217Learning about Learning Disabilities© 2012 Elsevier Inc.

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Instructional Interventions for Students with Mathematics Learning DisabilitiesMargo A. Mastropieri1, Thomas E. Scruggs1, Clara Hauth1, and Dannette Allen-Bronaugh21George Mason University, Fairfax, VA 22030-4444, USA2James Madison University, Harrisonburg, VA 22807, USA

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Chapter Contents

Introduction 217Learning Disabilities and Mathematics Achievement 221

Effective Math Interventions 222Behavioral Interventions on Computation 223

Early Behavioral Interventions: Improving Computation and Basic Skills 223Metacognitive Instruction in Math 224Metacognitive Instruction to Improve Computation 224Use of Manipulatives to Enhance Concreteness and Metacognitive Training 225Metacognitive Problem Solving with Visual Diagrams and Schemas 227Algebra and Metacognition 229

Response to Intervention Research in Math 230Tier 2 RtI Math Studies 231

What Are Evidence-Based Math Practices? 233Discussion and Future Directions 235References 236

CHAPTER

INTRODUCTION

Mathematics education is in the forefront of our national discussion on education reform. Recent laws mandate education accountability mea-sures for all students in reading and mathematics (NCLB, 2001). The focus of both the No Child Left Behind Act of 2001 and the recent Elementary and Secondary Education Act: Blueprint for Reform (USDOE, 2010) incorporate similar requirements of strong academic performance by all students and accountability measures that reflect that performance (NCLB, 2001). The ESEA blueprint has an added emphasis on preparation for college and career training (USDOE, 2010). In addition, NCLB (2010)

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emphasizes the use of evidence-based practices and expresses the goal for K-12 teachers to use scientifically based research to help direct and guide instructional practices and interventions in the classroom. Another criti-cal legislative piece is the Individuals with Disabilities Education Act of 2004, (IDEA, 2004), which mandates that students with disabilities have access to the same curriculum and are taught using best practices and high standards of learning. The National Council of Teachers of Mathematics (NCTM) also incorporates these same tenets of equal access for all stu-dents in their principles and standards in mathematics education.

Mathematics instruction in the United States is guided by the National Council of Teachers of Mathematics (NCTM) which is a national orga-nization for mathematics educators whose stated focus is “to ensure equitable mathematics learning of the highest quality for all students” (http://www.nctm.org). NCTM has identified six principles and ten stan-dards for students in kindergarten through 12th grade to succeed in math-ematics education (NCTM, 2000).

The principles for school mathematics include equity, curriculum, teaching, learning, assessment and technology. These principles help to pro-vide teachers, schools, and district and state administrators with the tools needed to guide educational decisions in mathematics for all students.l Equity Principle: Excellence in mathematics education requires equity,

high expectations and strong support for all students.l Curriculum Principle: A curriculum is more important than a col-

lection of activities: it must be coherent, focused on important mathematics, and well articulated across grades.

l Teaching Principle: Effective mathematics teaching requires under-standing what students know and need to learn and then challenging and supporting them to learn well.

l Learning Principle: Students must learn mathematics with under-standing, actively building new knowledge from experience and prior knowledge.

l Assessment Principle: Assessment should support the learning of important mathematics and furnish useful information to both teachers and students.

l Technology Principle: Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances student learning (www.nctm.org/standards/content; NCTM, 2000).

These principles engage educators in the overarching discussion regarding the design of curriculum for all students in K-12 mathematics education.

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In addition, NCTM emphasizes the incorporation of content and process standards to address the actual concepts and instruction indicative of an excellent mathematics education. These standards are used throughout the United States as state and local school leaders plan and create programs of study for mathematics.

The mathematics standards are delineated by content and process areas incorporating the skills, knowledge and understanding that students in grades preK-12 should attain in the educational system. The five con-tent area standards include numbers and operations, algebra, geometry, measurement, and data analysis and probability. The five process standards include problem solving, reasoning and proof, communication, connec-tion, and representation (NCTM, 2000).

The content standards incorporate guidelines for curricula which are expanded upon throughout the four grade bands of prekindergarten-grade 2, grades 3–5, grades 6–8 and grades 9–12. As an example for the number and operations standard, all students should have an understand-ing of number sense which may begin in preK programs as simple count-ing and ordering of natural numbers {1,2,3…} and which continues to develop across the grade bands to understanding the relationships between number systems {irrational, rational, integers…}. The content area stan-dards are embedded in each level of mathematics instruction, scaffolding to more complex problem sets.

