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https://doi.org/10.1177/0731948721994843

Learning Disability Quarterly 2021, Vol. 44(4) 248 –260 © Hammill Institute on Disabilities 2021 Article reuse guidelines: sagepub.com/journals-permissions DOI: 10.1177/0731948721994843 journals.sagepub.com/home/ldq

Article

Math word problems are linguistically presented arithmetic problems that require students to construct a problem model to solve the problem (Fuchs et al., 2006; Fuchs & Fuchs, 2007). Word problems require students to use linguistic information to identify relevant information for solution accuracy, construct the appropriate number sentence, and calculate the problem accurately. Students with or at risk for learning disabilities (LD) experience considerable difficulty with word problems as they involve processes beyond basic math skills (Swanson, 2006). In addition, students with LD perform significantly lower in math than age-equivalent peers, with the gap widening as each academic year passes (Cawley et al., 2001).

Given the considerable difficulty with word-problem- solving (WPS) students with LD face, it is important to identify effective instructional practices. One approach to identifying valuable instructional practices is to conduct a synthesis of WPS intervention studies for students with LD. Meta-analysis allows for the comparison of treatment effect sizes across studies to address specific research questions in addition to examining studies by instructional variables (Glass, 1977). Two previous meta-analyses (Gersten et al., 2009; Kroesbergen & van Luit, 2003) investigated the effect of general math intervention (e.g., calculation, mathematics

proficiency, basic skills, problem-solving strategies) to enhance the math achievement of students with mathemat- ics difficulties. To the authors’ knowledge, only four meta- analyses (Lein et al., 2020; Xin & Jitendra, 1999; Zhang & Xin, 2012; Zheng et al., 2013) to date have investigated specifically WPS interventions for students with LD in grades K to 12 broadly.

Xin and Jitendra (1999) investigated WPS interventions for students in elementary to postsecondary grades with “learning problems” at risk for math failure. Learning prob- lems (LP) were defined as mild disabilities such as learning disabilities, mild mental retardation, and emotional disabili- ties and at risk for mathematics failure. A total of 25 inter- vention studies (14 group-design, 12 single-subject) were included in the study. One study included both group and single-subject design. Moderator variables that may have

994843LDQXXX10.1177/0731948721994843Learning Disability QuarterlyKong et al. research-article2021

1Chapman University, Orange, CA, USA 2California State University, Fullerton, USA 3University of New Mexico, Albuquerque, USA

Corresponding Author: Jennifer E. Kong, Attallah College of Educational Studies, Chapman University, One University Drive, Orange, CA 92866, USA. Email: [email protected]

Word-Problem-Solving Interventions for Elementary Students With Learning Disabilities: A Selective Meta-Analysis of the Literature

Jennifer E. Kong, PhD1 , Christy Yan, PhD2, Allison Serceki, MS1, and H. Lee Swanson, PhD3

Abstract This meta-analysis assessed the effect of word-problem-solving interventions on the word-problem-solving accuracy of students identified as having a learning disability (LD) or at risk for an LD in kindergarten to the sixth grade. Eighteen randomized control group designed studies met the inclusion criteria. Overall, word-problem-solving interventions yielded a significant positive effect on the word-problem-solving accuracy of students in elementary grades with LD (effect size [ES] = 1.08). Instructional components that underlie effective studies were also identified. Results suggest that peer interaction and transfer instructions yielded large effects on treatment outcomes. Results also suggested that intensive interventions (50-min sessions, 34 total sessions) in Grade 3 regardless of instructional setting yielded the largest ESs. These findings support the need to develop and implement quality evidence-based instruction in classroom settings (Tier 1 instruction) prior to utilizing additional resources for more intensive and individualized intervention.

Keywords word-problem-solving intervention, elementary, at risk, math disabilities

Kong et al. 249

affected the overall treatment effect were also examined. These variables included student characteristics (IQ, grade groups—elementary, secondary, postsecondary, and clas- sification groups—LD, mixed disabilities, and at-risk), instructional characteristics (intervention approach, length of intervention, deliverer of intervention), and methodolog- ical features (published/unpublished, group assignment). Computer-assisted instruction in group-design studies was found to be most effective, yielding a mean effect size of 1.80, followed by representation techniques (d = 1.77), and strategy training (d = 0.74). An analysis of moderator vari- ables revealed a low mean effect size for studies with ele- mentary students (d = 0.78) and a large effect size for the postsecondary group (d = 1.68), but no significant differ- ence between the groups. The authors also presented a median PND (percentage of non-overlapping data) score to determine intervention effectiveness. Results from the single-subject design studies indicated a median PND of 89% (range, 11%–100%). That is, a median of 89% of data from the intervention phases were higher than any data point in the baseline phases. Analysis of moderator vari- ables in these single-subject studies revealed a significant advantage for interventions teaching representation tech- niques. The intervention effect sizes did not differ signifi- cantly across grade groups.

As a follow-up to the meta-analysis conducted by Xin and Jitendra, Zhang and Xin (2012) included studies that were published from 1996 to 2009 in their meta-analysis of WPS interventions for students with math difficulties. As in the previous review, studies that included students with “learning problems in mathematics” from kindergarten to 12th grade were included in the meta-analysis. A distinction in this study from their previous analysis was that students who were identified as those with learning problems were operationally distinguished from students identified with a learning disability via the discrepancy model. Twenty-nine group-design studies and 10 single-subject studies (39 total studies) were included in the follow-up meta-analysis. Moderator variables were analyzed in group studies only. Moderator variables included LD definition (at risk, dis- crepant LD), class setting (inclusive, special education), instructional features (intervention strategy, type of assess- ment), and types of word problems (algebraic thinking/ arithmetic, real-world/simple-structured). The researchers found that WPS interventions had a large effect on students’ performance in problem-solving accuracy, with an overall effect size of 1.85. Single-subject designs yielded a PND of 95%. Moderator analyses on group studies indicated that interventions provided in inclusive settings were more effective than in special education settings. Results also indicated that while all intervention strategies (problem structure representation, cognitive strategy training, strate- gies involving assistive technology) produced positive effects, problem structure representation techniques yielded

the highest effect sizes. Problem structure representation techniques include schema-based explicit instruction. In addition, no significant differences were found between simple-structured problem-solving and real-world problem- solving. Finally, there were no significant differences between effect sizes from students diagnosed with discrep- ant LD and at-risk students.

