Manipulatives are concrete or virtual objects (e.g., blocks and chips) often used in elementary grades to illustrate abstract mathematical concepts. We conducted a systematic review to examine the effects of interventions delivered with manipulatives on the learning of children with mathematics learning disabilities (MLD). The outcomes observed in the sample (
According to the American Psychiatric Association in the Diagnostic and Statistical Manual of Mental Disorders, DSM5 [
Studies have revealed that MLD is manifested by difficulties mastering number sense, number facts, or calculation, as well as difficulties with mathematical reasoning, and cannot be explained by intellectual disabilities, uncorrected visual or auditory acuity, other mental or neurological disorders, psychosocial adversity, or lack of proficiency in the language of academic instruction [
Our work focuses on the effects of using manipulatives in mathematics instruction on children’s learning and transfer. “Manipulatives” are concrete or virtual objects and are intended to reify central concepts in the mathematics curriculum. Students and teachers can configure and manipulate the objects, whether they are concrete or virtual, in ways that reflect the ideas at the heart of a lesson. Some research has indicated that manipulatives can be effective for the development of children’s conceptual and procedural knowledge of mathematics [
There is some evidence to suggest that children with MLD can benefit from instruction with manipulatives (e.g., [
The objective of this review is to evaluate the impact of using manipulatives—i.e., concrete materials such as blocks or plastic chips or virtual representations of similar objects—on the mathematics learning of children with MLD. We are not only interested in the effects of interventions that involve manipulatives, but also in the instructional contexts in which they are used. This research will contribute to current understandings of how external representations in mathematics could be beneficial for learning, maintenance, and transfer in this population. Moreover, the pedagogical implications for special educators are significant, as there is at present no consensus on the most effective ways to use concrete, or virtual, representations for students with MLD.
A handful of researchers have investigated the effectiveness of interventions for children with MLD and mathematics difficulties. Methe et al [
In another review, Jitendra et al. [
Bouck and Park [
The second way in which our review differs from that of Bouck and Park’s review is that their analysis primarily targeted students’ immediate learning, whereas our analysis also included the outcomes measures of maintenance and transfer. The goal of any instructional program in mathematics goes beyond the immediate replication of the content given during instruction; ultimately, the aim is for students’ longterm gains as well as the capacity to transfer new knowledge to other tasks and contexts. Indeed, it is believed that the ultimate test of conceptual understanding is the ability to use it to solve a novel problem (see [
Furthermore, it is particularly important to examine maintenance and transfer effects in the MLD population. Children with MLD have persistent deficits that are resistant to intervention [
Finally, we argue that it may be shortsighted to investigate learning without also taking the interrelated constructs of maintenance and transfer into account. When children learn such that they can transfer their knowledge, they are more likely to experience lasting effects of the instruction they were provided [
In this paper, we present a systematic review of the research examining the impact of interventions with manipulatives on the mathematics learning of children with MLD. We included studies that examined a wide variety of instructional approaches, ranging from instruction in inclusive classrooms to more targeted interventions outside the classroom for students with MLD [
These procedures resulted in a large initial sample, which we then narrowed down using the following exclusion criteria to focus the review on interventions for students with MLD. We excluded from our sample any study that focused only on children with disabilities other than MLD (i.e., intellectual disabilities, autism spectrum disorder, and emotional disorders). Some studies placed children with MLD and intellectual disabilities together in one instructional group; we excluded these studies from our review because we were unable to isolate the children with MLD from those with intellectual disabilities. For singlecase studies (we use the term “singlecase study” to refer to both singlecase and multiplecase studies), we focused the analyses only on the participants in the samples with MLD and not on those with other disabilities.
In the present review, we aimed to extend the results of Bouck and Park [
In sum, the review addressed the three following research questions:
What are the instructional contexts for the interventions with manipulatives? Specifically, what skills were targeted by the interventions, what were the characteristics of the interventions, and what types of manipulatives were used?
Can interventions that include manipulatives be considered evidencebased for children with MLD in terms of immediate learning, maintenance, and transfer?
Do the research designs used in the studies allow us to conclude that the manipulatives themselves added value to the interventions and do the designs allow for causal conclusions to be drawn about the interventions with manipulatives?
