564220
research-article2014
LDXXXX10.1177/0022219414564220Journal of Learning DisabilitiesWatt et al.
Article
Teaching Algebra to Students With Learning Disabilities: Where Have We Come and Where Should We Go?
Journal of Learning Disabilities 1–11 © Hammill Institute on Disabilities 2014 Reprints and permissions: sagepub.com/journalsPermissions.nav DOI: 10.1177/0022219414564220 journaloflearningdisabilities.sagepub.com
Sarah J. Watt, PhD1, Jessie R. Watkins, MA1, and Jason Abbitt, PhD1
Abstract This review investigates effective interventions for teaching algebra to students with learning disabilities and evaluates the complexity and alignment of skills with the Common Core State Standards in math. The review includes the results of 10 experimental and 5 single-subject designs (N = 15) producing a moderate overall effect size (g = 0.48). A total of five interventions were identified and analyzed across the studies using effect size data. Keywords strategies, instruction, mathematics The federal STEM education effort combined with the adoption of the Common Core State Standards (CCSS; Common Core State Standards Initiative, 2010) has changed the expectations for all students in mathematics. Although the CCSS in math includes fewer standards than states previously held, the content is more rigorous and requires students to have a greater conceptual understanding of basic math skills. The science, technology, engineering, and math (STEM) initiative has placed greater emphasis on math in the K–12 settings and has challenged schools to broaden the participation of subgroup populations in STEM education and employment. Overall, the United States is showing growth in the area of mathematics; however, large achievement gaps continue to exist for students with disabilities and other subgroup populations. The achievement gap in math is among the largest on the algebra subtest, according to the National Assessment of Educational Progress. In 2013, there was a 46-point difference in the average score scale for the subtest between 8th grade students with and without disabilities and a 40-point difference between 12th grade students in these respective groups. Items on the algebra subtests for 8th grade students ranged from identifying graphs of exponential growth to solving algebraic inequalities. The 12th grade subtest included items requiring students to analyze conjunction and disjunction inequalities, use algebraic expressions to model scenarios, and analyze properties of logarithmic function. The significant gaps in math performance on these subtests and overall math composition scores warrant the need for research on effective practices for supporting students with learning disabilities (LD) in areas of math that also align with the CCSS. Particular
emphasis needs to be placed on the skills, content, and instructional practices that create strong math education for students with LD, including components that support the learning of a range of algebra-based concepts. The CCSS include standards on operations and algebraic thinking as early as kindergarten. In addition, the completion of algebra I is now mandatory in most school districts for students to receive a diploma (Witzel, 2005). As a result, sixth and seventh grade students are now learning prealgebra and many eighth graders are already enrolled in Algebra I. The completion of algebra is critical for many reasons, one being that it serves as a gatekeeper to a postsecondary education. In addition, the STEM report to Congress (Gonzalez & Kuenzi, 2012) indicates that jobs, including vocations that do not require 2- and 4-year degrees, are hiring those with a strong background in mathematics. To prepare our students with LD for postsecondary options and the workforce, it is critical that they are competent with basic-level algebra.
The Influence of the Common Core State Standards Educating teachers to interpret and implement the standards using evidence-based practices may help to increase the rigor and quality of math instruction provided to students 1
Miami University, Oxford, OH, USA
Corresponding Author: Sarah J. Watt, Miami University, 201 McGuffey Hall, Oxford, OH 45056, USA. Email: [email protected]
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with LD (McLaughlin & Overturf, 2012). Frequently, elementary students with LD spend much of their time focusing on fact fluency and execution of processes versus building conceptual understanding of the relationships between math concepts. The CCSS in math are organized in domains, clusters, and standards and are created to provide coherence among the conceptual understandings, problemsolving skills, and both computational and procedural knowledge of students (Kendall, 2011). In addition, the CCSS in math include critical area learning progressions that organize the content around the big ideas in each domain based on a systematic way students learn. For example, in eighth grade the big ideas focus on algebraic concepts and progress from students solving linear equations to understanding the concept of a function to application of the Pythagorean theorem. These learning progressions can help special educators identify the key areas in which students with LD need to understand before moving on to more difficult content. Aligning individualized education program (IEP) goals to these standards can help ensure we are using the CCSS to support students with LD in the general education math curriculum (McLaughlin & Overturf, 2012).
Evidence-Based Practices in Algebra The call for teachers to provide effective instruction for all learners, including those with LD, has encouraged educational leaders to research effective practices that support student’s learning of algebra. The last review of the literature on effective practices for teaching algebra to students with LD was conducted by Maccini, McNaughton, and Ruhl in 1999. This review identified six published studies and one dissertation, and the findings described interventions with large effects on the algebra achievement of secondary and postsecondary students with LD. The instructional content among these six studies included solving algebra-based word problems, combining like terms, and solving for one unknown variable. The studies examined the use of self-monitoring (Huntington, 1994; Hutchinson, 1989, 1993), the concrete-representational-abstract sequence (CRA; Huntington, 1994), diagramming and translating information into numerical equations (Zawaiza & Gerber, 1993), identifying relational statements (Hutchinson & Hemingway, 1987), computer-assisted instruction (CAI; Kitz & Thorpe, 1995), and the use of team teaching and common planning times (Rosman, 1994) on the overall student achievement in algebra. A subsequent review by Maccini, Mulcahy, and Wilson (2007) examined effective interventions for teaching content to secondary students with LD but was not limited to algebra. This review identified a total of 23 studies. Of the studies, seven included algebra content (Bottge, 1999; Bottge, Heinrichs, Chan, & Serlin, 2001; Bottge, Rueda, Serlin, Hung, & Kwon, 2007; Maccini & Hughes, 2000;
Maccini & Ruhl, 2000; Witzel, 2005; Witzel, Mercer, & Miller, 2003) and one study examined prealgebra skills (Shimabukuro, Prater, Jenkins, & Edelen-Smith, 1999). The studies that examined algebra or prealgebra content found the use of the CRA sequence, self-monitoring, and enhanced anchored instruction (EAI) to be highly effective interventions for teaching algebra-based concepts to students with LD. Shortly after this review, the National Mathematics Advisory Panel (2008) was commissioned to make instructional recommendations informed by high-quality research on math education. In preparation for the panel’s recommendations and final report, Gersten et al. (2009) produced a meta-analysis synthesizing the existing quantitative intervention research for students with LD in math. This technical report found five essential components for effective math curriculum and instruction: (a) explicit instruction, (b) the use of heuristics, (c) student verbalizations of their mathematical reasoning, (d) the use of visual representations to solve problems, and (e) sequencing or providing a range of examples (i.e., easy to hard, concrete to abstract). In addition to these five components, formative assessment, adequate feedback, and peer learning were also found to be highly effective practices within math education. The strong research base for these five effective math components suggests that when included in instruction for students with LD, overall math achievement will improve significantly. However, the research is insufficient on the application of these components to algebra instruction due to the limited number of studies including instruction of algebra-based concepts. Given the critical nature and expansive range of algebra competencies among learners, it is important that we continue to research and examine effective practices for teaching algebra and understand how these interventions relate to the complexity of the critical area learning progressions (Common Core State Standards Initiative, 2010). The primary purpose of the current review was to identify effective interventions for teaching algebra to students with LD and extend the findings of the previous reviews conducted in these areas (Maccini et al., 1999; Maccini et al., 2007). A secondary purpose of this review was to investigate the complexity of algebra concepts being taught to students with LD and to determine effective interventions within the CCSS domains.
Method The studies that were included in this review met specific criteria for selection. Studies had to (a) include students identified as having an LD, (b) contain algebra content described under the Operations and Algebraic Thinking (Grades K–5), Expressions and Equations (Grades 6–8), or Algebra (Grades 9–12) domains, (c) examine effective instructional interventions on student achievement of
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Watt et al. students with LD in the identified CCSS domains, (d) use experimental, quasi-experimental, or single-subject designs, and (e) have been published from 1980 to 2014. A comprehensive search of literature was conducted. First, a computer database search of Academic Search Complete, PsycINFO, and EBSCOhost from 1980 to 2014 was completed. For the purpose of finding studies that met the criteria, the following descriptors (listed in alphabetical order) were used in numerous combinations: algebra, learning disability, learning disorder, math, prealgebra, and special education. The abstracts of these articles were reviewed before study selection to eliminate non-data-based studies. This search yielded 21 articles. Second, a manual search was conducted of journals that would most likely publish studies involving math interventions for algebra students. Therefore, the following journals were hand searched from 1980 to 2014: Journal for Research in Mathematics Education, Journal of Special Education, Learning Disabilities Research & Practice, Journal of Learning Disabilities, Remedial and Special Education, and Learning Disabilities Quarterly. This search yielded one article (Strickland & Maccini, 2013). Finally, an ancestral search of the located studies was conducted, yielding no new articles. In total, the search produced 19 articles that met our initial criteria. The researchers reviewed the articles in depth to ensure that they met the criteria. There were a total of 6 articles excluded because after closer examination they did not meet the inclusionary criteria. The search produced a total of 13 articles; 2 of the articles contained multiple studies, yielding a total of 15.
Data Analysis: Coding Procedures Once the 15 studies had been identified for inclusion in the review, they were coded for participant information (grade, age, gender, disability type, socioeconomic status, and ethnicity), design (methodology, participant assignment, and treatment fidelity), intervention characteristics (CCSS domain and standard, instructional components, core/supplemental, setting,) setting (general education classroom, resource room, or other), and dependent measures (immediate, delayed, generalization/transfer, measures of social validity). A graduate student working on the review, the primary researcher, and an outside researcher independently coded all of the studies (N = 15) for the aforementioned inclusionary criteria. An interrater agreement of 100% was established (agreements / [agreements + disagreements]) (Cohen & Swerdlik, 2005).
