a follow-up meta-analysis for word-problem-solving interventions for students with mathematics...
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This article was downloaded by: [University of North Carolina]On: 05 October 2014, At: 15:02Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK
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A Follow-Up Meta-analysis for Word-Problem-Solving Interventions for Students with MathematicsDifficultiesDake Zhang a & Yan Ping Xin ba Rutgers Universityb Purdue UniversityPublished online: 02 Aug 2012.
To cite this article: Dake Zhang & Yan Ping Xin (2012) A Follow-Up Meta-analysis for Word-Problem-SolvingInterventions for Students with Mathematics Difficulties, The Journal of Educational Research, 105:5, 303-318, DOI:10.1080/00220671.2011.627397
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The Journal of Educational Research, 105:303–318, 2012Copyright C© Taylor & Francis Group, LLCISSN: 0022-0671 print / 1940-0675 onlineDOI:10.1080/00220671.2011.627397
A Follow-Up Meta-analysis forWord-Problem-Solving Interventions
for Students with MathematicsDifficulties
DAKE ZHANGRutgers University
YAN PING XINPurdue University
ABSTRACT. Following a meta-analysis study conductedby Y. P. Xin and A. Jitendra (1999), the authors carriedout a follow-up meta-analysis of word problem-solving in-terventions published from 1996 to 2009 for students withlearning problems in mathematics. The authors examined theinfluence of education reforms as moderator variables on in-tervention effects, including inclusive movement, responseto intervention model, standard movement, and mathemat-ics education reform. The researchers analyzed 29 group-design studies and 10 single-subject-design studies that metthe criteria for inclusion. Separate analyses were performedfor group-design studies and single-subject studies using stan-dardized mean change and percentage of nonoverlapping data(PND), respectively. The overall mean-weighted effect size(d) and PND for word problem-solving instruction were pos-itive across the group-design studies (d = 1.848) and single-subject studies (PND = 95%). Implications for policymakersand researchers were discussed within the contexts of inclu-sive education, standard based movement, the response tointervention model, and mathematics education reform.
Keywords: intervention, mathematics difficulties, meta-analysis
I mproving mathematics problem-solving skills of allstudents including those with learning disabilities(LD) or learning problems (LP) is one of the prior-
ities of the United States (National Council of Teachers ofMathematics [NCTM], 2000; National Mathematics Advi-sory Panel, 2008). In fact, the recent National Assessment ofEducational Progress (NAEP; 2008) shows that while otherstudents made progress in mathematics over the years, stu-dents with LP did not show significant improvement. Stu-dents with LP manifest severe problems in word-problemsolving (Cawley, Parmar, Foley, Salmon, & Roy, 2001). Inword problems, implied mathematics operations (e.g., add,multiply) are represented by the actions in problem situationthrough language (Mastropieri & Scruggs, 2007). Given thedifficulties experienced by many students with LP in word-
problem solving, one must take a close look at recent inter-vention studies in solving word problems for students withLP.
Meta-analysis is a statistical technique that providesa quantitative summary of findings and characteristics ofmany empirical studies (Arthur, Bennett, & Huffcutt,2001; Glass, 1977). Xin and Jitendra (1999) published ameta-analysis study that synthesized all interventions ofword-problem solving for students with identified disabili-ties or those receiving remedial mathematics instruction.This meta-analysis (Xin & Jitendra, 1999) examinedmoderator variables that may affect the treatment effectsizes, including students’ characteristics (e.g., students IQand age), intervention characteristics (e.g., the length ofinterventions, intervention strategies, implemented byschool teachers or researchers), and other research char-acteristics (e.g., published journal papers or unpublisheddissertations). The computer-assistive technology thatintegrated effective instructional design was identified asmost effective. It reported that the long-term interventionswere more effective than short-term interventions.
Since 1999, a few meta-analysis or research synthesis stud-ies have been published (e.g., Kroesbergen & Van Luit, 2003;Maccini, Mulcahy, & Wilson, 2007; Gersten et al., 2009).All these meta-analyses or research syntheses were from aperspective of cognition and instruction. Kroesbergen andVan Luit (2003) first conducted a meta-analysis of 58 studiesin mathematics interventions for elementary students withspecial education needs. This meta-analysis evaluated inter-ventions in the area of preparatory arithmetic, basic skills,and problem solving. For the problem-solving instructionin particular, results indicated that (a) peer tutoring andcomputer-assisted instructions were less effective than otherintervention methods such as teacher-delivered instruction
Address correspondence to Dake Zhang. (E-mail: [email protected])
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and direction instruction, (b) interventions for students withmild mental retardation were more effective that those forstudents with LD, and (c) interventions for students withmixed disabilities are less effective than interventions forlow performers. Additionally, a review by Maccini et al.(2007) extended a previous review (Maccini & Hughes,1997) on mathematical interventions for secondary schoolstudents with LD. It described existing effective interven-tion strategies (e.g., mnemonic strategy instruction, gradu-ated instructional approach, cognitive strategy instruction,contextualized video disc instruction). Most recently, Ger-sten et al. (2009) conducted a meta-analysis study that in-cluded 42 mathematics intervention studies for students withLD. Gersten et al. focused on approaches to instruction orcurriculum design, formative assessment data and feedbackto teachers about students’ performance or to students ontheir performance, and peer-assisted mathematical instruc-tion. Heuristics and explicit instruction yielded practicallyand statistically important increase in effect size.
The purpose of present meta-analysis study was somewhatdifferent. During the past several years, substantial changeshave taken place in the field of special education and ed-ucation in general. Such changes include the No ChildLeft Behind Act (NCLB; 2002), the Individuals with Dis-abilities Education Act (IDEA; 2004), the inclusion move-ment (IDEA, 2004), the standards-based movement (NCLB,2002), and the response to intervention (RtI) model (IDEA,2004). Reforms also have taken place in the field of math-ematics education, such as emphasis on real world problemsolving (NCTM, 2000) and algebra readiness in elementaryschool curriculum (NCTM, 2000). No study has examinedhow these changes have affected intervention practice orword-problem solving for students with LP. In the presentarticle we discuss these areas relative to a review of the lit-erature and subsequently specify the present study’s researchquestions.
Inclusion Movement
Inclusive education or mainstreaming entails the processof integrating students with disabilities into general edu-cation classes to address the requirement of the least re-strictive environment (IDEA, 2004). Despite “the rally cry-ing today for inclusive schools” (D. Fuchs & Fuchs, 1994,p. 295), justification of the inclusive movement remainsunclear (Cook, Cameron, & Tankersley, 2007). A meta-analysis in 1980 (Carlberg & Kavale, 1980) reported thatspecial education settings were significantly superior to regu-lar classrooms for students with LD. Additionally, a researchsynthesis (Scruggs, Mastropieri, & Casto, 1987) reportedthat only a minority of teachers agreed that the general ed-ucation classroom was the best environment for studentswith special needs and that many teachers’ concerns werethat students with disabilities have a negative effect on theclassroom environment. Overall, two major questions have
been voiced over the years. First, do students with LP learneffectively in inclusive education settings? Second, do stu-dents with disabilities interfere with the normal-achievingstudents’ learning progress (Barton, 1992)? One of the inter-ests of this meta-analysis was to examine whether placementfrom the special education setting to the inclusive classroomaffects the intervention effectiveness.
