Mathematics Intervention for Children with Special Educational Needs Kroesbergen en Van Luit...

97R E M E D I A L A N D S P E C I A L E D U C A T I O N

Volume 24, Number 2, March/April 2003, Pages 97–114

Mathematics Interventions for Childrenwith Special Educational NeedsA Meta-Analysis

E V E L Y N H . K R O E S B E R G E N A N D J O H A N N E S E . H . V A N L U I T

A B S T R A C T

This article presents the results of a meta-analysis of 58 studies of mathematics interventions for elementary studentswith special needs. Interventions in three different domains wereselected: preparatory mathematics, basic skills, and problem-solving strategies. The majority of the included studies describedinterventions in the domain of basic skills. In general, these inter-ventions were also the most effective. Furthermore, a few spe-cific characteristics were found to influence the outcomes of thestudies. In addition to the duration of the intervention, the particu-lar method of intervention proved important: Direct instruction and self-instruction were found to be more effective than medi-ated instruction. Interventions involving the use of computer-assisted instruction and peer tutoring showed smaller effects than interventions not including these supports.

STUDENTS WITH DIFFICULTIES IN LEARNING

mathematics can be found in almost every classroom. About5% to 10% of the students in schools for elementary generaleducation have difficulties with mathematics (Rivera, 1997).The seriousness of these difficulties can vary from temporarydifficulties in one domain (i.e., a particular area of the mathcurriculum) to severe learning disabilities affecting severaldifferent domains. The difficulties can also manifest them-selves at different points in a child’s school career, not onlyin the learning of basic facts or in learning to apply previ-ously acquired knowledge but also in the learning of suchpreliminary mathematics skills as counting and seriation (Vande Rijt & Van Luit, 1998). The potential causes of these dif-

ficulties are numerous and can partly be explained by suchchild characteristics as intellectual functioning, motivation,problem-solving skills, memory skills, strategy acquisitionand application, and vocabulary. Another important cause ofmath difficulties may be a poor fit between the learning char-acteristics of individual students and the instruction theyreceive (Carnine, 1997). In the case of such a poor fit, theinstruction must be adapted to the students’ needs. In otherwords, all students with mathematics difficulties require spe-cial attention (Geary, 1994). These students have special edu-cational needs, need extra help, and typically require sometype of specific mathematics intervention, which is the focusof the present meta-analysis.

Although the group of students with difficulties in learn-ing math is very heterogeneous, some general characteris-tics of this group can be described (Goldman, 1989; Mercer,1997; Rivera, 1997). In general, these students have memorydeficits leading to difficulties in the acquisition and remem-bering of math knowledge. Moreover, they often show in-adequate use of strategies for solving math tasks, caused byproblems with the acquisition and the application of bothcognitive and metacognitive strategies. Because of theseproblems, they also show deficits in generalization and trans-fer of learned knowledge to new and unknown tasks. It isoften recommended to provide these students with direct,explicit instruction and to make the different steps needed fora given task as overt and explicit as possible (Carnine, 1997).Indeed, research has consistently shown direct instruction tobe very effective for students with math learning difficul-

98 R E M E D I A L A N D S P E C I A L E D U C A T I O N

Volume 24, Number 2, March/April 2003

solving mathematics word problems, it is not always clearjust which operation to apply or strategy to adopt, which musttherefore be learned. In keeping with these steps, a distinctioncan be made between interventions that focus on (a) prepara-tory arithmetic, (b) automatization of basic math facts, or (c) mathematical problem-solving strategies.

In the present meta-analysis, interventions will be dis-cussed in terms of the domains and skills that they address.Interventions can nevertheless differ from each other withregard to a number of other factors: the use of a computer, thesize of the groups used for intervention, the duration and fre-quency of the intervention, the instructional procedures, thesubdomains targeted, and so forth. The numerous interven-tions available today clearly reflect the many possibilities forintervention. Different children also need different interven-tions, but just which intervention and which characteristics ofthe intervention are most effective for which children is stillunclear.

In recent years, several reviews, research syntheses, andmeta-analyses have been published on the topic of math inter-vention. Mastropieri, Bakken, and Scruggs (1991) and Miller,Butler, and Lee (1998) have published reviews of the mathinterventions available for students with mental retardationand students with learning disabilities, respectively. Jitendraand Xin (1997; Xin & Jitendra, 1999) have conducted inter-esting research on interventions aimed at the word problem–solving skills of students with mild disabilities. Swanson andcolleagues (Swanson & Carson, 1996; Swanson & Hoskyn,1998, 1999) have published impressive meta-analyses of in-tervention studies for students with learning disabilities.

In contrast to the aforementioned reviews, interventionspertaining to the three different domains of math-related skillsare distinguished in the present analysis because we thinkthat certain interventions may be more effective in one do-main than in the other. Another difference between the afore-mentioned studies and the present study is the method ofanalysis used. Although some of the previous studies werequantitative, they nevertheless only compared weighted meaneffect sizes to estimate the contributions of different variablesand simply assumed that other characteristics were equallydistributed across the different groups being compared. Themethod of analysis used in the present study, namely, multi-level regression analysis of variance, clearly takes the amountof within- and between-group variance into consideration.The method used in the present study also makes it possibleto combine baseline and experimental designs within thesame analysis, whereas these different types of designs werestudied separately in the other reviews. A final difference isthat interventions for all groups of students with difficultiesin learning math are combined in the present study: interven-tions for students with mild disabilities, learning disabilities,and mental retardation. Because these children need similartypes of instruction (Kavale & Forness, 1992), it seems rea-sonable to combine them in one analysis.

ties to attain both automaticity and problem-solving skills.However, reforms in the mathematics curriculum have called for more implicit teaching (National Council of Teachers ofMathematics, 1989, 2000). New forms of instruction, there-fore, ask students to construct their own knowledge underguidance of the teacher. In this study, special attention isgiven to the effects of different forms of instruction that arerelated to the recent developments in mathematics education.

Given that difficulties can be encountered at differentages and in different mathematical domains, interventionmay be called for at different points in a child’s school careerand in different domains. Studies have shown that most mathdifficulties have a relatively early onset (Schopman & VanLuit, 1996). During kindergarten and first grade, childrentypically develop number sense, which then grows along thelines of the various Piagetian operations (e.g., number con-servation, classification, seriation) and in combination withvarious counting skills. A basic understanding of arithmeticoperations is established at this time (Correa, Nunes, & Bry-ant, 1999). The first category of interventions thus focuses onthese preparatory arithmetic skills. For example, Malabonga,Pasnak, Hendricks, Southard, and Lacey (1995) have studiedthe effects of specific seriation and classification instructionversus traditional instruction using academic materials (e.g.,counting, shape, and color knowledge) on children’s reason-ing and math achievement.