Computation of Effect Sizes Quantitative designs. The standardized mean differences were calculated in one of three ways: (a) using the means
and standard deviations of the control and treatment groups, (b) using sample size and p values, or (c) calculating the mean change in each group using within-group values and sample size. A total of 10 effect sizes (ESs) were created using quantitative designs. Each of the 10 ESs was calculated and scaled to Hedges’s g to account for the overestimation that occurs when calculating ES using studies with small sample sizes (Hedges, 1982). Effect magnitudes for Hedges’s g may be interpreted using Cohen’s (1988) convention of small (0.2), medium (0.5), and large (0.8), and therefore the effects in this study are compared using this magnitude scale. A follow-up test of homogeneity was conducted to determine if differences in ESs were due to sampling error alone or if further examination of study characteristics was warranted. Due to the minimal amount of studies using delayed, maintenance, or transfer measures, ESs were calculated using only the immediate dependent measure that most closely aligned to the purpose of our analysis. Studies were also analyzed by grade level (K–5, elementary; 6–8, middle school; and 9–12, high school) considering the relevance of the CCSS in relationship to grade level expectations. If studies spanned more than one of these grade categories, the study was coded based on the grade level(s) with the majority of the participants. These measures are listed in Table 1. All analyses were conducted using the Comprehensive Meta-Analysis Software Version 2 (Borensten, Hedges, Higgins, & Rothstein, 2005). Single subject designs. Based on the research of the percentage of nonoverlapping data (PND) as an acceptable metric for visually inspecting effects of single subject research over the past 25 years (Scruggs & Mastropieri, 2012), PND was calculated for all single subject studies included in this review (values in Table 2). However, due to more recent critiques of the metric and its inability to correct for positive trends in baseline, we also calculated tau-U values for each of the single subject designs included in this analysis (Parker, Vannest, Davis, & Sauber, 2011). These values were calculated using the tau-U ES calculator (www.singlecaseresearch.org.). Interrater reliability of effect size calculations. A graduate student in school psychology, with training calculating ESs, and the first author, a professor with training in metaanalytic calculations and extensive work with students with math difficulties, calculated effects using the methods listed above. They came to 100% agreement on the final effects. A researcher not included in the review and with expertise in the area of statistical analysis then calculated all ESs and came to 95% agreement. The first author and outside researcher worked together to come to 100% agreement on the final ES calculations.
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Table 1. Quantitative Study Characteristics. Study Bottge et al. (2001) Bottge et al. (2007, Study a) Bottge et al. (2007, Study b) Calhoon and Fuchs (2003) Fuchs et al. (2009) Ives (2007, Study a) Ives (2007, Study b) Witzel (2005) Witzel, Mercer, and Miller (2003) Xin et al. (2011)
Grade
Dependent measure
8 6–12 6–12 9–12 3 7–12 7–12 6–7 6–7 3–5
RG problem-solving test RG problem-solving test RG problem-solving test MCAT–Revised Find X algebra test RG problem solution test RG problem solution test RG equations test RG equations test Solving equation probe
g
SE
95% CI
p
0.351 1.020 0.934 0.581 0.373 0.099 0.287 0.431 0.850 0.831
0.256 0.223 0.247 0.211 0.212 0.356 0.829 0.133 0.251 0.390
−0.150, 0.853 0.583, 1.457 0.451, 1.418 0.167, 0.995 −0.043, 0.626 −0.599, 0.798 −0.288, 1.428 0.170, 0.692 0.359, 1.341 0.066, 1.596
.169 .000 .000 .006 .079 .780 .193 .001 .001 .033
Note. MCAT–Revised = Math Concepts and Applications Test–Revised; RG = researcher generated.
Table 2. Single Subject Study Characteristics. Study
Grade
Dependent measure
PND
Tau-U (Variance)
Tau-U p
Tau-U 95% CI
Strickland and Maccini (2013) Maccini and Hughes (2000) Maccini and Ruhl (2000) Scheuermann, Deschler, and Schumaker (2009) Xin, Wiles, and Lin (2008)
8–9
Domain and lesson probe Skill maintenance Skill transfer Strategy use Problem representation Problem representation Problem solution Instructed problems Uninstructed problems Solving equation probe Addition and subtraction Multiplication and division
100 100 100 100 100 100 100 100 67.4
1.00 (.31) 1.00 (.45) 1.00 (.45) 1.00 (.25) 1.00 (.31) 1.00 (.29) 1.00 (.29) 1.00 (.13) 0.82 (.13)
.002 .03 .03 .001 .001 .001 .001 .000 .000
100 100
1.00 (.26) 1.00 (.29)
.000 .000
0.3840, 1.6160 0.1154, 1.8846 0.1154, 1.8846 0.3999, 1.6001 0.3999, 1.6001 0.4238, 1.5762 0.4238, 1.5762 0.8179, 1.1821 0.6352, 1.0006 0.4971, 1.5029 0.4303, 1.5697
9–12 8 6–8 4–5
Results Participants Across all the studies, there were a total of 827 participants: 398 males (53%) and 359 (47%) females. One study did not report gender, and one study had missing data for two participants (Witzel et al., 2003). In all, 12 studies reported age, ranging from 8 to 19. The grades ranged from 3rd to 12th. There were 142 (17%) students in 3rd grade, 17 (2%) students in 4th grade, 8 (1%) students in 5th grade, 151 (18%) students in 6th grade, 285 (34%) in 7th grade, 91 (11%) in 8th grade, 75 (9%) in 9th grade, 24 (3%) in 10th grade, 26 (3%) in 11th grade, and 8 (1%) in 12th grade. Only 8 studies reported IQ scores, which ranged from 79 to 143. Ethnicity was reported in 10 studies. Of the 322 (39%) students for whom ethnicity was reported, 141 (44%) were African American, 133 (41%) were Caucasian, 42 (13%) were Hispanic, 5 (2%) were Asian, and 1 (0.3%) was multiracial. The disability status was reported in all 15 studies, with some students classified with multiple disabilities.
There were 224 students reported with LD and 10 students with behavioral/emotional disorders. There were 100 students classified with other disabilities with the article not specifying a disability category.
Overall Findings The primary purpose of this review was to identify effective interventions for teaching algebra to students with LD. Following the systematic search process, 15 studies were reviewed, coded, and analyzed for the purpose of examining the effects of these interventions. Of the 15 studies, 5 implemented single subject methods (PND range of 67.5 to 100; tau-U range of 0.82 to 1.00) and 10 used quasiexperimental or experimental procedures producing an overall ES g = 0.48 (CI = 0.329, 0.626). See Table 1 for individual study characteristics. Given the large amount of single subject designs, and no significant differences in the variance of the quantitative designs (Q = 15.93, p = .06), studies were compared based
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Watt et al. on individual study effects. With the exception of intervention type and grade level, other moderator variables identified in the coding phase were not compared statistically but rather were discussed descriptively in narrative form. Although limitations exist in the statistical analysis of studies at the intervention type and grade level, the ES helps us to compare interventions and provides a stronger discussion of the implications for further research and classroom implementation. Grade level was specifically analyzed as a result of the direct link between the CCSS and grade level expectations. Providing rich descriptions of the interventions, along with the quantitative index, will more effectively inform practice (Odum et al., 2005; Scruggs & Mastropieri, 2013). Among the 15 studies, we identified 5 interventions that were the primary focus of the study: (a) CRA, (b) cognitive strategy instruction, (c) EAI, (d) tutoring, and (e) graphic organizers. Tables 1 and 2 provide additional study characteristics, including grade level, ESs, confidence intervals, and dependent measures included in the analysis. As noted in the tables, only three studies, one single subject, examined algebra instruction at the elementary (3–5) level, with moderate effects on student achievement (g = 0.48, p = .011). Seven studies, three single subject, examined the effects of algebra-based interventions at the middle school (6–8) level, with moderate to large effects on student performance (g = 0.64, p = .000). In addition, there were four studies, one single subject, that examined high school interventions in algebra for students with LD, also producing moderate effects (g = 0.45, p = .012). The range of PND and tau-U values associated with grade level for single subject studies can be found in Table 2. The secondary focus of this review was to investigate the complexity of algebra content being taught to students with LD. Therefore the CCSS domain, standard, and cluster are identified within the summary of each study.
Concrete-Representational-Abstract (CRA) Instructional Sequence The most common intervention used within the reviewed studies was the CRA instructional sequence. All of the studies including the CRA instructional sequence produced high effects (g = 0.53, tau-U = 1.00) on student achievement in algebra. CRA is a gradual instructional method that moves students from the concrete (manipulatives) stage of learning to the representational stage (pictures), and finally transitions students to the use of abstract numbers and symbols. These stages are all interrelated, with the purpose of creating meaning between the use of manipulatives and the abstract stage of learning. The studies included in this review show students moving from the concrete phase of understanding algebra to the abstract phase. However, it is critical that students easily move back and forth between
these phases to develop a strong mathematical understanding of a skill (Witzel, Riccomini, & Schneider, 2008). Witzel et al. (2003) investigated the effectiveness of the CRA instructional sequence on the performance of sixth and seventh grade students with LD or identified at risk to solve inverse operations and transformations (CCSS.Math. Content.HSA-SSE.B.3). Unique to this study was that the intervention was conducted in an inclusive setting where the interventionist was the general education math teacher. A pre–post follow-up design was used to determine the effectiveness of using the CRA sequence to teach more advanced algebra concepts. Instruction in both control and treatment groups included the use of teacher modeling and guided and independent practice. Although the comparison group was taught the same content, instruction was limited to repeated abstract lessons as opposed to the use of pictorial representations and concrete objects. Both groups showed significant growth from pretest to posttest (g = 0.351), but those students in the CRA group outperformed the comparison group on follow-up tests. An error pattern analyses also indicated that students in the CRA group made less computational errors. The implications reported in the study suggest that the use of concrete objects and pictorial representations can be effective at the secondary level, in addition to benefiting elementary-age students. Furthermore, the researchers concluded that the CRA approach can easily be implemented within general education settings and that it may be appropriate for students with and without disabilities in mathematics. Witzel (2005) examined the use of the CRA instructional sequence in general education math settings. His study involved six general education math teachers and 231 middle school students, both general and special education. Of the students, 49 were identified as having an LD. Each teacher taught one of the inclusion math classes using the CRA method and the other class using traditional abstract instruction. The algebra unit they taught consisted of 19 lessons ranging from solving inverse operations to solving linear functions with unknowns on one or both sides of the equal sign (CCSS.Math.Content.6.EE.C.9). To be consistent with research on effective practices, both control and treatment group lessons consisted of (a) teacher modeling, (b) guided and independent practice, and (c) the use of visual cues (i.e., advanced organizer). Although both groups made adequate growth between pretest to posttest, students in the CRA instructional group scored higher on both immediate (g = 0.43) and follow-up tests. Scheuermann, Deschler, and Schumaker (2009) explored the effects of the CRA sequence on middle school students with LD. The use of an explicit inquiry routine (inquiry dialogue and the CRA model) was used to develop the understanding of solving one-variable equations imbedded in word problems (CCSS.Math.Content.6.EE.B.7). The most important instructional variable examined was the
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verbalization of mathematical understandings taking place among students and teachers during the instructional phase. The study was designed to examine a student’s ability to (a) abstractly solve problems over time, (b) concretely solve problems over time, (c) transfer skills to more complex equations, and (d) transfer skills learned in word problems to those found in a student’s algebra textbook. The researchers indicated that students were able to accurately solve problems over time with high accuracy (uninstructed problems, tau-U = 0.82; instructed problems, tau-U = 1.00) and that the more familiar students became with the instructional format and dialogue, the easier it was for them to generalize skills to more complex equations. The students maintained increased levels of performance for 11 weeks following the instructional intervention. Strickland and Maccini (2013) also examined the effectiveness of the CRA instructional sequence to teach algebra to students with LD. Using a multiple probe design, three high school students were taught to multiply linear expressions within area problems using the CRA sequence with a graphic organizer imbedded in the abstract phase of instruction (CCSS.Math.Content.HSA-APR.A.1). A portion at the beginning of each lesson was also spent reviewing previous learned concepts to help activate prior knowledge. The total intervention took place over three 40-min sessions. All students showed increased ability to multiply linear expressions from baseline to intervention phase (tau-U = 1.00) and were all able to maintain the skill for up to 6 weeks following intervention.