Process area standards were developed to guide educators as they teach the content area in the classroom. The process area standards integrate crit-ical thinking methodology by teaching math content via the processes of problem solving, reasoning and proof, communication, connections to the real world, and the use of alternate representational models. These areas are fostered by teachers as they introduce, teach and assess learners in the four grade bands. For example, with the problem solving standard, teachers are encouraged to teach with various problem solving techniques and to encourage a range of problem solving skills which may include diagrams, patterns, and manipulatives to discover content area concepts (NCTM, 2000). Educators must be cognizant of both content and process standards to fully develop curricula which will meet the needs of all learners.

The NCTM provides a focus and direction for mathematics educa-tion, which is also validated by the recent National Math Advisory Panel (NMAP, 2008) government report (USDOE, 2008). The National Math Advisory Panel report emphasizes the need for consistent high stan-dards in math education with an emphasis on equity for all students in

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mathematics. The report was commissioned by the U.S. government to address the growing concerns regarding our nation’s falling standings in mathematics achievement. Mathematics is a critical skill area for students as they graduate and enter into an increasingly global economy. They placed an increased emphasis on preparation for and instruction of algebra as a critical area for increasing American student achievement in math-ematics for higher education and career readiness. Much like the NCTM standards, the NMAP findings place a strong emphasis on curriculum standards and learning processes for all students. The report also included recommendations, or areas of focus, for the future of mathematics edu-cation in the United States. The main findings and recommendations include an increased focus on (a) curricular content; (b) learning processes; (c) teachers and teacher education; (d) instructional practices; (e) instruc-tional materials; (f) assessment; and (g) research policies and mechanisms to report findings (USDOE, 2008).

Students with disabilities often have difficulties meeting the academic benchmarks outlined by the NMAP report and the NCTM content and process standards. These students also have difficulty passing high stakes standardized tests (Thurlow, Altman, Cormier, & Moen, 2008). To meet the needs and differentiated learning styles of students with disabilities, evidence-based practices (EBP) and intervention are the focus for special educators in the mathematics classroom (Gersten, Chard, Jayanthi, Baker, Morphy, & Flojo, 2008).

The criteria for effective math instruction are, however, another area of noted importance as educators search for and implement strategies for the students they teach. The No Child Left Behind Act (NCLB, 2001) rec-ommends that schools use evidence-based practices when educating all students. Section 6516 of public law 110 encourages schools to:

…implement a comprehensive reform program that has been found through scientifically based research to scientifically improve the academic achievement of students participating in such program … or a program has been found to have strong evidence that such program will scientifically improve the academic achievement of students.

(Public law 107–110)

Evidence-based practices and scientifically-based research are impor-tant to help direct and guide instructional practices as well as interventions in the classroom.

Although the NCTM standards are presently exerting substantial influence on mathematics education reform in the United States, many

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special education professionals have expressed concern about the NCTM standards as applied to students with disabilities (e.g., Montague, 1996c; Rivera, 1997). These concerns are essentially based on fears that meth-ods (e.g., “discovery,” “inquiry”) that require independent insight on the part of the learner will not be effective for students with disabilities, for whom insight or deductive inference can be relative weaknesses. However, historically special education professionals have perhaps overemphasized rote learning of facts and procedures, to the extent that students have had little opportunity to experience and practice mathematical reasoning (e.g., Cawley, Miller, & School, 1987).

LEARNING DISABILITIES AND MATHEMATICS ACHIEVEMENT

Research has documented that students with learning disabilities can lag far behind other students in the area of mathematics. Previous and current research has consistently identified that students with learning disabilities score well below their typical peers on standardized math tests (e.g., Fuchs & Fuchs, 2003; Scruggs & Mastropieri, 1986; Parmar, Cawley, & Frazita, 1996). For example, McLeskey and Waldron (1990) reported that 64% of 906 students with learning disabilities in the state of Indiana from ages 5–19 were achieving below grade level in mathematics.

Recent research refers to students with math learning difficulties (MLD) as those individuals who perform lower in math who may or may not have been identified as having learning disabilities ( Jordan, Hanich, & Kaplan, 2003). Geary (2004) estimated that between 5 and 8% of school children may have MLD. Findings across math studies conclude that stu-dents with MLD and LD frequently struggle learning math, including basic skills and more conceptually based problem solving (e.g., Fuchs & Fuchs, 2002; Geary, 1993; 2003; Swanson & Beebe-Frankenberger, 2004). Evidence also indicates that when children experience early difficulties in math, challenges may persist well into upper elementary grades (e.g., Jordan, Kaplan, Ramineni, & Locuniak, 2009).