The next step is to learn the four basic mathematicaloperations (i.e., addition, subtraction, multiplication, and di-vision). Knowledge of these operations and a capacity toperform mental arithmetic play an important role in thedevelopment of children’s later math skills (Mercer & Miller,1992; Van Luit & Naglieri, 1999). Most children with mathlearning difficulties are unable to master the four basic oper-ations before leaving elementary school and, thus, need spe-cial attention to acquire the skills. A second category ofinterventions is therefore aimed at the acquisition and auto-matization of basic math skills. The domain of basic mathskills is very large and constitutes an important aspect of ele-mentary math teaching. The studies analyzed here address thelearning of such simple addition facts as 5 + 3 (e.g., Beirne-Smith, 1991) but also more complicated operations such asfour-digit addition or division facts (e.g., Skinner, Bamberg,Smith, & Powell, 1993).

Mastery of the basic operations, however, is not suffi-cient: Students must also acquire problem-solving skills inaddition to the basic computational skills (Carnine, 1997;Goldman, 1989). Mathematics frequently involves the solu-tion of both verbal and nonverbal problems through the appli-cation of previously acquired information (Mercer & Miller,1992). For mathematical problem solving, that is, childrenmust not only possess the basic mathematical skills but alsoknow how and when to apply their knowledge in new andsometimes unfamiliar situations. The third category of inter-ventions addresses such problem-solving skills. Moreover, in



RESEARCH QUESTIONS

The focus of the present study is on the characteristics of themost effective interventions. An intervention is judged effec-tive when the students acquire the knowledge and skills beingtaught and thus appear to adequately apply this informationat, for example, posttest. The question, then, is what makes a particular mathematics intervention effective. In order toanswer this question, a meta-analysis was undertaken of thosestudies concerned with mathematics intervention for studentswith special educational needs in both general and spe-cial elementary education. The effects of a number of vari-ables were analyzed. The following research questions wereaddressed:

1. Which domain (preparatory skills, basic skills,problem solving) is most investigated, andwhich domain produces the highest effectsizes?

2. Is there a trend in outcomes as a function of

• study characteristics, such as year of pub-lication and design?

• sample characteristics, such as number ofparticipants, age, and special needs?

• treatment parameters, such as duration,total instruction time, and content?

• treatment components, such as directinstruction, self-instruction, computer-assisted instruction, and peer tutoring?

• treatment components related to recentreforms in the mathematics curricula,such as mediated/guided instruction andrealistic mathematics education?

3. Which variables can explain the largest part ofthe between-studies variance in

• the total sample?

• the three different domains separately?

METHOD

Search Procedure

To answer the research questions, a search procedure wasundertaken to find as many studies as possible that reportedon an empirical study on the effectiveness of mathematicsinterventions for students with special educational needs ingeneral and special elementary education. An intervention is

defined as a specific instruction for a certain period to teacha particular (sub)domain of the mathematics curriculum.Mathematics instruction includes all educational settings thatare meant to improve students’ math knowledge and skills.Students with special educational needs in mathematics aredefined as students who have more trouble with learningmath than their peers, who perform at a lower level than theirpeers, or who need special instruction to perform at an ade-quate level.

To find such studies, the following search procedure wasused. The articles examined in the meta-analysis includedempirical studies published between 1985 and 2000. A com-puterized search was conducted of the following databases:Current Contents, ERIC (Educational Resources InformationCenter), PsychLit, and SSCI (Social Sciences Citation Index).The search descriptors included math(ematics), arithmetic,addition, subtraction, multiplication, division, or number con-cepts; intervention, instruction, training, or teaching method;primary/elementary (education) or children; and disabilities,difficulties, mild/educable mental retardation, disadvantaged,at-risk, underachieving, low-performing, below-average, orlagging in cognitive development. The adequacy of this set ofsearch terms was checked by a hand search of recent volumesof the most well-known journals in which most of the empir-ical studies in the field of special education are published,namely Remedial and Special Education, Journal of Learn-ing Disabilities, The Journal of Special Education, Excep-tional Children, Learning and Instruction, and LearningDisabilities Research & Practice. Furthermore, the referencelists of other recently published research syntheses, meta-analyses, and reviews were carefully checked (Miller et al.,1998; Swanson & Hoskyn, 1998, 1999; Xin & Jitendra,1999). Only English-language articles were included. Theinclusion of only published journal articles may have weak-ened the external validity of the present study due to a ten-dency to publish only studies with significant positive effects(White, 1994) and should therefore be kept in mind wheninterpreting the results of the meta-analysis.

Selection Criteria

The initial computerized search produced 656 references,including 264 articles that did not have mathematics inter-vention as the main topic or did not address mathematicsintervention in the manner we expected. Moreover, 172 of thearticles were either review articles themselves or not based onempirical analyses. Exclusion of the aforementioned refer-ences resulted in the retention of 220 publications. The pub-lications to be used in the meta-analysis were then selectedusing the criteria outlined hereafter. Although we agreed on95% of the articles, the other 5% were discussed to obtain fullagreement on the selection of articles. The following criteriawere used to select the set of articles for the meta-analysis.



The first criterion for inclusion in the meta-analysis wasthat the study be concerned with elementary mathematicsskills, as most math difficulties appear to have their origin inthe acquisition and automatization of these skills (Tissink,Hamers, & Van Luit, 1993). The focus of the meta-analysiswas therefore on interventions with children in kindergartenand elementary school, and studies with a mean participantage higher than 12 were excluded (33 references excluded).

The second criterion was that the study reported on anintervention involving mathematics instruction. This domaincould vary from “addition up to ten” to “third-grade math.”Seventy-one of the references did not report on an interven-tion and simply described, for example, the solution proce-dures used by children, without indicating the instructionthey received. These 71 studies were therefore excluded. In-struction aimed at other skills, such as planning or meta-cognition, and interventions based on homework or parenttraining were also not included (15 references excluded).

The third criterion was that the study reported on chil-dren with mathematical difficulties, based on the descriptiongiven. These difficulties varied from being at risk for disabil-ities or lagging behind to having mathematical learning dis-abilities. Studies reporting on participants with mathematicaldifficulties as a consequence of severe mental disabilitieswere also not included (12 references excluded).

The fourth criterion was that only those studies using abetween-subjects or within-subjects control condition wereincluded in the meta-analysis. All studies therefore had eitherat least an experimental and a control condition or a repeated-measures design (8 studies excluded). For statistical reasons,studies with less than three children were also excluded (6 ref-erences), and sufficient quantitative information had to be re-ported to allow the calculation of effect sizes according to ourmethodology (15 references excluded).

