Response to Intervention in Math
Publication Year: 2010
DOI: http://dx.doi.org/10.4135/9781452219356
Subject: Teaching AtRisk Students, AtRisk, Mathematics
 Chapters
 Front Matter
 Back Matter
 Subject Index

 Chapter 1: What is RTI, and why is it Important?
 Overview of RTI
 Assessment
 Instructional Tiers
 Key Research Support for RTI and Mathematics
 Increased Instructional Time and Supports
 SmallGroup Instruction
 Explicit Methods of Instruction
 The Use of Concrete and Pictorial Representations
 Strategy Instruction for Problem Solving
 Focus on Basic Facts and Problem Solving
 Aligning Tier 1 and Tier 2 Instruction
 Screening and Progress Monitoring to Target Deficit Areas
 Summary
 Chapter 2: The RTI Process for Math: Getting Started
 Selection of the RTI Team Members
 Belief System
 Core Belief #1: All Students can be Mathematically Proficient
 Core Belief #2: All Students Need a HighQuality Mathematics Program
 Core Belief #3: Effective Mathematics Programs Must Teach Conceptual Understanding, Computational Fluency, Factual Knowledge, and ProblemSolving Skills
 Core Belief #4: Effective Instruction Matters and Significantly Impacts Student Learning
 Common Models of Implementation
 Standardized Protocol Model
 ProblemSolving Model
 Hybrid Model Approach
 Assessment
 Universal Screening
 Progress Monitoring
 Diagnostic Assessments
 National Center on Student Progress Monitoring
 Importance of the Core Mathematics Program
 Summary
 Chapter 3: A Tiered Approach to More Effective Mathematics Instruction
 Tier 1 Instruction and Curriculum
 Tier 2 Intervention and Curriculum
 Size of the Instructional Group
 Mastery Requirement of Content
 Frequency of Progress Monitoring
 Instructor Considerations
 Tier 3 Instruction and Curriculum
 Summary
 Chapter 4: Mathematics Interventions Overview
 Who Needs Intervention?
 What Do I Teach for the Intervention?
 Who Should Intervene?
 Where?
 How Long?
 How do I Organize my Curriculum?
 What Types of Curricular Strategies should be used for Tier 2 and Tier 3 Interventions?
 Summary
 Chapter 5: Number Sense and Initial Math Skills
 Assessments of Number Sense
 Instructional Delivery of Number Sense
 Curricular Elements of a Number Sense Intervention
 Number Recognition
 Magnitude Comparison
 Strategic Counting
 Addition: Counting On
 Subtraction: Counting Down
 Subtraction: Counting the Difference
 Adding to Multiplication: Skip Counting
 Addition and Place Value: Plus 10
 Place Value: Equals 10
 Fact Fluency or Number Combinations
 In Context
 Summary
 Chapter 6: Building Students' Proficiency with whole Numbers
 Importance of Proficiency in whole Numbers
 General Recommendations for Building Proficiency
 Specific Criterion for Introducing New Facts
 Intensive Practice on Newly Introduced Facts
 Systematic Practice on Previously Introduced Facts
 Adequate Instructional Time
 Progress Monitoring and Record Keeping
 Motivational Procedures
 Building Automatic Recall of Basic Facts with the Mastering Math Facts Program
 Mastering Math Facts Overview
 Building Proficiency with whole Numbers through PALS Math
 PALS Math Overview
 Summary
 Chapter 7: Fractions and Decimals
 Fractions in the Standards
 Assessment for Fractions and Decimals
 When are Calculators Sufficient?
 Teaching the “What” with Fractions and Decimals
 Fractions
 Decimals
 Teaching the “How to Compute” with Fractions and Decimals
 Fractions
 Decimals
 Teaching Fluency with Fractions and Decimals
 In Context
 Summary
 Chapter 8: Teaching Problem Solving Strategically
 ProblemSolving Programs
 Hot Math
 Pirate Math
 Solving Math Word Problems: Teaching Students with Learning Disabilities Using SchemaBased Instruction
 Summary
 Chapter 9: The Importance of Teaching Mathematical Vocabulary
 The Importance of Teaching Mathematical Vocabulary
 General Guidelines for Teaching Mathematical Vocabulary
 Specific Guidelines for Teaching Mathematical Vocabulary
 Instructional Activities to Promote Learning of Essential Mathematical Vocabulary
 Technology Applications and Resources
 Directed Journaling Activities
 Teaching Word Parts and Origins
 Preteaching Vocabulary Prior to Instructional Lesson
 Practice Activities to Build Fluency with Important Mathematical Vocabulary
 Graphic Organizers
 Assessing Students' Knowledge of Mathematical Vocabulary
 Summary
 Chapter 10: Next Steps in the RTI Process
 Professional Development
 Reconsidering the Tier 1 Mathematics Curriculum
 Why is this Important for Educators Implementing an RTI Math Model?