We will address the second and third research questions by assessing the methodological quality (i.e., [
In conducting the review, we used the Preferred Reporting Items for Systematic Reviews and MetaAnalyses (PRISMA) statement [
We conducted the search for studies between September 15 and November 20, 2017. The selection procedure is presented in Figure
A total of 306 studies were identified through these searches. After excluding duplicates (
The process of determining eligibility resulted in 17 fulltext articles, which were read by the first author in their entirety. The bibliographies of these articles were examined, and an additional 28 articles were found, which were also read by the first author. The same eligibility criteria were applied again to the 45 studies, and seven were excluded, resulting in a final sample of 38 articles.
Two members of the research team (the first and third authors) extracted information about participants and outcomes (Table
Summary of participant information and outcome measures: immediate learning, maintenance, and transfer.
Authors (date)  Participants: number, math level, grade  Immediate learning  Maintenance  Transfer 

Bouck et al. [ 
3 children with disabilities, grades 678: one with LD, one with Di George syndrome, and one with mild intellectual disability  Improvement  Maintenance of improvement for 2 children 2 weeks after intervention  / 
Bouck, Chamberlain, and Park [ 
3 children with disabilities, grades 678: one with LD and two with intellectual disability  Improvement  /  Improvement with no manipulatives 
Shin and Bryant [ 
3 children with MLD, grades 678  Improvement for one child  /  / 
No improvement for two children  
Satsangi et al. [ 
3 children with MLD, grades 11–12  Improvement  /  Improvement with no manipulatives 
Bouck et al. [ 
11 children, grade 7–8: 10 with LD and 1 with emotional disability  Improvement  /  / 
Satsangi and Bouck [ 
3 children with MLD, grades 9 and 11  Improvement  Maintenance of improvement in perimeter 2 weeks after intervention; maintenance of improvement in area for 2 children  Improvement in area and perimeter word problem solving (with no accompanying visual illustration of the shape described) 
Flores et al. [ 
3 children with MD, grade 3  Improvement  Maintenance of improvement between 2 and 4 weeks after intervention for 3 children in subtraction and for 1 child in multiplication  / 
Fuchs et al. [ 
243 children at risk (with MLD), 254 children with lowrisk of MLD, grade 4  Improvement  /  / 
Yang et al. [ 
57 children, grade 1 : 33 without difficulties, 14 lowSES and lowmath children, 10 lowSES children without math difficulties  Improvement  /  Improvement in interest and confidence in math 
Fuchs et al. [ 
259 children at risk of MLD, 282 children with lowrisk of MLD, grade 4  Improvement  /  / 
Effect size of intervention: 1.82 for comparing fractions, 1.09 for fraction number line, 0.92 for NAEPTotal 

Watt [ 
32 children with MD, grade 6  Improvement in problem solving  Maintenance of improvement 2 weeks after intervention  / 
Effect size of intervention: 1.71 between intervention students and no intervention students  Effect size of intervention: 0.74 between intervention students and no intervention students  
No improvement in basic skills  
Mancl et al. [ 
5 children with LD, grade 3–5  Improvement  /  / 
Sealander et al. [ 
8 children, grade 1–2: 3 with LD, 2 with LD and language impairment, 3 with emotional disability  Improvement  Maintenance of improvement 4 weeks after intervention  Improvement in word problems 
Strickland and Maccini [ 
3 children, with LD  Improvement  Maintenance of improvement for 2 children 3–6 weeks after intervention  Improvement for 1 child in word problems and equations on volume 
Miller and Kaffar [ 
24 children, grade 2 : 6 with LD and 18 without math difficulties  Improvement in addition  /  Improvement in problem solving 
Flores [ 
6 children with math difficulties, grade 3  Improvement  Maintenance of improvement 6 weeks after intervention  / 
Powell and Fuchs [ 
80 children with math difficulties, grade 3  Improvement  /  Improvement in nonstandard open equations 
Effect size of intervention: 2.35 for Equalsign tasks between combined tutoring students and word problem students (both with manipulatives), 2.34 for Equalsign tasks between combined tutoring students and control students. No difference in standard open equations. 0.22 for Story problem between combined tutoring students and word problem students (both with manipulatives), 0.63 for Story problem between combined tutoring students and control students.  Effect size of intervention: 0.67 for nonstandard open equations between combined tutoring students and word problem students (both with manipulatives), 1.06 for nonstandard open equations between combined tutoring students and control students.  