Finally, articles describing mathematics or arithmeticinstruction without reporting the systematic use of instruc-tional strategies were excluded (2 references). The applica-tion of the aforementioned selection procedure yielded a totalof 58 studies for inclusion in the meta-analysis.

Calculation of Effect Sizes

For the studies with a between-group design, Cohen’s d wascalculated by dividing the difference between the scores forthe control groups and experimental groups at posttest by thepooled standard deviation. For the studies with a repeated-measures design, the baseline scores were treated as controlscores and, thus, subtracted from the treatment mean score,with the difference then divided by the pooled standard devi-ation. Although several authors have raised reservations aboutthis method (see Scruggs & Mastropieri, 1998), mainly be-cause it does not take into account the regression of behavioron time, we think it is still the best method to use when theeffects of single-subject designs are combined with theeffects of group design studies. Because the effect sizes of

single-subject research are usually higher than those of groupdesign studies, we included the design of the study as adummy variable in the analyses to correct for a possibleeffect of design. Some studies showed very high effect sizes(up to 12 or 14; Jitendra & Hoff, 1996; Van Luit & Van derAalsvoort, 1985). In light of the fact that differences betweeneffect sizes higher than 3 are essentially meaningless (Scruggs& Mastropieri, 1998), we adopted a maximum effect size of3 for such studies.

For studies that did not report means and standard devi-ations, the effect sizes were calculated on the basis of otherstatistical information (for exact procedures, see Rosenthal,1994). When the total N was reported but not the ns for theexperimental and control groups, we simply divided the totalN by two. When a t test was conducted but no means andstandard deviations or t values were reported, we used thesignificance level given to calculate the effect size.

Whereas many of the studies examined the effects ofintervention on a variety of tests, including tests of mathe-matics performance, motivation, perceived competence, andtransfer, we used only the scores for mathematics perfor-mance (and not for motivation, general achievement, or attri-bution) in the meta-analysis. When more than one test orsubtest was used to measure mathematics performance, wecalculated the effect sizes for all tests and then used the meaneffect size in the meta-analysis. This is because we wouldotherwise have had several outcome measures for some stud-ies and, in consequence, unequal weightings across studies(Rosenthal & Rubin, 1982; Swanson & Carson, 1996). Sev-eral studies also used more than one posttest. To calculate theeffect sizes, however, we used only the scores for the firstposttest. In other words, one effect size was calculated foreach study, unless the study made use of more than oneexperimental condition. In that case, and provided the exper-imental conditions differed significantly from each other, wecalculated one effect size for each experimental condition.This procedure was also followed for studies conducted withdifferent groups, such as children with learning disabilities(LD) and children with mild mental retardation (MMR).These procedures resulted in 61 effect sizes for the 58 re-ported studies.

Coding of the Studies

Three categories of coding variables were distinguished. Thefirst category of report identification and methodology in-cluded the following variables:

1. year of publication;

2. design (single-subject versus group); and

3. control (whether the control condition alsoreceived intervention or not).

The second category was sample description and in-cluded the following variables:



4. number of participants in the experimental andcontrol groups (if necessary, these numberswere estimated on the basis of the datareported);

5. mean age of the participants (some of the stud-ies did not report the mean age for the partici-pants, and other studies reported only gradelevel; in those cases, the mean age was esti-mated on the basis of the available data); and

6. type of special needs (low performing or atrisk; learning disabilities; mild or educablemental retardation; mixed groups or other dis-orders, such as behavior or attention disorders).

The third category was treatment and included the fol-lowing variables:

7. duration of the intervention;

8. total intervention time;

9. content (preparatory skills, such as countingskills or number sense; basic facts, such asaddition/subtraction and multiplication/division; or problem-solving strategies);

10. method (direct instruction, self-instruction, ormediated/assisted performance models; for adescription, see Goldman, 1989);

11. use of computer-assisted instruction (CAI) ornot;

12. peer tutoring or not; and

13. characteristics of Realistic Mathematics Edu-cation (RME; principles such as guided rein-vention, phenomenological exploration, theuse of self-developed models and meaningfulcontexts, student contribution, and interactiv-ity; for a more detailed description, seeGravemeijer, 1994).

Statistical Analyses

For the meta-analysis, we used a random effects model asdescribed by Raudenbush (1994), by means of the programVKHLM (Bryk, Raudenbush, & Congdon, 1994). This modelassumes that study outcomes vary across studies not onlybecause of random sampling effects but also because thereare real differences between the studies. The advantage ofusing multilevel regression analysis lies in the flexibility ofthe method and the ease with which the mean outcomes andvariances can be estimated (Hox, 1995; Hox & De Leeuw,1997). The parameter variance found for the studies is com-pared to the residual variance in the model including othercharacteristics of the studies. This results in a measure esti-

mating the amount of variance explained by different studycharacteristics. In other words, the effects of different ex-planatory variables are calculated while also taking a numberof other characteristics of the studies into consideration, justas in other regression analysis methods.

RESULTS

Table 1 presents an overview of the descriptive informationfor each study, including the authors, participants, proceduralinformation, and results. A total of 2,509 children withspecial mathematics needs was studied. First, the researchquestion on the distribution across the different domains isaddressed. Second, the research question on the effects of theseparate variables is answered (models with only one variableincluded). Finally, to address the third research question, theresults of the multilevel meta-analyses are described (modelswith more variables included).

Question 1: Distribution Across Domains

The studies were distributed across the three domains of ele-mentary mathematics as follows: 13 interventions for pre-paratory arithmetic, 31 for basic facts, and 17 for problemsolving. The domain of basic facts has been investigatedmost, χ2(2, N = 61) = 7.897, p = .019. However, no signifi-cant differences were found between the effect sizes of thethree different domains.

Question 2: Effects of Single Variables

Table 2 provides an overview of the distribution across thedifferent variables, together with the single weighted effectsof these variables. Of the 61 empirical investigations includedin the meta-analysis, 21 had single-subject designs and 40had group designs. The single-subject design studies had sig-nificant higher effect sizes than the group design studies.Studies in which the control condition received a differentintervention (n = 30), showed higher effect sizes than studieswithout a different intervention in the control condition (n =31). No effect of year of publication was found, γ = –0.020,p > .10.