Zheng et al. (2013) also conducted a selective meta-anal- ysis of intervention studies on WPS for students with math disabilities (MD). Students were identified as MD if partici- pants’ scores fell below the 25th percentile on a standard- ized math test (e.g., Wechsler Individual Achievement Test [WIAT], Wide Range Achievement Test-Third Edition [WRAT-3]). A total of 15 studies (seven group-design, eight single-subject) were included in the meta-analysis. WPS interventions were determined to be effective for students with MD, yielding an effect size of 0.78 for group-design studies (compared to students with MD who did not receive instruction). The average ES for single-subject studies across participants after removing outliers Rosenthal’s (1994) formula was 0.90. All studies were also coded for the occurrence of various instructional components. Studies that significantly improved students’ WPS skills included instructional components that incorporated advanced orga- nizers, skill modeling, explicit practice, task difficulty con- trol, elaboration, task reduction, questioning, and providing strategy cues. Also, small-group instruction was found to be an effective approach for students with MD.

In a more recent study, Lein et al. (2020) reviewed cur- rent studies on WPS interventions for K–12 students with learning disabilities and math difficulties. LD was defined as students who were identified based on a discrepancy model or through the school district evaluation (non- responsiveness to intervention). Math difficulties were defined as students who scored at or below the 35th percen- tile on a standardized mathematics test. A total of 31 group- design studies were included in this meta-analysis, which found that WPS interventions yielded a moderate mean effect size (g = 0.56). This study found that there was no significant difference in the magnitude of effect sizes by LD and at-risk status. In addition, interventions for students in elementary grades (g = 0.63) were found to yield higher effect sizes than those in secondary grades (g = 0.33). Finally, this study investigated intervention models as a moderating variable of overall effects. The results indi- cated that interventions that included a schema broadening and transfer instruction model yielded highest effect sizes (g = 1.06).

The present meta-analysis extends upon the previous reviews in several ways. All prior meta-analyses examined a broad range of ages, from kindergarten to postsecondary. This may be problematic for making instructional recom- mendations for a specific age or grade group. Xin and Jitendra’s meta-analysis included 11 studies with participants

250 Learning Disability Quarterly 44(4)

in kindergarten to Grade 6 (five group-design, six single- subject) out of 25 studies. Xin and Jitendra reported large effect sizes for postsecondary students (d = 1.68), moderate effects for secondary students (d = 0.78), and low effect size for the elementary students (d = 0.47). An analysis of spe- cific grades as a significant moderator of treatment outcomes was not conducted. Zhang and Xin’s study included 18 ele- mentary experimental studies (15 group-design, three single- subject) out of a total of 39 studies. Zheng and colleagues’ meta-analysis included 10 elementary studies (five group- design, five single-subject) out of 15 total studies. The study by Lein et al. included five studies conducted on elementary students and six with secondary studies. Furthermore, the meta-analyses conducted by Zhang and Xin (2012) and Zheng et al. (2013) did not report information on how inter- vention effects may have diverged for elementary and sec- ondary students. Lein et al. (2020) also did not investigate specific instructional components within interventions, but rather utilized general schema and models of intervention. Thus, specific conclusions about treatment effects of WPS interventions and instructional recommendations drawn from these meta-analyses may not be appropriate for elementary students specifically.

In addition, as evidenced by the studies above, variabil- ity exists in how students are identified with LD in research and practice. Earlier studies have included students with “learning problems” more broadly, including students with learning disabilities (Xin & Jitendra, 1999; Zhang & Xin, 2012) or have relied on a model that includes a discrepancy between IQ and achievement (Discrepancy model; for example, Hallahan et al., 2014). Researchers have also uti- lized the term “at risk for LD” to identify children who may be at risk for academic failure and benefit from inter- vention, but have not yet been identified as LD. For exam- ple, performance below the 25th percentile cut-off score on standardized measures has been commonly used to identify children at risk (e.g., Fletcher et al., 1989; Siegel & Ryan, 1989; Swanson et al., 2013). With the growing need to deliver interventions to students who are most at risk as early as possible, it will be important to clarify the role definitions play in instructional outcomes. Specifically, the present meta-analysis will examine interventional components such as the intensity of intervention, setting, and specific instructional features in relation to student characteristics (e.g., grade, LD identification).

Finally, earlier studies have not investigated the possible moderating effect of interventions for students who are English learners (ELs). ELs, in particular, may experience more difficulty in comparison to monolingual children with math problem-solving because of the need to preserve information while processing information in a second lan- guage (e.g., Swanson et al., 2019). The National Center for Education Statistics (NCES, 2019) reports that 41% of ELs score below basic in mathematics, compared with 16% of

their non-EL peers scoring below basic. Given that ELs are a rapidly increasing demographic in U.S. public schools, research to identify effective instructional strategies for problem-solving is critical.

The present meta-analysis will focus on group-designed intervention studies conducted with elementary-aged (K– 6) students in an attempt to make more detailed recommen- dations for effective interventions for this age group. The current study will add to the current research by including samples of students who have been identified as LD via the discrepancy model and “at risk” for MD and investigating the effects of specific instructional components within interventions rather than global procedures on experimen- tal studies conducted with elementary participants.

