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Practical Guidelines for Elementary Teachers "This little book provides readers with a plethora of ideas and strategies to produce the most effective instructional programs." —Alfred S. Posamentier, dean of the School of Education, Mercy College, New York
—Gale Yanish, elementary math specialist, Sage Elementary, Elko, Nevada
Proponents call the response to intervention (RTI) process a fundamental paradigm shift in education. RTI in Math describes how this three-tiered system of interventions, which has proven effective in reading instruction, can be applied to primary and elementary mathematics education. Authors William N. Bender and Darlene Crane offer an accessible, detailed map of the RTI process to help all students succeed in mathematics. This practical guidebook: •
Explains the three tiers of intervention and when each should be used
Provides research-based strategies for implementation and instruction
Explores numerous case studies to analyze common student difficulties in mathematics and suggests specific steps to address them
Includes reproducible tools for student and staff assessment, data collection, and documentation of RTI procedures
Visit go.solution-tree.com/rti to download the reproducibles in this book.
Practical Guidelines for Elementary Teachers
"RTI in Math contains information all teachers need to read and implement in their regular classrooms. The strategies are an absolute must for all teachers to utilize, and they will benefit all students. . . . The research grounding this book is excellent and up to date. I would highly recommend this book to every teacher!"
Practical Guidelines for Elementary Teachers
Copyright © 2011 by Solution Tree Press Materials appearing here are copyrighted. With one exception, all rights are reserved. Readers may reproduce only those pages marked "Reproducible." Otherwise, no part of this book may be reproduced or transmitted in any form or by any means (electronic, photocopying, recording, or otherwise) without prior written permission of the publisher. 555 North Morton Street Bloomington, IN 47404 800.733.6786 (toll free) / 812.336.7700 FAX: 812.336.7790 email: firstname.lastname@example.com solution-tree.com Visit go.solution-tree.com/rti to download the reproducibles in this book. Printed in the United States of America 14 13 12 11
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Library of Congress Cataloging-in-Publication Data Bender, William N. RTI in math : practical guidelines for elementary teachers / William N. Bender, Darlene Crane. p. cm. Includes bibliographical references and index. ISBN 978-1-934009-54-3 (perfect bound) -- ISBN 978-1-935249-32-0 (library edition) 1. Mathematics--Study and teaching (Elementary) 2. Curriculum planning. I. Crane, Darlene Barrientos, 1948- II. Title. QA135.6.B458 2010 372.7--dc22 2010020707 Solution Tree Jeffrey C. Jones, CEO & President Solution Tree Press President: Douglas M. Rife Publisher: Robert D. Clouse Vice President of Production: Gretchen Knapp Managing Production Editor: Caroline Wise Senior Production Editor: Suzanne Kraszewski Copy Editor: Rachel Rosolina Proofreader: Elisabeth Abrams Text and Cover Designer: Amy Shock
Table of Contents About the Authors
chapter 1 The RTI Process in Elementary Mathematics
What Is RTI?
The Three-Tier RTI Pyramid
Foundations of the RTI Process
RTI Implementation in Schools
Why Is RTI in Mathematics Necessary?
RTI and Standards-Based Instruction
Efficacy of RTI for Addressing Educational Standards
RTI and School Improvement
Advocacy for RTI
Teacher Understanding of the RTI Process
Implementation Issues
chapter 2 Tier 1 Mathematics Instruction: A Critical Component of RTI
Challenges to Effective Tier 1 Mathematics Instruction
Universal Screening in Tier 1 Mathematics
Benchmarking for Tier 1 Mathematics
High-Quality Instructional Approaches for Tier 1
Ten Explicit Tactics for Tier 1 Instruction
chapter 3 RTI Procedures for Number Sense and Early Mathematics Skills
What Is Number Sense?
Mathematics Curricula for Young Children
What Does an Early Mathematics RTI Look Like?
Important Issues for RTI in Early Math
Diagnosing a Learning Disability in Early Mathematics
chapter 4 RTI Procedures in Elementary Mathematics
Mathematics Curricula and Progress Monitoring for Elementary Grades
Case Study: An RTI for Automaticity in Multiplication Facts
Case Study: An RTI for Finding Common Denominators
Diagnosing a Learning Disability in Elementary Mathematics
chapter 5 RTI and Problem Solving
Mental Processing of Story Problems
The Problem With Story Problems
Enhancing Story Problem Instruction
Effective Tier 1 Instruction for Problem Solving
Assessment Options for Problem-Solving Skills
Effective Tier 2 and 3 Interventions for Problem Solving
Case Study: An RTI for Problem Solving
chapter 6 PLC Support Strategies for School- and Districtwide RTI Procedures in Mathematics
State- and District-Level Leadership
School-Level Planning
Professional Learning Communities
Table of Contents
appendix Reproducibles
References and Resources
About the Authors William N. Bender, PhD, is an international leader in instructional tactics with an emphasis on differentiated instruction, response to intervention, brain-compatible instruction, and classroom discipline. He is an accomplished author and presenter who consistently receives accolades for his workshops from educators at every level. Dr. Bender began his education career teaching in a junior high school resource classroom, working with adolescents with behavioral disorders and learning disabilities. He earned his doctorate in special education from the University of North Carolina. He has taught at Rutgers University in New Jersey and Bluefield State College in West Virginia, and most recently he served as a professor of education at the University of Georgia. He consults full time, writes professional development books, and presents workshops for educators throughout North America. Bender has written over sixty research articles. His numerous bestselling titles include Response to Intervention: A Practical Guide for Every Teacher; Beyond the RTI Pyramid: Solutions for the First Years of Implementation; Differentiating Instruction for Students With Learning Disabilities (second edition); Introduction to Learning Disabilities (sixth edition); Relational Discipline: Strategies for In-Your-Face Kids (second edition); and Reading Strategies for Elementary Students With Learning Difficulties: Strategies for RTI (second edition). Darlene Crane, MA, has served in the education profession for twenty years. Her career began as a middle school special education teacher and has included serving as an instructor at the postsecondary setting, coordinator of district-level improvement plans for early childhood education, classroom teacher, and most recently as a regional response to intervention specialist for the West Virginia Department of Education. As a response to intervention specialist, Crane has worked side by side with district and ix
school leaders and classroom teachers as they work to implement the tiered instructional model to support student achievement. In addition to her experience as a classroom teacher and RTI specialist, her work has included the development of support strategies for students with ADHD and learning disabilities at the postsecondary setting, coordinating regional-level parent involvement for parents of children with special needs, and training hundreds of teachers and administrators on various elements of implementation of response to intervention in the areas of math and reading. Crane's passion for supporting educators as they work to provide high-quality educational experiences for all students arises from a combination of personal and professional experiences.