The average number of participants in the studies was41.1, with a range of 3 to 136. A negative effect was found ofnumber of participants, γ = –0.007, p < .05, which indicatedthat small studies showed higher effects than large studies.The mean age was 8.6 years (SD = 1.8), with a range of 5 to12 years. Interventions with older students had more effectthan interventions with younger students, γ = 0.106, p < .10.Also, an effect of the special needs of the students was found,because the interventions for students with learning disabili-ties showed higher effect sizes than the interventions for the other groups; 23 studies described interventions for low-performing/at-risk children, 23 for children with learningdisabilities, 8 for children with mild or educable mental

102

TABLE 1. Overview of Studies Included in the Meta-Analysis

Procedure/ Sessions/Reference M N Special needs design Intervention duration Results/effect size

Ainsworth, 7.3 29Wood, & O’Malley (1998)

Beirne-Smith 8.7 30(1991)

Case, Harris, 11.3 4& Graham (1992)

Cassel & 9.1 4Reid (1996)

Dunlap & 11 3Dunlap (1989)

Fantuzzo, 10.6 47Davis, & Ginsburg (1995)

Fantuzzo, 10.5 68King, & Heller (1992)

Fuchs et al. 9.4 40(1997)

Low performing

LD, low performing

LD, low performing

LD, MMR

LD

At risk/lowperforming

At risk/lowperforming

Low performingwith andwithout LD

Pretest–intervention–posttest; randomizedblock design

Pretest–intervention–posttest; randomassignment

Baseline, Phase1 and 2probes, gen-eralization,maintenance

Multiple base-line across students

Multiple base-line across students

Pretest–intervention–posttest;randomassignment


Pretest–intervention–posttest

Computer intervention toteach that one problemcan have many differentcorrect solutions (experi-mental condition); useof concrete and every-day knowledge. Controlcondition: single answer

Single-digit addition facts:Counting on peer tutor-ing, rote memorizationpeer tutoring, and con-trol

Self-regulated strategydevelopment for addi-tion and subtractionword problems

Self-regulated strategyintervention; word prob-lem solving

Self-monitoring checklists;subtraction

RPT versus control; alsoeffect of parent interven-tion on computationskills

Effects of offering structurein RPT; also effect ofrewards on computationproblems

Effects of task-focusedgoals

2 × 35-minsessionsin 2weeks

20a × 30-min ses-sions in 4 weeks



13 sessionsin 3weeks

20 × 45-min sessionsin 10weeks

50a × 45-min ses-sions in5 months


Both conditions improvedsignificantly; conditionwith multiple answersimproved more thancondition in whichsingle answers wererequired, F(1, 50) =5.912, p < .019, d = 0.89

Peer-tutored students per-formed higher than stu-dents who received nopeer tutoring, p < .01,d = 0.82

Students’ overall perfor-mance improved,d = 1.56

Performance increased to80% mastery level,maintained 8 weeks,d = 1.14

All students showed imme-diate and dramatic gainsduring interventionperiod; d = 2.57

RPT higher scores thancontrol, p < .05; d =0.52

Structure + reward highestscores; main effect forreward, not for structure;d = 0.48

LD students performedlower than low-achieving students, nomain effect for treat-ment; d = 0.40

(table continues)

103


Ginsburg- 10.5 40Block & Fantuzzo (1997)

Ginsburg- 9.6 104Block & Fantuzzo (1998)

Greene 10.7 23(1999)

Harris, Mil- 8.3 13ler, & Mer-cer (1995)

Hasselbring 8 24& Moore (1996)

Heller & 9.9 54Fantuzzo (1993)

Ho & Cheng 7.0 45(1997)

Jaspers & 10.3 5Van Lie-shout (1994)

Jitendra et al. 10.4 34(1998)

At risk, low per-forming

Low performing

LD

LD

EMR, LD, SED

At risk

Low performing

EMR

LD, EMR, SED



Pretest–intervention–posttestrandomassignment

Baseline, pre-test, lessons1–10, post-test, lessons11–21



Pretest–intervention–posttest;matchedgroups

Multiple base-line designacross stu-dents

Pretest–intervention–posttest; randomizedblock design

2 conditions: RPT andpractice control, work-ing in dyads on compu-tational problems

NCTM standards-basedintervention: problemsolving, peer collabora-tion, both, or nothing(control). Computationand word problems

Mnemonic instructionversus more traditionalinstruction on multipli-cation facts

Multiplication facts 0 to81; concrete to represen-tational to abstractsequence

Contextualized learningenvironments versusdirect instruction. Wordproblem solving

Reciprocal peer tutoring,also effect of parentinvolvement; mathcomputation

Training in place-valueconcepts; 15 poor arith-metic children got train-ing; two control groups(good/poor)

CAI; word problem solvingability; focused on theconstruction of adequateproblem representation

Schema training versus tra-ditional word problemsolving


14 × 30-min ses-sions in7 weeks

5 × 20-minsessionsin 1week

10 lessonsin 8weeks

86 × 30-minlessons




20 × 45-min ses-sions in4a weeks

RPT higher math achieve-ment, F(1, 37) = 15.38,p < .001, d = 0.72

Positive results for peercollaboration, p < .01,d = 0.34, and for prob-lem solving, p < .05,d = 0.51, no interactioneffect

Mnemonic training mosteffective, F(1, 21) =79.74, p = .000, d = 1.19

Significant improvement; d = 2.71

Experimental groupshowed more improve-ment than control group;d = 0.27

RPT higher scores thancontrol, p < .01, d =1.00

Positive effects on place-value understanding,addition, and subtrac-tion, p < .05, d = 0.85

Significantly increased per-formance from baselineto training phase; d =1.26

Significant main effect forgroup; schema > tradi-tional, F(1, 31) = 1.8,p = .005, d = 0.64

(table continues)

(Table 1 continued)

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Jitendra & 10.1 3Hoff (1996)

Keogh, Whit- 10.6 16man, & Maxwell (1988)

Knapczyk 10 3(1989)

Koscinski & 9.5 6Gast (1993)

Lin, Podell, & 8.4 58Tournaki-Rein (1994)

Maag, Reid, 9.3 6& DiGangi (1993)

Malabonga, 5.8 17Pasnak,Hendricks,Southard,& Lacey (1995)

Marsh & 9 3Cooke (1996)

Mattingly & 11.6 4Bott (1990)

LD, low per-forming

MMR

LD

LD/ADHD

MMR

LD

At risk

LD

LD, EMR, BD

Multiple base-line

Pretest–intervention–posttest 1, 2;randomassignment

Multiple base-line acrossstudents

Multiple base-line


Multiple base-line



Multiple probedesign

Schema-based directinstruction strategy;word problem solving

Addition with regrouping;self-instruction vs. exter-nal instruction in smallgroups

Generalization of questionasking; videotape watch-ing; structuring opportu-nities for questionasking; providing feed-back

CAI constant time delay inteaching unknown multi-plication facts

CAI (drill and practice)versus paper–pencil;addition/subtractionfacts; MMR versusnondisabled

Effects of self-monitoringof attention, accuracy, orproductivity

experimental: seriation,classification trainingcontrol: training on aca-demic subjects

