................xv 1 Critical Components of Meaningful and Effective Mathematics Instruction for Students with Disabilities and Other Struggling Learners ........................1 I Identify and Understand the Mathematics 2 The Big Ideas in Mathematics and Why They Are Important ............15 3 Children’s Mathematics: Learning Trajectories .........................41 II Learning the Needs of Your Students and the Importance of Continuous Assessment 4 Barriers to Mathematical Success for Students with Disabilities and Other Struggling Learners .......................69 5 Math Assessment and Struggling Learners ............................97 III Plan and Implement Responsive Instruction 6 Making Flexible Instructional Decisions: A Continuum of Instructional Choices for Struggling Learners ......... 137 7 Essential Instructional Approaches for Struggling Learners in Mathematics ................................155 8 Changing Expectations for Struggling Learners: Integrating the Essential Instructional Approaches with the NCTM Mathematics Teaching Practices ...................... 217 9 Mathematics MTSS/RTI and Research on Mathematics Instruction for Struggling Learners .....................239 iii iv Contents 10 How to Intensify Assessment and Essential Instructional Approaches within MTSS/RTI .................247 11 Intensifying Math Instruction Across Tiers within MTSS: Evaluating System-Wide Use of MTSS ...............................269 IV Bringing It All Together 12 The Teaching Mathematics Meaningfully Process .....................281 References ...........................................................299 Appendices A Take Action Activities ............................................. 313 B ARC Assessment Planning Form ...................................337 C Peer-Tutoring Practice Activity .....................................341 D Using a Think-Aloud ..............................................343 E Case Study .......................................................345 Index ...............................................................375 About the Activities and Forms Purchasers of Teaching Mathematics Meaningfully: Solutions for Reaching Struggling Learners, Second Edition, are granted permission to photocopy and/or print out the forms and activities provided in Appendices A–D for professional and educational purposes. None of the forms may be reproduced to generate revenue for any pro- gram or individual. Photocopies and/or printouts may only be made from an origi- nal book. You can also find these materials online at www.brookespublishing.com/ allsopp/materials. v About the Authors David H. Allsopp, Ph.D., Professor of Special Education, College of Educa- tion, University of South Florida, 4202 East Fowler Avenue, EDU 105, Tampa, Florida 32620 Dr. Allsopp is Assistant Dean for Education and Partnerships in addition to being the David C. Anchin Center Endowed Chair and Director of the David C. Anchin Center at the College of Education at the University of South Florida. He is also Professor in the Department of Teaching and Learning—Special Education Programs. Dr. Allsopp holds degrees from Furman University (B.A., Psychology) and the University of Florida (M.Ed., Learning Disabilities; Ph.D., Special Education). Dr. Allsopp teaches at both the undergraduate and doctoral levels, and his scholarship revolves around effective instructional practices, with an emphasis on mathematics, for students with high-incidence disabili- ties (e.g., specific learning disabilities, attention-deficit/hyperactivity disorder, social-emotional/behavior disorders) and other struggling learners who have not been identified with disabilities. Dr. Allsopp also engages in teacher edu- cation research related to how teacher educators can most effectively prepare teachers to address the needs of students with disabilities and other struggling learners. Dr. Allsopp began his career in education as a middle school teacher for students with learning disabilities and emotional/behavioral difficulties in Ocala, Florida. After completing his doctoral studies at the University of Florida, Dr. Allsopp served on the faculty at James Madison University for 6 years. He has been a member of the faculty at University of South Florida since 2001. LouAnn H. Lovin, Ph.D., Professor of Mathematics Education, Department of Mathematics and Statistics, James Madison University, 800 South Main Street, MSC 1911, Harrisonburg, Virginia 22807 Dr. Lovin began her career teaching mathematics to middle and high school stu- dents before making the transition to Pre-K through Grade 8. For over 20 years, she has worked in elementary and middle school classrooms. Then and now, Dr. Lovin engages with teachers in professional development as they implement a student-centered approach to teaching mathematics. At the time of this publica- tion, she focused her research concerning teachers’ mathematical knowledge for teaching on the developmental nature of prospective teachers’ fraction