Tutoring Two studies examined the effects of tutoring using explicit instruction. One-on-one tutoring or peer assistance is a common intervention for students with LD. This tutoring is often provided by preservice teachers, special education teachers, paraprofessionals, or peers. Peer-assisted tutoring can range from cross-age tutoring to within-class dyads. The first study (Fuchs et al., 2009) used teacher-led tutoring groups, and the following study (Calhoon & Fuchs, 2003) investigated peer-assisted learning groups. The findings of these studies suggest tutoring, using explicit instruction, as an effective strategy (g = 0.40). Fuchs and colleagues (2009) investigated the efficacy of tutoring third graders with math difficulties in the area of algebraic number calculations (i.e., solving for x in a simple equation, x – y = c) to improve transfer of understanding to algebraic word problem outcomes and procedural calculations (CCSS.Math.Content.3.OA.A.4). The 133 students who qualified for participation were randomly assigned to one of three tutoring conditions (a) no treatment (control) condition, (b) word problem-solving condition, or (c) number calculations condition. Following 16 weeks (48 sessions of 20–30 min) of tutoring, students in the number
calculation condition showed enhanced computational fluency (g = 0.53), compared to the control group, and small gains in procedural calculations (g = 0.27). However, there was no difference in groups on word problem outcomes. Students in the word problem condition showed increased gains in computational fluency (g = 0.62) compared to the control condition, which transferred to large gains in knowledge of procedural calculations (g = 0.53) and overall word problem outcomes (g = 0.36). Calhoon and Fuchs (2003) examined the effects of PALS math combined with curriculum-based measurement (CBM) to teach algebra readiness skills to high school students with disabilities. Using a quasi-experimental design, 92 students in 10 classrooms across three high schools were randomly assigned to the intervention group. The intervention took place over 30 sessions of 30 min and consisted of peer learning dyads working together to complete problems involving operations, data analysis, problem solving, geometry, and algebra (range of skills from CCSS.Math. Content.4.OA to CCSS.Math.Content.6.EE). Students in the intervention groups were also introduced to a reinforcement system to help motivate dyads by earning points toward tangible items (i.e., fast-food coupons, sporting event tickets). The control condition consisted of explicit instruction through the use of a skill-based workbook without the inclusion of a reinforcement system. Students in the treatment (PALS + CBM) significantly increased in their overall math computation performance but showed no significant growth compared to the control group in their ability to apply these concepts to algebraic applications (g = 0.58). The medium effects of this intervention indicate that although PALS can be a useful math intervention for students in high school, more intensive focus on application concepts is necessary.
Cognitive Strategy Instruction Cognitive strategy instruction is the use of various tools to help students to organize and process information. Strategies often come in the form of a mnemonic or a heuristic to help students remember the steps to solve a specific problem. These strategies are meant not to be problem specific but to assist students in solving a range of problem types. Providing students with multiple cognitive strategies and guided discussion behind the differences will allow students to choose a strategy that works best for them (Woodward, 2006). The findings from the studies examining the use of cognitive strategy instruction support previous research (Gersten et al., 2009) that this is a highly effective intervention (g = 0.83, tau-U = 1.00). Maccini and Hughes (2000) continued to examine skill areas that have an extensive impact on algebra performance. Specifically, their study investigated the effects of incorporating the STAR learning strategy (Maccini, 1998), within
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Watt et al. the CRA sequence to teach students with LD to solve word problems involving integers (CCSS.Math.Content.7.NS.A.1). A multiple-probe design across six students, in Grades 9 to 12, was used to investigate individual performance over time. During each instructional phase of the CRA sequence the research would (a) model two to three problems while thinking aloud, (b) provide up to five problems with guided practice, and (c) present five problems for independent practice. All participants improved on mean percentage accuracy on immediate measures from baseline to the abstract instructional phase of CRA (problem representation, tau-U = 1.00; problem solution, tau-U = 1.00). In addition, all students showed increased ranges across integer problem types on near- and far-transfer generalization measures (Maccini & Hughes, 2000). The implications from this study indicate that students need more time to learn addition and subtraction of integers as opposed to multiplication and division. Maccini and Ruhl (2000) replicated a similar study design involving the use of the STAR strategy and the CRA sequence to teach word problems involving integers (CCSS.Math.Content.7.NS.A.1). Participants were three eighth grade students with LD. Essential components of the instructional procedures included (a) the use of teacher modeling and think aloud, (b) inclusion of students in the think-aloud process, (c) guided and independent practice, (d) corrective feedback, and (e) use of an advanced organizer. Findings from the study yielded similar results as those of Maccini and Hughes (2000), with all three students increasing their mean percentage accuracy on problem solutions from baseline to the abstract phase of instruction (tau-U = 1.00). Results indicated that students were able to master the mathematical objectives that were taught in a relatively short amount of time with the ability to generalize what they learned to more complex math problems. Xin, Wiles, and Lin (2008) assessed the functional relationship between word problem-solving interventions and the prealgebra math performance of five fourth and fifth grade students with math difficulty using an adapted multiple-probeacross-participants design. The intervention consisted of teaching word problem solving that represented symbolic algebraic expressions (CCSS.Math.Content.3.OA.D.8) using conceptual model-based problem solving (COMPS) versus an arithmetic-oriented approach. In addition students were taught a cognitive learning strategy (DOTS—Detect, Organize, Transform, Solve; Xin et al., 2008) to guide them through the problem-solving process. All five students showed increased ability to solve addition, subtraction, multiplication, and division word problems following the intervention. But more important for the purpose of this review was the increase in accuracy and overall performance for solving unknown variables in algebraic expressions following the intervention compared to baseline. The performance among students at baseline ranged from 0% to 33% accuracy, whereas following the
intervention students’ scores ranged from 67% to 100% accuracy on algebraic measures (tau-U = 1.00). Xin et al. (2011) extended the findings from Xin et al.’s (2008) study on the use of COMPS using an experimental design to clarify their findings on the effectiveness of COMPS. The current study had a total of 29 in fourth and fifth graders, of whom 16 were students with LD and 13 were identified by their schools as at-risk learners. Students were stratified by grade, gender, and pretest scores and then randomly selected for the COMPS instructional group or the control group. Within both instructional groups students were taught to solve word problems by writing equations with variables and to find the value of the unknown variable in the expression (CCSS.Math.Content.6.EE.B.6). The control group consisted of instruction using a general instructional heuristic, SOLVE (Search the question, Organize the problem, Look for a strategy, Visualize then work the problem, Evaluate your answer). Two researcher-generated measures of algebra readiness, tests of model expressions and solving equations, were used to compare groups. On both measures students in the COMPS group significantly improved (g = 0.859) compared to students in the control group.
Enhanced Anchored Instruction (EAI) EAI is an inquiry-based learning intervention that uses video anchors of real-life situations that provides students with a problem to solve or a task to complete. Each EAI lesson typically embeds the use of technology and hands-on activities to reinforce math concepts. In addition, research suggests that the combination of explicit teaching of skills and EAI has a greater impact on student achievement. The results of these studies indicate that the use of EAI is also highly effective for teaching algebra (g = 0.80). Bottge et al. (2001) examined the benefits of linking algebra instruction to video anchors with applied tasks (i.e., EAI). Four math classrooms, one remedial math and three prealgebra classes, were assigned to treatment (EAI; n = 34) or control (traditional instruction; n = 41) conditions. Each condition contained students with learning and behavioral disabilities (n = 19) and students at risk for failure in math. The instructional content for both groups focused on calculating distance, rate, and time (CCSS.Math.Content.6.EE.C.9). Students in the EAI treatment condition applied the algebraic skills to develop soapbox derby cars anchored in the problem depicted on the videodisc. The students in the comparison condition participated in a more traditional instruction approach to learning to solve word problems using the formulas to calculate distance, rate, and time. Overall results indicated that students in all instructional settings made large gains from pre- to posttest. However, students in remedial math classes using EAI made the largest gains on the problem-solving measures (g = 0.35).
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Bottge et al. (2007) continued to examine the effects of EAI on the overall algebra achievement of adolescents with LD. A total of 128 seventh grade students were participants, 13 with disabilities, 12 identified with LD. Four self-contained classrooms were randomly selected to one of two conditions, traditional algebra instruction or EAI. Students were taught a range of standards found in the 6–8 Expressions and Equations domain (e.g., interpreting data, constructing graphs and tables, recognizing relationships among data) but focused primarily on estimating and calculating distance, rate, and time (CCSS.Math.Content.6.EE.C.9). After the completion of the initial study, it was replicated with the same student population but a reversal of previously assigned conditions. Both studies found large effects on student’s problem-solving ability for participants using EAI (Study a, g = 0.93; Study b, g = 1.02). Given possible carryover effects between studies, caution should be used when interpreting the effects from study b.
Graphic Organizers Graphic organizers can be used in a variety of ways to support the retention of new and complex math problems. Graphic organizers can also be used to support vocabulary development, help students organize their work into individual steps, and help students make connections between new and previously taught content. The results of the studies using graphic organizers within this review indicate this intervention is highly effective (g = 0.57). Ives (2007) evaluated the effectiveness of using a graphic organizer to support the instruction of teaching high school students algebra. Using an experimental design, 30 students were randomly assigned to a control or intervention group. Both groups were instructed in how to solve linear equations with two unknown variables (CCSS.Math.Content. HSA-REI.C.6) through the use of explicit instruction. Although both groups used the same instructional materials, had the same amount of sessions, and received equal amount of time on each topic, the treatment group was provided with a graphic organizer to help them solve for each variable. The results of a researcher-generated probe indicated no significant differences between groups, but effects in favor of the treatment (g = 0.09). The author systematically replicated this study using 20 different high school students with LD. The results of this second study also did not reveal significant differences but continued to favor the treatment group (g = 0.19). The results of this study are difficult to interpret as a result of a small sample size, and the low effects warrant further research in this area before drawing conclusions.
