Table Of ContentRtI in Math
Learn how to help K–8 students who struggle in math. Now in its second edition, this book
provides a variety of clear, practical strategies that can be implemented right away to boost
student achievement. Discover how to design lessons that work with struggling learners,
implement math intervention recommendations from the Institute of Education Sciences
Practice Guides, the National Center on Intensive Intervention, and CEC, use praise and
self-motivation more effectively, develop number sense and computational fluency, teach
whole numbers and fractions, increase students’ problem-solving abilities, and more! This
edition features an all-new overview of effective instructional practices to support aca-
demic engagement and success, ideas for intensifying instruction within tiered interven-
tions, and a detailed set of recommendations aligned to both CCSSM and CEC/CEEDAR’s
High-Leverage Practices to help support students struggling to meet grade-level expecta-
tions. Extensive, current examples are provided for each strategy, as well as lesson plans,
games, and resources.
Linda Forbringer is a Professor in the Department of Teaching and Learning at Southern
Illinois University Edwardsville, USA.
Wendy Weber is a Professor in the Department of Teaching and Learning, and Program
Director for the undergraduate Special Education Program at Southern Illinois University
Edwardsville, USA.
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Coding as a Playground: Programming and Computational
Thinking in the Early Childhood Classroom
Marina Umaschi Bers
Implementing Systematic Interventions: A Guide for
Secondary School Teams
Hank Bohanon, Kelly Morrissey, and Lisa Caputo Love
Developing Teacher Leaders in Special Education:
An Administrator’s Guide to Building Inclusive Schools
edited by Daniel M. Maggin and Marie Tejero Hughes
Teaching Adolescents and Young Adults with Autism Spectrum Disorder:
Curriculum Planning and Strategies
Dianne Zager and Michael L. Wehmeyer
RtI in Math
Evidence-Based
Interventions
Second Edition
Linda Forbringer and Wendy Weber
Second edition published 2021
by Routledge
605 Third Avenue, New York, NY 10158
and by Routledge
2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
Routledge is an imprint of the Taylor & Francis Group, an informa business
© 2021 Taylor & Francis
The right of Linda Forbringer and Wendy H. Weber to be identified as authors of
this work has been asserted by them 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 u tilised
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.
First edition published by Routledge 2014
Library of Congress Cataloging-in-Publication Data
Names: Forbringer, Linda L., author. | Weber, Wendy H., author.
Title: RTI in math : evidence-based interventions / Linda Forbringer and
Wendy H. Weber.
Other titles: Response to intervention in math
Identifiers: LCCN 2020053261 (print) | LCCN 2020053262 (ebook) | ISBN
9780367444709 (hardback) | ISBN 9780367818999 (paperback) | ISBN
9781003010739 (ebook)
Subjects: LCSH: Mathematics–Study and teaching (Elementary)–United
States. | Mathematics–Remedial teaching. | Response to intervention
(Learning disabled children) | School failure–Prevention.
Classification: LCC QA13 .F67 2021 (print) | LCC QA13 (ebook) | DDC
372.70973–dc23
LC record available at https://lccn.loc.gov/2020053261
LC ebook record available at https://lccn.loc.gov/2020053262
ISBN: 9780367444709 (hbk)
ISBN: 9780367818999 (pbk)
ISBN: 9781003010739 (ebk)
Typeset in Palatino
by KnowledgeWorks Global Ltd.
Access the Support Material: www.routledge.com/9780367818999
Contents
Meet the Authors ��������������������������������������������������������������������������������������������������vii
Introduction ������������������������������������������������������������������������������������������������������������ix
1 Overview of Response to Intervention in Mathematics ............................................1
What Is Response to Intervention? 1
Models of Implementation 5
Summary 6
2 Using Assessment to Make Instructional Decisions ...................................................7
Step 1: Problem Identification 8
Step 2: Problem Analysis 10
Step 3: Intervention Plan Development 13
Step 4: Plan Implementation 15
Step 5: Plan Evaluation 16
Summary 20
3 Overview of Evidence-Based Practices for Teaching Mathematics .........................21
Evidence-Based Practices for the Core Curriculum (Tier 1) 21
Interventions to Support Students Who Struggle in Mathematics (Tiers 2 & 3) 25
Summary 30
4 Setting the Stage: Increasing Motivation .................................................................33
Why Focus on Motivation? 33
Increasing Motivation by Creating Meaningful, Engaging Lessons 34
Using Self-Monitoring and Goal-Setting to Increase Motivation 35
Effective Use of Praise 36
Rewards 38
Summary 45
5 Explicit Instruction ......................................................................................................47
Instructional Considerations for Struggling Learners 47
The Explicit Instruction Lesson 50
How Explicit Instruction Improves Motivation 61
Summary 61
6 Concrete and Visual Representation .........................................................................63
Research on the Concrete-Pictorial-Abstract (CPA) Sequence 64
Recommendations for Implementing the CPA Sequence 69
Summary 71
7 Developing Number Sense .........................................................................................73
Counting and Representing Whole Numbers 73
Magnitude Comparison 80
Place Value 81
Intensifying Instruction 86
♦ v
8 Operations with Whole Numbers: Addition and Subtraction .................................89
Developing Conceptual Understanding of Addition and Subtraction 89
Developing Computational Fluency with Basic Facts 96
Solving Multi-Digit Addition and Subtraction Problems 96
Intensifying Instruction during Interventions 102
Summary 103
9 Operations with Whole Numbers: Multiplication and Division ............................105
Developing Conceptual Understanding of Multiplication 105
Developing Fluency with Multiplication and Division Facts 109
Multiplication with Rectangular Arrays 110
The Standard Algorithm for Multiplication 112
Multi-Digit Division 115
Intensifying Instruction During Interventions 117
Summary 118
10 Fact Fluency ...............................................................................................................119