Researchers have identified potential difficulties many students with learning disabilities with math disabilities may exhibit (Montague, 2007; 1996a, b; see also Swanson & Jerman, 2006). Students with LD and MLD have difficulties with memory, working memory, memory for numbers, and number sense (e.g., Swanson & Sachse-Lee, 2001, Fuchs et al., 2005). Such challenges may differentially affect mathematics performance. Language

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and communication difficulties may also impact math performance when students are required to read and discuss math problems (e.g., Montague, 1996). Insufficient metacognitive strategies in solving math problems have also been related to students’ weaker math performance (e.g., Woodward & Montague, 2002). Finally, low motivation can impact individuals’ affect and attitude toward the content area of math (Montague, 1996). Given the potential learning challenges, what math intervention strategies have been shown to be effective with students with learning disabilities?

Effective Math InterventionsAlthough math intervention research with students with learning disabilities has been conducted, there is considerably less research than in other areas such as literacy. For example, Mastropieri et al. (2009) reported that of all the research published in special education major journals over a 19-year period only 15.9% represented intervention research. Of that percentage 49% represented literacy research while only 15% were math intervention studies. Fortunately over the years reviews of research have synthesized much of that research. Earlier reviews (e.g., Jitendra & Xin, 1997; Mastropieri, Scruggs, & Shiah, 1991) and more recent reviews (e.g., Gersten et al., 2008; Kroesbergen, & Van Luit, 2003; Swanson & Jerman, 2006) have described effective research practices and discussed the importance of strong research designs for the provision of evidence-based math practices. The purpose of this chapter is to update previous reviews and research on mathematics performance of students with learn-ing disabilities, in grades K-12 that have occurred since earlier research reviews.

This review provides information on mathematics interventions for students with LD. The studies considered in this review fell broadly into behavioral interventions in computation and meta cognitive strategies for problem solving. Strategies for problem-solving research has focused on meta cognitive strategies instruction, self-regulation, schema-based strat-egies, and embedded number combination practice within tutoring for problem solving. The most efficacious studies appeared to include: explicit instruction, self-regulation including verbalization, visualization of strate-gies, concrete manipulatives, and multiple examples, including real world applications. Finally, future issues including evidence-based practices and use of Response to Intervention (RTI) to prevent long-term math diffi-culties are presented.

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BEHAVIORAL INTERVENTIONS ON COMPUTATION

The vast majority of early research on mathematics with students with learning disabilities (LD) focused on methods for improving computation skills. The studies included methods for developing proficiency in basic numeration concepts, facts, and operations (addition, subtraction, multipli-cation, and division) with whole numbers, fractions, or decimals. A small selected sample of these is now described.

Early Behavioral Interventions: Improving Computation and Basic SkillsEarly research employed behavioral approaches, including reinforce-ment and direct instruction to improve computational skills. For example, Pavchinski, Evans, and Bostow (1989) described an intervention with a 12-year-old student with learning disabilities to increase basic reading and math skills. Using a changing criterion design and a token reinforcement system, the student was presented with Dolch sight words, and a second set of measures consisting of a list of 220 simple arithmetic problems. Tokens earned for meeting the target criterion could be exchanged for privileges. Similarly, Hastings, Raymond, and McLaughlin (1989) used task-analysis and direct instruction procedures and successfully trained seven students, including two with learning disabilities, to count money rapidly.

Other researchers varied presentation formats for students with learn-ing disabilities (e.g., Albers & Greer, 1991). For example, Cooke, Guzaukas, Pressley, and Kerr (1993) investigated the effects of interspersed drill and practice and found that more drill and practice facilitated recall. Koscinski and Gast (1993b) used a 4-second constant time delay procedure (cor-rect response was provided four seconds after the stimulus presentation, see Koscinski & Gast, 1993a; Koscinski & Hoy, 1993) to teach multiplication facts. Houten (1993) compared rote drill with a rule learning procedure in learning subtraction.

Several early studies examined the effects of reinforcement for improv-ing math homework completion (e.g., O’Melia & Rosenberg, 1994; Patzelt, 1991). Studies typically concluded that behaviorally oriented intervention improved basic computation skills, although most were single subject designs conducted over relatively short intervals involving very few participants.

Peer mediation during instruction was examined to determine whether the computational skills of students with LD would be improved.

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For example, Beirne-Smith (1991) investigated the effects of peer tutor-ing on acquisition of addition facts for elementary age students with LD. Hawkins et al. (1994) investigated the use of peer guided practice or inde-pendent practice during instructional pauses from lecture in arithmetic computation. Fuchs et al. (1996) investigated the quality and effective-ness of students’ (grades 2–4) mathematical explanations to 20 tutees with learning disabilities as a function of the ability of the tutor in mathemat-ics. Their results indicated that high-achieving tutors were rated higher on conceptual and procedural characteristics as well as on overall quality, as compared with tutors of average ability. This study provided foundational guidance for later peer mediation research in math.