Standards-Based Movement and Evidence-Based Interventions
Standards-based reform is a predominant challenge pub-lic schools must face. The standards-based reform movementcalls for clear, measurable standards for all school studentsand calls for evidence-based interventions to facilitate stu-dents to meet the standards. According to NCLB (2002),public schools receiving federal Title 1 funding but failingto meet the criteria of Adequate Yearly Progress for twoconsecutive years must offer choice or provide supplemen-tal services, measured by standardized tests. The standards-based movement has undoubtedly influenced specialeducation. Consequently, schools desire effective researchinterventions measured by such standardized high-stakes as-sessments to assure schools make adequate annual progress.The majority of previous research has adopted self-developedprobes to measure the effectiveness of an intervention; how-ever, such self-developed probes are not as reliable and validas standardized test (Kuncel & Hezlett, 2007). In addition,these self-developed probes may exactly reflect what weretaught during the intervention program, and thus could bemore sensitive to students’ short-term growth of the specif-ically taught academic skills. Therefore, it is important toquestion the effectiveness of such interventions on students’performance on standardized high-stakes assessment. In thepresent meta-analysis we were interested in examining if (a)present interventions have provided standardized measures(especially high-stakes assessment) for validating the effec-tiveness of interventions, or if present interventions stillstick with researchers’ self-developed measurement; and (b)there is a difference between the effect sizes measured withself-developed probes or standardized testing.
IDEA (2004) calls for implementing evidence-based prac-tice instruction and curriculum and providing services thatare based on peer-reviewed research to help students withspecial needs meet the standards. Evidence-based interven-tions refer to instructional practice that is supported by em-pirical research evidence of statistically significant effective-ness as treatments for specific problems (Morris & Mather,2008). A statistical meta-analysis can be used to draw astronger conclusion than a single research study about theeffectiveness of an intervention or a treatment when it iscompared to other treatments addressing the same problem(Odom et al., 2005). An interest of this meta-analysis was toexamine the effectiveness of different intervention strategiesfor teaching mathematics problem solving to students withLP.
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The RtI Model and Modification of the LD IdentificationProcedure
The traditional definition of LD is based on the discrep-ancy hypothesis, which means that a child’s actual achieve-ment is far below his or her expected achievement, as statedby the U.S. Office of Education (1977). The weakness ofsuch a definition is that children who were clearly strug-gling as early as kindergarten or Grade 1 had to wait untilthey failed to meet the discrepancy requirement (1- or 2-SDdiscrepancy between their IQ and achievement; Cutting,Clements, Lightman, Yerby-Hammack, & Denckla, 2003).Another concern about the discrepancy model regards ex-clusionary variables. For example, some students may beinstructional casualties resulting from poor instruction, in-stead of their real cognitive deficiencies (Shaywitz, Morris,& Shaywitz, 2008).
To address limitations of the discrepancy model(Shaywitz et al., 2008), IDEA was revised in December2004; the RtI model was recommended to identify childrenwith LD (D. Fuchs & Fuchs, 2006; Gresham, 2002). The RtImodel is based on students’ achievement and their responseto intervention (D. Fuchs, Mock, Morgan, & Young, 2003).With the RtI model, students are identified as having LDwhen their response to generally effective instruction is dra-matically inferior to that of normal-achieving students (D.Fuchs, Fuchs, & Compton, 2004). The present movementaway from discrepancy-based models of diagnosing an LD(U.S. Department of Education Office of Special Educationand Rehabilitative Services, 2002) might be beneficial forlow-achieving children, because they would be eligible forearly intervention, even when no discrepancy between IQand achievement is demonstrated (Cutting et al., 2003).
The change of LD definitions may have resulted in des-ignating different participants of the intervention studies.This may lead to an increasing number of interventionstudies focused on students at risk. Additionally, the at-riskor low-achieving students should have a better responseto interventions than students with real LD, a constructdeemed to be more related to cognitive deficits rather thaninstruction. In the realm of reading, a meta-analysis hasfound an overlap between the discrepant children and sim-ply low-achieving poor readers on reading-related constructs(Stuebing et al., 2002). Francis, Shaywitz, Steubing, Shay-witz, and Fletcher (1996) reported that the students withtraditional LD and those at risk for failure in readingdemonstrate comparable growth rates in word readingduring the school years. Unfortunately, no meta-analyseshave examined if the two populations share comparablefeatures during learning mathematics. According to theRtI model, the at-risk population should have a betterresponse to intervention than the students with real LD. Inthe present meta-analysis we explored if the discrepant LDmodel reflects the nonresponding nature of the construct ofreal LD. In other words, do the children with discrepant LDrespond poorer than at-risk students?
Reform in Mathematics Education
NCTM’s call for teaching real-world problem solving. TheNCTM standards (2000) calls for mathematics reform,which emphasizes thorough understanding and practical ap-plications. Teaching real-world problems helps students gaincompetence in daily life and work. In real life, problem fea-tures are often ill-defined. For instance, some problems mayinclude extra or irrelevant information; some may need mul-tiple steps to solve the problems; some need a round-up orround-down of the solution for reaching a meaningful answer(Xin & Zhang, 2009). In the present study we explored ifthe effects of interventions for solving real-world problemsdiffer from the effects of interventions for solving simple-structured essential problems.
NCTM’s call for algebra readiness. Algebra readiness hasbeen characterized as serving a gate-keeping function for sec-ondary and postsecondary education (Cai & Knuth, 2005;Maccini, McNaughton, & Ruhl, 1999). In fact, NCTM(2000) endorsed algebra as a K–12 enterprise and the Na-tional Research Council (Kilpatrick, Swafford, & Findell,2001) encouraged teachers to introduce basic algebraic con-cepts in early elementary grades to lay a foundation for al-gebra instruction in later years. For example, instructionalprograms should start from prekindergarten to enable stu-dents to obtain early experiences with patterns or with mod-eling problem situations with unknown quantities (NCTM,2000). These calls for algebra readiness or early introductionof algebra concepts in elementary curriculum have beenapplied to general education (Earnest & Balti, 2008; Xin,2007). However, concerns were voiced from educationalpractitioners doubting whether LD students could masteralgebraic concepts, which are more abstractive and more dif-ficult than arithmetic approaches (Maccini et al., 1999). Thepresent study was to evaluate the effectiveness of teachingalgebraic problem solving compared to teaching arithmeticproblem solving.
Research Questions
In the present meta-analysis we aimed to use the meta-analysis technique to investigate the effects of the moderatorvariables pertinent to the previously mentioned educationreforms on interventions for teaching word-problem solv-ing to students with LP in mathematics. Specific researchquestions included the following:
Research Question 1: Inclusion movement: (a) Were the in-terventions in inclusive classroom as effective as in spe-cial education settings for students with LP? (b) Werethe interventions in the inclusive classroom equivalentlyeffective for students with and without LP?
Research Question 2: Standards-based movement andevidence-based interventions: (a) What was the rela-tive effectiveness of various intervention strategies in
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teaching word-problem solving to students with LP? (b)Did the intervention effects vary with different measures,such as the high-stakes testing, researcher-developed test-ing, or other standardized tests?
Research Question 3: The RtI model and modification ofLD identification procedure: (a) Were more interventionsfocusing on at-risk students than 13 years ago? (b) Didstudents with traditional diagnosis of LD virtually respondto interventions less effectively than at-risk students?
Research Question 4: Reform in mathematics education: (a)Did the effects of interventions for solving real-worldproblems differ from the effects of interventions for solvingsimple-structured essential problems? (b) Were interven-tions for algebraic or algebraic-oriented problem solvingas effective as arithmetic problem solving for students withLP?
Method
Literature Search Procedures
Search procedures consisted of three steps: (a) an elec-tronic search in the library database through ERIC, Psych-INFO, Wilson Select, ProQuest, and Dissertation AbstractsInternational from 1996 to 2009. Descriptors for the com-puterized search included the following combinations: dis-abilities, disable, special education, learning disability, at risk,learning problems, word problem, story problem, problem solv-ing, intervention and instruction; (b) a hand-search of recentissues of refereed journals in special education and mathe-matics education, including Exceptional Children, Journal ofLearning Disabilities, The Journal of Special Education, LearningDisability Quarterly, Learning Disabilities Research & Practice,and Journal for Research in Mathematics Education; and (c) anancestral search using reference sections of identified articlesobtained through the previous steps.
