# Adapting Mathematics Core Curricula to Meet the Needs of Students with Disabilities

David H. Allsopp, Jennie L. Farmer, David Hoppey, and Brandi Leigh Kamp

University of South Florida and Clemson University

For the past 2 years, Mrs. Perez, a third-grade teacher, and Mr. Williams, a special education teacher, have co-taught a diverse class of 22 students, including 10 students with disabilities. Although they enjoy their collaborative relationship, Mrs. Perez, an 11-year veteran, would prefer to spend her day teaching reading and early writing instruction; similarly, Mr. William's love for reading and his desire to support students who struggle with reading and writing skill development motivated him to become a special education teacher. Although neither dislikes mathematics, it does not come as naturally to them. Mrs. Perez also feels a great deal of anxiety from the pressure of state assessments in a content area that had been a struggle for her as a learner. It is not surprising that for guidance they both cling to the district-adopted mathematics textbook and its supporting resources. The "toolbox" of resources Mr. Williams has collected and developed over the years for teaching reading, combined with Mrs. Perez's own reading strategies treasure trove, could serve as an accredited series independent of district resources. However, in mathematics, their concern about the content limits them to the core textbook and two resources suggested for struggling learners. Having nearly completed a second year together, they are increasingly concerned about what they are not doing, what the textbook does not provide, and their need for additional supports and solutions to help them truly meet the needs of their 22 students, especially those with disabilities.

Teaching mathematics can be challenging, particularly in diverse classrooms where students have different mathematical learning needs. In today's 21st-century classroom, mathematics instruction is increasingly structured around STEM (science, technology, engineering, mathematics) initiatives, even in the primary and elementary grades. In the context of the global economy, students need to develop literacy in each of the STEM areas if they are going to be competitive in the job market as adults. For example, mathematical literacy is more than simply being able to compute and recall formulas for the purpose of finding the correct answer to a given problem: Mathematical literacy has to do with making sense of mathematics and how it is represented in the world, and how it can be utilized to solve real-life problems. Mathematical literacy is integral to all the STEM areas because science, technology, engineering, and, of course, mathematics require individuals who can utilize, manipulate, and communicate mathematical representations and ideas.

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RTI is a prevention and intervention model for struggling learners commonly used for reading and mathematics. D. Fuchs, Fuchs, and Vaughn (2008) identified a framework of prevention and intervention consisting of primary, secondary, and tertiary instruction. Primary prevention (Tier 1) focuses on universal design, or instruction that benefits all students. Secondary efforts (Tier 2) incorporate adaptations that are feasible to implement, often in the general education classroom, for struggling learners in need of additional supports. Finally, the intervention stage (Tier 3) is intensive and individualized, often involving services provided by a specialist (e.g., special education teacher). The essential components of RTI include direct efforts towards universal screening, ongoing progress monitoring, a multilevel prevention system, and a continuous data-based decision-making process that seeks to improve student outcomes while being culturally responsive and integrating evidence-based interventions. In this article, we focus on Tier 1 and Tier 2 instruction with the realization that effective instruction applies across tiers and thus would be applicable to Tier 3 as well.

Regardless of how mathematics is framed, general and special education teachers, especially at the elementary level, have historically received limited preparation in mathematics education, including preparation related to mathematics content and effective mathematics instructional practices (Newton, Leonard, Evans, & Eastburn, 2012). A lack of preparation can lead to teachers’ having less confidence in teaching mathematics, particularly for students with disabilities. Like Mrs. Perez and Mr. Williams, these teachers are less likely to believe that they can use effective mathematics instructional practices when they are not confident teaching mathematics (Gresham, 2008; Smith, 1996; Borton Kahl, 2008), resulting in an overreliance on adopted mathematics texts/programs by schools (Clements, 2007).