Flores [ 
4 children with math difficulties, grade 3  Improvement  Maintenance of improvement 4 weeks after intervention  / 
Scheuermann et al. [ 
14 children with MLD, grades 6–8  Improvement  Maintenance of improvement 11 weeks after intervention  Improvement in noninstructed word problems and in more complex problems 
Smith and Montani [ 
12 children with math difficulties (special education), grades 3–4  Improvement  /  / 
Tournaki et al. [ 
45 children with LD, grade 1  Improvement  /  / 
Witzel [ 
231 children, grades 6–7: including 49 children MLD  Improvement  Maintenance of improvement 3 weeks after intervention  / 
Effect size of intervention: 0.56 between intervention students and no intervention students  No effect size  
Butler et al. [ 
50 children with MLD, grades 678  Improvement  /  / 
Effect size of intervention: 0.265 between CRA students and RA students for the 5 fraction measures  
Cass et al. [ 
3 children with LD, grades 7910  Improvement  Maintenance of improvement 2 weeks after intervention  Improvement in problem solving 
Witzel et al. [ 
68 children with LD or atrisk, grades 6–7  Improvement  Maintenance of improvement 3 weeks after intervention  / 
Effect size of intervention: 0.245 between CRA students and classroom students  No effect size  
Wisniewski and Smith [ 
4 children with special education in math, grades 34  Improvement  /  / 
Maccini and Hughes [ 
6 children with LD, grades 91012  Improvement  Maintenance of improvement 10 weeks after intervention  Improvement in problem solving 
Maccini and Ruhl [ 
3 children with LD, grade 8  Improvement  Maintenance of improvement 3 weeks after intervention  Improvement in problem solving 
Jordan et al. [ 
125 children, grade 4 : 5 with LD, 2 with emotional handicap, 1 with visual impairment, 4 with speech/hearing impairment, 18 gifted children, and 97 without difficulties  Improvement  Maintenance of improvement 3 weeks after intervention  / 
Miller et al. [ 
123 children, grade 2 : 12 children with LD, 1 with an emotional disability, 11 low achievers, and 99 normally achievers  Improvement  /  / 
Marshe and Cooke [ 
3 children with LD, grade 3  Improvement  /  / 
Harris et al. [ 
123 children, grade 2 : 12 children with LD, 1 with an emotional disability, and 99 normally achievers  Improvement  /  / 
Miller and Mercer [ 
9 children: 8 children with LD, 1 with an emotional disability, grade 123–5  Improvement  /  / 
Mercer and Miller [ 
109 children, 
Improvement  Maintenance of improvement 1 week after intervention  / 
Miller et al. [ 
15 children, grades 1–5: 10 with LD, 3 at risk for LD, 1 with mental handicap, and 1 with emotional disability  Improvement  Maintenance of improvement 3–5 days after intervention  / 
Peterson et al. [ 
3 children with MLD, grades 1, 2, and 4  Improvement  /  / 
Hudson et al. [ 
3 children with MLD, 8 and 11 years old  Improvement  Maintenance of improvement 1 week after intervention  / 
Peterson et al. [ 
24 children with LD, 8–13 years old  Improvement  Maintenance of improvement 1 week after intervention  / 
Characteristics of mathematics interventions with manipulatives in the sample.
Authors (date)  Primary learning objective (math topic)  Secondary learning objective  Delivery characteristics  Manipulatives 

Bouck et al. [ 
More advanced mathematics (equivalent fractions)  /  15 weeks, 12 lessons per week, 10–15 minutes, individual  VRA sequence teaching (virtual manipulatives: app 
Bouck et al. [ 
Precursor skills (placevalue) and arithmetic computation (whole numbers; subtraction)  /  3 phases (concrete manipulatives, virtual manipulatives, no manipulatives) and 1 phase with best condition, 5 lessons by phase, individual  Manipulatives (base 10 blocks) and virtual manipulatives (app 
Shin and Bryant [ 
More advanced mathematics (fractions; problem solving)  /  13 weeks, twice per week, 9–10 lessons for each child, 20 minutes, individual  Virtual manipulatives: 
Satsangi et al. [ 
More advanced mathematics (algebra: linear equations)  /  3 phases (concrete manipulatives, virtual manipulatives, no manipulatives) and 1 phase with best condition, 10 sessions per phase, 10–15 minutes, individual  Manipulatives (an Ohaus® school balance, chips, and canisters) and virtual manipulatives ( 
Bouck et al. [ 
More advanced mathematics (geometry: area and perimeter)  /  3 days, 3 lessons  Virtual manipulatives: 
Satsangi and Bouck [ 