Preface Newly devised response to intervention (RTI) procedures are transforming primary and elementary classrooms across the United States as teachers gear up their efforts to provide this intensive instructional support to students (Bender, 2009a). The RTI process involves targeting specific areas in which students are struggling and applying increasingly intensive research-proven interventions until the threat to learning is alleviated (Bender, 2009a). Nearly all of the research on RTI has taken place in the area of reading, because reading and literacy have been major national priorities since the Clinton presidential administration (Bender, 2009a). However, teachers are now beginning to use RTI procedures in mathematics. One 2009 survey of educational administrators indicates that over 50 percent of U.S. districts are currently implementing or soon intend to implement RTI for mathematics instruction (Spectrum K12 School Solutions, 2009). It is critical that teachers understand the similarities and differences It is critical that teachers understand in RTI implementation between reading and the similarities and differences in RTI implementation between reading and mathematics. mathematics. This book addresses the immediate need for information on how RTI procedures might be implemented in primary and elementary mathematics. It is intended for various groups of educators in the kindergarten through upper elementary years, including: yyTeachers seeking practical guidance on how to implement an RTI process for a student who may be struggling in mathematics yyAdministrators seeking to enhance schoolwide educational endeavors and more adequately address state mathematics standards yyProfessional learning communities (PLCs) providing leadership on the RTI process in mathematics In most of the chapters, we focus on implementation issues for RTI procedures in mathematics at the classroom level. In chapter 1, however, we provide some background on RTI and the development of RTI procedures themselves. Both chapter 1 and chapter 6 focus on implementation at the school district and the school level,
and they provide guidance for PLCs or other leadership teams to assist in guiding the RTI implementation process. More specific information on each chapter follows. In chapter 1, we discuss the overall RTI process and provide a brief description of the existing research on RTI procedures in mathematics. We present the RTI pyramid as the basis for supplemental interventions in the general education classroom. After a brief history of RTI, we describe problems many students today encounter in mathematics and how RTI procedures can address these problems. Next, we describe an RTI procedure and discuss progress-monitoring issues in a variety of mathematics areas. The standards of the National Council of Teachers of Mathematics (NCTM) as stated in their Principles and Standards for School Mathematics (2000) and the report of the National Mathematics Advisory Panel (NMAP), Foundations for Success (2008), provide the basis for this discussion of mathematics instructional techniques. We conclude chapter 1 by identifying issues that arise from implementation of RTI procedures in mathematics, including the primary issue of teacher time. Because a solid foundation in mathematics is critical for subsequent success in math, the instruction students receive in general education is also critical. In chapter 2, we describe Tier 1 instruction appropriate for the general education classroom. Universal screening and benchmarking of mathematics performance in the general educational classroom are likely to become the responsibility of the general education teacher. We then describe how mathematics teachers might use the RTI process in the context of differentiated instruction and scaffolded instruction in mathematics. Finally, we present a variety of practical instructional ideas for enhancing Tier 1 mathematics instruction. In chapter 3, we focus on early mathematics readiness in number sense and initial mathematics skills and the RTI process as implemented in kindergarten and first grade. We also discuss several curricula that may be appropriate for those grades. After a brief discussion on the brain basis for learning mathematics and number sense, we explore a case study of an RTI procedure addressing early mathematics skills. Finally, we discuss using the RTI process to diagnose learning disabilities in mathematics during the early school years. In chapter 4, we focus on various mathematics areas in the upper elementary grades and discuss several curricula that may be appropriate for those grades. In this chapter, we also provide case study RTI examples in various elementary mathematics areas, including several areas stressed by NMAP's 2008 report. Because that report emphasizes automaticity with math facts and intensified instruction in fractions, we present RTI case studies in these areas. We also discuss how teachers might use RTI procedures in the middle and upper elementary grades for diagnosis of learning disabilities in mathematics. In chapter 5, we discuss RTI procedures and intervention strategies for teaching problem solving. The field of mathematics has witnessed an increased emphasis on
problem solving, as various national reports have identified problem solving as one area that consistently proves challenging for many students. We present a variety of instructional procedures (such as specific learning strategies and project-based learning) for teaching problem solving at the Tier 1 and Tier 2 levels. We place these instructional procedures into the context of a case study on RTI that focuses on solving story problems. In chapter 6, we provide information for a PLC, a district-based leadership team, or a schoolwide RTI task force to assist with RTI planning and implementation in mathematics at the school level. We also include useful tips and descriptive notes on helpful websites and various forms for planning and implementing the RTI procedure in mathematics. Visit go.solution-tree.com/rti for downloadable versions of the reproducibles found in the appendix as well as for live links to the websites mentioned in text.
he instructional process referred to as response to intervention (RTI) has quickly taken root in the United States (Buffum, Mattos, & Weber, 2009; Kavale & Spaulding, 2008; Spectrum K12 School Solutions, 2009). In fact, proponents of RTI, as well as several advocacy groups and educational support organizations, have indicated that RTI may represent a fundamental paradigm shift in education (Fuchs & Deshler, 2007; Fuchs & Fuchs, 2007; Gersten & Dimino, 2006). For example, the National Association of State Directors of Special Education (NASDSE, 2006) indicates that RTI represents a "dramatic redesign" of general and special education. RTI has been implemented in reading and literacy in school districts across the United States, and in a survey of directors of special education, over half of the school districts reported implementation of RTI in mathematics as well (Spectrum K12 School Solutions, 2009). Clearly, there is a great deal of emphasis on RTI, and RTI is resulting in a refocusing of educational endeavors. Furthermore, fundamental changes in teaching practices are stemming from the RTI initiative, and these changes will probably impact most teachers in kindergarten through grade 12 (Bender, 2009a; Kame'enui, 2007; Spectrum K12 School Solutions/Council of Administrators of Special Education [CASE], 2008). For that reason, these changes must be considered carefully by school leadership teams and PLCs within the school as educators move into RTI implementation in mathematics. This chapter presents an overview of the RTI process in mathematics, with a focus on advocacy for RTI as a school improvement initiative. While this book is not primarily an introduction to RTI or a review of research on the RTI process, leadership personnel serving as proponents of RTI in mathematics will need some understanding of the research basis that provides the rationale for RTI procedures. We will also address questions, concerns, and problems that have arisen in the implementation of RTI in the areas of reading and literacy; these issues are likely to arise
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The RTI Process in Elementary Mathematics
What Is RTI? Response to intervention is a network of support intended to assist all students in their learning by providing increasing levels of intensive, research-proven instructional interventions targeted to the needs of the individual student (Kame'enui, 2007; NASDSE, 2006). For students who are struggling in any academic area or struggling behaviorally, the RTI process results in timely interventions that should assist in overcoming those threats to learning and development (Gersten & Dimino, 2006; Kame'enui, 2007). During the RTI process, student progress is monitored repeatedly in order to document how students respond to instruction. Response to intervention is a netThat evidence is then used for data-based decision work of support intended to assist making relative to students' educational programs. all students in their learning by proThrough the practice of universal screening, which viding increasing levels of intensive, is typically conducted at least three times each research-proven instructional interyear, it is likely that the recent emphasis on RTI ventions targeted to the needs of the for meeting mandated standards will impact every individual student. student in the public schools.
The Three-Tier RTI Pyramid In most of the literature on RTI, as well as in most state department of education guidelines, the RTI process is depicted as a pyramid of increasingly intensive, specifically targeted instructional interventions—referred to as intervention tiers— such as those presented in figure 1.1 (Berkeley, Bender, Peaster, & Saunders, 2009). Each tier within the pyramid contains more specifically targeted interventions. The pyramid shape suggests that fewer students require the more intensive interventions that appear at the top. This three-tier pyramid of interventions model was first used in the public health arena (Kame'enui, 2007), but has since been adapted to various areas in education, including reading interventions such as the national Reading First initiative, and interventions to reduce problem behaviors in schools (Bender, 2009a). While various states have adopted slightly different models, the most commonly used RTI model is the three-tier RTI pyramid (Berkeley et al., 2009; Howell, Patton, & Deiotte, 2008; NASDSE, 2006; Spectrum K12 School Solutions, 2009). For example, in a 2008 survey on RTI among special education administrators, 73 percent of the respondents indicated that their state had implemented a three-tier RTI pyramid (Spectrum K12 School Solutions/CASE, 2008), whereas only 22 percent of respondents indicated that their state had implemented a four-tier model. Given this rather robust implementation of the three-tier pyramid, that model is the basis for this text.
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when RTI is implemented in mathematics as well. We introduce these issues here to suggest possible solutions.
The RTI Process in Elementary Mathematics
5% Individual Intervention
Tier 2
15% Small-Group Intervention
Tier 1
80% General Education Activities
Figure 1.1: The three-tier RTI pyramid.
In the three-tier RTI model, as well as in some of the four-tier RTI models (such as those used in North Carolina and Georgia), the lower tier of the pyramid represents effective instruction provided for all students within the general education classroom (Fuchs & Fuchs, 2005, 2006; Howell et al., 2008; NASDSE, 2006). Instruction delivered in this tier may include whole-class, smaller-group, or tutorial instruction. AÂ wide variety of research-proven instructional procedures are typically implemented as the basis for Tier 1 instruction, such as grouping students for instruction, scaffolding instruction, and using group projects, learning centers, cooperative instruction, or differentiated instruction. The general education teacher is the primary facilitator of instruction for Tier 1.