Discussion The primary purpose of the current review was to identify effective interventions for teaching algebra to students with
LD and to extend the findings of the previous reviews (Maccini et al., 1999; Maccini et al., 2007). The initial review by Maccini and colleagues (1999) identified six studies examining effective algebra interventions for students with LD; this review included 15 studies not included in the previous analysis. The current review also identified 5 studies not included in the follow-up review by Maccini and colleagues (2007).
Analysis of Effective Interventions The overall analysis of the 15 studies identified five interventions for teaching algebra to students with LD: (a) CRA, (b) cognitive strategy instruction, (c) EAI, (d) tutoring, and (e) the use of graphic organizers. Although the interventions in many studies produced large effects, there was often a combination of instructional components (e.g., CRA + use of heuristic) among studies making it difficult to attribute the effects to only a single component (e.g., CRA). The overall ES for studies using EAI (g = 0.80) and cognitive strategies (g = 0.83, tau-U = 1.00) suggests that these are highly effective practices for teaching algebra content to students with LD. Studies using the CRA instructional sequence also produced large effects (g = 0.53, tau-U = 1.00). The use of this intervention was supported by seven studies within this review, all indicating increased student performance in math. In addition, the CRA approach to teaching algebra was replicated across the largest range of instructional content, from solving equations with unknown variables to multiplying linear expressions. Two of the seven studies using the CRA sequence used a cognitive strategy to support the retention and generalization of student learning. Although studies investigating the use of graphic organizers (g = 0.57) and individualized tutoring (g = 0.40) produced moderate to large effects on student learning in algebra, only two studies were included in each of the instructional categories, and more research is needed to support the interpretation of the findings. Although no statistical analysis of the instructional components found within each intervention was conducted, for the purpose of future implementation and better understanding of the educational implications from this review, the unique components have been summarized. The metaanalysis conducted by Gersten and colleagues (2009) identified several research-based practices for teaching students with LD or those who are struggling in math. Among these seven were the use of explicit instruction, multiple examples, verbalizing decisions and solutions, visually representing information, heuristic strategies, formative assessments, and peer-assisted instruction. After closely examining the studies included in this review, it was evident that each study included the use of explicit instruction. Research has consistently found explicit instruction to be linked to increased student achievement (Gersten et al., 2009). No other instructional component was found in all of
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Watt et al. the studies, but the use of visual representations within the intervention was present in 12 of the studies, formative assessment data were used to guide instruction in 8 of the included studies, and the use of heuristics or strategies was found within 6 of the studies reported in this review. It should also be noted that each study included an intervention that contained two or more of the instructional practices identified by Gersten and colleagues (2009) as effective components of instruction. Further research examining the statistical impact of each of these instructional factors is warranted.
Nature of Instructional Content A secondary focus of this review was to evaluate the complexity of the content being taught within the identified studies. Since the 1999 review (Maccini et al., 1999) the field of special education has seen a steady increase in the number of math studies investigating evidence-based practices for teaching algebra-based concepts. The initial review identified six studies that emphasized a range of skills. The review covered content including relational word problems (Huntington, 1994; Hutchinson, 1989, 1993; Zawaiza & Gerber, 1993), combing like terms (Kitz & Thorpe, 1995), and graphing inequalities (Kitz & Thorpe, 1995; Rosman, 1994). In each of these studies the math content was below the grade level of the student participants. The conclusion of the review called for additional focus on interventions addressing more complex algebraic skills. The findings of our current review reflect growth in the research on instruction of higher level math content among students with LD. Of the 15 studies reviewed, 13 instructed using standards from the intermediate and secondary domains of the CCSS. In addition to the use of higher level standards, 9 of the 15 studies taught students grade level content. Despite the growth reflected in the findings, the limited range of higher level standards taught across the studies included in the review should be noted. For example, in the sixth grade domain, Expressions and Equations, there are nine standards. The majority of the standards taught within the studies included in this review were from the sixth grade domain; however, among those studies only three standards were addressed. This finding illustrates the limited research across the algebra domains. Therefore, this impacts the knowledge base of effective practices for students with LD in math.
Educational Setting Despite the increase in complexity of content and the overall student achievement of students with LD instructed at grade level, only five of the studies in this review took place in general education settings (Bottge et al., 2001; Bottge et al., 2007 [Studies a and b]; Witzel, 2005; Witzel et al.,
2003). In all, 10 of the included studies took place during core math instruction, indicating that a large number of students with LD are receiving their content instruction in more restrictive educational settings. Based on this information and our earlier indication that nine studies were teaching grade level content, many students with LD are receiving grade level instruction outside of the general education classroom. Although this may suggest that students have other individual needs that impact their participation in math in the general education setting, it may also suggest that continued work on supporting inclusion for all learners in math is warranted. Maccini et al. (1999) identified one study that took place in general education compared to the five included in the current review. Although an increase in inclusive placements is evident, it is critical that greater emphasis be placed on the evaluation of instructional practices that support students with LD in general education math settings.
Limitations There are three main limitations to the conclusions of this review. First, given the large amount of single subject studies included in this review, quantitative analysis was limited. In addition, it is difficult to compare the magnitude of effects between the single subject and quantitative designs. Second, the majority of the measures were researcher generated, and reliability alphas were reported for only three studies. As we continue to prepare students with LD to participate in the general education curriculum, the use of standardized measures to assess their performance in math is necessary. Finally, many studies contained general education students and students with other identified disabilities, making it not possible to calculate effects for only students with LD. As a result of these limitations, findings should be interpreted with some caution.
Overall Implications As we move forward, research on best practices for students with disabilities in math needs to continue to place emphasis on early intervention of prealgebra skills and higher level content standards for students at the intermediate and secondary levels. A change in how we deliver core instruction needs to occur for all students to have access to grade level math content and the skills necessary to complete higher level math courses. The use of formative assessment data to match instruction to students’ individual needs and the increased use of standardized measures to analyze student growth are critical. This review also indicates the importance of the use of explicit instruction in combination with other effective instructional components.
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Research in this area needs to provide strong descriptions of students and intervention characteristics. Replications of studies indicating highly effective practices with diverse student populations need to be conducted across multiple settings. Research examining a larger range of the CCSS standards and measuring student achievement using reliable and valid measures is critical to the development and interpretation of effective practices for students with LD. Finally, and most important, research findings need to be shared with practitioners in the field through quality professional development. Declaration of Conflicting Interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding The author(s) received no financial support for the research, authorship, and/or publication of this article.
References *References marked with an asterisk indicate studies included in the analysis. Borenstein, M., Hedges, L., Higgins, J., & Rothstein, H. (2005). Comprehensive meta-analysis version 2. Englewood, NJ: Biostat. Bottge, B. A. (1999). Effects of contextualized math instruction on problem solving of average and below-average achieving students. Journal of Special Education, 33, 81–92. doi:10.1177/002245599903300202 *Bottge, B. A., Heinrichs, M., Chan, S. Y., & Serlin, R. C. (2001). Anchoring adolescents’ understanding of math concepts in rich problem-solving environments. Remedial and Special Education, 22, 299–314. doi:10.1177/074193250102200505 *Bottge, B. A., Rueda, E., Serlin, R. C., Hung, Y. H., & Kwon, J. M. (2007). Shrinking achievement differences with anchored math problems challenges and possibilities. Journal of Special Education, 41, 31–49. *Calhoon, M. B., & Fuchs, L. S. (2003). The effects of peerassisted learning strategies and curriculum-based measurement on the mathematics performance of secondary students with disabilities. Remedial and Special Education, 24, 235– 245. doi:10.1177/07419325030240040601 Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum. Cohen, R. J., & Swerdlik, M. E. (2005). Psychological testing and measurement: An introduction to tests and measurement. Boston, MA: McGraw-Hill. Common Core State Standards Initiative. (2010). Common core state standards for mathematics. Retrieved from http://www. corestandards.org/ *Fuchs, L. S., Powell, S. R., Seethaler, P. M., Cirino, P. T., Fletcher, J. M., Fuchs, D., & Zumeta, R. O. (2009). Remediating number combination and word problem deficits among students with mathematics difficulties: A randomized
control trial. Journal of Educational Psychology, 101, 561– 576. doi:10.1037/a0014701 Gersten, R., Chard, D. J., Jayanthi, M., Baker, S. K., Morphy, P., & Flojo, J. (2009). Mathematics instruction for students with learning disabilities: A meta-analysis of instructional components. Review of Educational Research, 79, 1202–1242. Gonzalez, H. B., & Kuenzi, J. J. (2012). Science, technology, engineering, and mathematics (STEM) education: A primer. Washington, DC: Library of Congress, Congressional Research Service. Hedges, L. V. (1982). Fitting continuous models to effect size data. Journal of Educational and Behavioral Statistics, 7, 245–270. Huntington, D. J. (1994). Instruction in concrete, semi-concrete, and abstract learning disabilities (Unpublished doctoral dissertation). University of Georgia, Athens. Hutchinson, N. L. (1989). Strategies for teaching learning disabled adolescents algebraic problems. Reading, Writing, and Learning Disabilities, 24, 292–303. Hutchinson, N. L. (1993). Effects of cognitive strategy instruction on algebra problem solving of adolescents with learning disabilities. Learning Disability Quarterly, 16, 34–63. doi:10.2307/1511158 Hutchinson, N., & Hemingway, P. (1987). Teaching representation and solution of algebraic word problems to learning disabled adolescents. Retrieved from ERIC. *Ives, B. (2007). Graphic organizers applied to secondary algebra instruction for students with learning disabilities. Learning Disabilities Research & Practice, 22, 110–118. Kendall, J. S. (2011). Understanding common core state standards. Alexandria, VA: ASCD. Kitz, W. R., & Thorpe, H. W. (1995). A comparison of the effectiveness of videodisc and traditional algebra instruction for collegeage students with learning disabilities. Remedial and Special Education, 16, 295–306. doi:10.1177/074193259501600506 Maccini, P. (1998). Effects of an instructional strategy incorporating concrete problem representation on the introductory algebra performance of secondary students with learning disabilities (Unpublished doctoral dissertation). Pennsylvania State University, University Park. *Maccini, P., & Hughes, C. A. (2000). Effects of a problemsolving strategy on the introductory algebra performance of secondary students with learning disabilities. Learning Disabilities Research & Practice, 15, 10–21. doi:10.1207/ SLDRP1501_2 Maccini, P., McNaughton, D., & Ruhl, K. L. (1999). Algebra instruction for students with learning disabilities: Implications from a research review. Learning Disability Quarterly, 22, 113–126. doi:10.2307/1511270 Maccini, P., Mulcahy, C. A., & Wilson, M. G. (2007). A follow-up of mathematics interventions for secondary students with learning disabilities. Learning Disabilities Research & Practice, 22, 58–74. *Maccini, P., & Ruhl, K. L. (2000). Effects of a graduated instructional sequence on the algebraic subtraction of integers by secondary students with learning disabilities. Education and Treatment of Children, 23, 465–489. McLaughlin, M., & Overturf, B. J. (2012). The common core: Insights into the K–5 standards. Reading Teacher, 66, 153–164.