Forty-five states and three U.S. territories recently have adopted the Common Core State Standards (CCSS). The CCSS for mathematics emphasize depth of understanding, while addressing fewer concepts/skills overall. The CCSS for mathematics bring to the fore conceptual understanding, relating abstract representations to real world contexts, and problem solving. A primary emphasis is on “understanding” mathematics including “the ability to justify … why a particular mathematical statement is true or where a mathematical rule comes from” (CCSS Initiative, 2012, p. 4). In other words, teachers need to possess in-depth conceptual understandings of the mathematics that they teach and abilities to teach students deeper understandings while enhancing their abilities to think critically and to connect abstract representations to the real world. Difficulty adjusting to these new standards and their emphasis on promoting deeper levels of mathematical understanding with students in turn may result in teachers relying on adopted texts and programs even more than they do now. Since the reauthorization of the Individuals with Disabilities Education Act (IDEA), the passage of No Child Left Behind (NCLB), and the implementation of initiatives such as RTI, this overreliance on prepackaged texts and programs has been complicated by the further integration of students with disabilities in the general education classroom. This has created a “catch-22” situation, wherein both parents and educators seek to further the integration of all students in general education despite the reality that using adopted mathematics texts and programs often limits the ability of teachers to individualize and differentiate instruction to meet the needs of struggling learners, especially those with identified disabilities.

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## Evidence-Based Practices, RTI, and Core Mathematics Curricula

The very essence of RTI seeks to integrate the use of evidence-based practices and the differentiation of these practices to address the needs of students who are not responding to core instruction (Voltz, Sims, & Nelson, 2010). Yet even with these positive inclusionary initiatives, mathematics outcomes for students with disabilities continue to be substantially below those of their typically developing peers. The 2011 National Assessment of Educational Progress (NAEP; National Center for Education Statistics, 2011) indicated that 55% of fourth grade students with disabilities were performing at or above a basic level in mathematics, compared to 85% of students without disabilities; 35% of eighth-grade students with disabilities were performing at or above a basic level, compared to 77% of students without disabilities (National Center for Education Statistics, 2011). These findings indicate that there is a critical need to support teachers and schools to address the mathematical learning needs of students with disabilities.

The use of scientifically based core curricula is a critical component of effective tier-based instruction (Vaughn, Wanzek, Woodruff, & Linan-Thompson, 2007). Tiered scientifically based curriculum is often assumed to be part of prepackaged mathematics textbooks and programs adopted by school districts for classroom implementation. Unfortunately, the extent to which the use of these core curricula actually leads to increased mathematics achievement outcomes generally is unclear, much less for students with disabilities.

To assist educators and curriculum planners in identifying appropriate scientifically based tiered interventions, the federal government’s What Works Clearinghouse (WWC) was developed to share unbiased data-based reviews of published interventions across multiple content areas, including mathematics. The intent of the WWC is for the districts, schools, and educators involved in making curriculum decisions for students to be able to review these findings and determine educational approaches and possible programs effective for tier-based application. However, this is based on the assumption that scientifically based programs exist.

We recently reviewed PK–8 mathematics texts and programs evaluated by the WWC and found that of the 31 texts and programs evaluated only two were rated as having both at least a medium to large evidence base and potentially positive or positive effects on mathematics achievement (see Resources, Table 1). Although the list of mathematics texts and programs reviewed by the WWC is not exhaustive, the database does include products from many leading educational publishers. Our review suggests that districts are quite limited in acquiring mathematics texts or programs that are proven to be effective for tier-based interventions. For a content area where many teachers, especially those working with struggling learners, rely on prepackaged texts and programs, this finding is of grave concern and challenges the foundational concepts of RTI, which guide much of mathematics instruction for the struggling learner.

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Herein lies a significant problem for students with disabilities. RTI assumes that, through the use of published core mathematics texts and programs, a significant majority of students receive scientifically based mathematics instruction at the Tier 1 level. However, the data do not support this assumption. In addition, many teachers are not well prepared to adapt mathematics curricula to address the needs of struggling learners and those with disabilities (Newton et al., 2012; Wilson, Floden, & Ferrini-Mundy, 2002), making it unlikely that these students are receiving instruction that addresses their mathematical learning needs. One way to address this problem would be for curriculum developers to design their texts and programs to be more accessible for students with disabilities. The Mathematics eText Research Center is currently studying the use of technology, e-texts, and the principles of universal design for learning to enhance the accessibility of mathematics texts. Although this work is exciting, necessary, and has promise for the future, accessibility is only part of the solution. (See also CAST's video, "UDL At a Glance", at http://youtu.be/bDvKnY0g6e4).