Tier 1 Instruction
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Tier 3
Tier 1 instruction is the foundation for all instructional interventions in mathematics, and it should be considered the single most important tier in the intervention pyramid, since effective instruction at this level greatly reduces the number of students requiring more intensive instruction at other levels of the pyramid. Proponents of the three-tier model suggest that this instructional tier should meet the needs of perhaps 80 percent of the students in a classroom (Boyer, 2008; Bradley, Danielson, & Doolittle, 2007; Fuchs & Fuchs, 2007). Because of the critical importance of effective mathematics Tier 1 instruction is the foundation for instruction in Tier 1, chapter 2 focuses exclusively all instructional interventions in mathematics, and it should be considered on effective instructional and performance moni- the single most important tier in the toring procedures in Tier 1 mathematics for intervention pyramid. primary and elementary classes.
Tier 2 Interventions
Tier 3 Interventions Tier 3 interventions are described within the three-tier RTI model as highly intensive educational interventions targeted at either one student or a very small group of students (Boyer, 2008; Bradley et al., 2007; Fuchs & Fuchs, 2007). These interventions have become the basis of some confusion as districts move to implement RTI. Proponents of RTI generally agree that if the instruction during Tier 2 intervention does not result in a student moving forward academically, that student will require a more intensive intervention within the framework of general education in order for his or her needs to be addressed. In contrast, some suggest that Tier 3 interventions should take place only after a student has been declared eligible for special education or other specialized services (Fuchs & Fuchs, 2006). Those other services might include programs for English learners or programs for migrant students. Fuchs and Fuchs (2006, 2007), for example, suggest that Tier 3 interventions take place after eligibility for specialized services has been documented.
Tier 3 interventions are highly intensive educational interventions targeted at either one student or a very small group of students.
There is no right or wrong standard of practice; however, most state models of RTI indicate that Tier 3 interventions will be provided prior to actual eligibility decisions, or as a component of the eligibility decision-making process (Berkeley et al., 2009; NASDSE, 2006). This level of intensive intervention is frequently described as one‑to‑one instruction and is designed to meet the needs of the remaining 3
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The interventions in Tier 2 of the RTI pyramid have been described as targeted, supplemental, systematic interventions for a small group of students who are struggling in mathematics. Unlike the occasional small-group instruction in Tier 1, Tier 2 supplemental instruction is more targeted, and it takes place over a longer period of time, possibly a grading period or two (Cummings, Atkins, Allison, & Cole, 2008; Fuchs & Fuchs, 2006, 2007). While 20 to 25 percent of the student population may require supplemental instruction in mathematics at this level of intensity, this intervention tier is typically described as meeting the needs of perhaps 15 percent of the school population. Therefore, in a typical class of twenty-four students, one might expect that five or six students will struggle in mathematics and require a level of supplementary instruction that is more intensive than the instruction offered to the entire class in Tier 1. In some school districts, the general education teacher is expected to deliver this supplemental instruction, but there is wide variety as to how to deliver instruction in Tier 2 interventions are targeted, supthe higher tiers of the pyramid (Berkeley et al., plemental, systematic interventions 2009). Subsequent chapters explore a variety of for a small group of students who are options concerning who delivers instruction in struggling in mathematics. various intervention tiers.
The RTI Process in Elementary Mathematics
Given the wide variety of RTI models and implementation procedures in use throughout the United States (Berkeley et al., 2009), it is somewhat difficult to discuss generally interventions in mathematics from the RTI pyramid. For purposes of this book, we define and describe the tiers in the same way as the majority of U.S. educators—a three-tier model in which all three of the intervention tiers take place as a function of general education and prior to special education eligibility decisions (Berkeley et al., 2009; NASDSE, 2006). Of course, educators working in districts that employ models other than this should interpret our suggestions and guidelines appropriately within the context of their own district policy. Most state departments of education present some type of RTI model on their state department website, and in many cases, the information on RTI applications for reading and mathematics is quite extensive. Educators should check that source to see the type of model their state uses, as well as any guidelines on implementation of RTI in mathematics versus implementation in reading.
Foundations of the RTI Process In almost all state models for RTI, five foundational principles undergird the RTI process, as shown in the following feature box (Bender, 2009a; Boyer, 2008; Buffum et al., 2009). First, almost all models are rooted in the concept of universal screening to identify students struggling in various subjects. This concept suggests that given the critical importance of both reading and mathematics, all students should be screened frequently (usually three times each year) to ensure that they are meeting their benchmark goals in each area. To date, this principle has been more widely applied in the primary and elementary grades, since benchmarking basic skills is more directly related to the content in those grades. However, universal screening is likely to gain emphasis across all grade levels as middle and high schools gear up to implement RTI procedures.
Foundational Principles of the RTI Process 1. Universal screening to identify students struggling in mathematics 2. A multitier model of increasingly intensive educational interventions 3. Research-based curriculum in each tier 4. Frequent monitoring of each child's performance related to benchmarks 5. Data-based decision making involving a collaborative team effort
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to 5 percent of the school population (Fuchs & Fuchs, 2005; Howell et al., 2008). Therefore, in our example of the classroom with twenty-four students, perhaps one or two students would require supplemental instruction in mathematics at this level of intensity.
Most RTI models specifically emphasize the use of research-based curriculum, and in that sense, RTI is similar to other national initiatives such as No Child Left Behind (Kame'enui, 2007). Also, frequent monitoring of each individual student's progress within that curriculum based on benchmarks is required in the RTI process. In addition to the universal screening three times each year, most districts require more frequent benchmarking (or progress monitoring) assessments during Tier 2 and Tier 3 interventions (Bender, 2009a). Finally, all RTI models emphasize data-based decision making involving a collaborative team in order to accurately interpret a student's progress relative to curricular standards in mathematics. Using actual data about the student's performance allows educators to make time-critical decisions about an individual student's educational planning.
RTI Implementation in Schools In 2004, federal legislation was enacted that allowed the use of RTI progressmonitoring data to become the basis for learning disability diagnosis (Kame'enui, 2007; NASDSE, 2006). For that reason, some educators have associated RTI with special education. While information from the RTI process can help determine eligibility for students with learning disabilities and can be used in other eligibility procedures for students with other disabilities (such as behavioral disorders), that is not the primary purpose of RTI. Rather, RTI is a support network of universal assessments in critical areas, paired with increasingly intensive interventions in those areas, for students having difficulty in any academic or behavioral area (NASDSE, 2006). Furthermore, in all states, RTI interventions take place primarily within the context of general education and not special education. In all states, RTI interventions take Thus, the RTI initiative will significantly impact place primarily within the context of general education teachers in profound ways, and general education and not special it should not be viewed as a special education education. Thus, the RTI initiative will initiative. This makes a school's RTI implementasignificantly impact general education process an extremely appropriate topic for tion teachers in profound ways, and consideration by the school PLC (DuFour, DuFour, it should not be viewed as a special education initiative. & Eaker, 2008; DuFour, DuFour, Eaker, & Karhanek, 2004) or school leadership teams. To date, discussions of RTI implementation have focused primarily on reading assessments and interventions in the primary and elementary grades, and only recently have proponents of RTI begun to provide information on how this procedure might work in the area of mathematics (Ardoin, Witt, Connell, & Koenig, 2005; Fuchs,
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Almost all RTI models depict different educational interventions in various levels or tiers (Boyer, 2008; Buffum et al., 2009) and at increasing levels of intensity, with an array of options for providing targeted, explicit instruction to struggling students in their areas of difficulty.
The RTI Process in Elementary Mathematics
In fact, some school districts have decided to implement RTI exclusively in reading initially and concentrate on implementation of RTI in mathematics and other areas later. West Virginia, for example, initiated RTI procedures in reading in thirty-six pilot schools in 2005, but it took a few years for those schools to implement RTI in mathematics (Boyer, 2008). With that noted, all schools in the nation will eventually implement some version of RTI for a wide variety of academic subjects, and one of the first will probably be mathematics. Teachers will require an understanding of the RTI process as well as how that process applies in mathematics.
Why Is RTI in Mathematics Necessary?