Criteria for Inclusion
To be included in the review, the article must have had (a)included K–12 students with LP in mathematics; (b) used anexperimental, quasiexperimental, or single-subject design;(c) examined effects of an instructional intervention; and(d) included word-problem solving as one of the dependentmeasures. Exclusion criteria in the present study involvedthe following: study did not involve word-problem solvingas an independent measure (Brosvic, Dihoff, & Epstein,2006; Calhoon & Fuchs, 2003; Kroesbergen, Van Luit, &Maas, 2004; Mayfield & Vollmer, 2007; Witzel, Riccomini,& Schneider, 2008) or was not empirical (e.g., Bottge, 2001;Garcia, Jimenez, & Hess, 2006; Naglieri, & Johnson, 2000).Other studies targeted children with other disabilities thanLD (Bottge, Rueda, & Skivington, 2006; Chung & Tam,2005; Morin & Miller 1998) or did not specifically report theperformance of children with LP (L. S. Fuchs et al., 2006).Additionally, studies that did not involve K–12 students
(Sullivan, 2005) or studies published in 1996 but werereviewed in the meta-analysis of Xin and Jitendra (1999;Cassel & Reid, 1996; Jitendra & Hoff, 1996) were excluded.
The literature search and selection procedures identified33 published studies and six unpublished doctoral disserta-tions. The final sample of 39 studies yielded 29 group studiesand 10 single-subject studies.
Coding Study Features
The following information was coded from each article todescribe the main characteristics of the individual study.
Inclusion movement. Two variables were coded under thismovement. The first, the setting variable, refers to where theintervention was implemented, either in an inclusive class-room or a special education setting (i.e., remedial class, re-source room, self-contained room, or pull-out settings). Thesecond regards student labels in inclusive settings. For stu-dents within the inclusive settings, participants were furthercategorized into two groups: (a) normal-achieving studentsor (b) students with LP in mathematics.
Standard movement. Two variables were coded under thismovement. The first, intervention strategy used, refers tothe orientation of each intervention strategy. Strategieswere coded into one of the four categories: problemstructure representation techniques, cognitive strategytraining, assistive technology, or traditional instruction.Such grouping does not indicate mutually exclusiveinterventions. On the contrary, it often happens thatapproaches overlap and share some similar components.In such cases, the researchers determined the codingcategory on the basis of the primary study’s orientation oremphasis of the intervention examined. Problem structurerepresentation techniques refer to the explicit instructionthat focuses on helping learners understand and representthe problem in schema-based diagrams (e.g., Jitendra &Hoff, 1996) or in mathematical models (e.g., Xin, Wiles,& Lin, 2008). For cognitive strategy training, studies werecategorized as cognitive interventions if they included sometype of cognitive or metacognitive strategies, graduatedinstructional sequence, or mnemonic strategy instruction.Assistive technology pertained to intervention employingcomputer-aided or video disc programs, or any technicalaccommodations such as cassettes and calculators.
The second variable, measures, refers to whether thedependent measure was researcher-designed testing, high-stakes testing, or other standardized testing. Both high-stakes assessments (e.g., Indiana State Test of EducationProficiency [ISTEP+], Pennsylvania System of School As-sessment [PSSA]) and other commercially published stan-dardized assessments (e.g., Stanford Achievement Test[SAT; Harcourt Brace, 2003], KeyMath [Connolly, 1998],Woodcock-Johnson Test of Achievement) belong to
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standardized testing. Some studies (e.g., Bottge, Rueda, Ser-lin, Hung, & Kwon, 2007; L. S. Fuchs, Fuchs, Hamlett, &Appleton, 2002; Maccini & Hughes, 2000) involved stu-dents’ standard test scores as students’ demographic infor-mation but did not involve standard testing as one of theposttest measures. Such cases were not coded as using high-stakes testing.
RtI. One variable, the targeted population, was codedto investigate the influence of the RtI targeted popula-tion. This variable refers to two levels: students with anLD who were identified by the discrepancy model or stu-dents with an LP or who were at risk. A study coded asthe discrepancy model should explicitly define students ashaving an LD due to discrepancy between IQ and lowachievement scores. A study coded as falling in the at-riskmodel may focus on low-achieving students or students whowere referred by school teachers at risk of failure in math-ematics. Some studies reported that low-achieving partici-pants included several students with an LD, without spec-ifying how they defined LD. Such cases were coded as atrisk.
Mathematics education reform. The first variable, algebrainstruction, refers to instructional strategies that focus onalgebraic thinking in solving either arithmetic or algebraic(prealgebraic) problems. Specifically, the (pre)algebraicproblem-solving instruction was coded where students usedsymbols to represent numbers and use equations to repre-sent numerical relations. Otherwise, the interventions werecoded as nonalgebra instruction. For example, for teachingthe same problem, “Emily has 3 books. Emily and John have7 books altogether. How many books does John have?”, thealgebraic interventions would encourage students to thinkin terms of an expression of mathematical relations includ-ing known and unknown quantities (e.g., 3 + x = 7, 3+ = 7), whereas arithmetic interventions would justfocus on the arithmetic (7 − 3). It is important to notethat some problems could only be solved with an algebraicapproach (e.g., Bottge, Rueda, Serlin, et al., 2007), so theywere coded as algebraic although some interventions didnot specify the algebraic thinking. Some other tasks can besolved through either arithmetic or algebraic/prealgebraic-oriented approaches (Linchevski, 1995), and only those thatspecifically addressed the algebraic thinking were coded as(pre)algebraic.
The second variable, tasks, refers to real-world problemsor simple-structured essential word problems. Simple-structured essential problems were defined as problemswithout irregular context. Tasks with irregular contextsor structures were coded as real-world problems, such asill-defined problems, problems with irrelevant informationor graphs, multistep problems, problems requiring back-ground information, and problems requiring rounding up orrounding down the final answer.
Effect Size Calculation
Group design studies. Consistently following Xin and Ji-tendra (1999), the standardized mean change was used tocalculate intervention effects (Becker, 1988). The standard-ized mean change computed for each sample was deemedto be an appropriate measure of effect size for within-subject comparisons. The reason for that was because themajority of the studies compared the effects of two ormore intervention approaches, rather than the effects ofone type of intervention versus no intervention. The stan-dardized mean change allows for the direct comparison ofdata from studies using different designs, and, thus, stud-ies without control groups need not be omitted (Becker,1988).
Consistent with Xin and Jitendra (1999), Cohen’s d wascomputed as the difference between the posttest and pretestmeans for a single sample. Cohen’s d was usually calculatedas the difference between the means of the experimentalgroup and control group, divided by the pooled standarddeviation (Hedges & Olkin, 1985). As for the measureswithin the same group (i.e., pretest and posttest), Dunlop,Cortina, Vaslow, and Burke (1996) convincingly claimedthat the formula of Cohen’s d for independent groups (i.e.,unpaired experimental group vs. control group) could beused to compute effect size for correlated designs (i.e., pretestvs. posttest).
Weighted Cohen’s d was used to measure the average ef-fect size within a group of interventions sharing a commoncharacteristic. When combining a group of studies sharingcommon characteristics, it is necessary to note that studieswith larger sample sizes are weighted more than those fromstudies with smaller sample sizes. Through grouping studentsby the variables the researchers coded, we calculated theweighted Cohen’s d as effect size for each subgroup and the95% confidence intervals of the effect sizes. Then the re-searchers were able to make a general comparison betweenthe effect sizes of subgroups.