 An Alternative Approach: A Two Tier 1 Core Mathematics Program
 Summary

Copyright
Copyright © 2010 by Corwin
All rights reserved. When forms and sample documents are included, their use is authorized only by educators, local school sites, and/or noncommercial or nonprofit entities that have purchased the book. Except for that usage, no part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher.
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Library of Congress CataloginginPublication Data
Riccomini, Paul J.
Response to intervention in math/Paul J. Riccomini and Bradley S. Witzel.
p. cm.
Includes bibliographical references and index.
ISBN 9781412966351 (pbk.)
1. Mathematics—Study and teaching (Elementary) 2. Mathematics—Study and teaching (Middle school) 3. Effective teaching. 4. Curriculum planning. 5. Learning disabled children—Education. 6. Mathematical ability—Testing. I. Witzel, Bradley S. II. Title.
QA135.6.R53 2010
372.7—dc22
2009035719
This book is printed on acidfree paper.
10 11 12 13 14 10 9 8 7 6 5 4 3 2 1
Acquisitions Editor: David Chao
Editorial Assistant: Sarah Bartlett
Production Editor: Jane Haenel
Copy Editor: Cheryl Rivard
Typesetter: C&M Digitals (P) Ltd.
Proofreader: Susan Schon
Indexer: Terri Corry
Cover and Graphics Designer: Michael Dubowe
Preface
Purpose of this BookThe purpose of this book is to introduce the overarching principles of an effective response to intervention (RTI) framework in mathematics with the primary focus on instructional recommendations for teachers to improve their daytoday instruction in mathematics. More and more schools, school districts, and state departments of education continue to expand current RTI models in reading to mathematics to help struggling students. Teachers and administrators are seeking guidance on how best to teach mathematics to students who are struggling and/or have learning disabilities. The eventual result of any effective RTI model is to increase the number of students successful (e.g., increased proficiency) in the general education mathematics classroom, Tier 1. This book is designed to provide instructional recommendations for teaching mathematics effectively to students who have traditionally struggled and draws upon currently available researchbased evidence for teaching mathematics.
The National Mathematics Advisory Panel's (NMAP) final report, Foundations for Success (2008), clearly states that effective mathematical programs must simultaneously develop (a) conceptual understanding, (b) computational fluency, (c) factual knowledge, and (d) problemsolving skills if students are to be successful in more advanced mathematics courses (e.g., algebra and geometry). Additionally, the NMAP (2008) specifically addresses the instructional needs of lowachieving students, students who struggle, and students with learning disabilities and provides instructional recommendations for teaching these groups of students. The Institute of Education Sciences (IES) released a Practice Guide, Assisting Students Struggling With Mathematics: Response to Intervention (RTI) for Elementary and Middle Schools, which puts forth nine recommendations to effectively deliver a math RTI model. Therefore, the recommendations contained in this book will address each of the areas highlighted in the NMAP's Final Report (2008) and the IES Practice Guide (2009) with the intent to provide instructionally relevant recommendations and assist educators who are responsible for improving the mathematics programs for students who are underperforming, struggling, and/or have learning disabilities.
Target Audience of this BookOne only has to look at a publisher's catalog to see the overwhelming number of RTI books for teachers and administrators; the list may total in the hundreds. Not surprisingly, almost all the published materials focus exclusively on the process (e.g., procedures and assessment) of setting up and implementing RTI models in reading; mathematics, if addressed at all, is given a cursory mention. We certainly recognize the importance of reading, because if students are unable to read at a functional level, they will struggle in many academic areas, including mathematics. However, mathematics education deserves a reform focused on student performance. That said, this book is written for any educator charged with the responsibility of teaching mathematics, particularly those who teach struggling students.