## Four "Buckets" of Effective Mathematics Practices for Students with Disabilities

*As part of their schools movement to a RTI framework, Mrs. Perez and Mr. Williams have attended multiple professional development sessions on RTI. Their school district places a great deal of emphasis on reading outcomes at the elementary level, so the associated RTI professional development has been devoted to research-supported reading instruction practices. As a result, Mrs. Perez and Mr. Williams continue to be uncertain about what to do with their mathematics instruction and what would be an effective Tier 1 or Tier 2 product, strategy, or intervention. They grasp the concept of RTI and how it should assist them as teachers, yet the lack of information about mathematics and RTI makes it difficult for them to determine what to do differently in order to better address the mathematical learning needs of their students. Although they\u2552ve done some Internet research on their own and have found some helpful resources (visit, for example, MathVIDS! and the Center on Instruction), there is just so much information it overwhelms them. Mrs. Perez and Mr. Williams have concluded that as learners they need a visual or at least an understanding of the big picture. They know what their limitations are; what they need are solutions within the RTI framework (which they have seen to work for reading) for mathematics. They are left scratching their heads and wondering what to do next.*

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The literature on students with disabilities and mathematics instruction (e.g., Allsopp, Kyger, & Lovin, 2007; L. S. Fuchs & Fuchs, 2001; Gersten, et. al, 2009; Gersten & Chard, 1999; National Council of Teachers of Mathematics, NCTM, 2013a; National Mathematics Advisory Panel, 2008; Swanson, 1999) suggests several interconnected domains of effective mathematics practice that can be utilized within an RTI framework:

- emphasize critical areas of mathematical knowledge;
- engage students in multiple ways of doing mathematics;
- emphasize research supported effective explicit teaching practices; and
- focus instruction on building understanding, proficiency, and maintenance of mathematics concepts/skills.

Incorporating practices within these essential domains—“the four buckets” (see Resources, Figure 1)—can lead to powerful mathematics teaching and learning for students at any tier. Understanding the contents of these buckets of effective mathematics practices for students with disabilities and associated support from the literature can help teachers like Mrs. Perez and Mr. Williams enhance their capacity to collaboratively address their RTI mathematics instruction in more effective ways for students with disabilities. If teachers understand and have access to the contents of each bucket, they will be able to incorporate critical practices when using adopted mathematics texts and programs to teach students with disabilities and other struggling learners, as well as recognize and respond to deficiencies that need to be further addressed.

## Bucket 1: Emphasize Critical Areas of Mathematical Knowledge

The PK–12 mathematics curriculum includes concepts and skills that build upon each other. The relationships between these concepts and skills become more complex as students move through the elementary and secondary grades. In addition, certain mathematics content areas seem to be more important than others for developing mathematical literacy among students, particularly during the elementary and middle school years (i.e., K–8). Three critical areas are number sense (or **numeracy**), number operations, and algebraic thinking (Gersten & Chard, 1999; National Mathematics Advisory Panel, 2008).

**Number sense**has to do with number recognition, understanding what numbers represent (e.g., magnitude), their relationships (e.g., that 6 is four more than 2), and how to flexibly and fluently use numbers to count, estimate, measure, and problem solve.**Number operations**has to do with using number sense to efficiently compute by adding, subtracting, multiplying, and dividing numbers including whole numbers, rational numbers, and integers.**Algebraic thinking**incorporates elements of both number sense and number operations where students make sense of numerical patterns, relations, and functions; represent and analyze mathematical situations and structures using mathematical symbols; and use mathematical models to represent and understand quantitative relationships and analyze change in various contexts.

Together, number sense, number operations, and algebraic thinking provide students with powerful mathematical thinking tools that allow them to effectively problem solve, represent mathematical situations, and reason. (The NCTM has developed web-based Illuminations and Focal Points that provide examples and tools to introduce and reinforce these concepts and skills.) An emphasis on these foundational concepts and skills has promise for increasing the mathematical competency of all students, especially students with disabilities.