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Fuchs, Compton, et al., 2007; Jordan, Kaplan, Locuniak, & Ramineni, 2007). With little guidance available on how to conduct RTI procedures in mathematics, school districts have been somewhat slower in implementing RTI in mathematics as compared to reading. In a March 2008 nationwide survey of special education administrators, 84 percent indicated that their school district had begun implementation of RTI procedures in reading, whereas only 53 percent indicated RTI implementation had begun in mathematics (Spectrum K12 School Solutions/CASE, 2008).
Researchers have begun to advocate for RTI procedures in mathematics for two specific reasons: (1) many students need help in mathematics, and they can benefit from RTI procedures, and (2) RTI-based instruction works. Some research suggests that as many as 25 percent of primary- and elementary-level students may experience some Researchers have begun to advocate difficulty in mathematics (Baskette, Ulmer, & for RTI procedures in mathematics for Bender, 2006; Fuchs, Compton, Fuchs, Paulsen, two specific reasons: (1) many stuBryant, & Hamlett, 2005). In addition, between dents need help in mathematics, and they can benefit from RTI procedures 6 and 10 percent of young children experience in mathematics, and (2) RTI-based challenges that are severe enough to be identified instruction works. as mathematics disabilities. Student difficulty in mathematics can manifest in a variety of ways. For example, Jordan and her colleagues have documented that young students with difficulties understanding basic number relationships perform less well on timed mathematics achievement calculation exercises ( Jordan et al., 2007; Locuniak & Jordan, 2008). Other students encounter problems at certain points in the curriculum in the early years of school when specific new mathematics content is introduced. These times can be thought of as "choke points" within the curriculum; students may do quite well in mathematics until these points, but they then fail to succeed in specific skill areas. In some cases, a bit of extra instructional time at these critical choke points can make the difference between success and failure in mathematics overall. For example, some students have success with early mathematics facts, such as addition and subtraction problems—even with two-digits—but once the concept of regrouping is introduced, they seem to choke. Other students do well through the
This body of research indicates that difficulties in mathematics are fairly common and that these difficulties must be addressed in some fashion if students are to succeed in the higher-grade mathematics curriculum. Clearly, RTI interventions in mathematics will be necessary for many students As the United States moves to imple(Spectrum K12 School Solutions/CASE, 2008). mentation of RTI as the primary way to As the United States moves to implementation support struggling students in general of RTI as the primary way to support struggling education, we cannot simply implestudents in general education, we cannot simply ment RTI in literacy and overlook stuimplement RTI in literacy and overlook students dents with mathematics disabilities; with mathematics disabilities; that would be that would be tantamount to denial of necessary services for those students. tantamount to denial of necessary services for those students.
Why Students Struggle in Mathematics There are several reasons that struggling students have not built a strong academic foundation in mathematics. First, the traditional instructional models in both reading and mathematics have relied on pull-out services to meet the needs of struggling students, such as with Title I instruction and special education services. Some educators have questioned the efficacy of pull-out programs, which tend to de-emphasize the general education teacher's role in mathematics instruction (Bender, 2009a; Boyer, 2008). A rigorous system of differentiated, supplemental instruction in mathematics within the context of the general education class reduces reliance on this pull-out instructional model. In addition, No Child Left Behind's focus on the achievement of individual subgroups of students (such as English learners, students with special needs, and minority students) has heightened our awareness of the increasing discrepancy of achievement among students in mathematics (NMAP, 2008). This focus on specific subgroups has left some general education teachers feeling inadequate about their own instructional skills in mathematics. Recent research has indicated that many elementary educators feel less confident with mathematics instruction than reading instruction (Burns, 2006; Enochs, Smith, & Huinker, 2000; Minton, 2007). Obviously, teacher knowledge of content and personal comfort in teaching mathematics will both impact the quality of instruction teachers provide.
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primary grades, but demonstrate extreme difficulty once fractions are introduced, usually in grade 3. Fractions are critically important in mathematics since success with fractions of all types (fractional parts, percentages, decimals) provides the basis for many aspects of the mathematics curriculum in later years (NMAP, 2008). Students who fail to master the content are severely handicapped in the higher grades. As struggling students get older, many of their problems in mathematics continue. Mabbott and Bisanz (2008) indicate that many students in the middle grades demonstrate serious mathematics disabilities in various operations or in solving mathematical problems.
The RTI Process in Elementary Mathematics
A strong facultywide emphasis on quality Tier 1 instruction as part of the RTI model, with appropriate support for RTI efforts in mathematics, has given many students the opportunity to participate in and benefit from the richness of the general education classroom. As educators become proficient with differentiated instruction and research-based instructional strategies of RTI, increasing numbers of struggling students have gained viable access to mathematics content.
RTI and Standards-Based Instruction Since the presidency of Bill Clinton, our national educational policy has focused on standards that would allow students in our schools to compete with students internationally in all academic areas, including mathematics (Kame'enui, 2007; NMAP, 2008). For this reason, any discussion of a curricular or instructional modification such as RTI must take place in the context of these curricular standards. The NCTM identified instructional standards and content standards for the public school mathematics curriculum in a document titled Principles and Standards for School Mathematics (2000). These standards are critically important because of the emphasis on high standards set forth in the political initiatives of the past several presidential administrations. Therefore, almost every state uses the NCTM standards to set curricular expectations in mathematics for all students. They include specific standards in ten broad areas across all grade levels, from kindergarten through grade 12 (Kemp, Eaton, & Poole, 2009). Those broad content areas are subdivided according to specific skills in each grade level. The ten areas of emphasis are as follows: 1. Numbers and operations 2. Algebra 3. Geometry 4. Measurement 5. Data analysis and probability 6. Problem solving 7. Reasoning and proof 8. Communication 9. Connections
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The inclusion instructional model has also contributed to students' struggles. The intention of inclusion is to re-engage students with exceptionalities and other learning difficulties in the general education environment in order to provide them with access to grade-level content. But inclusion models often leave general educators questioning their preparation for meeting student needs. In contrast, well-designed RTI models depend on high-quality general education instruction in mathematics coupled with a supplemental intervention support network, and thus RTI models empower general education teachers to better address the needs of diverse learners (Bender, 2009b).
10. Representation
Table 1.1: The National Council of Teachers of Mathematics Curriculum Focal Points in Mathematics Grade 1
Number and operations and algebra: Developing understandings of addition and subtraction and strategies for basic addition facts and related subtraction facts Number and operations: Developing an understanding of whole number relationships, including grouping in tens and ones Geometry: Composing and decomposing geometric shapes
Grade 2
Number and operations: Developing an understanding of the base-10 numeration system and place-value concepts Number and operations and algebra: Developing quick recall of addition facts and related subtraction facts and fluency with multidigit addition and subtraction Measurement: Developing an understanding of linear measurement and facility in measuring lengths
Grade 3
Number and operations and algebra: Developing understandings of multiplication and division and strategies for basic multiplication facts and related division facts Number and operations: Developing an understanding of fractions and fraction equivalence Geometry: Describing and analyzing properties of two-dimensional shapes
Grade 4
Number and operations and algebra: Developing quick recall of multiplication facts and related division facts and fluency with wholenumber multiplication Number and operations: Developing an understanding of decimals, including the connections between fractions and decimals Measurement: Developing an understanding of area and determining the areas of two-dimensional shapes
Grade 5
Number and operations and algebra: Developing an understanding of and fluency with division of whole numbers
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The NCTM modified the standards in 2006 when it published a document to identify the curriculum focal points within mathematics for each grade level (NCTM, 2006). Those curriculum focal points for grade levels range from preK through grade 8 (table 1.1). The complete description of each focal point can be found at the NCTM website (www.nctm.org).