SummaryEarly behavioral research provided information on the positive effects of a variety of behavioral techniques on the math computation perfor-mance of students with LD. These techniques have included presentation rate, drill and practice, direct instruction, cumulative review, reinforcement and behavioral contracting, commonly reported to be generally effective teaching strategies (e.g., Mastropieri & Scruggs, 2010). In addition, further information was provided on rule learning and peer mediation, includ-ing tutoring and cooperative homework teams. However, the early studies were typically short in duration, examined a single task (e.g., math facts), and failed to examine higher level math problem solving.

Metacognitive Instruction in MathMetacognitive interventions teach individuals the processes involved in learning by using verbal self-instructions, self-monitoring, self-evaluation and self-regulation as instructional components integrated within instruc-tion (e.g., Bandura, Gusec, & Menlove, 1966; Meichenbaum, 1977). Early studies examined basic computation and time on task, but later studies were extended to include manipulatives, problem solving strategies, and higher level math using more complex instructional packages

Metacognitive Instruction to Improve ComputationEarliest math research in LD examined self-instructions to improve time on task (e.g., Prater, Hogan, & Miller, 1992), which required students to check off their on-task or off-task behavior on hearing an audible tone. Other stud-ies employed variations of metacognitive training to enhance computation skills. For example, Dunlap and Dunlap (1989) investigated the effects of error

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self-monitoring to improve subtraction with regrouping. Laird and Winton (1993) compared the effectiveness of three self-instructional checking proce-dures on math performance. Wood, Rosenberg, and Carran (1993) studied the effects of tape-recorded self-instruction cues to solve addition and subtraction problems and Shiah et al. reported success with embedded self-instruction into computer-assisted problem solving. One notable study, for example, by Kamann and Wong (1993) investigated the effectiveness of self-statement instruction to reduce math anxiety in students with LD when solving problems involv-ing fractions. The trainer demonstrated a self-talk procedure, which consisted of positive, neutral, and negative statements. Oral and visual representations of poor self-talk, which affect performance, were presented to evoke awareness of students’ maladaptive styles of thinking. Next, coping strategies were intro-duced. The trainer modeled the strategies as outlined on two cue cards which described the steps in the coping process and the coping self-statements: situa-tion assessment, identifying and controlling negative thoughts, coping thought, and reinforcement. The confronting/coping/controlling self-statements included the following: Don’t worry. Remember to use your plan.

Take it step by step—look at one question at a time.Don’t let your eyes wander to other questions.Don’t think about what others are doing. Take it one step at a time.When you feel your fears coming on...take a deep breath, think, “I am doing just fine. Things are going well”

(Kamann & Wong, 1993, p. 632).

The earlier interventions investigating metacognitive strategies to improve math computation skills of students with LD were largely effec-tive, providing further evidence for successful training to improve per-formance in students with LD, but were relatively short in duration and taught only isolated skills. In the research presently reviewed, effective metacognitive training was implemented in the areas of self-monitoring of on-task behavior, error self-monitoring, making positive affective self-statements, self-instructional checking procedures, self-instruction of calculation procedures, and math anxiety.

Use of Manipulatives to Enhance Concreteness and Metacognitive TrainingUse of manipulatives can aid conceptual understanding during math instruction (NCTM, 2000) and studies have documented that the effec-tive use of manipulatives has enhanced the performance of students with LD (e.g., Marsh & Cooke, 1996). Miller and Mercer (1993a) examined

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an instructional sequence from concrete, using manipulatives, to abstract to determine the effectiveness of a sequence of concrete–semiconcrete-abstract (CSA) instruction of math facts (see also Mercer & Miller, 1992). The teacher provided a verbal organizer, demonstrated the skill by think-aloud and modeling, provided guided practice with prompts, cues, and feedback, and provided independent practice. The instruction sequences progressed from concrete, semiconcrete, to abstract, on each instructional level. The concrete level instruction employed manipulatives; illustrations were provided on the semiconcrete level; and on the abstract level, draw-ings were provided only when students could not recall the facts. The authors concluded that CSA was an effective means for teaching math computation. Miller and Mercer (1993b) replicated the effectiveness of a graduated word problem sequence strategy for teaching math problem solving. Each instructional level (concrete, semiconcrete, and abstract) contained four steps: (a) providing an advance organizer; (b) demonstrat-ing and having students model skills; and (c) guided; and (d) indepen-dent practice with feedback. The language used in the word problems matched the manipulative objects in the concrete and semiconcrete levels. For example, if students were learning to subtract using cubes, the word “cubes” was used in the problem:

4 cubes−2 cubes? cubes

During the abstract level, the difficulty of word problems increased gradually from simple words, phases, and sentences, such as:

8 pieces of candy−8 pieces of candy sold? are left (p. 172)

to more elaborate sentences:Jennie had 4 pensShe lost 2 of themShe has _____ pens left (p. 172)

to having the student created his/her own word problems. Post-test scores suggested that the intervention had been effective, although students predictably scored lowest in creation of their own problems.