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## Bucket 2: Engage Students in Multiple Ways of Doing Mathematics

The NCTM emphasizes that to increase competence students need to be exposed to multiple ways of doing math and of expressing their mathematical understandings. Although **mathematical competence** is often interpreted as computation and fact retrieval, it is much more than these skills; it includes a deeper, conceptual understanding of mathematics and being able to communicate such understanding effectively. To achieve mathematical competence, the NCTM advocates engaging all students in five processes for doing mathematics: problem solving, representations, communication, connections, and rationale/proof (visit the NCTM's Process Standards page for details on each of these processes). Teachers who engage students in these five processes are able to help them develop the conceptual and procedural understandings of mathematics. Unfortunately, students with disabilities often are not exposed to these different ways of doing mathematics, either because it is thought they cannot engage in these processes efficiently or because of an emphasis on procedural mathematics (e.g., computing). Engaging students with disabilities in multiple ways of doing mathematics can help to establish and deepen conceptual understandings of mathematics and provide them alternative ways to demonstrate their understandings.

## Bucket 3: Emphasize Effective Explicit Mathematics Teaching Practices

What are the critical instructional approaches to be integrated into our mathematics instruction, especially for struggling learners? Gersten and colleagues (2009) asked that question and through their analysis of research identified several promising teaching practices for mathematics: (a) use explicit instruction, (b) support student verbalization of mathematical reasoning, (c), provide visual representations, (d) provide a range of sequenced examples of target mathematics concepts and skills, and (e) provide students with data and feedback on their mathematics performance (Gersten et al., 2009). Other promising practices include teaching mathematics within authentic contexts (e.g., Bottge, Heinrichs, Metha, & Hung, 2002), engaging students in representing mathematics with concrete materials and pictures or drawings within a concrete-representational-abstract (C-R-A) sequence of instruction (L. S. Fuchs & Fuchs, 2001; Gersten, Jordan, & Flojo, 2005; Witzel, Riccomini, & Schneider, 2008), and using systematic and strategic instruction (Swanson, 1999). Research suggests that when these practices are utilized, students with disabilities do better in mathematics (e.g., Bottge et al., 2002; L. S. Fuchs & Fuchs, 2001; Gersten et al., 2009; Swanson, 1999). MathVIDS and the Center on Instruction-RTI/Mathematics) both provide extensive support in learning how to implement mathematics research-supported practices associated with these areas.

## Bucket 4: Focus Instruction to Build Understanding, Profiency, and Maintenance

Learning a new mathematical concept or skill is not an automatic process for most students, and particularly for students with disabilities. Most of us develop understanding of a new concept or skill and the ability to apply our understanding proficiently over time with repeated opportunities to apply our understanding, receive feedback, and use feedback to improve. Student learning follows this same process, in six stages (see Resources, Figure 2).

Initial stages of mathematical learning should focus on building an understanding of concepts (Allsopp, Kyger, & Lovin, 2007; Mercer & Mercer, 2001). The instructional focus at the **Entry and Acquisition** stages should be on **accuracy of understanding**. When students build accurate understandings of a concept or skill, they can begin building proficiency in applying it. The instructional focus at the **Proficiency** stage should be on both **accuracy and the rate** students are able to do mathematics. At the **Maintenance** stage students with disabilities require periodic and focused opportunities to do mathematics with which they have already developed proficiency. The instructional focus at this stage should be on **retention of adequate levels** **of accuracy and rate**. At this point, students are better equipped to generalize their understanding to different contexts and adapt existing mathematical knowledge to make new knowledge or enhance it. The instructional focus at the **Generalization and Adaptation** stages should be on **extension**, where students are engaged in using what they understand and can do mathematics in new ways. When teachers focus their instruction to facilitate students’ movement through each stage of mathematical learning, students are more likely to develop deep understandings of mathematics concepts and skills and the ability to use them efficiently over time and in different ways for different purposes.