The RTI Process in Elementary Mathematics
Grade 5
Geometry and measurement and algebra: Describing threedimensional shapes and analyzing their properties, including volume and surface area Grade 6
Number and operations: Developing an understanding of and fluency with multiplication and division of fractions and decimals Number and operations: Connecting ratio and rate to multiplication and division Algebra: Writing, interpreting, and using mathematical expressions and equations
Grade 7
Number and operations, algebra, and geometry: Developing an understanding of and applying proportionality, including similarity Measurement, geometry, and algebra: Developing an understanding of and using formulas to determine surface areas and volumes of threedimensional shapes Number and operations and algebra: Developing an understanding of operations on all rational numbers and solving linear equations
Grade 8
Algebra: Analyzing and representing linear functions and solving linear equations and systems of linear equations Geometry and measurement: Analyzing two- and three-dimensional space and figures by using distance and angle Data analysis, number and operations, and algebra: Analyzing and summarizing data sets
Source: NCTM, 2006
In 2008, NMAP issued a report that focused on mathematics instruction across the public school grades. The report began with the overall recommendation that all students should be required to pass algebra I and algebra II in order for the United States to continue to be a world leader in science and technology. The report called for more rigorous standards in mathematics, since most states and school districts require completion of algebra I only in their general education curriculum for high school graduation, coupled of course with one or two additional mathematics courses such as geometry, applied mathematics, or general mathematics. The report made recommendations for restructuring the mathematics curriculum from kindergarten up through high school, noting that preparation in the lower grades provides the groundwork for success in algebra I and algebra II (NMAP, 2008). It made more than forty distinct recommendations in virtually every area of mathematics instruction, such as textbook preparation and teacher preparation, and summarized critically important information on mathematics instruction for teachers
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Numbers and operations: Developing an understanding of and fluency with addition and subtraction of fractions and decimals
Table 1.2: Selected Recommendations of the 2008 Report of the National Mathematics Advisory Panel 1. The mathematics curriculum in grades 1 through 8 should be streamlined and recreated to emphasize critical topics in early grades that directly enhance learning and mastery of algebra in the high school grades. This emphasis is consistent with the Critical Focal Points report of the National Council of Teachers of Mathematics (2006). The Panel recommends use of benchmarks for mathematics in the elementary grades, and specifically stresses automaticity in mathematics facts and whole-number operations. This parallels the suggestions in the educational literature to focus instruction on the "power standards" that are essential for future success in the subject. 2. Fractions (including decimals, percents, and negative fractions) are critical for subsequent mathematics and algebra achievement. Thus, this mathematics content, including representation of fractions on a number line, should be stressed much more. Research suggests that this will have the broadest single impact on subsequent achievement in problem-solving performance. 3. The performance of groups that have been traditionally underrepresented in mathematics fields may be greatly enhanced by stressing social, affective, and motivational factors, as well as by providing explicit instruction in mathematics. 4. Schools should consider hiring full-time mathematics teachers in elementary grades as one option to enhance mathematics performance. If implemented, this recommendation may provide a mechanism for provision of Tier 2 or Tier 3 interventions, as well as Tier 1 instruction. 5. Research on instruction suggests that instruction should be based on both teacher-directed and student-centered activities; the research does not support the use of either one exclusive of the other. 6. Teachers should use various forms of tutoring and/or cooperative learning. Specifically, team-assisted instruction has been shown to increase students' computation skills. This is a highly structured strategy, employing heterogeneous groups of students helping each other through tutoring and/or other cooperative learning formats. 7. Teachers should regularly use formative assessment, and such assessment is likely to improve students' learning. While RTI is not mentioned in the Panel report, this recommendation may serve as a basis for emphasizing universal screening and repeated performance assessment in a multitiered intervention format for students struggling in mathematics. 8. Explicit instruction has been shown to work consistently for students struggling in mathematics, including students with learning disabilities and other disabilities. Explicit instruction employs teacher-directed instruction, many practice examples, encouragement for students to "talk aloud" during problem solving, and extensive feedback.
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in the primary, elementary, and middle school grades. Table 1.2 presents a synopsis of the recommendations that are most relevant to public school educators.
The RTI Process in Elementary Mathematics
10. The use of calculators has limited or no impact on computation skills, problem solving, or conceptual development; however, the Panel cautions that use of calculators may impede development of automaticity, and it suggests that calculators should not be used on assessment items that measure computational fluency. Source: NMAP, 2008
In addition to these selected recommendations, NMAP (2008) identified a series of eleven critical benchmarks that should guide instruction in elementary mathematics. Table 1.3 presents these benchmarks. A quick review of these benchmarks provides some insight into how instructional practices for mathematics may develop in the future. For example, none of these benchmarks deal with measurement topics; rather, they emphasize automaticity in mathematics facts, fractions, decimals, and problem solving. While it is too soon to tell the impact of this national report, it may lead to some modifications of the current mathematics curricula throughout the elementary grades. Table 1.3: National Mathematics Advisory Panel Benchmarks
Fluency With Whole Numbers 1. By the end of grade 3, students should be proficient with the addition and subtraction of whole numbers. 2. By the end of grade 5, students should be proficient with the multiplication and division of whole numbers.
Fluency With Fractions 3. By the end of grade 4, students should be able to identify and represent fractions and decimals and compare them on a number line or with other common representations of fractions and decimals. 4. By the end of grade 5, students should be proficient with comparing fractions and decimals and common percents and with the addition and subtraction of fractions and decimals. 5. By the end of grade 6, students should be proficient with multiplication and division of fractions and decimals. 6. By the end of grade 6, students should be proficient with all operations involving positive and negative integers. 7. By the end of grade 7, students should be proficient with all operations involving positive and negative fractions.
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9. Research on instructional software in mathematics has shown positive results on student achievement. Technology-based drill-and-practice examples can improve student learning in an array of areas.
Geometry and Measurement 9. By the end of grade 5, students should be able to solve problems involving perimeter and area of triangles and all quadrilaterals having at least one pair of parallel sides (such as trapezoids). 10. By the end of grade 6, students should be able to analyze the properties of two-dimensional shapes and solve problems involving perimeter and area, and analyze the properties of three-dimensional shapes and solve problems involving surface, area, and volume. 11. By the end of grade 7, students should be familiar with the relationship between similar triangles and the concept of the slope of a line. Source: NMAP, 2008
These standards, recommendations, and benchmarks, along with NCTM's ten curricular focal points, must provide the framework for our discussions of curricular or instructional modifications such as RTI. These standards, recommendations, and benchmarks are used as the basis for discussion of mathematics instruction throughout this text; however, we should note that additional changes in state mathematics standards are likely within the next five years (Clark, 2010). In July 2009, President Obama launched his Race to the Top education initiative, an initiative that promises additional funding to foster school reform and an increased emphasis on educational standards (Clark, 2010). In addition, in order to move toward a set of national curriculum standards that presumably would apply across state lines, the National Governors Association Center for Best Practices and the Council of Chief State School Officers from forty-eight states are working together to develop a set of core standards (Clark, 2010; see also www.corestandards.org). The curricular standards for kindergarten through grade 12 were released in February of 2010, and at this point, no one can predict the impact that these revised standards might have in mathematics instruction. Furthermore, even the NMAP recommendations of 2008 are quite recent and may have yet to be implemented in some states or school districts. Again, educators should check their state department of education website for the specific instructional standards and guidelines for mathematics instruction mandated by their state. Given the national emphasis on standards-based instruction and the identification of effective instructional practices, schools around the nation have focused considerable energy on making adequate yearly progress (AYP) toward meeting high educational standards. Schools across the United States have heard the demands from their communities to meet mandated curricular goals. In fact, AYP has been so heavily emphasized that teachers have begun to focus their instructional planning on specific AYP goals within their school.
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8. By the end of grade 7, students should be able to solve problems involving percent, ratio, and rate and extend this work to proportionality.
The RTI Process in Elementary Mathematics
One way schools have begun to address the issue of meeting high standards is through the implementation of RTI in various subject areas. If implementation of RTI results in schools meeting AYP goals toward higher standards, then RTI will enhance the overall school program and can be considered—and we argue should be considered—as a school-improvement technique. Research on the effectiveness of the RTI process indicates that implementation of RTI does assist schools in meeting AYP, and for that reason, we present some preliminary research in the following sections. Educators involved in RTI implementation should not only become aware of this growing research, but they should share this research with colleagues concerned with meaningful and substantive school reform efforts.