In a follow-up study Harris, Miller, and Mercer (1995) investigated the effect of using a concrete-representational-abstract teaching sequence to improve multiplication skills in mainstream classrooms. Understanding of multiplication concepts was taught (lesson 1–3) to students by using

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concrete manipulatives with language parallel to that used in the problems. On the representational level (lesson 4–6), students were taught to use pictures of objects and tallies to solve problems. The mnemonic acronym DRAW was taught in the seventh lesson. Students were encouraged to use the DRAW (Discover the sign, Read the problem, Answer (or draw), check, and Write the answer) mnemonic to solve problems without using manipulatives, pictures, or tallies for the following three lessons.

Word problems and increasing computation rate began at lesson 11. In this lesson, students were taught a mnemonic device, FAST DRAW, which combined the FAST (Find what you’re solving for, Ask yourself “What are the parts of the problem?”, Set up the numbers, and Tie down the sign) device and previous learned DRAW device, to set up and solve word problems. During lesson 12 to 21, students independently practiced word problems with or without extraneous information and filled in blank spaces to create their own word problems. Results indicated that students with disabilities substantially increased their math performance, although not to the level of the normally achieving students. In these studies instruction-using manipulatives was very explicit, systematic and included teacher instruction, guided practice and independent practice.

Metacognitive Problem Solving with Visual Diagrams and SchemasProblem solving has been the focus of many studies with students with LD that have included metacognitive training with self instruction, self-regulation, and the use of visual diagrams (e.g., Walker & Poteet, 1989–1990). Jitendra and Hoff (1996) evaluated the effects of schema-based instruction for one-step addition and subtraction word problem solv-ing. One third- and two fourth-grade students with learning disabilities whose difficulty involved using incorrect equations to solve word prob-lems participated in this study. During the problem schemata phase, stu-dents learned how to recognize relations in the problem and to distinguish three types of problems (change, group, and compare). Students mapped features of situations onto the appropriate schemata diagrams after read-ing story situations. For example, in a change problem, the procedure involved reading the change word (e.g., verb) to determine whether an increase or decrease had occurred to the beginning amount. When the ending amount was determined to be more than the beginning amount, the word “total” was written under the ending amount on the schema diagram; otherwise the word “total” was written under the beginning

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amount. When the total and one of the other numbers was known, using subtraction was appropriate; when the total was unknown but the other two numbers were known, using addition was appropriate. Students suc-cessfully applied this strategy to solve addition and subtraction word prob-lems. The results indicated that not only the overall percentage of correct word problems increased substantially over baseline, but also that at least some generalization was evident for all students. Students circled the key-word consistently, but were less likely to draw pictures. Students reported that they found the instruction beneficial.

Jitendra et al. (1998) and Xin, Jitendra, and Deatline-Buchman (2005) followed up on this study with variations of problems and schematic dia-grams and in a series of single subject designs (e.g., Jitendra et al., 1998; Jitendra et al., 2002). All findings replicated earlier results with one- and two-step word problems involving schematic instruction. In the Xin et al. study students were taught multiple schemata diagrams to solve multiple problem types. Then they were taught to read problems, identify which schema diagram and math procedure was required to solve the problem, write out the diagram, and solve the problem. The visual diagram and the specific explicit systematic instruction combined with self-monitoring checklists appeared to benefit students with LD.

Montague and colleagues have also examined problem-solving strat-egies using metacognitive strategies and Montague’s Solve It materi-als. Montague (1992) assessed the effects of cognitive and metacognitive strategy instruction (CMSI) on mathematical problem solving. Strategy training consisted of demonstration, guided practice, and testing sessions. Students learned seven cognitive processes (read, paraphrase, visualize, hypothesize, estimate, compute, and check) and the initial letters (RPV-HECC) by memory without being taught how to apply those processes. Metacognitive strategy training included self-instruction, self-questioning, and self-monitoring to monitor and control strategy usage. Students in the CMSI group learned to apply metacognitive activities to each cognitive process by using say, ask, and check activities without memorizing. Results indicated that the combination of cognitive and metacognitive strategies may be more effective than either cognitive or metacognitive strategies alone. Students improved their performance on mathematical problem solving.

In a follow-up study Montague, Applegate, and Marquard (1993) eval-uated the effectiveness of CMSI on 2- to 4-step math problem solving.

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One group received training on cognitive strategies for problem solv-ing; a second group received metacognitive training; while a third group received a combination of cognitive and metacognitive strategy training. After seven days of training, the first two experimental groups received an additional five days of training in the complementary component of the instructional program, while the third group received an additional five days of cognitive and metacognitive instruction. Although there were no differences on immediate tests, delayed tests indicated a slight advan-tage for students who received the combination cognitive and metacog-nitive strategy training. Findings across Montague’s studies demonstrated that when students were taught how to proceed, to ask themselves ques-tions while solving the problems, and to monitor by checking responses throughout the process they were more successful.