This study will address the following three research questions:

Research Question 1 (RQ1): Are WPS interventions effective for kindergarten to sixth-grade students with LD? Effective outcomes will be based on the magnitude of the ESs. The average ESs among the group designed studies in the previous syntheses was 1.18 for students in grades 1 to 12. Research Question 2 (RQ2): Do specific effect sizes in WPS interventions vary as a function of moderator vari- ables such as participant characteristics (EL status, LD definition, grade level)? Some of the previous syntheses have not reported the impact of sample characteristics on treatment outcomes and therefore generalization to chil- dren with specific learning difficulties is unclear. This meta-analysis attempts to characterize the sample found to benefit from problem-solving interventions. Research Question 3 (RQ3): Do effect sizes in WPS interventions vary as a function of specific instructional components? Previous synthesis have found that gen- eral instructional approaches, such as computer-assisted instruction, problem structure representation (i.e., schema-based explicit instruction), and instructional scaffolding (organizers, modeling, task reduction) con- tributed to significant improvements in students’ perfor- mance. These studies have focused on an array of children with learning problems (LD, mild mental retardation, emotional disabilities) in grades K–12. This study extends the literature by identifying instructional components of WPS interventions that are directed to elementary-aged (K–6) students identified as having a learning disability or at risk for a specific learning disability in math.

Method

Data Collection

The PsycINFO, Science Direct, and ERIC online databases were systematically scanned for studies from 1990 to 2019

Kong et al. 251

that met the inclusion criteria. Search terms describing word problem-solving (word problem-solving instruction or word problem-solving intervention or problem-solving instruction or story problem or math intervention), the pop- ulation (special education or learning disabled or learning disabilit* or at risk for math difficulty), and word-problem- solving outcomes were combined with these keywords: elementary school, efficacy, strategy instruction, schema- based instruction, scaffolded instruction, and peer interac- tion. This initial search generated approximately 1,592. Of these, 239 studies were selected for further review based on title and abstract review. The reference lists of prior meta- analyses (e.g., Gersten et al., 2009; Kroesbergen & van Luit, 2003; Xin & Jitendra, 1999; Zhang & Xin, 2012; Zheng et al., 2013) were also systematically scanned.

Study Eligibility Criteria

To be eligible for this analysis, each study had to meet the following criteria: (a) included students with or at risk for learning disabilities in Grades K to 6; (b) tested an inter- vention to improve WPS; (c) assessed students’ WPS accu- racy (measure included normed or experimental/researcher developed measures); (d) involved an experimental design with randomization, quasi-experiment with pre- and post- test data, or a within-subjects design (i.e., all students par- ticipated in both the treatment and comparison conditions); (e) provided data to permit the calculation of effect sizes and average weighted ESs; and (f) was published in English. Studies investigating the effectiveness of instruc- tion or improving only math calculation were not included. This procedure narrowed the search to 33 documents, 18 of which met inclusion criteria. Some studies had more than one WPS intervention, so 113 different ESs were calculated.

Inter-rater agreement. Two graduate students independently coded 22% of the articles for inclusion criteria and coding accuracy. Inter-rater agreement was calculated as a number of agreements divided by the number of agreements plus disagreements multiplied by 100. The mean inter-rater agreement for article inclusion was above 95%. The mean inter-rater agreement for coding of the 12 instructional components outlined below was also above 95%.

Coding of Study Features

The general categories of coding for each study included (a) year of publication, (b) sample characteristics (gender, grade, disability or risk, EL status), (c) intervention charac- teristics (number of sessions, number of minutes, group size, who delivered the instruction), and (d) components of instruction.

Categorization of treatment variables. Each study was coded on the occurrence or nonoccurrence of the following instruc- tional components. These instructional components have been linked to academic outcomes in earlier meta-analyses that have included students with learning disabilities (Dennis et al., 2016; Swanson & Hoskyn, 1998; Zheng et al., 2013). The instructional components coded are as follows:

1. Explicit instruction—statements in the treatment description included characteristics of explicit direct instruction (e.g., teacher/researcher directed instruc- tion, administering probes).

2. Technology—statements in the treatment descrip- tion about the use of technology tools such as com- puters, tablets, or other media to supplement or provide instruction.

3. Strategy cues—statements in the treatment descrip- tion about using strategies, multistep procedures, ver- balizations of procedures, metacognitive strategies, questioning, and think-alouds by teacher/researcher.

4. Peer interaction—statements in the treatment description about using peer interaction to complete activities to present, model, practice, or review instruction.

5. Instructional feedback—statements in the treatment description about providing participants with fre- quent instructional feedback and correction.

6. Visual aids—statements in the treatment description about the use of graphics, charts, diagrams, illustra- tions, visual aids, semantic mapping, or pictorial representations to supplement instruction.

7. Foundational skills—statements in the treatment description about providing participants with instruction and practice in foundational skills such as computation and fact fluency.

8. Schema instruction—statements in the treatment description about providing participants with explicit instruction of underlying structures of the word problem type, basic schema for problem type, and solving specific problem types.

9. Instruction to transfer—statements in the treatment descriptions about explicit instruction to transfer or generalize skills on novel problems.

10. Manipulatives—statements in the treatment descrip- tions about providing students with concrete materi- als, manipulatives, or other hands-on materials.

11. Behavioral reinforcement—statements in the treat- ment description about providing participants with praise, token economy, or reinforcement schedules.

12. Self-regulated learning—statements in the treatment description about students setting goals for their per- formance, self-monitoring, or self-evaluation.