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Watt et al. National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education. Odom, S. L., Brantlinger, E., Gersten, R., Horner, R. H., Thompson, B., & Harris, K. R. (2005). Research in special education: Scientific methods and evidence-based practices. Exceptional Children, 71, 137–148. Parker, R. I., Vannest, K. J., Davis, J. L., & Sauber, S. B. (2011). Combining nonoverlap and trend for single case research: Tau-U. Behavior Therapy, 42, 287–299. Rosman, N. J. S. (1994). Effects of varying the special educator’s role within an algebra class on math attitude and achievement. Retrieved from ERIC. *Scheuermann, A. M., Deschler, D. D., & Schumaker, J. B. (2009). The effects of the explicit inquiry routine on the performance of students with learning disabilities on one-variable equations. Learning Disability Quarterly, 32, 103–120. Scruggs, T. E., & Mastropieri, M. A. (2013). PND at 25: Past, present, and future trends in summarizing single-subject research. Remedial and Special Education, 34, 9–19. Shimabukuro, S., Prater, M., Jenkins, A., & Edelen-Smith, P. (1999). The effects of self-monitoring of academic performance on students with learning disabilities. Education and Treatment of Children, 22, 397–414. *Strickland, T. K., & Maccini, P. (2013). The effects of the concrete-representational-abstract integration strategy on the ability of students with learning disabilities to multiply linear expressions within area problems. Remedial and Special Education, 34, 142–153. doi:10.1177/0741932512441712
*Witzel, B. S. (2005). Using CRA to teach algebra to students with math difficulties in inclusive settings. Learning Disabilities—A Contemporary Journal, 3, 49–60. *Witzel, B. S., Mercer, C. D., & Miller, M. D. (2003). Teaching algebra to students with learning difficulties: An investigation of an explicit instruction model. Learning Disabilities Research & Practice, 18, 121–131. doi:10.1111/15405826.00068 Witzel, B. S., Riccomini, P. J., & Schneider, E. (2008). Implementing CRA with secondary students with learning disabilities in mathematics. Intervention in School and Clinic, 43, 270–276. Woodward, J. (2006). Developing automaticity in multiplication facts: Integrating strategy instruction with timed practice drills. Learning Disability Quarterly, 29, 269–289. *Xin, Y. P., Wiles, B., & Lin, Y. Y. (2008). Teaching conceptual model-based word problem story grammar to enhance mathematics problem solving. Journal of Special Education, 42, 163–178. *Xin, Y., Zhang, D., Park, J., Tom, K., Whipple, A., & Si, L. (2011). A comparison of two mathematics problemsolving strategies: Facilitate algebra-readiness. Journal of Educational Research, 104, 381–395. doi:10.1080/0022067 1.2010.487080 Zawaiza, T. R. W., & Gerber, M. M. (1993). Effects of explicit instruction on math word-problem solving by community college students with learning disabilities. Learning Disability Quarterly, 16, 64–79. doi:10.2307/1511159
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research-article2014
LDXXXX10.1177/0022219414564220Journal of Learning DisabilitiesWatt et al.
Article
Teaching Algebra to Students With Learning Disabilities: Where Have We Come and Where Should We Go?
Journal of Learning Disabilities 1–11 © Hammill Institute on Disabilities 2014 Reprints and permissions: sagepub.com/journalsPermissions.nav DOI: 10.1177/0022219414564220 journaloflearningdisabilities.sagepub.com
Sarah J. Watt, PhD1, Jessie R. Watkins, MA1, and Jason Abbitt, PhD1
Abstract This review investigates effective interventions for teaching algebra to students with learning disabilities and evaluates the complexity and alignment of skills with the Common Core State Standards in math. The review includes the results of 10 experimental and 5 single-subject designs (N = 15) producing a moderate overall effect size (g = 0.48). A total of five interventions were identified and analyzed across the studies using effect size data. Keywords strategies, instruction, mathematics The federal STEM education effort combined with the adoption of the Common Core State Standards (CCSS; Common Core State Standards Initiative, 2010) has changed the expectations for all students in mathematics. Although the CCSS in math includes fewer standards than states previously held, the content is more rigorous and requires students to have a greater conceptual understanding of basic math skills. The science, technology, engineering, and math (STEM) initiative has placed greater emphasis on math in the K–12 settings and has challenged schools to broaden the participation of subgroup populations in STEM education and employment. Overall, the United States is showing growth in the area of mathematics; however, large achievement gaps continue to exist for students with disabilities and other subgroup populations. The achievement gap in math is among the largest on the algebra subtest, according to the National Assessment of Educational Progress. In 2013, there was a 46-point difference in the average score scale for the subtest between 8th grade students with and without disabilities and a 40-point difference between 12th grade students in these respective groups. Items on the algebra subtests for 8th grade students ranged from identifying graphs of exponential growth to solving algebraic inequalities. The 12th grade subtest included items requiring students to analyze conjunction and disjunction inequalities, use algebraic expressions to model scenarios, and analyze properties of logarithmic function. The significant gaps in math performance on these subtests and overall math composition scores warrant the need for research on effective practices for supporting students with learning disabilities (LD) in areas of math that also align with the CCSS. Particular
emphasis needs to be placed on the skills, content, and instructional practices that create strong math education for students with LD, including components that support the learning of a range of algebra-based concepts. The CCSS include standards on operations and algebraic thinking as early as kindergarten. In addition, the completion of algebra I is now mandatory in most school districts for students to receive a diploma (Witzel, 2005). As a result, sixth and seventh grade students are now learning prealgebra and many eighth graders are already enrolled in Algebra I. The completion of algebra is critical for many reasons, one being that it serves as a gatekeeper to a postsecondary education. In addition, the STEM report to Congress (Gonzalez & Kuenzi, 2012) indicates that jobs, including vocations that do not require 2- and 4-year degrees, are hiring those with a strong background in mathematics. To prepare our students with LD for postsecondary options and the workforce, it is critical that they are competent with basic-level algebra.
The Influence of the Common Core State Standards Educating teachers to interpret and implement the standards using evidence-based practices may help to increase the rigor and quality of math instruction provided to students 1
Miami University, Oxford, OH, USA
Corresponding Author: Sarah J. Watt, Miami University, 201 McGuffey Hall, Oxford, OH 45056, USA. Email: [email protected]
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with LD (McLaughlin & Overturf, 2012). Frequently, elementary students with LD spend much of their time focusing on fact fluency and execution of processes versus building conceptual understanding of the relationships between math concepts. The CCSS in math are organized in domains, clusters, and standards and are created to provide coherence among the conceptual understandings, problemsolving skills, and both computational and procedural knowledge of students (Kendall, 2011). In addition, the CCSS in math include critical area learning progressions that organize the content around the big ideas in each domain based on a systematic way students learn. For example, in eighth grade the big ideas focus on algebraic concepts and progress from students solving linear equations to understanding the concept of a function to application of the Pythagorean theorem. These learning progressions can help special educators identify the key areas in which students with LD need to understand before moving on to more difficult content. Aligning individualized education program (IEP) goals to these standards can help ensure we are using the CCSS to support students with LD in the general education math curriculum (McLaughlin & Overturf, 2012).
Evidence-Based Practices in Algebra The call for teachers to provide effective instruction for all learners, including those with LD, has encouraged educational leaders to research effective practices that support student’s learning of algebra. The last review of the literature on effective practices for teaching algebra to students with LD was conducted by Maccini, McNaughton, and Ruhl in 1999. This review identified six published studies and one dissertation, and the findings described interventions with large effects on the algebra achievement of secondary and postsecondary students with LD. The instructional content among these six studies included solving algebra-based word problems, combining like terms, and solving for one unknown variable. The studies examined the use of self-monitoring (Huntington, 1994; Hutchinson, 1989, 1993), the concrete-representational-abstract sequence (CRA; Huntington, 1994), diagramming and translating information into numerical equations (Zawaiza & Gerber, 1993), identifying relational statements (Hutchinson & Hemingway, 1987), computer-assisted instruction (CAI; Kitz & Thorpe, 1995), and the use of team teaching and common planning times (Rosman, 1994) on the overall student achievement in algebra. A subsequent review by Maccini, Mulcahy, and Wilson (2007) examined effective interventions for teaching content to secondary students with LD but was not limited to algebra. This review identified a total of 23 studies. Of the studies, seven included algebra content (Bottge, 1999; Bottge, Heinrichs, Chan, & Serlin, 2001; Bottge, Rueda, Serlin, Hung, & Kwon, 2007; Maccini & Hughes, 2000;
Maccini & Ruhl, 2000; Witzel, 2005; Witzel, Mercer, & Miller, 2003) and one study examined prealgebra skills (Shimabukuro, Prater, Jenkins, & Edelen-Smith, 1999). The studies that examined algebra or prealgebra content found the use of the CRA sequence, self-monitoring, and enhanced anchored instruction (EAI) to be highly effective interventions for teaching algebra-based concepts to students with LD. Shortly after this review, the National Mathematics Advisory Panel (2008) was commissioned to make instructional recommendations informed by high-quality research on math education. In preparation for the panel’s recommendations and final report, Gersten et al. (2009) produced a meta-analysis synthesizing the existing quantitative intervention research for students with LD in math. This technical report found five essential components for effective math curriculum and instruction: (a) explicit instruction, (b) the use of heuristics, (c) student verbalizations of their mathematical reasoning, (d) the use of visual representations to solve problems, and (e) sequencing or providing a range of examples (i.e., easy to hard, concrete to abstract). In addition to these five components, formative assessment, adequate feedback, and peer learning were also found to be highly effective practices within math education. The strong research base for these five effective math components suggests that when included in instruction for students with LD, overall math achievement will improve significantly. However, the research is insufficient on the application of these components to algebra instruction due to the limited number of studies including instruction of algebra-based concepts. Given the critical nature and expansive range of algebra competencies among learners, it is important that we continue to research and examine effective practices for teaching algebra and understand how these interventions relate to the complexity of the critical area learning progressions (Common Core State Standards Initiative, 2010). The primary purpose of the current review was to identify effective interventions for teaching algebra to students with LD and extend the findings of the previous reviews conducted in these areas (Maccini et al., 1999; Maccini et al., 2007). A secondary purpose of this review was to investigate the complexity of algebra concepts being taught to students with LD and to determine effective interventions within the CCSS domains.