Use of concrete manipula-tives to teach mathemat-ical problem solving

Constant time delay proce-dure in teaching un-known multiplicationfacts



6 × 50-minsessions

6 sessionsin 2weeks

13 × 15-min sessionsin 1month



8 × 20-minsessions

6–19 ses-sions in2 weeks

Increased percentage ofcorrect solutions to wordproblems; d > 3

Self-instructional traininghigher results than exter-nal instruction, F(1, 12) =3.27, p = .05, d = 0.67

Gradual increase of accu-racy scores during treat-ment; d = 0.63

Results indicated that thisprocedure was an effec-tive method of teachingmultiplication facts toLD students; d = 2.18

No significant differencesbetween two conditions;d = –0.44

Self-monitoring of accu-racy most effective; d = 1.53

Posttest scores of theexperimental childrenwere higher than thoseof the control children,t(15) = 1.86, p < .05,d = 1.61

Significant improvementduring intervention; d > 3

Time delay procedure waseffective in teaching thetargeted multiplicationfacts; d > 3

(table continues)

(Table 1 continued)

105


Mevarech 7 204(1985)

Mevarech & 8 376Rich (1985)

Miller & 9.4 9Mercer (1993)

Naglieri & 11.9 4Gottling (1995)

Naglieri & 10.1 12Gottling (1997)

Pasnak (1987) 5 22

Pasnak, Hans- 5 64barger,Dodson,Hart, & Blaha (1996)

Pasnak, Holt, 5 57Campbell,& Mc-Cutcheon (1991)

At risk

At risk

LD, EMR

LD

LD

Low performing

Low performing

Low performing

Intervention–posttest

Intervention–posttests


Baseline–intervention

Baseline–intervention


Intervention–posttest;randomassignment

Intervention–posttest;restrictedrandomiza-tion

CAI in individualized ormore traditional training

CAI (TOAM): testing andpractice; individualizedinstruction

Concrete–semiconcrete–abstract teachingsequence for basic mathfacts and coin sums

Cognitive instruction inaddition and multiplica-tion that facilitated plan-ning

Cognitive mathematicsinstruction, planning;low/high planning

Classification, seriation,conservation trainingversus typical mathinstruction

Cognitive intervention:classification, seriation,conservation; controlchildren receivedinstruction on verbal andmathematics materials

Piacceleration curriculum:classification, seriation,number conservation;Control: standard cur-riculum

80 × 20-min ses-sions in1 yeara

80 × 20-min ses-sions in1 yeara


7 × 25-minsessions



40 shortlessonsin 3months

40a × 15-min ses-sions for3 months

Significant main effects forthe use of CAI, p <.001, d = 0.73, not forindividualized instruc-tion, d = –0.20

Experimental childrenscored higher on arith-metic achievement thanpupils receiving tradi-tional instruction,p < .0001; d = 0.54

CSA was effective foracquisition and short-term retention; d = 2.28

Cognitive facilitation inter-vention did not improvemultiplication perfor-mance for average plan-ners, but was effectivefor those with poor plan-ning scores; d = 0.96

Low planning studentsshowed most improve-ment; d = 0.71

Significant positive effectfor experimental condi-tion, F(1, 20) = 2.72,p < .05, d = 0.73

Experimental instructionproduced higher scoresthan control instruction,F(1, 42) = 6.20, p < .02;d = 0.53

Piacceleration curriculumwas significantly supe-rior to typical curricu-lum, Wilks’ λ = 8.43,p < .01; d = 1.09

(table continues)

(Table 1 continued)

106


Pearce & 10.5 8Norwich (1986)

Perry, Pas- 8.2 23nak, & Holt (1992)

Pigott, Fan- 11 12tuzzo, & Clement (1986)

Podell, 8.2 28Tournaki-Rein, & Lin (1992)

Schopman & 6 60Van Luit (1996)

Schunk, Han- I: I:son, & Cox 10.6 80(1987) II: II:

10.9 80

Shiah, Mas- 10.2 30tropieri,Scruggs, & Mushinski Fulk (1995)

Skinner, 11.5 3Bamberg,Smith, & Powell (1993)

LD

MMR

Low performing

MMR/LD

Developmentallag

Low performing

LD

BD

Baseline, teach-ing, mainte-nance;matchedgroup,randomassigned

Pretest–intervention–posttest;matchedpairs

Multiple base-line




Pretest–instruction–posttests 1,2; randomassignment

Within subjects,across problems,multiplebaseline

CAI versus direct teaching;simple number estima-tion skills

Classification and seriationversus verbal/numberconcepts

RPT with group reinforce-ment; basic arithmeticoperations

CAI/paper–pencil; automa-tization of basic skills inaddition

Preparatory arithmeticstrategies, directing andguiding versus control

Influence of attributes ofpeer models: rapid ver-sus gradual acquisitionof fraction skills

I: also effect of same- oropposite-gender model

II: also effect of one modelversus three models

CAI on math word problemsolving; two conditionswith cognitive strategytraining, with one usinganimated pictures

Cognitive cover, copy, andcompare intervention ondivision facts






6 × 40-minsessionsin 6 days

3 × 30-minsessionsin 1weeka


CAI group had higherscores than direct teach-ing group; direct teach-ing group had higherscores in maintenanceperiod; d = 0.13

Significant difference infavor of experimentalgroup, F(1, 16) = 2.71,p < .05, d = 0.25

During treatment, perfor-mance increased to aver-age level; d > 3

No significant differencesin accuracy; CAI >paper and pencil inresponse time, F(1, 48) =5.22, p < .05, d = 0.62

Experimental group scoredhigher than controlgroup, F(1, 56) = 35.57,p < .001, d = 0.89

Gradual acquisition modelsignificantly enhancedchildren’s performance;

I: F(1, 71) = 45.46, p <.001, d = 1.43

II: F(1, 71) = 10.49, p <.01, d = 0.91

Significant improvement inall conditions; no differ-ences between condi-tions; d = 0.32

Two of three studentsshowed increased per-formance; the third stu-dent needed feedbackand goal-setting; d = 2.63

(table continues)

(Table 1 continued)

107


Stellingwerf 11.3 100& Van Lieshout (1999)

Sugai & 10 7Smith (1986)

Swanson 9.5 3(1985)

Thackwray, 8.6 60Meyers,Schleser,& Cohen (1985)

Van de Rijt 5.5 136& Van Luit (1998)

Van Luit 11.4 52(1987)

Van Luit I: I:(1994) 11.7 16

II: II:9.7 28

Van Luit & I: I:Naglieri 12.7 42(1999) II: II:

10.8 42

LD, EMR

LD

SED, low per-forming

Low performing

Low performing

LD/EMR/impulsive

I: EMRII: LD

I: MMRII: LD

Pretest–instruction–posttest;matching,randomassignment


Multiple base-line


Pretest–intervention–posttest 1, 2;matchedgroups

Baseline–intervention–follow-up



Manipulatives (MAN) andnumber sentences(NUM) in CAI wordproblem solving

Demonstration plus model;equal addition methodof subtracting

Cognitive behavioral train-ing in addition and sub-traction problems

Specific self-instruction;general self-instruction;didactic instruction;addition problems

Guiding–structuring addi-tional early mathematicsprogram

Self-instructional arith-metic training program

Structural (SC) vs realistic(RC); addition and sub-traction with renaming

MASTER training pro-gram: strategy training;teacher assists childrenin selecting strategies;self-instruction. Multi-plication facts.