Using RTI to Meet AYP in Mathematics Research on RTI is fairly recent, and is certainly much more extensive in the areas of reading and literacy than in mathematics (Bender, 2009a). Still, the existing research indicates that RTI procedures help many students struggling in mathematics overcome their difficulties and may actually enable them to catch up with their peers in overall mathematics achievement (Bender, 2009b; Bryant et al., 2008; Chard et al., 2008; Fuchs et al., 2005; Fuchs et al., 2006; Fuchs, Fuchs, & Hollenbeck, 2007). For example, Bryant and her colleagues (2008) documented the impact of a Tier 2 supplementary intervention on first-grade students struggling in mathematics. Of the 161 students in the study, 42 were identified as needing a Tier 2 intervention. That group was provided with a supplemental intervention in mathematics by two trained tutors that involved explicit instruction for twenty minutes per day, four days per week, for twenty-three weeks. Data at the end of that time suggested that this level of intervention resulted in improvements in mathematics that were greater than would have been predicted by pretest scores for those forty-two struggling students. The research shows that the RTI process does alleviate mathematics difficulties for many struggling students (Bender, 2009b; Bryant et al., 2008; Chard et al., 2008; Fuchs et al., 2005). It is important to note that in the studies mentioned in this section, someone other than the general education teacher delivered the Tier 2 intervention. More research is needed before we can conclude that general education teachers, given their time constraints, would be as effective as support personnel who are able to focus completely on supporting mathematics instruction. However, these results do look quite promising. RTI procedures do help students meet standards in math, and meeting AYP in mathematics is often a concern for educators. This research collectively suggests that a Tier 2 intervention alleviated the problems in mathematics for between 30 and 60 percent of the students, and that a more intensive intervention may alleviate mathematics problems for the other
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Efficacy of RTI for Addressing Educational Standards
RTI Works for Students Struggling in Reading While our focus here is RTI in mathematics, educators at all levels will need to understand the positive research that undergirds the national RTI effort, and much of that is in the area of reading and literacy. Research has shown that the provision of intensive supplemental reading instruction can alleviate reading problems and put students back on track towards long-term reading success (Torgesen, 2007; Vellutino, Scanlon, Small, & Fanuele, 2006). In particular, the research demonstrates convincingly that students struggling in reading will benefit from supplemental, phonemically based, explicit reading instruction, and that the academic growth resulting from those interventions is maintained over time (Denton, Fletcher, Anthony, & Francis, 2006; Simmons et al., 2008; Torgesen, 2007). While some of these early studies investigated efficacy of specific Tier 2 interventions directly, others documented the efficacy of multitiered interventions in a complete RTI process. Hughes and Dexter (2008) reviewed the research on the efficacy of the entire RTI process and found that the process assists struggling students in improving their academic performance. While results vary significantly, the broad body of available research suggests that Tier 2 intensive supplemental intervention alleviates or eliminates between 40 and 60 percent of academic problems for students who are struggling in either reading or math (Hughes & Dexter, 2008; O'Connor, Fulmer, Harty, & Bell, 2005; Simmons et al., 2008; Torgesen, 2007).
RTI Works for English Learners Several specific subgroups of children face more challenges in school. One persistent concern among educators is provision of effective instruction for students with a primary language other than English. Research has shown that RTI is a very effective way to address the educational needs of these students (Kamps et al., 2007; Linan-Thompson, Cirino, & Vaughn, 2007; Lovett et al., 2008). The data show that within the broader RTI process, even these students will perform much better on various measures of academic achievement (Linan-Thompson, Vaughn, Prater, & Cirino, 2006; Rinaldi & Samson, 2008; Torgesen, 2007).
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students as well. Thus, many students who are struggling in mathematics show significant progress towards AYP when provided with supplemental instructional interventions in It is fair to say research supports the implementation of the RTI promathematics within the multitier RTI process. cess in mathematics and shows that Based on these data, it is fair to say research supRTI works for many struggling kids. ports the implementation of the RTI process in This is how schools should address mathematics and shows that RTI works for many the demands of meeting AYP in struggling kids. This is how schools should address mathematics. the demands of meeting AYP in mathematics.
The RTI Process in Elementary Mathematics
Other Evidence That Supports RTI
In addition, evidence shows that initiation of the RTI process does tend to reduce the number of students referred for special education services (Abernathy, 2008; Torgesen, 2007). One nationwide survey of public school administrators indicated that 39 percent of respondents believe that RTI implementation has reduced referrals for special education by at least 10 percent; however, 53 percent indicated data were insufficient to make a determination on that question in April of 2009 (Spectrum K12 School Solutions, 2009). Torgesen (2007) summarized data from a multitier Reading First initiative in Florida and indicated that the percentage of the school population in grades kindergarten through 3 diagnosed as learning disabled fell from 10.4 percent to 6 percent during the three years of that reading initiative. With fewer students requiring special educational assistance, the intensive resources required for those students can be applied elsewhere, and many districts are redirecting those resources into general education classes to support elements of tiered instruction. Some interesting anecdotal evidence has emerged on the efficacy of RTI in addressing another persistent concern among educators: disproportionality. Disproportionality is the long-standing concern for the overrepresentation of minority students in special education. Disproportionality might suggest some inadvertent racial or social bias in the current eligibility procedures, and educators have long sought a remedy for this problem.
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There are a number of other benefits of implementation of the RTI process. First, RTI empowers general education teachers (Bender, 2009a). Virtually every general education teacher has felt pressured to spend a little more time helping a student with his or her mathematics. With RTI, teachers can now expect to have a wider variety of supports available to them. These supports allow them to advocate for and spend more time with students who are struggling (Bender, 2009a).
At least one preliminary report from the districts that have implemented RTI suggests that disproportionality might be significantly reduced using the RTI process. For example, during the 2004/2005 academic year in New Hanover County, North Carolina, African American students were 1.7 times as likely to be placed in special education as were white students. However, after implementing RTI in 2006/2007, the African Early anecdotal evidence suggests that implementation of RTI has American students were only 1.2 times as likely reduced disproportionality in some to be placed in special education (Abernathy, school districts. 2008). This suggests a significant reduction in disproportionality.
RTI and School Improvement As the evidence suggests, the RTI process should be understood as a powerful, research-proven school improvement effort to enhance education for all students. As
We should note that RTI fits very well with other nationally recognized school improvement strategies, such as the PLC model (DuFour et al., 2004). Formative evaluation is recommended as a progress-monitoring tool within PLCs, and that emphasis is certainly characteristic of RTI efforts. PLCs emphasize creation of a schoolwide system of intervention supports, which is virtually synonymous with the RTI model. PLCs within the school are one vehicle for implementation of RTI.
Advocacy for RTI Educational leaders must serve as advocates for RTI. Strong support from school leaders can greatly enhance the willingness of faculty to undertake the process. Leaders should make sure various supports for RTI are available within the school, including instructional coaches or tutors to lead Tier 2 or Tier 3 interventions, or paraprofessionals to assist in general education classes when teachers are working with small groups of students in Tier 2 intervention. Overall, however, the most powerful advocacy for RTI typically involves teachers' sense of their own role in the academic lives of RTI should be presented to educators their students. RTI is a process that empowers as effective teaching for the 21st centeachers, and it should be presented to educators tury, an instructional paradigm that as effective teaching for the 21st century—an works for all students. instructional paradigm that works for all students.
Teacher Understanding of the RTI Process Many educators have begun using RTI in reading and are familiar with the process. Even so, it may be advantageous to have teachers complete a self-assessment on RTI to gauge their understanding. A sample Response to Intervention SelfReport Needs-Assessment for Teachers of Elementary Math is shown in table 1.4. (See pages 168-169 for a reproducible of this table. Visit go.solution-tree.com/rti
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the United States increasingly sets high standards in mathematics, educators will need bold new approaches to teaching to meet those standards, and as the research demonstrates, RTI is an effective instructional process for meeting those demands. Furthermore, this process works for many of the students that are most challenged by the highstandards curriculum, including English learners, students already diagnosed with disabilities, students from impoverished homes, and other groups that face educational challenges (Bender, 2009a; Torgesen, 2007). Meeting and exceeding AYP within these specific subgroups has proven to be highly challenging for schools. Initial evidence does suggest that the RTI process can be The RTI process should be understood one of the most effective ways to meet the chalas a powerful, research-proven school lenge of making adequate yearly progress toward improvement effort to enhance educahigh educational standards for all students—even tion for all students. those facing special challenges.