Algebra and MetacognitionAlgebra has also gained some research attention (e.g., Lang, Mastropieri, Scruggs, & Porter, 2004; Witzel, 2005; Witzel, Mercer, & Miller, 2003). In one of the first studies, Hutchinson (1993) investigated the effects of cognitive strategy instruction on algebra problem solving. Twenty adoles-cent students with learning disabilities were randomly assigned to either the instructional or comparison condition. Instructional condition stu-dents were provided with cognitive strategy training for representing and solving algebra word problems. Three types of word problems (relational, proportional, and two-variable) with each surface structure or story line (work, age, distance, money, and number) were used through the study. Each student in the instructional condition met with the instructor indi-vidually on alternate days for a period of four months. Students in the comparison condition received an equivalent amount of instruction in the resource room.

Students in the instructional condition were taught to apply the following self-questions for representing word problems:1. Have I read and understood each sentence: Are there any words whose

meaning I have to ask?2. Have I got the whole picture, a representation, for this problem?3. Have I written down my representation on the worksheet? (goal;

unknown(s); known(s); type of problem; equation).4. What should I look for in a new problem to see if it is the same kind

of problem? (Hutchinson, 1993, p. 39).

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For solving algebra word problems, students were taught to ask them-selves the following:1. Have I written an equation?2. Have I expanded the terms?3. Have I written out the steps of my solution on the worksheet? (col-

lected like terms; isolated unknown(s); solved for unknown(s); checked my answer with the goal; highlighted my answer).

4. What should I look for in a new problem to see if it is the same kind of problem? (Hutchinson, 1993, p. 39).

Students were also provided with a structured worksheet to assist with organizing, representing and solving problems. Although individual stu-dents varied on the amount of material mastered, post-tests, transfer, and maintenance tests all demonstrated very substantial gains relative to those of students in the comparison condition.

SummarySuccessful findings to date have added to the accumulating problem solving research using meta cognitive strategies and real world problems (e.g., Bottge, Rueda, LaRoque, Serlin, & Kwon, 2007; Bottge, Rueda, Grant, Stevens & Laroque, 2010). Teaching fact families, and providing schema, diagrams, visualizations, and training in multiple-step cognitive strategies proved to be very effective in increasing the problem-solving skills of stu-dents with learning disabilities in areas of arithmetic and algebra. Effective research shares common instructional features including: explicit instruc-tion, teaching systematically, embedding metacognitive instructions such as self-regulation, self-instructions or self-monitoring strategies, providing suf-ficient opportunities for guided and independent practice, and a wide range of examples to help encourage generalized learning of the concepts and principles. One area of recent investigation in math with MLD students has been in the Response to Intervention (RtI) arena.

RESPONSE TO INTERVENTION RESEARCH IN MATH

Response to Intervention (RtI) is a multi-tiered prevention of failure delivery system designed including general screening for all and increasing intensity tiers of instruction that begin in the general education inclusive classroom with Tier 1. All students are screened in the general class while the first instructional tier is intended to provide evidence-based math

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(or other content area) instruction to all students. Students who fail to make adequate progress in Tier 1 are moved into Tier 2 instruction, which is frequently smaller group intensified instruction. If students fail to make adequate progress in Tier 2, they are moved to Tier 3, which may be special education in some models (e.g., Fuchs, Fuchs, & Compton, 2012). Since students are assessed periodically to determine movement within the tier system, the model is referred to as RtI. Recent research in math using RtI has been conducted and is described below, because although samples are not directly identified as LD, students are MLD and or at risk for MLD and LD in math. Note throughout that the “pack-ages” of intervention in this research appear to contain explicit instruction, use of manipulatives or sequencing of instruction from concrete, semicon-crete to abstract or visuals or diagrams, multiple opportunities for practice, and embedded meta cognitive strategies, all of which appear to be highly related to efficacious practices.