12 × 30-minsessions

30 × 15-min sessions

22 × 20-minsessions

4 × 45-minsessionsin 2weeksa

26 × 30-minlessonsin 3months

50a × 30-min sessionsin 20weeks

I: 100a ×45-minsessionsin 6months

II: 140a ×45-minsessionsin 8months


No effect of MAN, signifi-cant effect of NUM, p <.05. MAN: d = 0; NUM:d = 0.43

Instruction effective forstudent’s computation ofsubtraction withregrouping; d = 2.88

Training favorablyimproved academic per-formance; d > 3

Main effect for instructiongroup on generalization;specific self-instructionscored highest, p < .05;d = 0.35

Experimental group scoredhigher than controlgroup, F(3, 132) = 7.19,p = .000; Guiding: d =1.04; Structuring: d =1.24

Significant improvement,t(1, 15) = 11.69,p = .000; d = 2.38

I: no significant differencesbetween groups; d = 0.09

II: SC > RC on trainedtasks, p = .002; d = 0.66

Experimental groupsshowed higher posttestscores than comparisongroups. I: F(1, 40) =54.44, p < .001, d =2.80. II: F(1, 40) =81.82, p < .001, d = 2.28

(table continues)

(Table 1 continued)

108


Van Luit & 6.2 124Schopman (2000)

Van Luit & 12 4Van der Aalsvoort (1985)

Waiss & 6 24Pasnak (1993)

R. Wilson, 10.2 4Majsterek,& Sim-mons (1996)

C. L. Wilson 9 62& Sindelar (1991)

D. K. Wood, 10 3Frank, & Wacker (1998)

D. A. Wood, 9.5 9Rosenberg,& Carran (1993)

Woodward & 10 38Baxter (1997)

Note. LD = learning disabilities; EMR/MMR = educable/mild mental retardation; SED = serious emotional disturbance; ADHD = attention-deficit/hyperactivity disor-der; BD = behavioral disorder; CAI = computer-assisted instruction; RPT = Reciprocal peer tutoring; NCTM = National Council of Teachers of Mathematics; CSA =concrete–semiconcrete–abstract.aEstimated on the basis of available information.

LD/MMR

EMR

Low performing

LD

LD

LD

LD

Low performing


Baseline, inter-vention,follow-up

Pretest–instruction–posttest;matched andrandomlyassigned

Single-subject,alternatingtreatment

Pretest–intervention–posttest 1, 2;stratifiedrandomassignment

Multiple base-line

Baseline–instruction–generaliza-tion; ran-domlystratified


Early numeracy based onperceptual gestalt theory

Self-instruction in solvingaddition and subtractionproblems

Experimental: classifica-tion, number conserva-tion

Control: verbal and arith-metic skills

CAI vs. teacher-directedinstruction; multiplica-tion facts

Strategy plus sequenceinstruction vs. strategyonly instruction vs.sequence only instruc-tion; addition and sub-traction word problems

Instructional package formultiplication, involvingcategorizing multiplica-tion facts, mnemonicstrategies, steps to becompleted

Individualized self-instruc-tion vs. observation ofself-instruction trainingvs. control

NCTM standards programversus traditionalapproach in third-gradecurriculum

50 × 30-min ses-sions in6 months


50 × 15-min sessionsin 20a

weeks

13–31 ×30-minsessions

14 × 30-minlessonsin 1montha

3 sessions

2 sessions

Daily ses-sions in1 year

Experimental group scoredhigher on posttest thancomparison group,t(124) = 3.29, p = .001,d = 0.66

At end of training, all stu-dents had reached goal:solve subtraction prob-lems with renaming; d > 3

Significant improvementfrom pretest to posttest;no significant differ-ences between groups; d = 0.06

All students mastered morefacts under teacher-directed intruction; d = 1.95

Strategy plus sequencescored higher thansequence only, F(1, 57) =7.49, p < .05; strategyonly higher than se-quence only, F(1, 57) =4.33, p < .05, d = 0.21

Substantial improvementduring intervention; d > 3

Both experimental groupsshowed higher scores onposttest than controlgroup; d > 3

For low-achieving studentsno significant differ-ences; d = –0.24

(Table 1 continued)



retardation, and 7 for a mixed group of children with severaldisabilities.

The duration of the interventions varied from 2 to 140sessions (M = 10.6, SD = 9.0); the duration of the sessionsranged from 10 minutes to 60 minutes, with a mean of 35minutes (SD = 13.8). The total instruction time varied from 1 week to 1 year. Both the instruction time, γ = –0.001,p < .05, and the duration, γ = –0.018, p < .05, showed a neg-ative correlation with effect sizes. No effect was found forcontent (preparatory, basic, of problem-solving skills).

Most of the studies (n = 35) used direct instruction in theintervention. However, self-instruction (n = 16) led to highereffect sizes. A total of 12 studies used computer-assistedinstruction (CAI), 10 studies peer tutoring. No effect of peertutoring was found, but the studies using CAI showed lowereffect sizes than studies in which the teacher instructed thestudents.

One last variable concerns reform-based characteristicsof interventions. Mediated or assisted instruction, contrary todirect instruction, requires students to discover and develop

their own math skills, with the assistance of a teacher. Tenstudies used mediated or assisted instruction models, but thisdid not lead to higher effect sizes. Fourteen studies used oneor more instructional principles from Realistic MathematicsEducation, such as guided reinvention, the use of self-developed models and meaningful contexts, or student con-tribution and interactivity. Although a trend was found infavor of more traditional interventions, no significant differ-ences were found.

Question 3: Multilevel Meta-Analysis

The aim of meta-analysis is to discover which variablesexplain at least part of the variance not explained by samplingvariance. The results were significantly heterogeneous (esti-mated parameter reliability for the 58 studies σ2 = 0.510,p = .000), which justifies the use of a random effects model.Because the differences between studies were large, it wouldbe wrong to assume that the students included were selectedfrom one population.