252 Learning Disability Quarterly 44(4)

Data Analysis

Effect size calculation. Effect sizes (ESs) were calculated uti- lizing pretest and posttest means and standard deviations. Hedges’s g was the measure of ES for this study, calculated as the difference between pretest-posttest means for the treatment group and the pretest-posttest means for the com- parison group. This difference score was then divided by the pooled within-group standard deviation of posttest scores. Hedges’s g was calculated as

X X X X

n s n s n n

post pre post pre1 1 2 2

1 1 2

2 2 2

1 21 1 2

−( )− −( ) − + − + −([ ] [ ] / [ ]])

where X pre1 and X pre2 were unadjusted pretest means, X post1 and X post2 were unadjusted posttest means, n1 and n2 were sample sizes, and s1 and s2 were unadjusted stan- dard deviations for the treatment and comparison groups, respectively.

Several planned tests that compared effect sizes as a function of intervention characteristics and grade levels utilizing a general linear model procedure were computed (Borenstein et al., 2009). Because of the variance between and within studies, the PROC Mixed (SAS, 2012) proce- dure was used to determine effect sizes as a function of instructional components. For this mixed analysis, the grand mean centered variable of “grade” was used as a covariate in the analysis. Due to the small sample in these comparisons, we employed a restricted maximum likeli- hood estimation (REML) with a Bonferroni correction (McNeish, 2017).

Results

Question 1: Are WPS Interventions Effective for Kindergarten to Sixth-Grade Students With LD?

To answer Research Question 1, a single-weighted ES for all 18 studies was calculated. To determine whether specific sample characteristics related to excess variability in ESs, general linear models categorizing between-class effects were analyzed. Grade, LD definition, and intervention char- acteristics, such as who delivered the instruction, group size, number of sessions, type of measure, and number of minutes (intensity of intervention), were examined for con- tributing excess variability in ES.

Table 1 provides a descriptive summary of the studies included in this synthesis. The total n refers to the total number of students who were included in the studies. The LD (Learning Disability) n is the number of LD students who received treatment. The EL (English Learner) n is the number of LD students who were identified as English learners. Table 1 also displays whether LD students were identified by the discrepancy model or considered at risk for

LD (below specified cut-off score), grade level, and type of research design.

All studies included in this synthesis were published in peer-reviewed journals, with publication dates ranging from 1998 to 2014. Fourteen of the 18 studies focused on only third-grade students. Participants’ grade levels ranged from 2 to 5. Eight studies included students designated as LD by discrepancy (e.g., Fuchs, Fuchs, Prentice, Burch, Hamlett, Owen, and Schroeter, 2003), while the other 10 studies included at-risk students (e.g., Moran et al., 2014).

Intervention. The number of intervention sessions ranged from 4 (Owen & Fuchs, 2002) to 60 (Jitendra, Dupuis, et al., 2013). The length of each session varied from 20 to approximately 140 min (Fuchs, Fuchs, Prentice, Burch, Hamlett, Owen, Hosp, et al., 2003). One study did not report the length of each intervention session (Owen & Fuchs, 2002). Eight studies reported administering the intervention in a whole group, general education setting. An intervention was conducted in a small-group setting in eight studies, and one study individually. Two studies reported multiple inter- ventions with both whole-group and small-group instruc- tions. General education teachers delivered the intervention in five studies, and research assistants/graduate students were responsible for delivering instruction in eight studies. Two studies reported administration of the intervention by researchers and a community member delivered one. Two studies reported multiple deliverers of intervention: one study included an instructional assistant and parent, while another utilized a teacher and graduate student. Finally, 12 studies used researcher-developed measures to assess WPS accuracy, and two of the 18 studies used norm-referenced tests on pretests, posttests, and transfer tests. Four studies utilized both researcher-developed and norm-referenced measures.

Overall, WPS interventions had a positive effect on WPS accuracy across all studies, Hedges’s g = 1.08 (K = 113, 95% confidence interval [CI] = [0.79, 1.37]). According to Cohen’s (1988) criterion, this is a large effect size. A homo- geneity statistic Q was computed to determine whether studies shared a common ES. The statistic Q has a distribu- tion similar to the distribution of chi-square with k − 1 degrees of freedom, where k is the number of ESs. As expected, there was significant heterogeneity in the find- ings, Q (df = 112) = 2,036.78, p < .001. Because homoge- neity was not achieved (which is usually the case), the variability of the ES as a function of moderator variables were analyzed. The results are shown in Table 2. Because the commonly reported Q statistic has been criticized, the I2 statistic (Higgins & Thompson, 2002) was computed, using the following formula:

I Q k

Q 2 1=

( ) − −

Kong et al. 253

The I2 indices of 25%, 50%, and 75% are classified as low, medium, and high heterogeneity, respectively (e.g., Higgins & Thompson, 2002). The I2 statistic was 0.95, sug- gesting an extremely high percentage of variability across the majority of measures.

Moderator variables. Table 2 shows the Hedges’s g mean effect sizes and 95% CIs for the moderator variables. There were several significant differences when comparisons were made within the various moderator variables. For example, there were significant differences in weighted effect sizes (Hedges’s g weighted by the reciprocal of the sampling variance) by the number of reported minutes per session, QB(df = 9) = 863.10, p < .001. The QB statistic is the weighted between-categories sum of squares of an analysis of variance (ANOVA). Fifty-minute sessions pro- duced the largest effect size relative to the other conditions whereas 25-min sessions produced the smallest effect size. There were also significant differences in weighted ESs as a function of the number of sessions, QB(df = 9) = 79.28, p < .05. Interventions with 34 sessions yielded the largest effect size.

There were significant differences in effect sizes by type of measure used QB(df = 1) = 244.54, p< .05. Effect sizes of researcher-developed measures were significantly larger than those of norm-referenced measures. Furthermore,

Table 1. Summary of Study Characteristics.