With the emergence of RTI at all levels, school accountability, and emphasis on preparing more students for algebra, the primary responsibility for addressing the instructional needs of all students is placed squarely on the shoulders of general education mathematics teachers: elementary and secondary. We contend that if the instructional strategies and recommendations contained in this book are systematically incorporated into the general education mathematics daytoday classroom instruction, the more likely all students will be successful.
We are certainly aware of the realities of the education system and recognize that much of the responsibility of teaching struggling learners within an RTI model often falls at the feet of special educators, curriculum coaches, interventionists, and even school psychologists. This book is also written for those of you who act in the capacity of supporting the general education teacher (e.g., coteaching, resource teachers, tutors) and/or delivering targeted interventions. Additionally, parents of young children are now both experiencing firsthand the shortfalls within their children's elementary mathematics programs and having to address these shortfalls with their children. We hope many parents who read this book find the information helpful as they research ideas to support their children's mathematical development.
Why is the Book Needed?Teaching mathematics effectively requires skillful planning and a deep understanding of not only mathematical concepts but also effective instructional pedagogy, especially when teaching students who are low achievers, struggling, and/or have learning disabilities. The NMAP's Final Report (2008) states unequivocally that research over many years clearly indicates that students who are low achievers and struggling to learn mathematics and/or have learning disabilities require regular access to explicit methods of instruction available on a regular basis. This conclusion, although evidenced by data, has been scrutinized by some people in education. As more and more students with learning disabilities and other significant instructional needs are being included in the general education mathematics classroom, general education teachers are being required to more effectively meet struggling students’ many and varied needs in the context of their daily instructional lessons. As researchers, we support the implementation of evidencesupported practices to the maximum extent possible and focus our recommendations in this book accordingly. As parents, we applaud the teachers who do the same in their classrooms.
Response to Intervention in Math provides educators with an understanding of the components of effective instructional design and delivery for students with diverse needs in the area of mathematics. Specifically, readers will learn procedures for teaching mathematics using systematic and explicit instruction as an approach to assessment, instructional planning, and evaluation. The instructional recommendations found in this book are aligned with the recommendations put forth by the NMAP's Final Report (2008; http://www.ed.gov/mathpanel), the IES Practice Guide written by Gersten and Colleagues (2009), and the research base on effective mathematics instruction, albeit relatively small compared to research available for reading.
The authors would also like to note that there is no one “thing” or “waving of a magic mathematics wand” that will address the many and varied issues impacting the learning or lack of learning in mathematics. Specific student differences are so widely diverse and often very complex, it is unlikely that the ideas in this book will address every student issue appearing in classrooms. As such, none of the recommendations in this book should be interpreted as “absolutes”, but rather as starting points for consideration in the context of your mathematics program and specific characteristics of your students. Moreover, we advocate that only a concerted effort at all levels and by all educators, both general and special education teachers, is an effective and efficient approach that will ultimately capitalize on efforts to improve your school's curriculum and instruction in the area of mathematics. Without unilateral support for student learning and improvements in mathematics, an RTI math effort is premature.
Chapter OverviewChapter 1, “What Is RTI, and Why Is It Important?” provides an overview of response to intervention (RTI) in general and how it specifically relates to teaching mathematics. Topics covered in this chapter include an overview and description of RTI practices and procedures and common components in models of RTI, and it concludes with a brief overview of key research supporting RTI in mathematics.
Chapter 2, “The RTI Process for Math”, provides a description of the essential components to consider when designing and implementing an RTI model in math, more detailed description of the standard protocol model and problemsolving model, progress monitoring, and the importance of the core mathematics program.
Chapter 3, “A Tiered Approach to More Effective Mathematics Instruction”, differentiates different levels of instruction and intervention necessary for implementing RTI. Additionally, through a series of detailed selfstudies of curriculum, instructional delivery, and interventions along with some classroom examples, it becomes evident whether a school or district is ready to initiate RTI in mathematics.
Chapter 4, “Mathematics Interventions Overview”, describes who requires interventions and how to define the necessary interventions per each student's needs. Details about building an appropriate environment for interventions as well as choosing effective curriculum and instructional delivery are explained as well as setting the time frame for intervention and developing interventions. The chapter includes a list of mathematics interventions and programs to consider.