TABLE 2. Effects of Nominal Variables Included in the Study

Effect size

Variable N Sample size M SD Weighted effect size

Design**baseline 21 155 2.27 0.79 2.16experimental 40 2,354 0.68 0.73 0.62

Control condition**no intervention 30 701 1.87 1.01 1.51intervention 31 1,808 0.54 0.59 0.51

Special needs*low performing 23 1,608 0.83 0.78 0.74learning disabilities 23 416 1.65 1.11 1.36mild mental retardation 8 192 1.01 1.27 0.80mixed groups 7 293 1.09 1.12 0.73

Contentpreparatory 13 664 0.92 0.72 0.92basic facts 31 1,324 1.50 1.19 1.14problem solving 17 521 0.84 0.86 0.63

Method**direct instruction 35 1,671 1.13 0.94 0.91self-instruction 16 372 1.77 1.12 1.45mediated/assisted 10 466 0.52 0.97 0.34

Medium**teacher 49 1,635 1.32 1.07 1.05computer 12 872 0.64 0.79 0.51

Peer tutoringno 51 1,954 1.24 1.10 0.96yes 10 555 0.92 0.76 0.87

Realistic mathno 47 1,709 1.33 1.07 1.04yes 14 800 0.71 0.88 0.70

*p < .10. **p < .05.



Four variables were found to explain 69% of thebetween-studies variance. The best explanatory variable wasdesign, γ = 1.4, p = .000. Studies with a single-subject designproduced higher effect sizes than those with a group design.A second important explanatory variable was the durationof the intervention, γ = –0.01, p = .032. Interventions thatlasted longer had less effect than shorter interventions. Athird explanatory variable was the domain of intervention(preparatory—basic: γ = 0.24, p = .177; preparatory—problem solving: γ = 0.70, p = .004; basic—problem solving:γ = 0.46, p = .018). Interventions in the domain of problemsolving were found to be less effective than in both otherdomains. A final explanatory variable was the method used(direct instruction—self-instruction: γ = –0.42, p = .051;direct instruction—mediated/assisted instruction: γ = 0.24,p = .199; self-instruction—mediated/assisted instruction:γ = 0.66, p = .009). Self-instruction was found to be moreeffective than direct instruction or mediated/assisted instruc-tion. These results were also found when only analyzing thestudies with group designs. The studies with single-subjectdesigns were too few and too homogeneous to analyze sepa-rately.

Separate meta-analyses were conducted for each of thethree categories of intervention. Due to the high number ofvariables and small Ns for the category of preparatory arith-metic, the separate meta-analysis for this category was con-ducted with only a few variables.

Preparatory Arithmetic. Because of the low numberof studies in this category that used interventions with CAI(1), peer tutoring (0), self-instruction (1), and mediated in-struction (1), these variables were not analyzed. None of thestudies in this domain had a single-subject design. The analy-ses were conducted with the following variables: year of pub-lication, intervention in control group, number of students,mean age, duration, total time, and RME. For the interven-tions directed at preparatory tasks, the between-studies vari-ance was found to be low, σ2 = 0.262, p = .001. The variablesexplaining the most variance were the duration of the inter-vention, γ = –0.08, p = .001, and total instruction time, γ =0.01, p = .001. Together, these variables explained 99% of thevariance.

Basic Facts. In the category of interventions aimed atbasic skills, 85% of the between-studies variance (σ2 = 0.943,p = .000) could be explained by the following variables: inter-vention in control group, γ = 1.49, p = .000; peer tutoring,γ = –0.76, p = .011; mean age, γ = 0.39, p = .000; and method(direct instruction—self-instruction: γ = 0.76, p = .043; directinstruction—mediated/assisted instruction: γ = 1.55, p = .001;self-instruction—mediated/assisted instruction: γ = 0.79,p = .070). Thus, the main results are that studies in which thecontrol group also received an intervention produced lowereffect sizes than studies with no intervention in the controlgroup; that interventions using peer tutoring are less effective

than those not using this method; that the interventions forolder students proved more effective; and that direct instruc-tion is more effective than mediated/assisted instruction orself-instruction.

Problem Solving. The effect sizes for the studies con-cerned with problem solving were also found to be heteroge-neous, σ2 = 0.170, p = .005; 99% of the between-studiesvariance was explained by the following variables: interven-tion in control group, γ = 0.54, p = .052, year of publication,γ = 0.08, p = .061; type of special needs (low performing:γ = 0.77, p = .068; learning disabilities: γ = 0.32, p = .223;mild mental retardation: γ = 1.09, p = .143); peer tutoring,γ = –1.62, p = .011; and use of CAI, γ = –0.78, p = .025. Themain results for interventions aimed at problem-solving skillsare that studies with an intervention in the control groupshowed lower effect sizes than studies with no intervention inthe control group; that peer tutoring and computer-assistedinstruction were less effective than other intervention meth-ods; that interventions for children with mild mental retarda-tion were more effective than those for children with learningdisabilities; and that interventions for students with diverseand mixed problems were less effective than interventions forlow performers.

DISCUSSION

With the use of a random effects model in the present study,it was possible to determine which study characteristics ap-peared to be most important for the prediction of effect size.The single variable analyses showed that several variableshad a significant influence on the study outcomes. However,when analyzed together, only four variables were found toexplain a significant part of the variance in the effect sizes forall studies considered together. For instance, both the dura-tion of intervention and the total instruction time proved sig-nificant when analyzed separately. However, when analyzedtogether, it appeared that the variable duration alreadyexplained a significant part of the variance and that the totalinstruction time did not contribute any additional explainedvariance. Apparently, there was a strong relationship betweenthe total instruction time and the duration of the intervention.The same phenomenon was found for the variables design,control condition, age, and domain. Therefore, the resultsfrom the multiple variable analyses are taken as the basis forour conclusions. It should be noted that these conclusions arebased only on studies published in well-known journals,which may weaken the external validity of the findings and,therefore, call for caution in their interpretation.

The first conclusion is that the majority of the studiesexamined described an intervention in the domain of basicskills. The interventions in the domain of basic skills never-theless showed the highest effect sizes. The domain of basicmath skills is large and plays an important role in the devel-



opment of students’ later math skills (Mercer & Miller, 1992;Van Luit & Naglieri, 1999). Therefore, it is not surprising thatmany studies are concerned with this domain, which was alsofound in previous meta-analyses (e.g., Miller et al., 1998).Basic math skills also appears to be a domain in which inter-ventions are effective. It may be easier to teach basic skills tostudents with special needs than to teach problem-solvingskills. Moreover, because we selected only studies on ele-mentary students, it was expected that the area of problemsolving would be underrepresented. However, when com-pared to the research synthesis of Mastropieri et al. (1991), achange toward more research concerning problem solving isdetected.