Consistent with Xin and Jitendra (1999), the Q statisticsof Hedges and Olkin (1985) were used to explore the ho-mogeneity of effect sizes to determine if the sampling erroralone would cause the variances among effect sizes. If theeffect sizes of studies were not homogenous, then the vari-ance could not be interpreted by chance. Q statistics havean approximate chi-square distribution with k − 1 degreesof freedom, where k is the number of effect sizes (Hedges& Olkin, 1985). Using the Comprehensive Meta-analysisProgram (Borenstein, Hedges, Higgins, & Rothstein, 2005),we calculated Qb to estimate the homogeneity of effect sizesbetween groups with a fully random model. Within the fullyrandom model, a random-effects model was used to combinestudies within each subgroup; and a random-effects modelwas used to combine subgroups and yielded an overall effect(Borenstein et al., 2005). Alpha levels for all analyses ofhomogeneity of effect sizes in this meta-analysis were set atthe .05 level of significance.
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Single-subject studies. Consistent with Xin and Jitendra(1999), the percentage of nonoverlapping data (PND;Kavale, Mathur, Forness, Quinn, & Rutherford, 2000) wasused as an effect size measure for single-subject-design stud-ies. Specifically, PND was computed as the number of treat-ment data points that exceeded the highest baseline datapoints in an expected direction, divided by the total numberof points in the treatment session (Scruggs et al., 1987). Thedependent measure was the percentage of correct answers toword problems. The PND score for each subject of interven-tion was calculated independently within the study. Whena multiple baseline design across participants was used, theindividual PND score was calculated for each subject; a me-dian was deemed as the overall effect size of the intervention.A total of 13 PND scores were obtained from the 10 single-subject-design studies. Because PND scores are not alwaysnormally distributed, the researchers reported median scoresthat are less likely to be affected by outliers than mean scores(Scruggs et al., 1987).
Interrater Agreement
A second rater independently coded 30% of all the se-lected studies. Interrater reliability was defined as the to-tal number of agreement divided by the total number ofagreement and disagreements and multiplied by 100%. Theinterrater reliability for coding was 95%. A discussion washence held between raters; any disagreements were resolvedby further clarifying the coding definition.
Results
Overall Effect of Mathematics Word-Problem-SolvingInstruction in the Past 13 Years
All but six effect sizes in the reviewed studies werecalculated from the reported mean and standard deviationof the scores in pretests and posttests. Only six effect sizesof two studies (Hasselbring & Moore, 1996; Stellingwerf& Van Lieshout, 1999) were calculated according top values. Table 1 reports the unbiased effect sizes (d). Themean effect size before correction for sample size (d) was1.580 with a range from −0.54 to +11.82, whereas in Xinand Jitendra’s (1999) study the unbiased d was 0.89, witha range from −0.42 to +6.77. Table 1 suggests that thelargest effect size (d = 11.82) was from the study of L. S.Fuchs et al. (2004), and the three negative sizes were fromJitendra, Griffin, Deatline-Buchman, and Sczesniak (2007);Ginsburg-Block and Fantuzzo (1998); and Toppel (2004).
Moderator Influences on the Effect of Word-Problem Solving
Moderator variables were analyzed to explore theirinfluence on mathematical problem-solving instruction.These moderator variables were related to inclusive move-ment (e.g., inclusive classroom or special education set-
ting, normal-achieving students, students with mathemat-ics learning difficulties), RtI movement (e.g., students withdiscrepancy LD or at-risk students), and standard move-ment (e.g., high-stakes test, researcher-developed test, otherstandardized test) and mathematics education reforms (e.g.,arithmetic- or algebraic-oriented problem solving, simple-structured or real-world problems). Results are presented inTable 2.
Variables of inclusion movement. Results of the first vari-able, settings, are presented in Table 3. A total of 19 effectsizes from inclusive settings and 29 effect sizes from specialeducation settings were yielded. Analyses indicated a signif-icant difference of the unbiased effect sizes of interventionsin special education settings and in inclusive settings (Qb =9.938, p < .01). Specifically, interventions implemented inspecial education settings (d+ = 1.348) displayed a smallereffect size than those implemented in inclusive classrooms(d+ = 2.600).
For the second variable, student labels in inclusive set-tings, a further comparison was conducted between the effectsizes of interventions for normal-achieving students and forstudents with LP in inclusive settings. As seen in the analy-sis presented in Table 2, normal-achieving students seemedto yield a similar effect size (d+ = 2.993) to students withLP (d+ = 2.571) and no statistically significant differenceswere found (Qb = 0.521, p > .05).
Variables of standard movement. Six effect sizes wereyielded with other standardized tests, and 46 effect sizes werecalculated from studies with researcher-developed tests. Onestudy (Jitendra, Griffin, Haria, et al., 2007) did not employthe same standardized instruments on pre- and posttesting,and, thus, this study was excluded when calculating the ef-fect sizes on standardized measures.
Due to the small number of effect sizes yielded from high-stakes testing, it is risky to make any conclusions whetherinterventions yielded different effect sizes on high-stakestesting or other tests. On the other hand, the effect sizemeasured with standardized testing, if combining high-stakestesting and other standardized testing, was significantly lowerthan the effect size measured by the self-development testing(Qb = 5.142, p < .05). Specifically, the weighted effectsize for a self-developed test was 1.872, and the weightedeffect size for standardized testing was 0.597. Thus, thereis a significant difference between weighted effective sizesmeasured with two types of measurement. In other words,the measurement mediates the effect size of the intervention.
For the second variable, intervention strategies, signif-icant differences were found among the weighted effectsizes from the three intervention types (Qb = 9.510, p< .05). Sixteen treatments adopted problem structurerepresentation techniques and yielded a mean effect sizeof 2.637, the highest effect size among all four categories.Cognitive strategy training involved 12 treatments andyielded a mean effect size of 1.855. Assistive technology
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orth
Car
olin
a] a
t 15:
02 0
5 O
ctob
er 2
014
TA
BL
E1.
Sum
mar
yof
Gro
up-D
esig
n-St
udie
s
Stud
yLD
defin
itio
nSe
ttin
gM
easu
res
Inte
rven
tion
stra
tegi
esA
lgeb
rain
stru
ctio
nT
asks
Effe
ctsi
zes
Bot
tge
(199
9)D
iscr
epan
tLD
Spec
iale
duca
tion
Res
earc
her-
deve
lope
dte
stA
ssis
tive
tech
nolo
gyA
rith
met
icSi
mpl
ean
dre
al-w
orld
prob
lem
s0.
560
Bot
tge,
Hei
nric
hs,C
han,
&Se
rlin
(200
1)A
tris
kSp
ecia
ledu
cati
onR
esea
rche
r-de
velo
ped
test
and
stan
dard
ized
test
Ass
isti
vete
chno
logy
Alg
ebra
icR
eal-
wor
ldpr
oble
ms
4.03
6
Bot
tge,
Hei
nric
hs,M
ehta
,&H
ung
(200
2)A
tris
kIn
clus
ive
educ
atio
nR
esea
rche
r-de
velo
ped
test
Ass
isti
vete
chno
logy
Ari
thm
etic
Sim
ple
and
real
-wor
ldpr
oble
ms
0.32
3
Bot
tge,
Hei
nric
hs,C
han,
Meh
ta,&
Wat
son
(200
3)A
tris
kSp
ecia
ledu
cati
onR
esea
rche
r-de
velo
ped
test
Ass
isti
vete
chno
logy
Ari
thm
etic
Rea
l-w
orld
prob
lem
s0.
506
Ass
isti
vete
chno
logy
Ari
thm
etic
0.32
3B
ottg
e,R
ueda
,LaR
oque
,Se
rlin
,&K
won
(200
7)D
iscr
epan
tLD
Spec
iale
duca
tion
Res
earc
her-
deve
lope
dte
stan
dst
anda
rdiz
edte
st
Ass
isti
vete
chno
logy
Alg
ebra
icR
eal-
wor
ldpr
oble
ms
0.88
9
Ass
isti
vete
chno
logy
Alg
ebra
ic2.
593
Bot
tge,
Rue
da,S
erlin
,Hun
g,&
Kw
on(2
007)
Atr
isk
Incl
usiv
eed
ucat
ion
Res
earc
her-
deve
lope
dte
stA
ssis
tive
tech
nolo
gyA
lgeb
raic
Rea
l-w
orld
prob
lem
s4.