Tier 2 RtI Math StudiesFuchs and colleagues (Fuchs et al., 2005) have investigated Tier 2 math instruction designed to improve learning and understanding of word problem solving and computation. Sample interventions are multifac-eted, include explicit instruction, with modeling, guided and indepen-dent practice, on schema-based problem solving involving concrete to semi concrete to abstract representations of content. Students are taught to set up and solve various word problems, and to check and label responses. Counting strategies using fingers to assist in computation are also taught. Peer tutoring is employed during the process as a compo-nent of the math lesson and reinforcement with students earning tokens is a component. In a representative study, students in first grade were pro-vided regular math instruction plus 40-minute interventions consisting of 30 minutes of intensive small group instruction using a concrete rep-resentation to abstract instructional sequence with peer tutoring and 10 minutes working individually with computers on a math facts program. The tutoring component often emphasized number sense and opera-tion activities. In a third grade study students were taught RUN to solve word problems where R = read math problems, U = underline the ques-tion, and N = name the problem type. Findings have been generally posi-tive for students in experimental conditions across first and third graders (see Fuchs et al., 2002; Fuchs et al., 2004; Fuchs et al., 2010a, b), with less

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consistent findings on fluency measures at the first grade and on standard-ized math measures at the third grade.

Similarly, Bryant, Bryant and colleagues have investigated the effects of Tier 2 early numeracy training programs for first graders with math dif-ficulties (Bryant et al., 2008a, b; 2011). The first study examined the effects of early numeracy intervention covering number concepts, and operations, basic facts, and place value four days a week in 15-minute sessions over 18 weeks. However, no significant differences were obtained which the authors speculated may be due to an insufficient duration and intensity of the intervention (Bryant et al., 2008a). In a follow-up study the inter-vention was increased in intensity and duration and significant differences were obtained for the Tier 2 instruction (Bryant et al., 2008b). In a third investigation, additional details were provided on the interventions, which were split between two 10-minute sessions. Lessons included warm-up fluency activities and activities designed to teach conceptual knowledge using concrete manipulatives and visual representations and to learning meta cognitive strategies such as count on, and fact families to improve efficiency in solving problems more efficiently. All lessons were taught using explicit instruction using modeling, guided practice, and indepen-dent practice, progress monitoring and multiple opportunities for practice. These findings were mixed with positive effects on progress monitoring measures but not on problem-solving measures.

SummaryResearch in RtI math has promise and is beginning to demonstrate some effects for providing more intensive early preventative math instruction to at risk learners. The studies reviewed here all employ rigorous research designs and implement what most have described as components of effi-cacious practice for teaching math to students with LD, including explicit instruction, use of concrete manipulatives and a teaching sequence begin-ning with concrete and moving towards abstract as students gain under-standing, multiple practice opportunities, including a range of examples to build generalization, and metacognitive components including self- instruction, self-regulation, self-monitoring. These preliminary findings are based on research conducted with first and third graders and the field waits for replications and for extensions to the upper grade levels. Moreover, more Tier 1 and Tier 3 research is needed to provide additional guidance to the field and high quality research is needed that meets evidenced-based standards.

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WHAT ARE EVIDENCE-BASED MATH PRACTICES?

NCLB (2001) emphasized the importance of teachers using evidence-based practices to guide instructional practice and interventions within the K-12 setting. In addition there is an increased emphasis on teacher accountability for the progress of the students within their classrooms. New programs and cur-ricula inundate teachers; however the programs are not always evidence based. An unanswered question is where do teachers go to find evidence-based practices to use within their classroom? Most teachers are familiar with What Works Clearing House (http://ies.ed.gov/ncee/wwc/), and may utilize this resource to locate evidence-based practices to use in their classroom.

In order to facilitate the investigation of evidence-based practices in mathematics, the Institute of Education Sciences (IES), Council for Exceptional Children (CEC) and National Council for Mathematics (NCTM) websites were searched to identify each organization’s guidelines for evidence-based practices. After exploring these organizational websites, the criteria for evidence-based practices outlined by IES were utilized. The following four factors: randomized controlled trial or quasi-experimental design, statistical significance, a significant effect size to show that the strat-egy worked, and consistency in findings across studies, are required by IES in order for a study to be awarded a positive rating of effectiveness. The What Works Clearing House was utilized to see if any math curriculum received a positive ranking based on the IES factors. In addition to the cur-riculum reviews, articles listed on the What Works Clearing House website that investigated math were examined to determine whether any of them included students with disabilities. It was found that IES reviewed 361 stud-ies, and from these, only four met IES standards for evidence-based practices in middle school math for core comprehensive math curricula (http://ies.ed.gov/ncee/wwc/reports/middle_math/topic/, 2007). Of noted impor-tance, of these four studies, none of them involved students with disabilities.