Study Total n LD n EL n LD definition Grade Design

1 Fuchs et al. (2002) 40 30 0 D 4 RCT 2 Fuchs, Fuchs, et al. (2008) 243 243 4 AR 3 RCT 3 Fuchs, Fuchs, Finelli, et al. (2004) 351 33 4 AR 3 RCT 4 Fuchs, Fuchs, Prentice, Hamlett, et al. (2004) 366 57 2 AR 3 RCT 5 Fuchs, Fuchs, and Prentice (2004) 201 35a 4 AR 3 RCT 6 Fuchs, Fuchs, Prentice, Burch, Hamlett,

Owen, Hosp, et al. (2003) 375 52 8 AR 3 RCT

7 Fuchs, Fuchs, Prentice, Burch, Hamlett, Owen, and Schroeter (2003)

40 23 5 D 3 RCT

8 Fuchs, Seethaler, et al. (2008) 35 16 3 AR 3 RCT 9 Griffin and Jitendra (2009) 30 5 0 D 3 RCT 10 Jitendra, Dupuis, et al. (2013) 109 53 17 AR 3 RCT 11 Jitendra et al. (2007) 45 2 3 D 3 RCT 12 Jitendra et al. (1998) 34 17 0 D 2, 3, 4, 5 RCT 13 Jitendra, Rodriguez, et al. (2013) 135 71 63 AR 3 RCT 14 Moran et al. (2014) 72 49 2 AR 3 RCT 15 Owen and Fuchs (2002) 24 16 0 D 3 RCT 16 Swanson, Moran, et al. (2014) 82 62 0 AR 3 RCT 17 Wilson and Sindelar (1991) 62 21 0 D 2, 3, 4, 5 RCT 18 Xin et al. (2011) 29 16 0 D 3, 4, 5 RCT

Note. Total n = total number of students who were included in the study; LD n = LD students who received treatment; EL = reported number of students who were English learners receiving intervention; D = students identified by discrepancy model; AR = students at risk for LD or below percentile cutoff score; RCT = randomized control trial. aMD students only.

there were significant differences in the mean effects by grouping of students in the intervention, QB(df = 2) = 111.18, p < .05 and the deliverer of intervention, QB(df = 5) = 229.57, p < .05. Interventions that were delivered in whole groups (M = 2.85) and small groups (M = 2.39) yielded higher effect sizes than interventions delivered in individual settings (M = 0.92). Finally, interventions deliv- ered by the classroom teacher (M = 2.11) and university/ graduate students (M = 2.80) produced higher effect sizes when compared with researchers, instructional assistants, parent, or community higher (M = 0.50, 0.02, 0.07, 036, respectively).

In summary, WPS interventions had a positive and large effect for students who are in grades 2 to 5. Effect sizes for interventions that were delivered across 34 sessions yielded the highest effect size. In addition, interventions delivered in small- or whole-class instruction by classroom teachers or graduate students yielded the highest effect sizes.

Question 2: Do Specific Effect Sizes in WPS Interventions Vary as a Function of Participant Characteristics (EL Status, LD Definition, Grade Level)

To answer Research Question 2, a meta-regression analysis (Borenstein et al., 2009) was conducted on the moderator

254 Learning Disability Quarterly 44(4)

variables related to the sample description (LD definition, EL status, grade level) to determine whether three modera- tors accounted for excess variability in ESs.

There were significant differences in weighted ESs as a function of LD definition, QB(df = 1) = 183.97, p < .001. Mean effect sizes for students at risk for LD (M = 1.35) were higher than for students who were identified as LD through the school district (M = 0.74). There were also significant differences in effect sizes by ratio of students who were ELs QB(df = 1) = 373.03, p < .001. Effect sizes for interventions that included a higher ratio of students who were ELs reported higher mean effects (M = 1.40) than studies that did not include students who were ELs (M = 0.77). Finally, there were differences in the weighted ESs as a function of grade, QB(df = 3) = 70.38, p < .001.

The majority of the effect sizes that were computed in this review were for students in third grade (82 effect sizes). Effect sizes for interventions taught to third-grade students reported highest mean effects (M = 2.71). The mean effect sizes, Hedges’s g ES, Q and I2 statistics, and the 95% CIs as a function of the moderator variables are shown in Table 2.

In summary, effect sizes of WPS interventions were highest for students who are defined as “at risk” and in grade 3. Also, the mean ES of interventions that included students who are ELs was higher than interventions that did not include or did not report inclusion of ELs. It is impor- tant to note that only 11 out of 18 (61%) studies included this demographic information, so more research in this area may need to be conducted.

Table 2. Mean Effect Sizes and Confidence Intervals as a Function of Moderator Variables.

Moderator variable K ES SE 95% CI Q I2

LD definition Discrepancy 49 0.74 0.2 [0.34, 1.14] 377.56 0.87 At risk 63 1.35 0.21 [0.94, 1.77] 1,557.13 0.96 EL Studies with EL 56 1.4 0.23 [0.94, 1.86] 1,492.83 0.96 Studies without EL 57 0.77 0.18 [0.41, 1.12] 424.60 0.87 Grade 2 4 0.12 1.09 [−1.61, 1.85] 38.48 0.92 3 82 1.31 0.18 [0.95, 1.68] 1,705.92 0.95 4 15 0.77 0.25 [0.24, 1.31] 82.45 0.83 5 6 0.08 0.46 [−1.11, 1.27] 56.99 0.91 Duration of study 12 sessions 18 1.15 0.48 [0.15, 2.16] 281.24 0.94 18 sessions 8 0.01 0.34 [−0.80, 0.81] 41.21 0.83 20 sessions 14 0.21 0.17 [0.00, 0.42] 4.82 0.00 24 sessions 2 1.38 0.95 [−10.63, 13.39] 6.61 0.85 26 sessions 8 1.75 0.55 [0.45, 3.05] 211.51 0.97 32 sessions 4 0.65 0.38 [−0.57, 1.87] 5.06 0.41 34 sessions 8 3.24 0.73 [1.51, 4.96] 374.57 0.98 36 sessions 7 1.45 0.38 [0.53, 2.38] 93.70 0.94 60 sessions 6 0.12 0.08 [−0.08, 0.32] 0.00 0.00 Deliverer of instruction Researcher 6 0.35 0.11 [0.05, 0.64] 0.75 0.00 Teacher 37 1.23 0.25 [0.73, 1.74] 357.22 0.90 Instructional assistant 2 0 0 [−0.31, 0.31] 0.04 0.00 University student 64 1.15 0.21 [0.73, 1.58] 1,541.54 0.96 Parent 2 0 0.07 [−0.86, 0.86] 0.31 0.00 Community hire 2 0.36 0.01 [0.28, 0.44] 0.00 0.00 Grouping of students Large group 40 1.64 0.27 [1.09, 2.19] 703.63 0.94 Small group 69 0.78 0.17 [0.44, 1.13] 1,216.84 0.94 Individual 4 0.67 0.25 [−0.15, 1.49] 5.97 0.50 Type of measure Norm referenced 24 0.37 0.16 [0.03, 0.71] 69.98 0.67 Researcher developed 89 1.27 0.18 [0.92, 1.63] 1,858.26 0.95