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These four buckets of effective mathematics practices provide teachers with powerful ways to address the CCSS utilizing suggestions from research (e.g., Bottge et al., 2002; L. S. Fuchs & Fuchs, 2001; Gersten et al., 2009; Swanson, 1999), NCTM, and other leaders in mathematics education (e.g., the National Mathematics Advisory Panel):

- Similar to the CCSS, Bucket 1 emphasizes the need to focus on foundational concepts supporting key ideas that cut across the mathematics curriculum. The NCTM also stresses the need to emphasize a coherent set of big ideas, or
*focal points*, that serve as a unifying thread across the PK–12 mathematics curriculum. - Bucket 2 emphasizes multiple ways of doing mathematics, utilizing the five NCTM process standards. These five processes of doing mathematics are essential to students’ abilities to build deeper understanding and become mathematically literate, an important goal of the CCSS.
- Bucket 3 incorporates actual practices that are effective in helping students with disabilities understand mathematics at deeper levels and make meaning of abstract representations. The developers of the CCSS for mathematics conceded that the standards themselves could not address the range of needs of all students nor did they define “intervention methods or materials necessary to support students who are well below or well above grade-level expectations” (CCSS Initiative, 2012, p. 4). These research-supported effective explicit teaching practices provide students with cognitive “access” to understanding mathematical concepts related to grade-level expectations.
- Bucket 4 has to do with teachers’ “instructional focus” as it relates to how students with disabilities learn. This relates directly to NCTM’s equity principle, that instruction is accommodated to address student differences (NCTM, 2013b). The CCSS for mathematics also emphasizes this idea, that the curriculum and its implementation must “respect what is known about how students learn” (CCSS Initiative, 2012, p. 4).

When teachers pinpoint their instruction to help students move from little to no understanding of a concept to advanced understanding to proficiency and then to maintenance, they are respecting the learning needs of students with disabilities and other struggling learners. With regard to RTI, the practices within each bucket provide teachers with a foundation for enhancing their instruction for students with disabilities at Tier 1 and for intensifying mathematics interventions at Tiers 2 and 3. The four buckets provide teachers with a frame for conceptualizing effective mathematics practices for students with disabilities and understanding why they are important.

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*Mrs. Perez and Mr. Williams’s discussion of these four buckets of effective mathematics practice for students with disabilities helps them to think about their own practice in a more focused way. For example, the principle of Bucket 2 (engage students in multiple ways of doing mathematics) resonates with Mrs. Perez; she especially likes the idea of using language to communicate mathematical ideas. Mr. Williams is drawn towards Bucket 3 (effective teaching practices). He sees some familiar practices that he uses in reading with students with disabilities, such as explicit modeling and peer-mediated learning, and is interested in learning more about teaching problem-solving strategies. He knows strategy instruction is helpful for students with disabilities for reading comprehension and that they need support in developing *metacognitive awareness* (i.e., thinking about thinking and learning). With the four buckets in mind, Mrs. Perez and Mr. Williams begin to actively reflect on the extent to which they are emphasizing practices from each bucket and how this might enhance their Tier 1 mathematics instruction.*

## Using the Four Buckets to Evaluate and Adapt Core Mathematics Curricula

The four buckets of effective mathematics practices provide educators with a frame for conceptualizing research and, thus, reflect on the extent to which educators emphasize these practices in their own teaching. The next step is to use this knowledge to critically evaluate core mathematics texts and programs, identify strengths and potential weaknesses, and plan for interventions that integrate effective practice supporting or replacing adopted mathematics curricula.

Mathematics curricula and programs differ widely in how mathematics content is sequenced, the instructional practices emphasized, and the structure utilized to build mathematical understanding. Most important, curricula can vary greatly regarding the extent to which they integrate research-supported effective instructional practices that address the mathematical learning needs of students with disabilities. When educators have resources and practical ways to use information, they become powerful mathematical instruction decision makers. Teachers can use our guide and accompanying rating/evaluation system to assess different mathematics texts, products, and programs (see Resources, “Evaluating Mathematics Curricula for the Integration of Effective Mathematics Practices for Students With Disabilities”). This resource, which incorporates the concept of the four buckets and is based on earlier work by Allsopp, Alvarez McHatton, Ray, and Farmer (2010), can assist teachers when:

- comparing mathematics texts/programs when making decisions about adopting a text,
- evaluating an already adopted mathematics text/program to determine the extent to which it incorporates research supported mathematics instructional practices for students with disabilities, and
- planning how to adapt a core mathematics text/program to meet the needs of students with disabilities.