Method The studies that were included in this review met specific criteria for selection. Studies had to (a) include students identified as having an LD, (b) contain algebra content described under the Operations and Algebraic Thinking (Grades K–5), Expressions and Equations (Grades 6–8), or Algebra (Grades 9–12) domains, (c) examine effective instructional interventions on student achievement of
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Watt et al. students with LD in the identified CCSS domains, (d) use experimental, quasi-experimental, or single-subject designs, and (e) have been published from 1980 to 2014. A comprehensive search of literature was conducted. First, a computer database search of Academic Search Complete, PsycINFO, and EBSCOhost from 1980 to 2014 was completed. For the purpose of finding studies that met the criteria, the following descriptors (listed in alphabetical order) were used in numerous combinations: algebra, learning disability, learning disorder, math, prealgebra, and special education. The abstracts of these articles were reviewed before study selection to eliminate non-data-based studies. This search yielded 21 articles. Second, a manual search was conducted of journals that would most likely publish studies involving math interventions for algebra students. Therefore, the following journals were hand searched from 1980 to 2014: Journal for Research in Mathematics Education, Journal of Special Education, Learning Disabilities Research & Practice, Journal of Learning Disabilities, Remedial and Special Education, and Learning Disabilities Quarterly. This search yielded one article (Strickland & Maccini, 2013). Finally, an ancestral search of the located studies was conducted, yielding no new articles. In total, the search produced 19 articles that met our initial criteria. The researchers reviewed the articles in depth to ensure that they met the criteria. There were a total of 6 articles excluded because after closer examination they did not meet the inclusionary criteria. The search produced a total of 13 articles; 2 of the articles contained multiple studies, yielding a total of 15.
Data Analysis: Coding Procedures Once the 15 studies had been identified for inclusion in the review, they were coded for participant information (grade, age, gender, disability type, socioeconomic status, and ethnicity), design (methodology, participant assignment, and treatment fidelity), intervention characteristics (CCSS domain and standard, instructional components, core/supplemental, setting,) setting (general education classroom, resource room, or other), and dependent measures (immediate, delayed, generalization/transfer, measures of social validity). A graduate student working on the review, the primary researcher, and an outside researcher independently coded all of the studies (N = 15) for the aforementioned inclusionary criteria. An interrater agreement of 100% was established (agreements / [agreements + disagreements]) (Cohen & Swerdlik, 2005).
Computation of Effect Sizes Quantitative designs. The standardized mean differences were calculated in one of three ways: (a) using the means
and standard deviations of the control and treatment groups, (b) using sample size and p values, or (c) calculating the mean change in each group using within-group values and sample size. A total of 10 effect sizes (ESs) were created using quantitative designs. Each of the 10 ESs was calculated and scaled to Hedges’s g to account for the overestimation that occurs when calculating ES using studies with small sample sizes (Hedges, 1982). Effect magnitudes for Hedges’s g may be interpreted using Cohen’s (1988) convention of small (0.2), medium (0.5), and large (0.8), and therefore the effects in this study are compared using this magnitude scale. A follow-up test of homogeneity was conducted to determine if differences in ESs were due to sampling error alone or if further examination of study characteristics was warranted. Due to the minimal amount of studies using delayed, maintenance, or transfer measures, ESs were calculated using only the immediate dependent measure that most closely aligned to the purpose of our analysis. Studies were also analyzed by grade level (K–5, elementary; 6–8, middle school; and 9–12, high school) considering the relevance of the CCSS in relationship to grade level expectations. If studies spanned more than one of these grade categories, the study was coded based on the grade level(s) with the majority of the participants. These measures are listed in Table 1. All analyses were conducted using the Comprehensive Meta-Analysis Software Version 2 (Borensten, Hedges, Higgins, & Rothstein, 2005). Single subject designs. Based on the research of the percentage of nonoverlapping data (PND) as an acceptable metric for visually inspecting effects of single subject research over the past 25 years (Scruggs & Mastropieri, 2012), PND was calculated for all single subject studies included in this review (values in Table 2). However, due to more recent critiques of the metric and its inability to correct for positive trends in baseline, we also calculated tau-U values for each of the single subject designs included in this analysis (Parker, Vannest, Davis, & Sauber, 2011). These values were calculated using the tau-U ES calculator (www.singlecaseresearch.org.). Interrater reliability of effect size calculations. A graduate student in school psychology, with training calculating ESs, and the first author, a professor with training in metaanalytic calculations and extensive work with students with math difficulties, calculated effects using the methods listed above. They came to 100% agreement on the final effects. A researcher not included in the review and with expertise in the area of statistical analysis then calculated all ESs and came to 95% agreement. The first author and outside researcher worked together to come to 100% agreement on the final ES calculations.
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Table 1. Quantitative Study Characteristics. Study Bottge et al. (2001) Bottge et al. (2007, Study a) Bottge et al. (2007, Study b) Calhoon and Fuchs (2003) Fuchs et al. (2009) Ives (2007, Study a) Ives (2007, Study b) Witzel (2005) Witzel, Mercer, and Miller (2003) Xin et al. (2011)
Grade
Dependent measure
8 6–12 6–12 9–12 3 7–12 7–12 6–7 6–7 3–5
RG problem-solving test RG problem-solving test RG problem-solving test MCAT–Revised Find X algebra test RG problem solution test RG problem solution test RG equations test RG equations test Solving equation probe
g
SE
95% CI
p
0.351 1.020 0.934 0.581 0.373 0.099 0.287 0.431 0.850 0.831
0.256 0.223 0.247 0.211 0.212 0.356 0.829 0.133 0.251 0.390
−0.150, 0.853 0.583, 1.457 0.451, 1.418 0.167, 0.995 −0.043, 0.626 −0.599, 0.798 −0.288, 1.428 0.170, 0.692 0.359, 1.341 0.066, 1.596
.169 .000 .000 .006 .079 .780 .193 .001 .001 .033
Note. MCAT–Revised = Math Concepts and Applications Test–Revised; RG = researcher generated.
Table 2. Single Subject Study Characteristics. Study
Grade
Dependent measure
PND
Tau-U (Variance)
Tau-U p
Tau-U 95% CI
Strickland and Maccini (2013) Maccini and Hughes (2000) Maccini and Ruhl (2000) Scheuermann, Deschler, and Schumaker (2009) Xin, Wiles, and Lin (2008)
8–9
Domain and lesson probe Skill maintenance Skill transfer Strategy use Problem representation Problem representation Problem solution Instructed problems Uninstructed problems Solving equation probe Addition and subtraction Multiplication and division
100 100 100 100 100 100 100 100 67.4
1.00 (.31) 1.00 (.45) 1.00 (.45) 1.00 (.25) 1.00 (.31) 1.00 (.29) 1.00 (.29) 1.00 (.13) 0.82 (.13)
.002 .03 .03 .001 .001 .001 .001 .000 .000
100 100
1.00 (.26) 1.00 (.29)
.000 .000
0.3840, 1.6160 0.1154, 1.8846 0.1154, 1.8846 0.3999, 1.6001 0.3999, 1.6001 0.4238, 1.5762 0.4238, 1.5762 0.8179, 1.1821 0.6352, 1.0006 0.4971, 1.5029 0.4303, 1.5697
9–12 8 6–8 4–5
Results Participants Across all the studies, there were a total of 827 participants: 398 males (53%) and 359 (47%) females. One study did not report gender, and one study had missing data for two participants (Witzel et al., 2003). In all, 12 studies reported age, ranging from 8 to 19. The grades ranged from 3rd to 12th. There were 142 (17%) students in 3rd grade, 17 (2%) students in 4th grade, 8 (1%) students in 5th grade, 151 (18%) students in 6th grade, 285 (34%) in 7th grade, 91 (11%) in 8th grade, 75 (9%) in 9th grade, 24 (3%) in 10th grade, 26 (3%) in 11th grade, and 8 (1%) in 12th grade. Only 8 studies reported IQ scores, which ranged from 79 to 143. Ethnicity was reported in 10 studies. Of the 322 (39%) students for whom ethnicity was reported, 141 (44%) were African American, 133 (41%) were Caucasian, 42 (13%) were Hispanic, 5 (2%) were Asian, and 1 (0.3%) was multiracial. The disability status was reported in all 15 studies, with some students classified with multiple disabilities.
There were 224 students reported with LD and 10 students with behavioral/emotional disorders. There were 100 students classified with other disabilities with the article not specifying a disability category.
Overall Findings The primary purpose of this review was to identify effective interventions for teaching algebra to students with LD. Following the systematic search process, 15 studies were reviewed, coded, and analyzed for the purpose of examining the effects of these interventions. Of the 15 studies, 5 implemented single subject methods (PND range of 67.5 to 100; tau-U range of 0.82 to 1.00) and 10 used quasiexperimental or experimental procedures producing an overall ES g = 0.48 (CI = 0.329, 0.626). See Table 1 for individual study characteristics. Given the large amount of single subject designs, and no significant differences in the variance of the quantitative designs (Q = 15.93, p = .06), studies were compared based
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Watt et al. on individual study effects. With the exception of intervention type and grade level, other moderator variables identified in the coding phase were not compared statistically but rather were discussed descriptively in narrative form. Although limitations exist in the statistical analysis of studies at the intervention type and grade level, the ES helps us to compare interventions and provides a stronger discussion of the implications for further research and classroom implementation. Grade level was specifically analyzed as a result of the direct link between the CCSS and grade level expectations. Providing rich descriptions of the interventions, along with the quantitative index, will more effectively inform practice (Odum et al., 2005; Scruggs & Mastropieri, 2013). Among the 15 studies, we identified 5 interventions that were the primary focus of the study: (a) CRA, (b) cognitive strategy instruction, (c) EAI, (d) tutoring, and (e) graphic organizers. Tables 1 and 2 provide additional study characteristics, including grade level, ESs, confidence intervals, and dependent measures included in the analysis. As noted in the tables, only three studies, one single subject, examined algebra instruction at the elementary (3–5) level, with moderate effects on student achievement (g = 0.48, p = .011). Seven studies, three single subject, examined the effects of algebra-based interventions at the middle school (6–8) level, with moderate to large effects on student performance (g = 0.64, p = .000). In addition, there were four studies, one single subject, that examined high school interventions in algebra for students with LD, also producing moderate effects (g = 0.45, p = .012). The range of PND and tau-U values associated with grade level for single subject studies can be found in Table 2. The secondary focus of this review was to investigate the complexity of algebra content being taught to students with LD. Therefore the CCSS domain, standard, and cluster are identified within the summary of each study.