Note. ES = effect sizes; CI = confidence interval; LD = learning disabilities; EL = English learner; K = number of effect sizes.

Kong et al. 255

Question 3: Which Specific WPS Interventions/ Components of WPS Intervention Are Effective With Kindergarten to Sixth-Grade Students With LD?

To answer Research Question 3, a multilevel random effect analysis of covariance was conducted to determine whether significant effects in weighted ESs existed between studies

that included instructional components and those that did not (McNeish, 2017). Mean centered grade was utilized as a covariate in the analysis. A multilevel analysis of covari- ance (ANCOVA) model included a random effects variance within and between studies.

Table 3 displays a summary of the occurrence of instruc- tional components in each study and mean effect sizes for each study. All studies included explicit instruction and

Table 3. Summary of Reported Use of Instructional Components.

Study

Instructional components

IC1 IC2 IC3 IC4 IC5 IC6 IC7 IC8 IC9 IC10 IC11 IC12

1 Fuchs et al. (2002) Mean ES = 1.24

X X X X — X — X X — X —

2 Fuchs, Fuchs, et al. (2008) Mean ES = 0.67

X — X X X X X X X X X X

3 Fuchs, Fuchs, Finelli, et al. (2004) Mean ES = 3.24

X — X X X X X X X — — —

4 Fuchs, Fuchs, Prentice, Hamlett, et al. (2004)

Mean ES = 3.31

X — X X X X — X X — — —

5 Fuchs, Fuchs, and Prentice (2004) Mean ES = 2.09

X — X — — X X X X — — X

6 Fuchs, Fuchs, Prentice, Burch, Hamlett, Owen, Hosp, et al. (2003)

Mean ES = 0.66

X — X X X X — X X — — —

7 Fuchs, Fuchs, Prentice, Burch, Hamlett, Owen, and Schroeter (2003)

Mean ES = 1.28

X — X — — — X X X — — X

8 Fuchs, Seethaler, et al. (2008) Mean ES = 1.05

X — X — X X X X X X X X

9 Griffin and Jitendra (2009) Mean ES = -0.06

X — X X X X X — — X — X

10 Jitendra, Dupuis, et al. (2013) Mean ES = 0.00

X — X — — X — — — X — X

11 Jitendra et al. (2007) Mean ES = −0.002

X — X X X X — X — X — X

12 Jitendra et al. (1998) Mean ES = 0.52

X — X — X X — X X — — X

13 Jitendra, Rodriguez, et al. (2013) Mean ES = 0.36

X — X — — X — X — — — X

14 Moran et al. (2014) Mean ES = 0.53

X — X — X — — X — — — —

15 Owen and Fuchs (2002) Mean ES = 3.48

X — X X — X — — X — — X

16 Swanson, Moran, Lussier, and Fung (2014) Mean ES = 0.16

X — X — X — X — — — — —

17 Wilson and Sindelar (1991) Mean ES = −0.01

X — X — — — — X X — — X

18 Xin et al. (2011) Mean ES = 0.01

X X X X X X — — — — — X

18 2 18 9 11 14 7 13 11 5 3 12

Note. ES = effect size; IC1 = explicit instruction; IC2 = technology; IC3 = strategy cues; IC4 = peer interaction; IC5 = instructional feedback; IC6 = visual aids; IC7 = foundational skills; IC8 = schema instruction; IC9 = instruction to transfer; IC10 = manipulatives; IC11 = behavior reinforcement; IC12 = self-regulated learning.

256 Learning Disability Quarterly 44(4)

strategy cues as instructional component. Fourteen of the 18 (78%) of the interventions included visual aids, while 72% (13 out of 18) of the studies included schema instruction. Twelve out of 18 studies (67%) included self-regulated learning in descriptions of interventions. Sixty-one percent of the studies included descriptions of instructional feed- back and instruction to transfer. Peer interaction was reported in 50% of the studies. Instruction and practice in foundational skills such as computation and fact fluency was reported in 39% of the studies. Twenty-eight percent of the studies included concrete math materials and manipula- tives. Behavior reinforcements were reported in 17% of the studies. Finally, only one of the studies included technology tools in the study.

Table 4 shows the fixed effects of studies that included and did not include each instructional component. A Bonferroni correction for multiple comparisons was uti- lized to determine significance (p = .004). A multilevel ANCOVA revealed that studies that included instructional component 4—peer interaction (F1,64 = 13.50, p = .0005) and instructional component 9—instruction to transfer (F1,64 = 10.11, p = .002) yielded significant contrasts when compared with studies that did not included these compo- nents. Studies that included descriptions of peer interaction in the intervention (M = 1.70) yielded significantly higher effect sizes than studies that did not include peer interaction (M = 0.24). Finally, studies that included descriptions of instruction to transfer (M = 1.61) yielded higher effect sizes when compared with studies that did not include transfer instruction (M = 0.42).