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The guide is organized according to the four buckets of effective mathematics practice; the teacher evaluates the text/program on the extent that it includes the identified research-supported practices. The rating guide provides suggestions for what the results might mean with respect to determining the overall suitability of a particular mathematics text or program for students with disabilities and provides them with a system for comparing different products to determine which one better incorporates effective mathematics practices. The rating system also helps educators evaluate specific bucket areas, and decide whether and how to adapt the text or program to particular practices.

By understanding areas of effective practice that each of the buckets represent and the significance they have individually and collectively to the mathematics learning needs of students with disabilities, teachers can prioritize where to focus their initial instructional decision making. Because we assume that teachers will be more familiar with certain practices than others, to begin, teachers might decide to focus on integrating practices with which they are familiar. They also can target practices they want to learn more about, and develop a plan with their grade-level, department, or school leadership teams for further professional development.

## Evaluation in Practice: Mrs. Perez and Mr. Williams

*Let’s consider Mrs. Perez and Mr. Williams one final time. Realizing they need to invest time, they decide to use our process to evaluate their district’s adopted program for mathematic instruction, with the goal of improving their teaching during the current year and in the future. After completing the evaluation (see Resources, Figure 3), Mrs. Perez and Mr. Williams review their scores. Their total score of 20 places the text in the middle with respect to integration of effective practices, meaning that their text incorporates some effective practices but could benefit from adaptations. Reviewing a chart they made to help visualize their ratings (see Figure 4; see Resources for a blank chart), the teachers decide to focus on Bucket 3 (effective explicit mathematics teaching practices).*

*Mrs. Perez and Mr. Williams pinpoint three explicit mathematics teaching practices: explicit instruction and continuous progress monitoring/instructional decision making (each of which the teachers rated a 2), and multiple response opportunities provided for each learning objective (which they rated a 3). Their text’s daily lesson sequence provides a framework for explicit systematic instruction and attempts to bridge new content with prior background knowledge and recently taught concepts. Lessons incorporate brief focus activities, teacher modeling, guided practice through a combination of teacher and peer support (e.g., cooperative learning), and independent practice. Mrs. Perez and Mr. Williams agree that the overall structure reflects a systematic instruction sequence, but feel the text is lacking in terms of explicitness. For example, the modeling section relies on an animated video that accompanies each lesson. The videos provide students with visual representations, but they do not allow students an opportunity to interact with the content (e.g., to manipulate objects or drawings as the video models the lesson’s target mathematics concept or skill)—except for responding to predetermined questions that are imbedded in the video.*

*Mrs. Perez and Mr. Williams also observe that their text does not apply a C-R-A sequence of instruction consistently across lessons and there is a lack of explicit teaching of metacognitive strategies to problem solve and to self-monitor progress. They would like to see more use of teacher “think-alouds” when lessons have teachers model a mathematical task, and more of an emphasis on teaching students to “think aloud” themselves as they problem solve (maybe including teaching students to use graphic organizers). Also, they feel that the guided practice section of the text does not provide students with enough scaffolded support; typically there are only three to four problems or response tasks in this section. Although each unit and lesson includes a list of possible accommodations and modifications to differentiate instruction for students with disabilities or English language learners, these recommendations are generally very basic surface-level accommodations that do not address specific disability-related characteristics (e.g., “play card game using only the basic facts” or “use a calculator to skip count”). Mrs. Perez and Mr. Williams believe that it would be more helpful if the resource provided information on specific classroom accommodations that address common disability-related characteristics that impact mathematical learning (see MathVids Learner Accommodations page). For example, students who have visual and spatial processing difficulties can find it difficult to line up digits when computing. Teachers can show these students how to use graph paper or how to turn notebook paper horizontally in order to use the lines to organize their computations.*

*With respect to continuous progress monitoring and instructional decision making, Mrs. Perez and Mr. Williams explore the assessment package included with the text. Having worked with this text over the past 2 years, they are pretty familiar with the regular assessments such as chapter and unit tests as well as the ongoing daily formative assessment probes, called “quick checks.” As they look more thoroughly at the curriculum, they realize that the assessments generally focus on abstract level computations and lack appropriate universal screening tools and diagnostic assessments (e.g., informal C-R-A assessments, error pattern analysis, flexible interviewing) that could help them better target individual students’ needs and assist in developing intervention plans using research-supported mathematics instructional practices*.