The RTI Process in Elementary Mathematics
for downloadable versions of all reproducibles found in this book.). A self-Âassessment survey such as this provides lead teachers, administrators, and RTI facilitators with a snapshot of exactly where educators in their building are in relation to RTI implementation in mathematics and reading.
Table 1.4: Response to Intervention Self-Report Needs-Assessment for Teachers of Elementary Math
Circle one numeral for each descriptive indicator. 1 = I have little knowledge and want additional in-service on this. 2 = I have some knowledge, but some additional in-service will be helpful. 3 = I have a good understanding of this, but I need to put this into practice this year. 4 = I have complete understanding and have reached proficiency at this practice. N/A = Not applicable for our state, school, or school district
General Understanding of RTI 1. The pyramid of intervention in our state and district
2. The problem-solving model for our state or district
3. The tiers of intervention in our school
4. The intervention timelines for each intervention tier
5. How RTI applies to all students
6. RTI in mathematics
General Knowledge of Universal Screening and Progress-Monitoring Procedures 1. The mathematics screening tests used in our school
2. The early mathematics standards
3. The individual mathematics assessments used in our school
4. Progress monitoring in mathematics during Tier 2 and Tier 3 interventions
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A self-assessment survey provides lead teachers, administrators, and RTI facilitators with a snapshot of exactly where educators in their building are in relation to RTI implementation in mathematics and reading.
6. The benchmark scores in mathematics across the grade levels
7. The data-management system for RTI
Knowledge of Interventions to Facilitate Student Progress 1. The supplemental mathematics RTI programs used in our school
2. Frequency and intensity of mathematics interventions
3. How to use flexible grouping for Tiers 1, 2, and 3
4. The technology-based mathematics programs used in our school
5. How to use creative staffing to make time for interventions
My Contributions and Suggestions for RTI in Our School 1. In what RTI areas should we plan further staff development?
2. What suggestions can you offer for making RTI work better for students in our school?
Source: Adapted from Bender, 2009a, pp. 13–14.
Implementation Issues A number of concerns have surfaced as educators have started implementing RTI in reading and math; some of these concerns have become serious problems that need to be addressed during the implementation process (Bender, 2009a; Berkeley et al., 2009; Boyer, 2008; Spectrum K12 School Solutions/CASE, 2008). This section identifies the most pressing implementation concerns and provides guidance on how to deal with problems.
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5. Data-gathering procedures (weekly or daily) for RTI
The RTI Process in Elementary Mathematics
Protecting the General Education Teacher's Time
As RTI has been expanded to include areas such as mathematics, behavioral interventions, and writing, this issue of teacher time becomes even more critical. It is not realistic to think that general education teachers can make the time to do a wide variety of interventions in reading, mathematics, and other subject areas without the resources to assist them (Bender, 2009a). This section presents some options for allowing general education teachers the time they need for these interventions.
Co-Teachers for RTI Interventions One option many districts have used to assist general education teachers is arranging for someone other than the teacher to do the Tier 3 intervention, and perhaps the Tier 2 intervention (Bender, 2009a). For example, in a co-teaching instructional model, a general education teacher and a special education teacher are teaching in the same classroom at the same time for a portion of the school day. This provides an opportunity for either of those teachers to conduct a Tier 2 intervention while the other teacher is instructing the remaining students in the class. This strategy has the added benefit of allowing special education teachers to become acquainted with struggling students, a few of whom may one day require special education services.
Other Professionals for RTI Interventions Another option is redefinition of roles for current educators. Various schools have redefined the roles of instructional coaches, lead teachers, or even speech-language pathologists to allow those persons to conduct some of the required interventions. While it seems more common to modify the roles of these individuals and subsequently make them available for conducting the more intensive interventions in Tier 3, some districts may consider using these persons for Tier 2 interventions under certain conditions (Bender, 2009a). For example, when a speech-language pathologist is implementing a curriculum on language production, stressing letter/sound formation or articulation,
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When general education teachers are introduced to the concept of RTI, their biggest concern is finding the time to do the necessary interventions (Bender, 2009a). Initially, the RTI intervention process was implemented for reading in Reading First schools across the United States (Bender, 2009a; Boyer, 2008). In those schools, the general education teacher was expected to implement Tier 1 instruction for all students, provide Tier 2 intervention in reading, and provide Tier 3 intervention for a limited number of students facing severe difficulty. Of course, this raised a critical question: what were the remaining members of the class doing when the general education teacher was conducting a Tier 2 intervention with four to six students, not to mention a Tier 3 intervention with one or two students with significant needs? Resources of some type had to be made available for those general education teachers to make the interventions possible, such as paraprofessional assistance and computer-based instructional interventions.
Another option may soon exist that involves using certified personnel for RTI interventions. The 2008 report of the NMAP specifically recommends that certified mathematics teachers take over most mathematics instruction in all public school grades. While it is too early to tell if this recommendation will be implemented, doing so would provide another certified teacher who could greatly assist with not only Tier 1 instruction, but also with Tier 2 and Tier 3 interventions. Again, this is merely a recommendation at this point, but with a new presidential administration in office since January of 2009, implementation is a possibility.
Paraprofessionals for RTI Interventions or for Supporting the General Education Teacher In some districts, paraprofessionals have been hired specifically to make RTI interventions possible (Bender, 2009a). Depending on the expertise of these paraprofessionals and the curriculum selected for the Tier 2 or Tier 3 interventions, the paraprofessional may either conduct the daily intervention activities under the supervision of the general education teacher (while that teacher conducts activities for other class members), or the paraprofessional can monitor classwide activities while the general education teacher works on Tier 2 intervention with a small group. This option holds some financial advantage, since paraprofessionals can be employed for a smaller salary than certified teachers or other certified personnel.
Technology for RTI Interventions There are now many scientifically validated computer-based curricula in both reading and mathematics available that offer highly targeted, intensive instruction that would be quite appropriate for Tier 2 or Tier 3 interventions (Bender, 2009a, 2009b). Using a computer-based mathematics intervention, for example, a teacher could place a number of Tier 2 intervention students on different computers in the class while continuing to conduct activities for the other class members. Specific examples of these programs are included in chapters 2, 3, and 4 of this text.
Class Sharing In some districts, none of the previously mentioned options may be feasible. Those schools should explore a simple class-sharing concept. In this arrangement, teachers with similar classes share the responsibility for Tier 2 and Tier 3 interventions (Bender, 2009a). For example, if two third-grade teachers in classrooms across the hall from each other each need to provide a Tier 2 mathematics intervention to five students, the teachers could share their classes such that one teacher leads a Tier 2 intervention for all ten students who need that level of intensive intervention, while the other teacher teaches the remaining class members from both classes (Bender,
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that intervention might be very appropriate as an early reading intervention for some young children (Bender, 2009a).
The RTI Process in Elementary Mathematics
An Intervention Class Finally, many districts have devised various schedules that allow for offering Tier 2 and Tier 3 interventions during a separate intervention period (Bender, 2009a). While this is more common in middle and high schools, support intervention classes are also offered in some elementary schools as well. For example, semester-long intervention classes in some elementary schools allow students struggling with particular aspects of the mathematics curriculum to switch to an intervention class for a semester or two. In most cases, the intervention classes are somewhat smaller than the other general education classes, so students will be receiving more individual attention.