Discussion

The purpose of this meta-analysis was to determine whether WPS interventions are effective for improving WPS accu- racy in students with LD in elementary grades and if so, determine whether effect sizes vary as a function of partici- pant and/or instructional components. Three important find- ings emerged. First, problem-solving interventions had a positive effect on WPS accuracy overall. These results were qualified in that the largest effect sizes occurred in intensive interventions (50-min sessions and 34 total sessions). Second, effect sizes for students at risk for LD were higher than for students who were identified as LD through the school dis- trict. Effect sizes for interventions that included a higher ratio of students who were ELs yielded higher mean effects than studies that did not include students who were ELs. Finally, peer interaction and transfer instructions yielded large effects on treatment outcomes relative to the other conditions.

We will now address the three questions that directed this study.

Question 1: Are WPS Interventions Effective for Kindergarten to Sixth-Grade Students With LD?

Generally, WPS interventions were effective for students with LD in elementary grades, resulting in a weighted ES of 1.08 across 18 studies. Two previous studies (Lein et al., 2020; Xin & Jitendra, 1999) that included students in Grades K–12 research has reported divergent effect sizes for ele- mentary grades (g = 0.63 and d = 0.47, respectively). These studies included 11 and 12 studies for elementary- aged students in their respective meta-analyses. This study suggests that recent research in elementary grades have shown that WPS interventions are highly effective for stu- dents with LD. In addition, the results indicated that 50-min sessions and 34 total sessions yielded the highest effect sizes when compared with other reported time durations. Intervention effects were highest in small- and whole-group instructions (compared with individual instruction). In addition, interventions delivered by the classroom teacher and university students yielded highest effect sizes. These results should be interpreted with caution however, as the majority of participants were in third grade and a large num- ber of studies utilized researcher-developed measures.

This finding is consistent with previous research (e.g., Gersten et al., 2009; Zheng et al., 2013) that has found that intensive interventions are effective for students with learn- ing disabilities. Although the results indicated that interven- tions that included 34 total sessions and 50-min sessions yielded the highest effects, these figures are not prescrip- tive, per se. What this seems to reflect is the sentiment that intensive interventions are effective for students with LD. Gersten and colleagues (2009) found a negative correlation

Table 4. Fixed Effects of Instructional Components.

Included Did not include Contrast

Estimate SE Estimate SE F Ratio p Value

IC1 1.09 0.21 0.68 1.28 0.10 .75 IC2 0.54 0.66 1.15 0.22 0.76 .39 IC3 1.17 0.21 0.28 0.66 1.66 .20 IC4 1.70 0.24 0.39 0.26 13.50 .0005a

IC5 0.90 0.29 1.27 0.29 0.84 .36 IC6 1.40 0.24 0.53 0.32 4.73 .03 IC7 1.10 0.35 1.08 0.26 0.00 .96 IC8 1.14 0.24 0.94 0.39 0.19 .66 IC9 1.61 0.25 0.42 0.28 10.11 .002a

IC10 0.58 0.52 1.18 0.22 1.14 .29 IC11 1.02 0.57 1.10 0.22 0.02 .89 IC12 0.68 0.42 1.21 0.23 1.22 .27

Note. IC1 = explicit instruction; IC2 = technology; IC3 = strategy cues; IC4 = peer interaction; IC5 = instructional feedback; IC6 = visual aids; IC7 = foundational skills; IC8 = schema instruction; IC9 = instruction to transfer; IC10 = manipulatives; IC11 = behavior reinforcement; IC12 = self-regulated learning. aBonferroni correction; p = .004.

Kong et al. 257

between the number of treatment sessions in general math instruction and effect size but did not specify the number of sessions. However, these studies may not be directly com- parable as the effects of general math instruction and problem-solving intervention may differ.

This study did not find that there was a significant differ- ence in effect sizes for interventions administered in smaller groups or whole class inclusive settings, though either of these settings yielded higher effects that individual instruc- tion. In addition, the results of this study revealed that effects of interventions delivered by classroom teachers and university students yielded similarly high effects. In previ- ous research (Zhang & Xin, 2012), the issue of administer- ing interventions in special education settings (small group) or inclusive classroom settings (whole class) has been debated. The results of the meta-analysis reveal that for WPS interventions specifically, either of these particular settings did not appear superior in terms of yielding higher effect sizes. We speculate that it is possible that the severity of students’ disabilities may differ in various instructional settings in schools, with students with more severe needs requiring more intensive interventions (to be discussed below under Question 2). However, these findings support the importance of providing quality evidence-based instruc- tion in Tier 1 general class instruction before the need for intensive interventions in smaller groups is needed. WPS interventions delivered in general class instruction may have great potential for students with learning disabilities and students at risk alike, bolstering the need for quality Tier 1 instruction.

Of the 18 studies included in this study, 12 studies uti- lized researcher-developed tests, two used standardized assessments, and four used both. Results indicated that effect sizes on researcher-developed measures were signifi- cantly higher than standardized measures, which seems consistent with previous research that have indicated the possibility of alignment of the intervention materials and researcher-developed probes, which mirrors curriculum- based measures that are more sensitive to changes (Zhang & Xin, 2012). This finding, however, is particularly of interest for teachers of students with LD who may be receiv- ing special education services in schools. This finding affirms the importance of utilizing curriculum-based mea- sures to monitor progress and evaluate intervention effec- tiveness for specific skills that are taught in the classroom.

Question 2: Do Specific Effect Sizes in WPS Interventions Vary as a Function of Moderator Variables Such as Participant Characteristics (EL Status, LD Definition, Grade Level)?