Chapter 5, “Number Sense and Initial Math Skills”, details the basic components of number sense and early numeracy as defined by educational programs and related assessments. More important, from number recognition, to magnitude, to counting strategies for basic facts, instruction delivery and interventions are described, with illustrations that may be used to teach number sense to students who are struggling in mathematics.
Chapter 6, “Building Students’ Proficiency With Whole Numbers”, provides a rationale for the importance of teaching students to proficiency with basic whole number operations. Instructional strategies will be provided for building understanding, relationships, and fluency with whole numbers. Peerassisted learning strategies (PALS) in math are also described for kindergarten and Grades 2–6.
Chapter 7, “Fractions and Decimals”, acknowledges the major struggles that students have with fractions, decimals, and percents. These struggles are worse for those with math difficulties. The failure to succeed in fractions has an ill effect on performance in secondary mathematics, particularly algebra. In this chapter, gradelevel expectations are set along with illustrated ways on how to instruct and intervene with the teaching of fractions.
Chapter 8, “Teaching Problem Solving Strategically”, will present teaching problem solving strategically through three problemsolving programs that have been used as Tier 2 instructional programs.
Chapter 9, “The Importance of Teaching Mathematical Vocabulary”, focuses exclusively on mathematical vocabulary and how it influences mathematical proficiency. Five general guidelines for teaching vocabulary and seven mathspecific recommendations for teaching mathematical vocabulary are described. Additionally, five instructional activities to facilitate deeper understanding are described and how to assess student's vocabulary knowledge.
Chapter 10, “Next Steps in the RTI Process”, explores the next steps as models continue to be refined and expanded to secondary settings, other student groups such as gifted and talented, and implications for changing systems. Additionally, an alternative approach to Tier 1 instructional programs is described for future consideration.
Acknowledgments
Corwin gratefully acknowledges the contributions of the following individuals:
Rachel Aherns, Level 1 Special Education/SixthGrade Collaboration Teacher
West Des Moines Community School District
West Des Moines, IA
David Allsopp, Professor
Department of Special Education
University of South Florida
Tampa, FL
Judith Filkins, K–8 Math Curriculum Coordinator
Lebanon School District
Lebanon, NH
Russell Gersten, Professor Emeritus
Executive Director
College of Education Instructional Research Group
University of Oregon
Eugene, OR
Kent Johnson, Founder and Director
Morningside Academy
Seattle, WA
About the Authors
Paul J. Riccomini, PhD, began his career as a dualcertified general education mathematics teacher of students with learning disabilities, emotional and behavioral disabilities, and gifted and talented students in Grades 7–12 in inclusive classrooms. His teaching experiences required a strong content knowledge in mathematics and the development and maintenance of strong collaborative relationships with both general and special educators. He earned his doctorate in special education from The Pennsylvania State University and his master's degree in education and Bachelor of Arts in mathematics at Edinboro University of Pennsylvania. Currently, he is an Associate Professor of Special Education at Clemson University. His research focus is on effective instructional approaches, strategies, and assessments for students who are low achievers and/or students with learning disabilities in mathematics. He has written several research and practitioner articles related to effective strategies for teaching mathematics to students who struggle and has coauthored two math intervention programs targeting fractions and integers. As a former middle and high school general education and special education mathematics teacher, Dr. Riccomini knows firsthand the challenges and difficulties teachers experience every day when working with struggling students, a motivation for writing this book. You can email Dr. Riccomini at pjr146@clemson.edu.
Bradley S. Witzel, PhD, is an experienced and decorated teacher of students with disabilities and atrisk concerns. He has worked as a classroom teacher and before that as a paraeducator in inclusive and selfcontained settings. Dr. Witzel received his BS in psychology from James Madison University and his master's degree in education and his PhD in special education from the University of Florida. He currently serves as an associate professor, coordinator of the three special education programs, and assistive department chair of curriculum and instruction at Winthrop University in Rock Hill, South Carolina, where he recently received the 2009 Winthrop Graduate Faculty Award. In higher education, Dr. Witzel has taught undergraduate and graduate courses in special and general education methods as well as a variety of other courses from transition to behavior support. He has written several research and practitioner articles, books, and book chapters on mathematics education and interventions, and served as a reviewer of the Final Report from the National Mathematics Advisory Panel. Recently he coauthored an IES practice guide on response to intervention in mathematics. You can email Dr. Witzel at witzelb@winthrop.edu.

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