Availability of Appropriate Assessments and Curricula in Early-Level Mathematics While assessment tools and supplemental intervention materials have long been available in both reading and math, the more recent research on early reading interventions has documented specific benchmarks within the kindergarten through grade 3 curriculum. In many ways, this body of research made RTI possible in early-level reading and literacy. For example, by the first month of kindergarten, prior to any formal training at all, students who can accurately detect phoneme similarities can be identified, and they are likely to progress normally in their reading skills. However, students who cannot hear those differences or similarities in sound have much more difficulty in the next set of early literacy skills, the association of sounds and specific letters. In short, that benchmark—recognition of initial consonant sounds—will identify students who need a more intensive supplemental reading program. If students hit such benchmarks throughout the early grades, they will progress normally in reading; whereas if they do not, long-term reading progress is much less likely (Bender & Shores, 2007). The research on early mathematics literacy is not as advanced as it is in reading. For example, research on benchmarks in early mathematics for preK and kindergarten students is limited. In fact, educators have differing perspectives on the most basic terminology in this area of early mathematics. For example, to many psychoneurologists, the term number sense means exclusively the most basic numeration skills (the brain's recognition of the numbers 1, 2, or 3, for instance), whereas many teachers identify number sense as a higher-level skill, even in the context of determining the relative size of improper fractions (Bender, 2009b; Sousa, 2008). In that sense, it is accurate to say that educators do not really agree on what early mathematics skills are, and thus they do not agree what curricula should include for the youngest students.
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2009a). This option has been employed in various districts in Texas with some success, and all of the students requiring Tier 2 intervention received it. Furthermore, this option involves no additional costs to the school district, although it may require some creative scheduling on the part of teachers and administrators.
Each of these issues presents some concern when conceptualizing how the RTI process might be implemented in mathematics. While educators have dealt with these issues successfully in the elementary grades, lingering questions about early mathematics assessment may hamper RTI efforts at the very early elementary level (Bender, 2009b). While there are a very limited amount of number sense and early mathematics skills assessments available, none have yet to receive wide acceptance (Bender, 2009a).
Conclusion This chapter presented the basics of the RTI process in mathematics within the context of the three-tier RTI pyramid along with a synopsis of research supporting RTI in both mathematics and reading. We presented material to assist school leaders in their advocacy for RTI procedures, as well as identified issues of concern for educators to consider as they implement the RTI process in mathematics. The foundation of the RTI process in mathematics is the quality of mathematics instruction in the general education classroom. The next chapter focuses on Tier 1 instruction specifically. Subsequent chapters will focus on interventions at the Tier 2 and Tier 3 levels in various areas of primary and elementary mathematics. Finally, chapter 6 presents an array of guidelines for schoolwide planning for implementation of RTI in mathematics.
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This lack of agreement on what constitutes appropriate number-sense curriculum is not an issue beyond the beginning of school. As discussed previously, the NCTM (2006) specified appropriate curricular content from the beginning of kindergarten through grade 8; however, that organization did not present a blueprint of the types of mathematics readiness skills that a child entering kindergarten should have previously mastered, so benchmarking mathematics readiness in the early months of kindergarten, as we are currently doing in reading, remains highly problematic. Chapter 3 focuses on implementation of RTI procedures in mathematics during the early school years and addresses this issue on early mathematics readiness assessment and curricula in more depth.
Practical Guidelines for Elementary Teachers "This little book provides readers with a plethora of ideas and strategies to produce the most effective instructional programs." —Alfred S. Posamentier, dean of the School of Education, Mercy College, New York
—Gale Yanish, elementary math specialist, Sage Elementary, Elko, Nevada
Proponents call the response to intervention (RTI) process a fundamental paradigm shift in education. RTI in Math describes how this three-tiered system of interventions, which has proven effective in reading instruction, can be applied to primary and elementary mathematics education. Authors William N. Bender and Darlene Crane offer an accessible, detailed map of the RTI process to help all students succeed in mathematics. This practical guidebook: •
Explains the three tiers of intervention and when each should be used
Provides research-based strategies for implementation and instruction
Explores numerous case studies to analyze common student difficulties in mathematics and suggests specific steps to address them
Includes reproducible tools for student and staff assessment, data collection, and documentation of RTI procedures
Visit go.solution-tree.com/rti to download the reproducibles in this book.
Practical Guidelines for Elementary Teachers
"RTI in Math contains information all teachers need to read and implement in their regular classrooms. The strategies are an absolute must for all teachers to utilize, and they will benefit all students. . . . The research grounding this book is excellent and up to date. I would highly recommend this book to every teacher!"
Practical Guidelines for Elementary Teachers
Published on Jul 3, 2013
This practical guide of the RTI process for mathematics offers background research, implementation guidance, tools for reflection and growth...


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IRBManager is now live university-wide!!
Quick access: https://gvsu.my.irbmanager.com/
Important: Additional information about the system and the roll-out process, including an instruction manual and details regarding the transfer of IRBNet data, can be found here.
Training sessions will continue to be offered throughout the fall semester to help users become familiar with the system. Please visit Sprout to view dates/locations and to register for a session if interested.
Questions? Please contact firstname.lastname@example.com or call 616-331-3197.
The Human Research Review Committee (HRRC), an entity of the Office of Research Compliance and Integrity, ensures that the basic rights and welfare of research participants are fostered and protected. The HRRC supports researchers through its coordinated activities in education, regulatory compliance oversight, and post approval-monitoring. All research involving human subjects performed at GVSU and/or by GVSU students, staff, and/or faculty must be reviewed and authorized by the HRRC prior to the beginning of the research.
The HRRC is comprised of GVSU faculty, staff and community members, who review proposed research studies involving living persons.
If you are looking for Epigeum training, click here:
Note: This will take you to a protected webpage showing instructions for how to setup an Epigeum account, as well as further information related to using Epigeum at GVSU. When prompted on the next screen, enter your GVSU user name and password to access this information.
Quick Links
Are you looking to conduct human research? The information below will quickly get you started.
IRBNet Instructions
- IRBNet User Guide
- IRBNet: Getting Started
- Instructions: Submitting a New Protocol
- Instructions: Editing a Study Package after Submitting
- Instructions: Signing a Research Protocol (Authorizing Official)
- Instructions: Revising a Tabled Study
- Instructions: Requesting Changes to an Approved Protocol
- Instructions: Submitting a Continuing Review
- Instructions: Submitting a Closure Form
- Instructions: Submitting an Unanticipated Problem/Serious Adverse Event Form
- IRBManager User Guide
Application and Other Forms
****Please right-click and select "Save Link As" to download the form(s) to your computer. If you are having difficulty opening the PDF documents, Microsoft Word versions of the forms are also available from the Office of Research Compliance & Integrity. Please contact us at email@example.com.****
- Human Subject Research Determination Form *Updated 1/4/17
- Human Research Closure Form *Updated 2/9/17
- Continuing Review Application Form
- Individual Investigator Agreement Form *Updated 8/31/17
- IRB Authorization Agreement (GVSU Relying) Form *Updated 6/15/17
- IRB Authorization Agreement (GVSU Providing) Form
- Unanticipated Problem/Serious Adverse Event Reporting Form
- Change In Approved Protocol Form
- HIPAA Authorization to Use and Disclose Individual Health Information for Future Recruitment Purposes Form
- HIPAA Authorization to Use and Disclose Individual Health Information for Research Purposes Form


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well this is actually my first fan-fic...so plzz tell me what you think of it so far...XoXo
What if amnesia is just a trouble that can jump on you anytime?
"And if you could not accept my love, then you could not accept anyone's."
I have a question
Um.... I am writting it as it goes so sorry if its horrid.
Gerard has a drinking problem, and Frankie keeps hurting Frerard....
Bob Bryar, Pancakes, and a toilet. What do you think?
When Frank finally meets that perfect guy, can he keep his real shitty life a secret and for how long until it all falls apart?
True story...my story...
Some MORE auditions! Enjoy =D
When you begin a journey of revenge, start by digging two graves; one for your enemy and one for yourself. Confucius
Alissa had to forget alot of things. but when her.....chapter 6 is up (: ~Full summary inside.~
Noooo this isn't some fanfic. Well.. It is. But it isn't about some random chick and Gerard falling in love it's just all these what ifs like if Bob didn't like cats *gassppp*


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The challenge of bringing people together to share stories is minimized by the opportunities ahead to allow genuine individual and collective voices of expression to emerge. We all need the time to talk and listen to eachother's in order to co-create alternative visions for the future beyond stereotypes, especially under challenging socio-economic circumstances and uncertain futures for all. Learning from each other's narratives is a gift but where do we start our journey?
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Learn how to criss cross your stitches to make interesting lines of work within your project.
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