796
Ass
isti
vete
chno
logy
Alg
ebra
ic2.
416
L.S.
Fuch
s,Fu
chs,
Cra
ddoc
k,H
olle
nbec
k,H
amle
tt,&
Scha
tsch
neid
er(2
008)
Atr
isk
Incl
usiv
ecl
ass
Res
earc
her-
deve
lope
dte
stan
dst
anda
rdiz
edte
st
Prob
lem
stru
ctur
ere
pres
enta
tion
Ari
thm
etic
Sim
ple
and
real
-wor
ldpr
oble
ms
4.03
0
L.S.
Fuch
s,Fu
chs,
Pren
tice
,H
amle
tt,F
inel
li,&
Cou
rey
(200
4)
Atr
isk
Incl
usiv
ecl
ass
Res
earc
her-
deve
lope
dte
stPr
oble
mst
ruct
ure
repr
esen
tati
onA
rith
met
icSi
mpl
ean
dre
al-w
orld
prob
lem
s11
.820
Prob
lem
stru
ctur
ere
pres
enta
tion
Ari
thm
etic
8.54
8
L.S.
Fuch
s,Se
etha
ler,
Pow
ell,
Fuch
s,H
amle
tt,&
Flet
cher
(200
8)
Atr
isk
Spec
iale
duca
tion
Res
earc
her-
deve
lope
dte
stan
dst
anda
rdiz
edte
st
Prob
lem
stru
ctur
ere
pres
enta
tion
Alg
ebra
icSi
mpl
epr
oble
ms
0.42
1
L.S.
Fuch
s,Fu
chs,
Ham
lets
,&
App
leto
n(2
002)
Dis
crep
ant
LDSp
ecia
ledu
cati
onR
esea
rche
r-de
velo
ped
test
Ass
isti
vete
chno
logy
Ari
thm
etic
Sim
ple
and
real
-wor
ldpr
oble
ms
3.32
7
Cog
niti
vest
rate
gyA
rith
met
ic4.
653
Ass
isti
vete
chno
logy
Ari
thm
etic
1.35
5L.
S.Fu
chs,
Fuch
s,Fi
nelli
,C
oure
y,&
Ham
lett
(200
4)A
tris
kIn
clus
ive
educ
atio
nR
esea
rche
r-de
velo
ped
test
Prob
lem
stru
ctur
ere
pres
enta
tion
Ari
thm
etic
Sim
ple
and
real
-wor
ldpr
oble
ms
3.46
3
Prob
lem
stru
ctur
ere
pres
enta
tion
Ari
thm
etic
1.85
6
L.S.
Fuch
s,Fu
chs,
Pren
tice
,B
urch
,Ham
lett
,Ow
en,
Hos
p,&
Janc
ek(2
003)
Atr
isk
Incl
usiv
ecl
ass
Res
earc
her-
deve
lope
dte
stPr
oble
mst
ruct
ure
repr
esen
tati
onA
rith
met
icSi
mpl
ean
dre
al-w
orld
prob
lem
s3.
909
Prob
lem
stru
ctur
ere
pres
enta
tion
Ari
thm
etic
1.60
4
Prob
lem
stru
ctur
ere
pres
enta
tion
Ari
thm
etic
3.01
8
(Con
tinue
don
next
page
)
309
Dow
nloa
ded
by [
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vers
ity o
f N
orth
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olin
a] a
t 15:
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5 O
ctob
er 2
014
TA
BL
E1.
Sum
mar
yof
Gro
up-D
esig
n-St
udie
s(C
onti
nued
)
Stud
yLD
defin
itio
nSe
ttin
gM
easu
res
Inte
rven
tion
stra
tegi
esA
lgeb
rain
stru
ctio
nT
asks
Effe
ctsi
zes
L.S.
Fuch
s,Fu
chs,
Pren
tice
,Bur
ch,
Ham
lett
,Ow
en,&
Schr
oete
r(2
003)
Atr
isk
Incl
usiv
ecl
ass
Res
earc
her-
deve
lope
dte
stC
ogni
tive
stra
tegy
Ari
thm
etic
Sim
ple
and
real
-wor
ldpr
oble
ms
2.33
5
Cog
niti
vest
rate
gyA
rith
met
ic1.
358
Gin
sbur
g-B
lock
&Fa
ntuz
zo(1
998)
Atr
isk
Spec
iale
duca
tion
Res
earc
her-
deve
lope
dte
stA
ssis
tive
tech
nolo
gyA
rith
met
icSi
mpl
epr
oble
ms
1.00
5A
ssis
tive
tech
nolo
gyA
rith
met
ic0.
200
Ass
isti
vete
chno
logy
Ari
thm
etic
−0.5
47G
riffi
n&
Jite
ndra
(200
9)D
iscr
epan
tLD
Incl
usiv
ecl
ass
Res
earc
her-
deve
lope
dte
stPr
oble
mst
ruct
ure
repr
esen
tati
onA
lgeb
raic
Sim
ple
prob
lem
s0.
8527
Gre
tche
n(2
003)
Atr
isk
Spec
iale
duca
tion
Res
earc
her-
deve
lope
dte
stA
ssis
tive
tech
nolo
gyA
rith
met
icR
eal-
wor
ldpr
oble
ms
0.56
62H
asse
lbri
ng&
Moo
re(1
996)
Atr
isk
Spec
iale
duca
tion
Res
earc
her-
deve
lope
dte
stan
dst
anda
rdiz
edte
stA
ssis
tive
tech
nolo
gyA
rith
met
icR
eal-
wor
ldpr
oble
ms
1.38
91
Jite
ndra
,Gri
ffin,
McG
oey,
Gar
dill,
Bha
t,&
Rile
y(1
998)
Dis
crep
ant
LDSp
ecia
ledu
cati
onR
esea
rche
r-de
velo
ped
test
Prob
lem
stru
ctur
ere
pres
enta
tion
Ari
thm
etic
Sim
ple
prob
lem
s1.
6599
Jite
ndra
,Gri
ffin,
Dea
tlin
e-B
utch
man
,&
Scze
niak
(200
7)A
tris
kIn
clus
ive
clas
sR
esea
rche
r-de
velo
ped
test
Prob
lem
stru
ctur
ere
pres
enta
tion
Ari
thm
etic
Sim
ple
prob
lem
sSi
te1:
1.41
3Pr
oble
mst
ruct
ure
repr
esen
tati
onSi
te2:
−0.1
74Ji
tend
ra,G
riffi
n,H
aria
,Leh
,Ada
ms,
&K
aduv
etto
or(2
007)
Atr
isk
Incl
usiv
ecl
ass
Res
earc
her-
deve
lope
dte
stan
dhi
gh-s
take
sst
anda
rdiz
edte
st
Prob
lem
stru
ctur
ere
pres
enta
tion
Ari
thm
etic
Sim
ple
prob
lem
s1.
4387
Jite
ndra
,Sta
r,St
aros
ta,L
eh,S
ood,
Cas
kie,
Hug
hes,
&M
ack
(200
9)A
tris
kIn
clus
ive
clas
sR
esea
rche
r-de
velo
ped
test
Prob
lem
stru
ctur
ere
pres
enta
tion
Alg
ebra
icSi
mpl
epr
oble
ms
0.58
0
Kon
old
(200
4)A
tris
kIn
clus
ive
educ
atio
nR
esea
rche
r-de
velo
ped
test
Cog
niti
vest
rate
gyA
lgeb
raic
Sim
ple
prob
lem
s1.
149
Lam
bert
(199
6)A
tris
kSp
ecia
ledu
cati
onR
esea
rche
r-de
velo
ped
test
Cog
niti
vest
rate
gyA
rith
met
icR
eal-
wor
ldpr
oble
ms
1.84
4La
ng(2
001)
Dis
crep
ant
LDSp
ecia
ledu
cati
onR
esea
rche
r-de
velo
ped
test
Cog
niti
vest
rate
gyA
lgeb
raic
Rea
l-w
orld
prob
lem
s0.