More advanced mathematics (geometry: area and perimeter)  /  5–9 lessons, 40 minutes, individual  Virtual manipulatives: 
Flores et al. [ 
Arithmetic computation (whole numbers; subtraction, multiplication)  /  3 months, 4 days per week, 25 minutes, until 40 lessons, individual  CRA sequence teaching (concrete manipulatives: base ten blocks) 
Fuchs et al. [ 
More advanced mathematics (fractions; conceptual understanding (measurement and partwhole) and procedural skills (addition and subtraction))  /  12 weeks, three times per week, 36 lessons, 30 minutes, in small group  Concrete materials: objects with shaded regions 
Yang et al. [ 
Precursor skills, arithmetic computation, word problem solving, and more advanced mathematics (general math achievement)  Mathematics interest and confidence  8 weeks, twice per week, 16 lessons, 40 minutes, small group  Concrete materials 
Fuchs et al. [ 
More advanced mathematics (fractions; conceptual understanding (measurement and partwhole) and procedural skills (addition and subtraction))  /  12 weeks, three times per week, 36 lessons, 30 minutes, in small group  Concrete materials: objects with shaded regions 
Watt [ 
Arithmetic computation (whole numbers and fractions; algebra: evaluating equations and simplifying fractions)  /  2 weeks, 5 times per week, 10 lessons, 30 minutes, in whole class  CRA sequence teaching: (concrete manipulatives: some examples are indicated, such as popsicle sticks, string, tongue depressors, and cups) 
Mancl et al. [ 
Arithmetic computation and word problem solving (whole numbers; computation fluency and problem solving)  /  11 times, 30 minutes, individual  CRA sequence teaching (concrete manipulatives: 3D plastic base ten blocks) 
Sealander et al. [ 
Arithmetic computation (whole numbers; subtraction)  /  9 lessons, 15 minutes, individual  CRA sequence teaching (concrete manipulatives: concrete materials that are not described) 
Strickland and Maccini [ 
Word problem solving and more advanced mathematics (whole numbers; algebra: equation)  /  3 lessons, 30–40 minutes, individual  CRA sequence teaching (concrete manipulatives: algebra blocks) 
Miller and Kaffar [ 
Arithmetic computation and word problem solving (whole numbers, addition computation and problem solving)  /  4 weeks, 16 lessons, 60 minutes, in whole class  CRA sequence teaching (concrete manipulatives: base ten blocks) 
Flores [ 
Arithmetic computation (whole numbers; addition and subtraction with regrouping)  /  3 times per week, 10 lessons, 30 minutes, individual  CRA sequence teaching (concrete manipulatives: base ten blocks) 
Powell and Fuchs [ 
Arithmetic computation and word problem solving (whole numbers; prealgebra, equalsign, and equations)  /  5 weeks, 3 times per week, 15 lessons, 25–30 minutes, individual  Manipulatives (bears and blocks) 
Flores [ 
Arithmetic computation (whole numbers; subtraction with regrouping)  /  10 lessons, individual  CRA sequence teaching (concrete manipulatives: base ten blocks) 
Scheuermann et al. [ 
More advanced mathematics (whole numbers; algebra, conceptual understanding of equation)  /  Many lessons, 55 minutes, in small groups  CRA sequence teaching (concrete manipulatives: 3D objects such as buttons, Unifix cubes) 
Smith and Montani [ 
Arithmetic computation (whole numbers; fluency)  /  7 months, each week, 11–13 lessons, 40 minutes, in whole class  Multisensory materials: base ten blocks, fractions stacks 
Tournaki et al. [ 
Arithmetic computation (whole numbers; addition and subtraction and fluency)  /  3 weeks, 15 lessons, 30 minutes, individual  Rekenrek, frame (5102 × 10), plastic counters 
Witzel [ 
More advanced mathematics (whole numbers; algebra, and equation)  /  19 lessons, 50 minutes, in whole class  CRA sequence teaching (concrete manipulatives: algebra material without indication) 
Butler et al. [ 
More advanced mathematics (fractions; conceptual understanding, computational fluency, and problem solving)  /  2 weeks, twice per day, 10 lessons, in whole class  CRA/RA sequence teaching (concrete manipulatives: fraction circles, small white dried beans, and fraction squares of construction paper) 
Cass et al. [ 
More advanced mathematics (whole numbers; geometry (area and perimeter), conceptual understanding, computational fluency, and problem solving)  /  Each day, 7 lessons, 15–20 minutes, individual  Geoboard (rubber bands and tape) 
Witzel et al. [ 