Eight studies included descriptions of students who were identified as LD via the discrepancy model and/or through the school district. These studies ranged from 1991 to 2009. With more recent efforts to address limitations to the

discrepancy model of identifying children with LD, the Response to Intervention (RtI) model has been recom- mended (Individuals with Disabilities Education Act, 2004). Students who are “at risk” for LD, or achieving below a designated cut-off point (e.g., 25th percentile), would be eligible to receive intervention to begin to remedi- ate any existing achievement gaps. Studies that included students “at risk” ranged from 2003 to 2014. The results of this study indicated WPS interventions were more effective for students at risk for LD than for students identified as LD through a discrepancy model. As mentioned earlier, it is possible that students who are diagnosed as LD via the dis- crepancy model may have more extensive needs. However, these findings seem to support the notion that the RtI model might be a start to differentiate between students who respond to intervention and were merely at risk for MD, and those who do not and may require more intensive support, all the while providing much-needed instruction to low- achieving students (Fuchs, Mock, et al., 2003).

One of the areas that is particularly difficult for EL stu- dents is solving math word problems (Bumgarner et al., 2013; Powell et al., 2020). The results of this study indicated that studies that included students who were ELs yielded higher effects than ones that did not. This supports the emerging research that demonstrates that problem-solving interventions are highly effective for elementary students who are ELs (Gersten & Baker, 2000; Kong & Swanson, 2019; Orosco et al., 2011; Swanson et al., 2019). However, these findings should be interpreted with caution, as some studies may have included participants who were ELs, but were not reported as such in the studies we reviewed. It is also worth noting the small percentage of students that were reported as ELs in the studies included in this meta-analysis (5.06%) compared with national averages (9.6% nationally in 2016; U.S. Department of Education, 2019). Previous meta-analyses on the effects of WPS for K–12 students with LD (Xin & Jitendra, 1999; Zhang & Xin, 2012; Zheng et al., 2013) have not included EL status as a moderating variable.

Finally, a majority of the studies included participants in the third grade. Effect sizes of interventions provided for third-grade students were significantly higher than the effect sizes for second-, fourth-, and fifth-grade students. No studies of WPS interventions for young children (K–1) were included in this study. Future studies could investigate story problem interventions for young children, as well as the continued effectiveness of problem-solving interven- tions for older elementary students.

Question 3: Which Specific WPS Interventions/ Components of WPS Intervention Are Effective With Kindergarten to Sixth-Grade Students With LD?

The results indicated that descriptions of peer interaction were reported in nine out of the 18 studies included in this

258 Learning Disability Quarterly 44(4)

meta-analysis. Of those nine studies, five included descrip- tions of students identified as LD via the discrepancy model, and four studies included students at risk. These studies that included descriptions of peer interaction (to present, model, practice, or review instruction) in the inter- vention yielded higher effect sizes than studies that did not include peer interaction. This indicated that WPS interven- tions for students with LD should include opportunities for students with LD to collaborate and interact with more skilled peers. This does not seem to support the existing literature on peer-assisted learning in math for students with LD (Gersten et al., 2009). Gersten and colleagues’ meta-analysis on math instruction for students with LD found that while studies that included cross-age tutoring yielded high effect sizes, studies that included peer-assisted learning or peer interaction within the class did not yield high effect sizes (g = 0.14). One point to consider, how- ever, is that this previous analysis included studies in all math interventions broadly and across all grade levels (K–12) and not WPS in elementary-age students specifi- cally. Further analysis on the moderating effects of grade and WPS interventions specifically were not considered. It may be possible that interventions of WPS that include peer interaction and mathematical discourse may be better suited for elementary grades or for WPS specifically. Learning via peer interaction is consistent with the social development theory (Vygotsky, 1978), in which children acquire knowl- edge through social and verbal experiences from a more knowledgeable individual. As suggested by Gersten and colleagues (2009), when provided explicit and structured guidelines and moderated by teachers, elementary-aged stu- dents with LD may perhaps be able to learn new WPS skills from interaction with their peers.

In addition, students with LD in elementary grades ben- efited from explicit instruction to transfer learned skills to novel problems. This finding supports the existing literature on instructional components that improve students’ WPS skills (Griffin et al., 1994; National Research Council, 2001; Zheng et al., 2013). Similar to other academic skills, it is important for young students to transfer knowledge of skills to novel situations. WPS may be a crucial medium to select and apply strategies to solve everyday problems.

Limitations

Although this synthesis provided information about stu- dents with LD in the elementary grades, the findings should be interpreted with caution. First, the criteria for determin- ing at-risk students varied across studies. Although we did attempt to categorize studies based on how students were identified, criteria differed even within those categories. Second, only group studies published in peer-reviewed arti- cles were included, excluding unpublished work, disserta- tions, and single-subject designs. These selection processes

reduce generalization of our findings. Finally, a majority of the studies included in this meta-analysis included partici- pants in third grade, which may limit the generalization of these findings.

Implications for Practice

The present meta-analysis found that WPS interventions, specifically those that include peer interaction and explicit instruction to transfer learned skills to novel problems, are effective for elementary students with LD. Elementary stu- dents with LD or at risk for LD may benefit from WPS interventions with opportunities to use language and inter- act with peers and instructors to transfer skills or schema to new problems. The results of this review suggest that these instructional components are more effective for students who are at risk for LD. In addition, students may also ben- efit from intensive intervention regardless of the instruc- tional setting. This supports the significance of delivering evidence-based instruction in the general classroom (Tier 1 instruction) before resources for small group instruction are utilized.

More research is needed to identify effective compo- nents of instruction for students in elementary school who are at risk for or identified as LD. Particularly, research should be conducted with students in primary grades (K–2), to identify possible precursors for WPS difficulty and early interventions. In addition, future studies should consider learner characteristics, particularly for those who are most at risk (ELs, low socioeconomic status, LD).

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.

ORCID iD

Jennifer E. Kong https://orcid.org/0000-0001-7520-8023

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