Developing Strategies for Solving Basic Facts 121
Automaticity 137
Summary 144
11 Representing Rational Numbers ..............................................................................147
Fractions 147
Decimals 165
Percent 171
Intensifying Instruction 171
Summary 173
12 Problem-Solving ........................................................................................................175
Problem-Solving in the Core Curriculum (Tier 1) 175
Teaching Problem-Solving During Interventions (Tier 2 & Tier 3) 178
Summary 197
13 Conclusion: Using RtI to Improve Achievement in Mathematics ..........................199
Selecting Materials for Core Instruction (Tier 1) 199
Selecting Materials for Interventions (Tiers 2 & 3) 202
Intensifying Instruction 204
Summary 206
References �����������������������������������������������������������������������������������������������������������207
vi ♦ Contents
Meet the Authors
Dr. Linda Forbringer is a professor in the Department of Teaching and Learning at Southern
Illinois University Edwardsville (SIUE). She teaches undergraduate and graduate courses
in methods of teaching mathematics, classroom management, and instructional methods
for students with disabilities, and has served as program director for the undergraduate
and graduate programs at SIUE. Before joining SIUE, she served as a special education
classroom teacher, case manager for students with emotional disabilities, and administrator
of a day treatment center serving kindergarten through 12th grade students. She consults
with schools nationally on effective interventions to support students who struggle with
mathematics or who exhibit challenging behavior. She has presented on math interven-
tions, differentiating instruction, motivation, and teacher education in webinars and at a
variety of national and international conferences. Her research interests include methods
of teaching mathematics, effective teaching practices, and strategies to support students in
inclusive settings.
Dr. Wendy Weber, PhD is a professor in the Department of Teaching and Learning at
Southern Illinois University Edwardsville. She served as the principal investigator of the
IHE Partnership, a multi-year State Personnel Development Grant focused on the integra-
tion of MTSS into educator preparation programs across Illinois. She is the program director
for the Undergraduate Special Education program at SIUE. Dr. Weber has earned the SIUE
Teaching Excellence Award in 2015 and was named one of SIUE’s Phenomenal Women in
2019. In addition to her passion for teaching, Dr. Weber provides professional develop-
ment and educational consulting to school districts in Illinois in the areas of Response to
Intervention/Multi-Tier System of Supports (MTSS), mindfulness practices, and maximizing
student engagement. Dr. Weber authored the book Mindfulness for Students: A Curriculum for
Grades 3-8 and has implemented her mindfulness curriculum, Mynd Time©, in numerous
schools. Her research interests include mindfulness, self-regulation, social-emotional learning,
and school improvement.
The authors wish to thank Melissa Landwehr for her wonderful contributions, and John
Ransom for his technical support.
♦ vii
Introduction
Students in the United States are struggling in mathematics. Results of the 2019 National
Assessment of Educational Progress (NAEP), often called the “nation’s report card,” show
that only 21 percent of 12th grade students are proficient in mathematics, while 40 percent of
students scored below the basic level (NCES, 2019.). Although the Every Student Succeeds
Act of 2015 states that all children will succeed, clearly many children have not been suc-
cessful in mathematics.
One model for providing early intervention and support for struggling learners is
Response to Intervention (RtI). RtI (also called Multi-Tiered System of Supports or MTSS)
is a multilevel prevention system that integrates data-based decision-making, high-qual-
ity instruction, and intervention matched to student needs in order to maximize student
achievement. The initial focus of RtI was primarily on improving reading achievement, but
schools have now expanded RtI to mathematics. While a multitude of books and articles
have been written about RtI, most of them describe the RtI process, recommendations for
universal screening and progress monitoring, and instruction and interventions for read-
ing. As schools begin to look beyond support in reading, there is a need for resources that
address evidence-based interventions for mathematics.
This book is for teacher educators, classroom teachers, special educators, math spe-
cialists, math coaches, teacher aides, administrators, related service providers, and other
professionals who directly or indirectly support students struggling to master mathemat-
ics. We begin with an overview of the RtI process and discuss how to use assessment to
make instructional decisions in mathematics. Chapter 3 provides an overview of the evi-
dence-based practices for teaching mathematics in the general classroom and the interven-
tions that support students who are struggling with core concepts. Because a large body of
research suggests that careful attention to both lesson design and motivational strategies
can significantly improve struggling learners’ mathematical achievement, we address these
foundational topics in Chapters 4 and 5. In the remaining chapters, we provide a detailed
description of interventions to help struggling learners master concepts and operations
involving whole numbers and rational numbers. Our goal is to clarify how instruction dur-
ing interventions differs from the core instruction provided in the general education class-
room. We include information about locating effective materials to use during interventions,
as well as ideas for adapting and supplementing other available materials to provide the
intensive instruction that is necessary to support students who require mathematical inter-
ventions. The evidence-based interventions discussed in this book follow the recommenda-
tions from the Institute of Education Sciences Practice Guide, Assisting Students Struggling
with M athematics: Response to Intervention (RtI) for Elementary and Middle Schools, the National
Center on Intensive Intervention, the Council for Exceptional Children, and other national
experts. Using these strategies can help prevent difficulties, support struggling learners,
and allow all students to be successful in mathematics.
♦ ix