927
Ow
en&
Fuch
s(20
02)
Atr
isk
Spec
iale
duca
tion
Res
earc
her-
deve
lope
dte
stC
ogni
tive
stra
tegy
Ari
thm
etic
Sim
ple
prob
lem
s1.
449
Cog
niti
vest
rate
gyA
rith
met
ic2.
128
Cog
niti
vest
rate
gyA
rith
met
ic4.
199
Stel
lingw
erf&
Van
Lies
hout
(199
9)A
tris
kSp
ecia
ledu
cati
onR
esea
rche
r-de
velo
ped
test
Ass
isti
vete
chno
logy
Ari
thm
etic
Sim
ple
prob
lem
s1.
006
Ass
isti
vete
chno
logy
Alg
ebra
ic0.
200
Ass
isti
vete
chno
logy
Alg
ebra
ic−0
.547
Ass
isti
vete
chno
logy
Ari
thm
etic
5.92
E-02
Top
pel(
2004
)D
iscr
epan
tLD
Spec
iale
duca
tion
Res
earc
her-
deve
lope
dte
stC
ogni
tive
stra
tegy
Alg
ebra
icSi
mpl
epr
oble
ms
0.16
4
Cog
niti
vest
rate
gyA
lgeb
raic
−0.1
12T
roff
(200
4)A
tris
kSp
ecia
ledu
cati
onSt
anda
rdiz
edte
stC
ogni
tive
stra
tegy
Alg
ebra
icR
eal-
wor
ldpr
oble
ms
3.38
7X
in,J
iten
dra,
&D
eatl
ine-
Buc
hman
(200
5)D
iscr
epan
tLD
Spec
iale
duca
tion
Res
earc
her-
deve
lope
dte
stPr
oble
mst
ruct
ure
repr
esen
tati
onA
lgeb
raic
Sim
ple
prob
lem
s2.
896
Not
e.LD
=le
arni
ngdi
sabi
lity.
310
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The Journal of Educational Research 311
TABLE 2. Summary of Effect Sizes in Group Studies (Fully Random Model)
Variable and class Qb k d+ 95% CI
Inclusive movementSetting 9.938∗∗
Special education 29 1.348 [0.863, 1.834]Inclusive education 19 2.600 [1.992, 3.207]
StudentsStudents with LP 0.521 18 2.571 [1.749, 3.392]Normal achieving 18 2.993 [2.192, 3.795]
Standard movementMeasures 5.142∗∗
Self-developed test 46 1.872 [1.479, 2.264]Standardized test 6 0.597 [−0.433, 1.627]
Intervention strategies 9.510∗∗
Problem structure representation 16 2.637 [1.960, 3.314]Cognitive strategy training 12 1.855 [1.074, 2.635]Assistive technology 20 1.218 [0.622, 1.814]
RtI modelDiagnostic approach
Discrepancy model 0.136 12 1.722 [0.913, 2.530]RtI model–At risk 36 1.898 [1.425, 2.869]
Mathematics educational reformAlgebraic problem solving 1.136
Arithmetic 32 2.006 [1.513, 2.499](Pre)algebraic 16 1.551 [0.876, 2.226]
Problem tasks 0.002Simple problems 36 1.801 [1.377, 2.225]Real-world problems 27 1.814 [1.335, 2.293]
Note. CI = confidence interval; LP = learning problem; RtI = response to intervention.
placed third, whereby 20 treatments yielded a mean effectsize of 1.218. Intervention strategies seemed to mediate theeffect size of the intervention.
Variables About RtI
One variable (i.e., target groups) was examined regardingRtI. In Xin and Jitendra (1999), 18 of 26 (69.23%) effectsizes were reviewed from group studies that targeted studentswith an already-diagnosed LD. In the present meta-analysis,only 12 of 48 (25%) effect sizes from group studies focusedon students with an already-diagnosed LD with the discrep-ancy model. Apparently, the percentage of studies with anemphasis on at-risk students has greatly increased.
Table 2 presents the results comparing the effect sizesfrom students who were at risk and students who were iden-tified with discrepant LD (Qb = 0.136, p > .05). The re-sults suggested no significant differences between at-risk stu-dents and students with discrepant LD. Results demonstratedquite similar average effect sizes from the two populations(1.722 vs. 1.898), suggesting the students with LD studentswho were identified with the discrepancy model actually re-spond to interventions similarly to students who were simplylow achievers.
Variables about mathematics education reform. For the firstvariable, algebra instruction, 32 effect sizes were yielded fromtreatments with an arithmetic-oriented instructional ap-proach, and 16 effect sizes were yielded from treatments ad-dressing (pre)algebraic approach or algebraic content. Anal-yses reported no significant difference (Qb = 1.136, p > .05)between the effect sizes of the arithmetic (d+ = 2.006) andthe (pre)algebraic problem solving (d+ = 1.551). Thus, theproblem-solving approach has not mediated the effect sizeof intervention.
For the second variable, problem tasks, 36 effect sizes wereyielded from treatments that focused on simple-structuredproblems. Twenty-seven effect sizes were from treatmentstargeting real-world problems. No results found any signifi-cant difference between the effect sizes of simple-structuredand real-world problems (Qb = 0.002, p > .05). Resultssuggested that the variable of problem tasks seemed not tomediate the effect size of the interventions.
Results for the Single-Subject-Design Studies
Table 3 summarizes the coding results of the 10 single-subject-design studies. A total of 13 effect sizes were reportedfor 10 single-subject studies. Different from findings of group
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TA
BL
E3.
Sum
mar
yof
Sing
le-S
ubje
ct-D
esig
nSt
udie
s
Stud
yT
arge
tgro
upSe
ttin
gM
easu
res
Inte
rven
tion
stra
tegi
esA
lgeb
rain
stru
ctio
nT
asks
PND
Cas
s,C
ates
,Sm
ith,
&Ja
ckso
n(2
007)
Dis
crep
antL
DSp
ecia
ledu
cati
onR
esea
rche
r-de
velo
ped
test
Cog
niti
vest
rate
gyA
rith
met
icR
eal-
wor
ldpr
oble
ms
93%
Jite
ndra
,Hof
f,&
Bec
k(1
999)
Dis
crep
antL
DSp
ecia
ledu
cati
onR
esea
rche
r-de
velo
ped
test
Prob
lem
stru
ctur
ere
pres
enta
tion
Ari
thm
etic
Sim
ple
prob
lem
sand
real
-wor
ldpr
oble
ms
83%
forr
egul
ar,9
0%fo
rrea
l-w
orld
prob
lem
sJi
tend
ra,D
ipip
i,&
Perr
on-J
ones
(200
2)
Dis
crep
antL
DSp
ecia
ledu
cati
onR
esea
rche
r-de
velo
ped
test
Prob
lem
stru
ctur
ere
pres
enta
tion
Alg
ebra
icSi
mpl
epr
oble
ms
95%
Mac
cini
&H
ughe
s(2
000)
Dis
crep
antL
DSp
ecia
ledu
cati
onR
esea
rche
r-de
velo
ped
test
Cog
niti
vest
rate
gyA
lgeb
raic
Sim
ple
prob
lem
s10
0%
Mac
cini
&R
uhl
(200
0)D
iscr
epan
tLD
Spec
iale
duca
tion
Res
earc
her-
deve
lope
dte
stC
ogni
tive
stra
tegy
Alg
ebra
icSi
mpl
epr
oble
ms
100%
Stau
lter
s(20
06)
Dis
crep
antL
DSp
ecia
ledu
cati
onR
esea
rche
r-de
velo
ped
test
Ass
isti
vete
chno
logy
Ari
thm
etic
Sim
ple
prob
lem
s90
%
van
Gar
dere
n(2
007)
Dis
crep
antL
DSp
ecia
ledu
cati
onR
esea
rche
r-de
velo
ped
test
Prob
lem
stru
ctur
ere
pres
enta
tion
Ari
thm
etic
Sim
ple
prob
lem
sand
real
-life
prob
lem
s10
0%
Xin
(200
8)D
iscr
epan
tLD
Spec
iale
duca
tion
Res
earc
her-
deve
lope
dte
stPr
oble
mst
ruct
ure
repr
esen
tati
onA
lgeb
raic
Sim
ple
prob
lem
s10
0%
Xin
,Wile
s,&
Lin
(200
8)A
tris
kSp
ecia
ledu
cati
onR
esea
rche
r-de
velo
ped
test
and
stan
dard
ized
test
Prob
lem
stru
ctur
ere
pres
enta
tion
Alg
ebra
icSi
mpl
epr
oble
ms
95%
onre
sear
cher
-de
velo
ped
test
,10
0%on
stan
dard
ized
test
Xin
&Zh
ang
(200
9)A
tris
kSp
ecia
ledu
cati
onR
esea
rche
r-de
velo
ped
test
and
stan
dard
ized
test
Prob
lem
stru
ctur
ere
pres
enta
tion
Alg
ebra
icR
eal-
wor
ldpr
oble
ms
97%
forr
esea
rche
r-de
velo
ped
test
,10
0%in
crea
seon
stan
dard
ized
test
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studies, most single-subject design studies remained focusedon students with identified LD. Specifically, 8 of 10 (80%)single-subject studies reviewed in the present meta-analysisonly targeted students who were identified as having dis-crepant LD. This ratio is quite similar to that in the previousmeta-analysis by Xin and Jitendra (1999), in which 10 of 12(83.33%) effect sizes were for students with discrepant LD.Additionally, no study adopted high-stakes test, only a few(2 of 10) studies adopted standardized test, and only fourstudies (40%) involved real-world problems during instruc-tion. No significant difference was found in all moderators,which may be because all interventions yielded a PND closeto 1, a ceiling effect.