The second conclusion is that the most important pre-dictor variable in the present study was found to be researchdesign. Similar to the findings of other studies (e.g., Xin &Jitendra, 1999), our analyses showed that studies with asingle-subject design show more powerful results than thosewith a group design. Several possible explanations can begiven. One of them is that children in a single-subject designoften complete the same test or parallel versions of the sametest and, therefore, grow accustomed to the test and performbetter over time. It should be noted that the effect sizes forsingle-subject studies were calculated in a slightly differentmanner than the effect sizes for (quasi-) experimental groupstudies. An intragroup comparison is made in single-subjectstudies, whereas intergroup comparisons are made in experi-mental group studies. The effect sizes for the differencebetween pretest and posttest in the single-subject studies gen-erally tended to be larger than the effect sizes for the differ-ences between the control and the experimental groups in thegroup studies. In most group designs, however, the controlgroup received some form of intervention; in single-subjectdesigns, in contrast, the baseline involves no intervention.This difference may also explain the varying effect sizes ob-served for the different study designs. The significant effectof intervention in the control group found in the preparatoryand basic skills interventions supports this explanation.Finally, it should be noted that the training in a single-subjectrepeated-measures study is often criterion based; that is,training continues until a set criterion is reached (usually80% correct). Given that the training is only stopped whenthe results are sufficiently high, the large effect sizes undersuch circumstances are not surprising.

A third conclusion concerns the effects of sample char-acteristics. Overall, no differences were found between stud-ies that reported interventions for students with differentspecial needs. However, in the studies concerned with prob-lem solving, the interventions for students with mild mentalretardation were more effective than those for students withlearning disabilities. This is consistent with the findings ofother meta-analyses (e.g., Xin & Jitendra, 1999). A possibleexplanation for this unexpected result may be found in thenature of the interventions. Children with mild mental retar-dation often receive intensive training across an extended

period of time, mainly focused on basic skills. If the in-tervention involves a new method or domain (in this case,problem solving), the children may become very motivated.Students with specific learning disabilities may already havea history of failure in the topic concerned and, as a result,have motivational problems.

The fourth conclusion is that the duration of the inter-vention correlated negatively with effect size, especially forthe interventions focused on preparatory tasks. We originallyexpected this correlation to be positive. A possible explana-tion (see also Xin & Jitendra, 1999) is that short interventionstend to focus on a very small and specific domain of knowl-edge, such as addition up to 10. Prior to such an intervention,the children score very low; after a short period of interven-tion, however, they have fully acquired the relevant knowl-edge and thus score quite high. Longer interventions, incontrast, may focus on a broader domain of knowledge, costmore time, and therefore produce smaller effect sizes thanshorter interventions. Moreover, the testing of an interventionin a broad domain of knowledge may be complicated bynumerous interacting variables.

The fifth conclusion, regarding the treatment compo-nents of interventions, is that it appears from the presentmeta-analysis that in general, self-instruction is most effec-tive. However, for the learning of basic skills, direct instruc-tion appears to be the most effective. This is in line with manystudies that have examined the effects of direct instruction(Carnine, 1997) and with other meta-analyses (e.g., Swanson& Carson, 1996; Swanson & Hoskyn, 1999). Another con-clusion we can draw is that the use of the computer as an aidto instruction cannot replace the teacher. The interventionswith computer-assisted instruction produced lower effectsizes than other interventions. This finding corresponds to thefindings of other studies (e.g., Hativa, 1994) and suggests thatthe computer is less effective than a human teacher. However,Xin and Jitendra (1999) found computer-assisted instructionto be most effective. A possible explanation for this differ-ence is that the majority of the studies they found used somekind of direct instruction. As we have seen, direct instructionis in general more effective, whether given with the aid of acomputer or not. Furthermore, interventions making use ofpeer tutoring were found to be less effective than other inter-ventions. This may be explained by a number of different fac-tors. One important factor is that peers are less capable ofperceiving the needs of other students than teachers. Also,young students are often not accustomed to working togetherand need experience to develop the necessary skills (Wilkin-son, Martino, & Camilli, 1994). The role of the teacher thusappears to be critical to help students and to evaluate theirprogress.

A final conclusion concerns the effects of reform-basedinterventions. Mediated/assisted instruction was found to beless effective than direct instruction or self-instruction. Noeffects were found on the variable realistic mathematics edu-cation (RME). In other words, these analyses confirm that the



recent changes in mathematics education do not lead to bet-ter performance for students with special needs (Jitendra &Hoff, 1996; Kroesbergen, Van Luit, & Maas, 2002; Wood-ward & Baxter, 1997).

IMPLICATIONS FOR PRACTICE

From this study, a few interesting conclusions can be drawnwith regard to math interventions. When choosing and orga-nizing an intervention, one should keep in mind the follow-ing findings. The first finding concerns the method used toteach students mathematics. Both self-instruction and directinstruction seem to be adequate methods for students withspecial needs. For the learning of basic math facts, directinstruction appears to be most effective. For the learning ofproblem-solving skills, self-instruction methods are alsoquite effective.

A second finding concerns the use of CAI, which can bevery helpful when students have to be motivated to practicewith certain kinds of problems. With the use of a computer, itis possible to let children practice and automatize math factsand also to provide direct feedback (e.g., Koscinski & Gast,1993). However, the computer cannot remediate the basic dif-ficulties that the children encounter. The results of the presentstudy show that in general, traditional interventions withhumans as teachers, and not computers, are most effective.

We often have children work together in order that theymight help and teach each other. It appears, however, thatchildren with special needs do not particularly profit fromthis strategy. Of course, peer tutoring may be helpful andeffective at times, but the present study shows that it cannotreplace or be as effective as instruction by an adult teacher.

Finally, this study suggests that not all of the changesproposed by math reformers are as effective as more tradi-tional approaches. However, it always takes time to adjust tochanges, and this variable should therefore be re-examinedthoroughly when more data become available. ■

EVELYN H. KROESBERGEN, PhD, is a researcher and teacher at theDepartment of Special Education at Utrecht University, The Netherlands.Her research focuses on mathematics education for special children.JOHANNES E. H. VAN LUIT, PhD, is an associate professor of specialneeds education at Utrecht University, The Netherlands. His current interestsinclude exceptional children, particularly those with mild mental retardation;mathematics education for special children; program and test developmentfor low-performing children in mathematics; and cognitive problem solving.Address: Evelyn H. Kroesbergen, Utrecht University, Department of SpecialEducation, PO Box 80140, 3508 TC Utrecht, The Netherlands; e-mail:[email protected]

AUTHORS’ NOTES

1. The funding of this project by the Netherlands Organization for Scien-tific Research is gratefully acknowledged. The research was supportedby Grant 575-36-002 from this organization’s Social Science ResearchCouncil.

2. The authors appreciate the helpful comments provided by J. J. Hox andA. Vermeer on an earlier draft of this article.

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Received August 23, 2001Revision received February 8, 2002

Second revision received July 8, 2002Final acceptance July 19, 2002

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