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*Regarding multiple response opportunities for each learning objective, Mrs. Perez and Mr. Williams note that the text includes independent practice and homework sections. However, these sections do not provide their students enough opportunities to respond to newly introduced concepts and skills—one reason they think many of their students have difficulties with applying previously covered material.*

*Based on their experiences examining and evaluating their mathematics text, Mrs. Perez and Mr. Williams realize that there is a lot that they can do to better integrate research-supported effective practices in conjunction with the Tier 1 core third-grade mathematics curriculum. They have a better appreciation for how their mathematics text is structured, its strengths, and its weaknesses with respect to the mathematical learning needs of students with disabilities. Mrs. Perez and Mr. Williams also have a conceptual frame (the four buckets) for reflecting on the areas of practice that they need to learn more about. Overall, they feel more empowered and confident about the direction they are taking compared to before they engaged in this evaluation and reflection process. For the first time, Mrs. Perez and Mr. Williams believe they have a strong sense of how to more effectively utilize their mathematics text, where they need to make instructional adaptations, and the specific research supported instructional practices they can emphasize to better meet the needs of their students.*

## Final Thoughts

Unlike other PK–12 STEM areas, mathematics instruction is increasingly integrated within the RTI framework for struggling learners and those with disabilities. Within this framework, educators seek to integrate evidence-based practices across their tier-based efforts. Although RTI offers a framework applicable to the needs of struggling learners and those with disabilities, the limited nature of the research base specific to mathematics and RTI, the extent to which educators overly rely on texts and programs for mathematics instruction within RTI, and the concern that teachers (especially at the elementary level) are limited in their mathematics comfort and expertise present problematic issues for providing effective mathematics instruction for struggling learners and those with disabilities. Findings from mathematics and special education researchers provide educators with an increasing foundation of promising effective mathematics instructional practices. Using these practices within tier-based mathematics instruction provides students with disabilities the best chance at mathematical success; mathematics outcomes for students with disabilities improve when research-supported effective mathematics practices are utilized in thoughtful and systematic ways (Bottge et al., 2002; L. S. Fuchs & Fuchs, 2001; Gersten et al., 2009; Swanson, 1999). Assisting both general and special education teachers to collaboratively evaluate and adapt their core mathematics curricula in an informed and systematic way to effectively integrate effective mathematics practices is an important aspect to improving mathematics instruction and mathematics outcomes for students with disabilities.

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## About the Authors

**David H. Allsopp**, Professor, Department of Special Education, College of Education, University of South Florida, Tampa. **Jennie L. Farmer**, Assistant Professor, Department of Teacher Education, Clemson University, Clemson, South Carolina. **David Hoppey**, Assistant Professor, Department of Special Education, College of Education, University of South Florida, Tampa. **Brandi Leigh Kamp**, Doctoral Student, Department of Teacher Education, Clemson University, Clemson, South Carolina.

Address correspondence concerning this article to David H. Allsopp, University of South Florida, 4202 E. Fowler Ave, EDU 105, Tampa, FL 33620 (e-mail: dallsopp@usf.edu).

TEACHING Exceptional Children, 45(4)

## Supplemental media files

Four Buckets of Effective Mathematics Practices for Students with Disabilities - (Figure1)

Stages of Learning and Relation to Instructional Focus - (Figure 2)

Sample Completed Mathematics Text Evaluation - (Figure 3)

Mrs. Perez and Mr. Williams' Mathematics Curriculum Evaluation - (Figure 4)

Evaluating Mathematics Curricula for the Integration of Effective Mathematics Practices for Students With Disabilities - (Resource 1)

Mathematics Curricula Evaluation Chart - Blank Chart - (Resource 2)

Ratings of PK-8 Mathematics Interventions and Curricula By the What Works Clearinghouse - (Table 1)