Discussion
Results of the present meta-analysis suggested that re-search during the past 13 years has identified interventionsthat have improved students’ performance in word-problemsolving. The overall effect for all treatment conditions was1.848 (SD = 0.203). The treatment effect for single-subject-design studies was 95%. Yet this review was limited in sev-eral ways. First, the effect size measure used in this study wasthe standardized mean change between preintervention andpostintervention. As a result, the effect size scores in thisstudy might have been larger than the traditional effect sizeGlass’s (1977), which is the standardized mean differencebetween the experimental group and the control group di-vided by the control group’s standard deviation. However,the limitation of the standardized mean change as an effectsize measure does not negate its potential for exploring therelationship between mediating variables and treatment ef-fectiveness (Xin & Jitendra, 1999). Second, the PND mea-sure for single-subject studies used in the present reviewcould suffer from a ceiling effect. The PND measure countsonly the percentage of data points in the intervention con-dition that either exceeds or overlaps with the highest datapoint in the baseline. As such, the PND measure may not besufficiently sensitive to evaluate the magnitude of treatmenteffectiveness (Parker & Vannest, 2009).
How Does the Inclusive Movement Influence Recent Research?
The present study was aimed to answer two controver-sial questions concerning inclusive movement: (a) Areinterventions in inclusive education as effective as in specialeducation setting for those with mathematic LP? (b) Is theintervention in an inclusive classroom equally effective forthose with and without disabilities?
Inclusion settings. Results from the inclusive educationmovement involved multiple contradictions (Cook et al.,2007; Estell, Johns, Pearl, Van Acker, Farmer, & Rodkin,2008; D. Fuchs & Fuchs, 1994; Norwich, 2008). The presentmeta-analysis found that interventions in inclusive class-
rooms seemed more effective than interventions in spe-cial education settings for students with LP (Qb = 9.938,p < .05). This result differs from the former meta-analysis(Carlberg & Kavale, 1980), which reported that the in-clusive education setting was superior to a special edu-cation setting for students with mental retardation butwas inferior to special education setting for students withLD. The differential conclusions could be due to the in-terventions reviewed in this study particularly designedfor students with LP. Perhaps, too, the treatments im-plemented in inclusive classroom were more likely to bewell-recognized interventions and thus were more likelyto obtain national funds to conduct large-scale interven-tions. In either case, the results suggested, as long as ap-propriate interventions are provided, students with LP gainsmore from inclusive education. Results support the NCTM(2000) standards that call for general education mathematicsclasses.
Students with and without LP in inclusive settings. Resultsalso suggested that placing students with LP in the inclu-sive classroom did not interfere with the learning progressof normal-achieving students. Results showed that data thateffect sizes from normal-achieving students seemed to be sim-ilar to those from students with LP (Qb = 0.521, p > .05).This indicated that the experimental interventions appearedto favor normal-achieving students as much as students withLP as in inclusive classrooms. Therefore, the inclusion ofstudents with LP does not interfere with the progress ofnormal achieving students; instead, these normal-achievingstudents also benefit from these evidence-based interven-tions as much as those students with LP. Therefore, thereshould be no concern about the effectiveness of implement-ing such interventions inclusive classrooms, and the previ-ous concern that involvement of students with disabilitiescould disrupt normal-achieving students’ progress can bedisproved.
How Does the Standard Movement Influence Recent Research?
The purpose of the present meta-analysis was to answerthree questions. First, were the interventions effective andwhat type of intervention type was the most effective? Sec-ond, how many studies employed high-stakes testing andhow was the effectiveness measured by a standardized test?
Intervention type. For the first issue concerning theevidence-based interventions, the present analysis suggestedthat all three intervention strategies yielded high effectsizes. In the mean time, significant differences were foundamong the weighted effect sizes from the three interventiontypes (Qb = 9.510, p < .05). The most effective interventionwas determined to be the problem structure representationtechniques examined by 22 treatments. Cognitive strategytraining was placed second, followed by the strategies
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involving assistive technology. Results indicated that, over-all, all three categories of intervention strategies enhancedstudents’ learning of word problems. Interestingly, congruentwith concerns from a previous meta-analysis (Kroesbergen& van Luit, 2003), the assistive technologies were notas effective as the two categories of human-deliveredinterventions. Such results were in alignment with someconcerns of school teachers and practitioners that assistivetechnologies are not as effective as commercially advertised(Kroesbergen & van Luit, 2003). Therefore, further exam-ination of the usefulness of certain assistive technologiesshould be one of the priorities for special educationresearchers.
It should also be noted that problem structure represen-tation techniques ranked the most effective word-problem-solving interventions for students with LP. The previousmeta-analysis (Xin & Jitendra, 1999) only found one study(Jitendra & Hoff, 1996) that employed this strategy (i.e., theschema-based intervention). Whereas the present reviewhas found 20 treatments with multiple variations of prob-lem structure representation techniques, such as schema-based instruction plus transfer by sorting (e.g., L. S. Fuchset al., 2004), or conceptual model-based problem solving(COMPS) with an algebraic approach (e.g., Xin et al., 2008;Xin & Zhang, 2009). In all, problem structure representa-tion techniques have yielded sufficient effectiveness in re-cent years.
Explicit instruction and conceptual modeling are essentialkeys to the success of these problem structure interventions.Using these structure representation interventions, theteachers either explicitly teaches children how to map theproblem features to the corresponding schema and then solvethe problems (e.g., Jitendra, Griffin, Deatline-Buchman,& Sczesniak, 2007; Jitendra & Hoff, 1996; Jitendra et al.,1998), or use equation models to represent mathematicalrelations in the problem (e.g., Xin, Jitendra, & Deatline-Buchman, 2005; Xin et al., 2008; Xin & Zhang, 2009).In particular, one distinguishable feature of the COMPS,recently developed by Xin and colleagues, is that it focuseson representing the word problem in a defined mathematicalmodel. By representing problems in mathematical modelequations (e.g., part + part = whole for additive problemsolving, unit rate x number of units = product for multiplica-tive problem solving), students did not have to memorizenumerous solution rules to make decisions on the choice ofoperation for finding the solution; rather, the mathematicalmodel provides the students with a defined algebraicequation for solution (to the unknown; e.g., Xin et al.,2008).