Concrete-Representational-Abstract (CRA) Instructional Sequence The most common intervention used within the reviewed studies was the CRA instructional sequence. All of the studies including the CRA instructional sequence produced high effects (g = 0.53, tau-U = 1.00) on student achievement in algebra. CRA is a gradual instructional method that moves students from the concrete (manipulatives) stage of learning to the representational stage (pictures), and finally transitions students to the use of abstract numbers and symbols. These stages are all interrelated, with the purpose of creating meaning between the use of manipulatives and the abstract stage of learning. The studies included in this review show students moving from the concrete phase of understanding algebra to the abstract phase. However, it is critical that students easily move back and forth between
these phases to develop a strong mathematical understanding of a skill (Witzel, Riccomini, & Schneider, 2008). Witzel et al. (2003) investigated the effectiveness of the CRA instructional sequence on the performance of sixth and seventh grade students with LD or identified at risk to solve inverse operations and transformations (CCSS.Math. Content.HSA-SSE.B.3). Unique to this study was that the intervention was conducted in an inclusive setting where the interventionist was the general education math teacher. A pre–post follow-up design was used to determine the effectiveness of using the CRA sequence to teach more advanced algebra concepts. Instruction in both control and treatment groups included the use of teacher modeling and guided and independent practice. Although the comparison group was taught the same content, instruction was limited to repeated abstract lessons as opposed to the use of pictorial representations and concrete objects. Both groups showed significant growth from pretest to posttest (g = 0.351), but those students in the CRA group outperformed the comparison group on follow-up tests. An error pattern analyses also indicated that students in the CRA group made less computational errors. The implications reported in the study suggest that the use of concrete objects and pictorial representations can be effective at the secondary level, in addition to benefiting elementary-age students. Furthermore, the researchers concluded that the CRA approach can easily be implemented within general education settings and that it may be appropriate for students with and without disabilities in mathematics. Witzel (2005) examined the use of the CRA instructional sequence in general education math settings. His study involved six general education math teachers and 231 middle school students, both general and special education. Of the students, 49 were identified as having an LD. Each teacher taught one of the inclusion math classes using the CRA method and the other class using traditional abstract instruction. The algebra unit they taught consisted of 19 lessons ranging from solving inverse operations to solving linear functions with unknowns on one or both sides of the equal sign (CCSS.Math.Content.6.EE.C.9). To be consistent with research on effective practices, both control and treatment group lessons consisted of (a) teacher modeling, (b) guided and independent practice, and (c) the use of visual cues (i.e., advanced organizer). Although both groups made adequate growth between pretest to posttest, students in the CRA instructional group scored higher on both immediate (g = 0.43) and follow-up tests. Scheuermann, Deschler, and Schumaker (2009) explored the effects of the CRA sequence on middle school students with LD. The use of an explicit inquiry routine (inquiry dialogue and the CRA model) was used to develop the understanding of solving one-variable equations imbedded in word problems (CCSS.Math.Content.6.EE.B.7). The most important instructional variable examined was the
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verbalization of mathematical understandings taking place among students and teachers during the instructional phase. The study was designed to examine a student’s ability to (a) abstractly solve problems over time, (b) concretely solve problems over time, (c) transfer skills to more complex equations, and (d) transfer skills learned in word problems to those found in a student’s algebra textbook. The researchers indicated that students were able to accurately solve problems over time with high accuracy (uninstructed problems, tau-U = 0.82; instructed problems, tau-U = 1.00) and that the more familiar students became with the instructional format and dialogue, the easier it was for them to generalize skills to more complex equations. The students maintained increased levels of performance for 11 weeks following the instructional intervention. Strickland and Maccini (2013) also examined the effectiveness of the CRA instructional sequence to teach algebra to students with LD. Using a multiple probe design, three high school students were taught to multiply linear expressions within area problems using the CRA sequence with a graphic organizer imbedded in the abstract phase of instruction (CCSS.Math.Content.HSA-APR.A.1). A portion at the beginning of each lesson was also spent reviewing previous learned concepts to help activate prior knowledge. The total intervention took place over three 40-min sessions. All students showed increased ability to multiply linear expressions from baseline to intervention phase (tau-U = 1.00) and were all able to maintain the skill for up to 6 weeks following intervention.
Tutoring Two studies examined the effects of tutoring using explicit instruction. One-on-one tutoring or peer assistance is a common intervention for students with LD. This tutoring is often provided by preservice teachers, special education teachers, paraprofessionals, or peers. Peer-assisted tutoring can range from cross-age tutoring to within-class dyads. The first study (Fuchs et al., 2009) used teacher-led tutoring groups, and the following study (Calhoon & Fuchs, 2003) investigated peer-assisted learning groups. The findings of these studies suggest tutoring, using explicit instruction, as an effective strategy (g = 0.40). Fuchs and colleagues (2009) investigated the efficacy of tutoring third graders with math difficulties in the area of algebraic number calculations (i.e., solving for x in a simple equation, x – y = c) to improve transfer of understanding to algebraic word problem outcomes and procedural calculations (CCSS.Math.Content.3.OA.A.4). The 133 students who qualified for participation were randomly assigned to one of three tutoring conditions (a) no treatment (control) condition, (b) word problem-solving condition, or (c) number calculations condition. Following 16 weeks (48 sessions of 20–30 min) of tutoring, students in the number
calculation condition showed enhanced computational fluency (g = 0.53), compared to the control group, and small gains in procedural calculations (g = 0.27). However, there was no difference in groups on word problem outcomes. Students in the word problem condition showed increased gains in computational fluency (g = 0.62) compared to the control condition, which transferred to large gains in knowledge of procedural calculations (g = 0.53) and overall word problem outcomes (g = 0.36). Calhoon and Fuchs (2003) examined the effects of PALS math combined with curriculum-based measurement (CBM) to teach algebra readiness skills to high school students with disabilities. Using a quasi-experimental design, 92 students in 10 classrooms across three high schools were randomly assigned to the intervention group. The intervention took place over 30 sessions of 30 min and consisted of peer learning dyads working together to complete problems involving operations, data analysis, problem solving, geometry, and algebra (range of skills from CCSS.Math. Content.4.OA to CCSS.Math.Content.6.EE). Students in the intervention groups were also introduced to a reinforcement system to help motivate dyads by earning points toward tangible items (i.e., fast-food coupons, sporting event tickets). The control condition consisted of explicit instruction through the use of a skill-based workbook without the inclusion of a reinforcement system. Students in the treatment (PALS + CBM) significantly increased in their overall math computation performance but showed no significant growth compared to the control group in their ability to apply these concepts to algebraic applications (g = 0.58). The medium effects of this intervention indicate that although PALS can be a useful math intervention for students in high school, more intensive focus on application concepts is necessary.
Cognitive Strategy Instruction Cognitive strategy instruction is the use of various tools to help students to organize and process information. Strategies often come in the form of a mnemonic or a heuristic to help students remember the steps to solve a specific problem. These strategies are meant not to be problem specific but to assist students in solving a range of problem types. Providing students with multiple cognitive strategies and guided discussion behind the differences will allow students to choose a strategy that works best for them (Woodward, 2006). The findings from the studies examining the use of cognitive strategy instruction support previous research (Gersten et al., 2009) that this is a highly effective intervention (g = 0.83, tau-U = 1.00). Maccini and Hughes (2000) continued to examine skill areas that have an extensive impact on algebra performance. Specifically, their study investigated the effects of incorporating the STAR learning strategy (Maccini, 1998), within
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Watt et al. the CRA sequence to teach students with LD to solve word problems involving integers (CCSS.Math.Content.7.NS.A.1). A multiple-probe design across six students, in Grades 9 to 12, was used to investigate individual performance over time. During each instructional phase of the CRA sequence the research would (a) model two to three problems while thinking aloud, (b) provide up to five problems with guided practice, and (c) present five problems for independent practice. All participants improved on mean percentage accuracy on immediate measures from baseline to the abstract instructional phase of CRA (problem representation, tau-U = 1.00; problem solution, tau-U = 1.00). In addition, all students showed increased ranges across integer problem types on near- and far-transfer generalization measures (Maccini & Hughes, 2000). The implications from this study indicate that students need more time to learn addition and subtraction of integers as opposed to multiplication and division. Maccini and Ruhl (2000) replicated a similar study design involving the use of the STAR strategy and the CRA sequence to teach word problems involving integers (CCSS.Math.Content.7.NS.A.1). Participants were three eighth grade students with LD. Essential components of the instructional procedures included (a) the use of teacher modeling and think aloud, (b) inclusion of students in the think-aloud process, (c) guided and independent practice, (d) corrective feedback, and (e) use of an advanced organizer. Findings from the study yielded similar results as those of Maccini and Hughes (2000), with all three students increasing their mean percentage accuracy on problem solutions from baseline to the abstract phase of instruction (tau-U = 1.00). Results indicated that students were able to master the mathematical objectives that were taught in a relatively short amount of time with the ability to generalize what they learned to more complex math problems. Xin, Wiles, and Lin (2008) assessed the functional relationship between word problem-solving interventions and the prealgebra math performance of five fourth and fifth grade students with math difficulty using an adapted multiple-probeacross-participants design. The intervention consisted of teaching word problem solving that represented symbolic algebraic expressions (CCSS.Math.Content.3.OA.D.8) using conceptual model-based problem solving (COMPS) versus an arithmetic-oriented approach. In addition students were taught a cognitive learning strategy (DOTS—Detect, Organize, Transform, Solve; Xin et al., 2008) to guide them through the problem-solving process. All five students showed increased ability to solve addition, subtraction, multiplication, and division word problems following the intervention. But more important for the purpose of this review was the increase in accuracy and overall performance for solving unknown variables in algebraic expressions following the intervention compared to baseline. The performance among students at baseline ranged from 0% to 33% accuracy, whereas following the
intervention students’ scores ranged from 67% to 100% accuracy on algebraic measures (tau-U = 1.00). Xin et al. (2011) extended the findings from Xin et al.’s (2008) study on the use of COMPS using an experimental design to clarify their findings on the effectiveness of COMPS. The current study had a total of 29 in fourth and fifth graders, of whom 16 were students with LD and 13 were identified by their schools as at-risk learners. Students were stratified by grade, gender, and pretest scores and then randomly selected for the COMPS instructional group or the control group. Within both instructional groups students were taught to solve word problems by writing equations with variables and to find the value of the unknown variable in the expression (CCSS.Math.Content.6.EE.B.6). The control group consisted of instruction using a general instructional heuristic, SOLVE (Search the question, Organize the problem, Look for a strategy, Visualize then work the problem, Evaluate your answer). Two researcher-generated measures of algebra readiness, tests of model expressions and solving equations, were used to compare groups. On both measures students in the COMPS group significantly improved (g = 0.859) compared to students in the control group.