Since the studies listed on the IES website did not demonstrate adequate resources for special educators, further investigation into CEC’s website was warranted. Like IES, CEC recognizes experimental and quasi-experimental studies as the gold standard, however they include criteria for single subject research to their category of research-based practice. In order to obtain a ranking of an evidence-based practice, CEC’s criteria include the following:a. experimental and quasi-experimental research must have a minimum

of four acceptable quality studies or two high quality studies (Gersten et al., 2005, p. 152) with

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b. a significance level of .05.In order for a single subject research study to receive a research-based

practice rating by CEC there must be:a. a minimum of five studies that meet quality indicators and document

experimental control (Horner et al., 2005, p 14)b. by a minimum of three different researchers across a minimum ofc. three different locations, with at leastd. twenty total participants between these studies.A report by NCTM (2000) calls for enhancing the rigorous math standards for all students. NCTM states that all rigorous mathematics curricula should include the following five content areas: numbers and operations, alge-bra, geometry, measurement, and data analysis and probability. In addition, NCTM also recommends that the following five process standards: problem solving, reasoning and proof, communication, connections, and represen-tations (Miller & Hudson, 2007) should be integrated within each of the content areas. Moreover, a literature search using ERIC and PsycInfo was conducted using the keywords: evidence-based practices, school mathemat-ics, and special education to locate articles written about evidence-based practices and middle school mathematics (Allen-Bronaugh & Hauth, 2010).

This search revealed limited curricular materials that have been researched and found to be evidence based for middle school mathematics. This is significant, because many times teachers go to professional devel-opment or trainings that focus on specific curriculum to use in order to teach mathematics to their students; however most of these materials lack research to determine if they are evidence based. More readily available are instructional practices teachers can incorporate into their teaching that have been studied and found to be evidence based. These practices include using visual and graphic depiction (Artus & Dyrek, 1989), systematic and explicit instruction (Xin, Jitendra, & Deatline-Buchman, 2005), student think-alouds (Shunk & Cox, 1986), structured peer assisted learning (Bahr & Rieth, 1991), and range and sequence instruction (Witzel, Mercer, & Miller, 2003).

It is evident from this search that the need persists for continued research and dissemination of evidence-based practices and remains a crit-ical area in special education. IES, CEC and NCTM are organizations that are available for teachers to utilize in order to stay up to date on evidence-based practices. These websites are continually changing in the digital era and educators need to frequently check these websites as well as profes-sional journals to access new information.

Instructional Interventions for Students with Mathematics Learning Disabilities 235

DISCUSSION AND FUTURE DIRECTIONS

A review of research on mathematics instruction involving students with learning disabilities revealed a variety of behavioral, cognitive, and meta-cognitive approaches, which have been found to be effective in improving both mathematical computation and mathematical problem solving in stu-dents with LD. Most metacognitive instruction has been combined with very explicit instructional strategies during which students are taught, and then provided ample opportunities of practice.

Research in metacognitive strategy training has revealed the effective-ness of such training across an expanding area of tasks, as well as types of training. Of particular interest in recent years is the research on explicit problem solving, including the expansion to higher-level math including algebra and research on affectively oriented self-instruction. This research has extended our knowledge of the potential breadth of metacognitive training, with respect to types of intervention as well as content area.

Similar to earlier reviews (Mastropieri, Scruggs, & Shiah, 1991), a num-ber of investigations in the present review were concerned with calcu-lation performance. More recently instructional packages have included practice on number combinations within instructional packages that also contained problem solving (see RtI section). This finding appears to con-trast strongly with the expressed views of the National Research Council (1989) and the National Council of Teachers of Mathematics (1989), who have repeatedly argued against the emphasis on computation over concep-tual development. However, it has been clearly demonstrated that students with LD and MLD frequently exhibit persistent difficulties mastering basic number facts and computational skills (Geary, 2004), as well as in simple verbal problem solving (Lerner & Johns, 2012).

Early word problem-solving interventions involved relatively simple and straightforward problems of the sort typically found in math work-books (for an exception see Hutchinson, 1993). Such problems do not generally correspond to the NCTM (1989) emphasis on “word problems of varying structures” (p. 20), such as problems that require analysis of the unknown, problems that provide insufficient or incorrect data, problems that can be solved in more than one way, or that have more than one cor-rect answer (see Baroody, 1987; Parmar & Cawley, 1996). However, more recent research with students with LD and MLD and in the areas of RtI appears to meet more recent NCTM (2000) standards. These research packages are multifaceted and integrate metacognitive problem-solving

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strategies, including computational practice and strategies, concrete to abstract teaching sequences using manipulatives, explicit instruction and multiple practice opportunities that use a range of exemplars to facilitate generalized learning (e.g., Bryant et al., 2011).

Nevertheless, finding evidence-based practices in math remains chal-lenging for teachers. Currently few studies meet established criteria by the What Works Clearing House which places teachers in the awkward posi-tion of searching for math programs. This is especially true for teachers of students with LD and MLD. Future research efforts would do well to conduct rigorous studies which could yield more available evidence-based practices and programs for students with LD and MLD. Overall, it can be stated that research in mathematics education for students with LD and MLD is progressing steadily. Future researchers and practitioners will be able to benefit greatly from the insights gained from the present research and look forward to more research in the future.

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