Measurement. For the second issue about measurement,the present review only found one effect size adoptedthe high-stakes testing, six effect sizes employed otherstandardized assessments and all other studies only usedresearcher-developed tests. Although the present studycannot make a conclusion whether interventions yielded
different effect sizes on high-stakes testing or other tests,results indicated that the obtained effect sizes measured bystandardized testing (i.e., the high-stakes testing and otherstandardized testing) were significantly lower than effect sizesmeasured by self-developed testing (Qb = 5.142, p < .01).It is possible that the researcher development probes weremore in alignment with the short-term curriculum adoptedin the specific intervention program. These probes can beconsidered curriculum-based assessments. Based on existingresearch, curriculum-based assessments are typically moresensitive than those large-scale standardized assessment(Deno, 1985). Cautions should be exercised when general-izing the findings from studies that use researcher-developedprobes.
In addition, the present meta-analysis only found ninestudies adopting standardized testing, including either pub-lished standardized assessments or high-stakes tests. A stan-dardized test usually has a better reliability and validity thanself-developed tests (Kuncel & Hezlett, 2007). A soundmeasurement is required for a high-quality experimental orquasiexperimental study (Gersten et al., 2004) and single-subject-design studies (Horner et al., 2004). As such, therare employment of standardized testing across the publishedquantitative research is worthy of future researchers’ atten-tion.
How Does Change of LD Definition Influence Recent Research?
Results suggested that with the initiation of the RtI model,more researchers have moved their interest from only thosewho have failed in mathematics, to those who have displayeddifficulties in learning mathematics at an early stage. In Xinand Jitendra (1999), 28 of 38 (73.68%) effect sizes (i.e.,18 out of 26 effect sizes from group studies, 10 out of 12effect sizes from single-subject studies) focused on studentsdiagnosed as having discrepant LD. In the present meta-analysis, only 20 of 58 (34.48%) effect sizes (i.e., 12 out of 48effect sizes from group-design studies, and 8 of 10 from single-subject studies) focused on students with discrepant LD.Obviously, an increasing trend is observed for the numberof studies with an emphasis on at-risk students, especiallyamong group-design studies.
Moreover, the results did not find any statistically signif-icant difference between the effect sizes from at-risk par-ticipants and students with discrepant LD. The results sug-gested that at-risk (d+ = 1.898) responded to interventionswith an almost equivalent growth rate to the students withdiscrepant LD (d+ = 1.722), and nonsignificant differencewas found (Qb = 0.136, p>.05). According to the RtI theory,two categories of students comprise the at-risk population:some of the students responded effectively to early tiers ofinterventions, and thus should not be titled with real LD;only those who do not respond well to first and second tiersof interventions should be considered with LD (D. Fuchs &Fuchs, 2006). Therefore, those with real LD should have apoorer response to interventions than those low achievers
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due to exclusionary factors such as inadequate instruction.Results in the present study suggested that students with tra-ditional discrepant LD demonstrated relatively similar re-sponses, compared with at-risk students, to evidence-basedinterventions. In short, with high-quality instruction, thosestudents diagnosed with discrepant LD could also learn withsatisfactory growth rate and responded effectively to the in-terventions. The results indicated a possibility that the dis-crepancy model does not reflect the nonresponding natureof the construct of the real LD. Instead, the discrepancybetween intelligence and achievement could be resultedfrom inadequate instruction, and, thus, the RtI model is sup-ported. Additionally, such results also indicate the necessityof introducing early interventions to those at-risk students,rather than not providing any intervention until they fail(L. S. Fuchs, Fuchs, & Hollenbeck, 2007).
How Does the Reform of Mathematics Education InfluenceRecent Research?
Two issues were discussed in the session. First, did the ef-fects of interventions for resolving real-world problems differfrom the effects of interventions for solving simple structuredessential problems? Second, were interventions for algebraicor algebraic-oriented problem solving as effective as arith-metic problem solving for students with LP?
Real-world problem solving. A total of 27 of 63 effect sizeswere yielded for the real-world problem solving. No signifi-cant difference was found between the effect sizes for simple-structured problem solving and real-world problem solving(Qb = 0.002, p > .05). In other words, the effects of interven-tions for solving real-world problems did not differ from theeffects of interventions for solving simple-structured essen-tial problems. Results indicated that through interventions,children with LP were able to solve real-world problems aswell as simple-structured problems. Concerning mathematiceducation reform, however, presently there were more stud-ies addressing real-world problem solving than there were13 years ago, but the majority of research reviewed still onlyfocused on regular problem solving. Such a small numberof studies involved real-world problem solving indicates thenecessity of continually promoting real-world problem solv-ing activities in present education research and practice toreflect the spirits of NCTM (2000).
Algebraic instruction. Were interventions for algebraic oralgebraic-oriented problem solving as effective as arithmeticproblem solving for students with LP? Regarding thisconcern, results showed that students yielded the similareffect size either with the arithmetic or (pre)algebraicproblem solving and no statistically significant differencewere found (Qb = 1.136, p > .05). It indicated that studentswith LP were capable of mastery of (pre)algebraic problemsolving approach as long as appropriate interventions were
provided. Inspiringly, compared to Xin and Jitendra’s (1999)meta-analysis, the number of studies with an algebraicinstructional approach has greatly increased; which reflectsthe alignment of the research and the education reform thatcalls for the (pre)algebra readiness in elementary curriculum.
Conclusions
The present meta-analysis may provide implications forfuture researchers and educational practitioners. In sum-mary, the effects of various word-problem-solving inter-ventions were encouraging on students with LP. Gener-ally speaking, the most influential educational reforms werefound to have generated positive effects on mathematicaleducation for students with learning difficulties. First, aslong as effective interventions are provided, children withand without LP can effectively learn in inclusive settings,therefore, the issue is not about where to teach, but moreabout how to teach. Similarly, students with LP benefit fromthe algebra and arithmetic approaches when solving eithersimple-structured or real-world problems as long as effectiveinstructions are provided.
Second, as for what effective instruction is, the struc-ture representation technique has been found to be mosteffective; in the meantime, the present meta-analysis re-flects the present debates about the effectiveness of assistivetechnology among educational practitioners. Such resultsmay provide information to help school educators decidewhat intervention to adopt as evidence-based interventionsin practices.
As for the influence of the RtI model, the present re-view has found more and more studies focused on studentsat risk, indicating that the initiation of RtI model is con-gruent with practical trends. More importantly, studentswith discrepant LD did not respond more poorly than at-risk students. In other words, when high-quality instructionis given, these students with discrepant LD can demonstrateimpressive growth rates. Such reports support the preventivefunction of interventions for at-risk students.
Unfortunately, although practitioners desire for interven-tions directly addressing improving students’ performance onhigh-stakes testing, rarely does such research come about. Itmay have also reflected the present debates about whetherschool daily instruction should target on high-stakes testing.Future researchers should examine the relationship betweenhigh-stakes testing and mathematics intervention program-ming in special education field.
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AUTHORS NOTE
Dake Zhang and is an Assistant Professor in Special Edu-cation at Rutgers University. Her current research interestsinclude assessment and intervention for students with learn-ing disabilities/difficulties in mathematics and sciences.
Yan Ping Xin is an Associate Professor of Special Ed-ucation at Purdue University. Her current research inter-ests include effective instructional strategies in mathematicsproblem solving with students with learning disabilities ordifficulties, conceptual model-based problem solving and al-gebra readiness, and computer-assisted tutoring system.
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