Enhanced Anchored Instruction (EAI) EAI is an inquiry-based learning intervention that uses video anchors of real-life situations that provides students with a problem to solve or a task to complete. Each EAI lesson typically embeds the use of technology and hands-on activities to reinforce math concepts. In addition, research suggests that the combination of explicit teaching of skills and EAI has a greater impact on student achievement. The results of these studies indicate that the use of EAI is also highly effective for teaching algebra (g = 0.80). Bottge et al. (2001) examined the benefits of linking algebra instruction to video anchors with applied tasks (i.e., EAI). Four math classrooms, one remedial math and three prealgebra classes, were assigned to treatment (EAI; n = 34) or control (traditional instruction; n = 41) conditions. Each condition contained students with learning and behavioral disabilities (n = 19) and students at risk for failure in math. The instructional content for both groups focused on calculating distance, rate, and time (CCSS.Math.Content.6.EE.C.9). Students in the EAI treatment condition applied the algebraic skills to develop soapbox derby cars anchored in the problem depicted on the videodisc. The students in the comparison condition participated in a more traditional instruction approach to learning to solve word problems using the formulas to calculate distance, rate, and time. Overall results indicated that students in all instructional settings made large gains from pre- to posttest. However, students in remedial math classes using EAI made the largest gains on the problem-solving measures (g = 0.35).
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Bottge et al. (2007) continued to examine the effects of EAI on the overall algebra achievement of adolescents with LD. A total of 128 seventh grade students were participants, 13 with disabilities, 12 identified with LD. Four self-contained classrooms were randomly selected to one of two conditions, traditional algebra instruction or EAI. Students were taught a range of standards found in the 6–8 Expressions and Equations domain (e.g., interpreting data, constructing graphs and tables, recognizing relationships among data) but focused primarily on estimating and calculating distance, rate, and time (CCSS.Math.Content.6.EE.C.9). After the completion of the initial study, it was replicated with the same student population but a reversal of previously assigned conditions. Both studies found large effects on student’s problem-solving ability for participants using EAI (Study a, g = 0.93; Study b, g = 1.02). Given possible carryover effects between studies, caution should be used when interpreting the effects from study b.
Graphic Organizers Graphic organizers can be used in a variety of ways to support the retention of new and complex math problems. Graphic organizers can also be used to support vocabulary development, help students organize their work into individual steps, and help students make connections between new and previously taught content. The results of the studies using graphic organizers within this review indicate this intervention is highly effective (g = 0.57). Ives (2007) evaluated the effectiveness of using a graphic organizer to support the instruction of teaching high school students algebra. Using an experimental design, 30 students were randomly assigned to a control or intervention group. Both groups were instructed in how to solve linear equations with two unknown variables (CCSS.Math.Content. HSA-REI.C.6) through the use of explicit instruction. Although both groups used the same instructional materials, had the same amount of sessions, and received equal amount of time on each topic, the treatment group was provided with a graphic organizer to help them solve for each variable. The results of a researcher-generated probe indicated no significant differences between groups, but effects in favor of the treatment (g = 0.09). The author systematically replicated this study using 20 different high school students with LD. The results of this second study also did not reveal significant differences but continued to favor the treatment group (g = 0.19). The results of this study are difficult to interpret as a result of a small sample size, and the low effects warrant further research in this area before drawing conclusions.
Discussion The primary purpose of the current review was to identify effective interventions for teaching algebra to students with
LD and to extend the findings of the previous reviews (Maccini et al., 1999; Maccini et al., 2007). The initial review by Maccini and colleagues (1999) identified six studies examining effective algebra interventions for students with LD; this review included 15 studies not included in the previous analysis. The current review also identified 5 studies not included in the follow-up review by Maccini and colleagues (2007).
Analysis of Effective Interventions The overall analysis of the 15 studies identified five interventions for teaching algebra to students with LD: (a) CRA, (b) cognitive strategy instruction, (c) EAI, (d) tutoring, and (e) the use of graphic organizers. Although the interventions in many studies produced large effects, there was often a combination of instructional components (e.g., CRA + use of heuristic) among studies making it difficult to attribute the effects to only a single component (e.g., CRA). The overall ES for studies using EAI (g = 0.80) and cognitive strategies (g = 0.83, tau-U = 1.00) suggests that these are highly effective practices for teaching algebra content to students with LD. Studies using the CRA instructional sequence also produced large effects (g = 0.53, tau-U = 1.00). The use of this intervention was supported by seven studies within this review, all indicating increased student performance in math. In addition, the CRA approach to teaching algebra was replicated across the largest range of instructional content, from solving equations with unknown variables to multiplying linear expressions. Two of the seven studies using the CRA sequence used a cognitive strategy to support the retention and generalization of student learning. Although studies investigating the use of graphic organizers (g = 0.57) and individualized tutoring (g = 0.40) produced moderate to large effects on student learning in algebra, only two studies were included in each of the instructional categories, and more research is needed to support the interpretation of the findings. Although no statistical analysis of the instructional components found within each intervention was conducted, for the purpose of future implementation and better understanding of the educational implications from this review, the unique components have been summarized. The metaanalysis conducted by Gersten and colleagues (2009) identified several research-based practices for teaching students with LD or those who are struggling in math. Among these seven were the use of explicit instruction, multiple examples, verbalizing decisions and solutions, visually representing information, heuristic strategies, formative assessments, and peer-assisted instruction. After closely examining the studies included in this review, it was evident that each study included the use of explicit instruction. Research has consistently found explicit instruction to be linked to increased student achievement (Gersten et al., 2009). No other instructional component was found in all of
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Watt et al. the studies, but the use of visual representations within the intervention was present in 12 of the studies, formative assessment data were used to guide instruction in 8 of the included studies, and the use of heuristics or strategies was found within 6 of the studies reported in this review. It should also be noted that each study included an intervention that contained two or more of the instructional practices identified by Gersten and colleagues (2009) as effective components of instruction. Further research examining the statistical impact of each of these instructional factors is warranted.
Nature of Instructional Content A secondary focus of this review was to evaluate the complexity of the content being taught within the identified studies. Since the 1999 review (Maccini et al., 1999) the field of special education has seen a steady increase in the number of math studies investigating evidence-based practices for teaching algebra-based concepts. The initial review identified six studies that emphasized a range of skills. The review covered content including relational word problems (Huntington, 1994; Hutchinson, 1989, 1993; Zawaiza & Gerber, 1993), combing like terms (Kitz & Thorpe, 1995), and graphing inequalities (Kitz & Thorpe, 1995; Rosman, 1994). In each of these studies the math content was below the grade level of the student participants. The conclusion of the review called for additional focus on interventions addressing more complex algebraic skills. The findings of our current review reflect growth in the research on instruction of higher level math content among students with LD. Of the 15 studies reviewed, 13 instructed using standards from the intermediate and secondary domains of the CCSS. In addition to the use of higher level standards, 9 of the 15 studies taught students grade level content. Despite the growth reflected in the findings, the limited range of higher level standards taught across the studies included in the review should be noted. For example, in the sixth grade domain, Expressions and Equations, there are nine standards. The majority of the standards taught within the studies included in this review were from the sixth grade domain; however, among those studies only three standards were addressed. This finding illustrates the limited research across the algebra domains. Therefore, this impacts the knowledge base of effective practices for students with LD in math.
Educational Setting Despite the increase in complexity of content and the overall student achievement of students with LD instructed at grade level, only five of the studies in this review took place in general education settings (Bottge et al., 2001; Bottge et al., 2007 [Studies a and b]; Witzel, 2005; Witzel et al.,
2003). In all, 10 of the included studies took place during core math instruction, indicating that a large number of students with LD are receiving their content instruction in more restrictive educational settings. Based on this information and our earlier indication that nine studies were teaching grade level content, many students with LD are receiving grade level instruction outside of the general education classroom. Although this may suggest that students have other individual needs that impact their participation in math in the general education setting, it may also suggest that continued work on supporting inclusion for all learners in math is warranted. Maccini et al. (1999) identified one study that took place in general education compared to the five included in the current review. Although an increase in inclusive placements is evident, it is critical that greater emphasis be placed on the evaluation of instructional practices that support students with LD in general education math settings.
Limitations There are three main limitations to the conclusions of this review. First, given the large amount of single subject studies included in this review, quantitative analysis was limited. In addition, it is difficult to compare the magnitude of effects between the single subject and quantitative designs. Second, the majority of the measures were researcher generated, and reliability alphas were reported for only three studies. As we continue to prepare students with LD to participate in the general education curriculum, the use of standardized measures to assess their performance in math is necessary. Finally, many studies contained general education students and students with other identified disabilities, making it not possible to calculate effects for only students with LD. As a result of these limitations, findings should be interpreted with some caution.
Overall Implications As we move forward, research on best practices for students with disabilities in math needs to continue to place emphasis on early intervention of prealgebra skills and higher level content standards for students at the intermediate and secondary levels. A change in how we deliver core instruction needs to occur for all students to have access to grade level math content and the skills necessary to complete higher level math courses. The use of formative assessment data to match instruction to students’ individual needs and the increased use of standardized measures to analyze student growth are critical. This review also indicates the importance of the use of explicit instruction in combination with other effective instructional components.
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Research in this area needs to provide strong descriptions of students and intervention characteristics. Replications of studies indicating highly effective practices with diverse student populations need to be conducted across multiple settings. Research examining a larger range of the CCSS standards and measuring student achievement using reliable and valid measures is critical to the development and interpretation of effective practices for students with LD. Finally, and most important, research findings need to be shared with practitioners in the field through quality professional development. Declaration of Conflicting Interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding The author(s) received no financial support for the research, authorship, and/or publication of this article.
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