knowledge. She has published articles in Teaching Children Mathematics, Mathematics Teaching in the Middle School, Teaching Exceptional Children, and the Journal of Mathematics Teacher Education. She coauthored the Teaching Student-Centered Mathematics Professional Development Series with John A. van de Walle, Karen Karp, and Jenny vii viii About the Authors Bay-Williams ( Pearson, 2013). Dr. Lovin is an active member of the National Coun- cil of Teachers of Mathematics, the Association of Mathematics Teacher Education, and the Virginia Council of Teachers of Mathematics. Sarah van Ingen, Ph.D., Assistant Professor of Mathematics Education, Depart- ment of Childhood Education and Literacy, College of Education, University of South Florida, 4202 East Fowler Avenue, EDU 202, Tampa, Florida 33620 Dr. van Ingen codirects the innovative and nationally recognized Urban Teacher Residency Partnership Program. In this role, she partners with Hillsborough County Public Schools’ teachers and administrators to improve the learning of both elementary students and prospective elementary teachers. She also teaches courses in mathematics education and teacher preparation at the undergraduate, masters, and doctoral levels. Dr. van Ingen holds a bachelor’s degree from St. Olaf College, a master of arts in teaching from the University of Tampa, and a doctoral degree from the University of South Florida. She was elected into membership in Phi Beta Kappa and was the recipient of the prestigious STaR fellowship in mathematics education. She taught mathematics for many years in urban, inclu- sive middle school classrooms before her work at the university level. Dr. van Ingen’s research agenda lies at the intersection of equitable math- ematics education and clinically rich teacher preparation. Her research interests include teachers’ use of research to inform practice, the use of mathematics con- sultations to meet the mathematics learning needs of students with exceptionali- ties, and the implementation of integrated STEM lessons in K–5 classrooms. She regularly publishes and presents her research to audiences who work in math- ematics education, special education, and teacher preparation. She is the principal investigator and coprincipal investigator for federally funded research and is active in leadership in her professional organizations. Preface Welcome to the second edition of Teaching Mathematics Meaningfully: Solutions for Reaching Struggling Learners. Since the first edition, additional research has been published detailing how students in general, students with disabilities, and other struggling learners progress in their reasoning and understanding in various mathematical domains. Also, with the advent of the Common Core State Standards (CCSS), along with reform movements such as the increased utilization of multi- tiered systems of support (MTSS) and response to intervention (RTI) models, the landscape of mathematics education in schools is changing. The second edition is substantially revised to reflect those new developments. Although this edition has the same intent as the first—to help teachers of struggling learners teach mathematics meaningfully—it differs substantially in organization and content. We hope readers of the first edition find this one to be a logical and helpful “next step.” For those who have not read the first edition, the second is our best attempt at integrating the current literature related to research and practice in mathematics instruction for struggling learners to provide you valuable insight and information into improving mathematics outcomes for them. This edition incorporates findings from updated research since the first edition, focusing on students’ learning trajectories in mathematics and integrating that information with effective instructional practices for struggling learners within the context of the CCSS and MTSS/RTI. This preface explains the purpose and intended audience of the book, its organization, and how it can be used to further your professional development. PURPOSE Meaning is the seminal theme of this book. As a consequence, its purpose is to help teachers facilitate struggling learners’ access to high-quality mathematics so those learners can make sense of the mathematics and become mathematically proficient. To facilitate such access, teachers must be effective problem solvers and decision makers. This book is organized to provide teachers of mathematics and those who want to become teachers of mathematics with an informed and practical process for doing this very important job. Two primary bodies of literature and practice—special education and math- ematics education—inform the book’s content. We represent both disciplines, and for this book to achieve its purpose, collaboration between these two disciplines is essential. Both content and pedagogy must be represented and integrated in an informed and accurate way if K–12 mathematics content is to be made accessible to struggling learners. ix