More advanced mathematics (whole numbers; algebra and equation)  /  19 lessons, 50 minutes, in whole class  CRA sequence teaching (concrete manipulatives: objects without indication) 
Wisniewski and Smith [ 
Arithmetic computation (whole numbers; addition, fluency)  /  14 weeks, each day, 20 minutes, in whole class 

Maccini and Hughes [ 
Word problem solving (whole numbers; conceptual understanding (representation), problem solving)  /  20–30 minutes, individual  CRA sequence teaching (concrete manipulatives: tiles) 
Maccini and Ruhl [ 
Word problem solving (whole numbers; conceptual understanding (representation), computational fluency, and problem solving)  /  6 lessons, individual  CRA sequence teaching (concrete manipulatives: tiles) 
Jordan et al. [ 
More advanced mathematics (fractions; conceptual understanding, computational fluency)  /  In whole class  CRA sequence teaching (concrete manipulatives: paper circles cut into fraction pieces and stripes) 
Miller et al. [ 
Arithmetic computation (whole numbers; multiplication basic facts and fluency)  /  21 lessons, in whole class  CRA sequence teaching (concrete manipulatives: paper plates and plastic discs) 
Marshe and Cooke [ 
Word problem solving (whole numbers)  /  13 lessons, 20 minutes, in whole class  Cuisenaire rods 
Harris et al. [ 
Arithmetic computation (whole numbers; multiplication basic facts, fluency)  /  21 lessons, in whole class  CRA sequence teaching (concrete manipulatives: paper plates and plastic discs) 
Miller and Mercer [ 
Arithmetic computation (whole numbers; addition, division, and fluency)  /  5–7 lessons, 20 minutes, individual  CRA sequence teaching (concrete manipulatives: checkers, popsicle sticks, buttons, pennies, dimes, etc.) 
Mercer and Miller [ 
Precursor skills and computational fluency (whole numbers; placevalue and multiplication basic facts)  /  21 lessons, 11 hours in all (30 minutes per lesson), in whole class  CRA sequence teaching (concrete manipulatives: checkers, cubes, buttons, etc.) 
Miller et al. [ 
Precursor skills, computational fluency, and word problem solving (whole numbers; placevalue and multiplication basic facts; word problems)  /  9 lessons, 20 minutes per day, individual  CRA sequence teaching (concrete manipulatives: no indication) 
Peterson et al. [ 
Precursor skills (whole numbers; placevalue)  /  9–15 days, 5 days per week, 15 minutes  CRA sequence teaching (concrete manipulatives: plastic cubes and placevalue strips) 
Hudson et al. [ 
Precursor skills (whole numbers; placevalue)  /  Individual  CRA sequence teaching (concrete manipulatives: plastic cubes and placevalue square strips) 
Peterson et al. [ 
Precursor skills (whole numbers; placevalue, conceptual understanding)  /  18 lessons, one lesson per day, 10–15 minutes, in whole class  CRA sequence teaching (concrete manipulatives: plastic Unifix cubes, placevalue sticks (popsicle sticks), placevalue strips) 
To determine the level of methodological quality of each study, we applied the quality indicators (QIs) outlined by Gersten et al. [
The same coding procedures were conducted with group studies and singlecase research. Using the appropriate set of QIs, we adopted Jitendra et al.’s [
Gersten et al. [
The two coders (i.e., the first and third authors) used the studies by Gersten et al. [
In this section, we begin with a description of the research methodology and characteristics of the participants across the sample. We then report the findings for each of the three research questions in turn.
In the sample, we found 16 group studies (singlecase designs or multiplegroup studies with either experimental or quasiexperimental designs) and 22 singlecase studies. Twentythree studies in the sample incorporated inferential statistics to test effects (15 group studies and eight singlecase studies), whereas the remaining 15 studies reported only descriptive statistics (one group study and 14 singlecase studies). As shown in Tables
In total, 2250 children were tested altogether across the 38 studies. Among these participants, 1131 were children with persistent mathematics difficulties, most of whom we classified as having an MLD either because the authors of the study used the DSM criteria in effect at the time the study was conducted or we ourselves made the determination using the current criteria outlined in the DSM5 [
No study focused on children aged 5 years or below, 15 on children aged 6 to 9 years (first through third grades), 16 on children aged 10 to 12 years (fourth through sixth grades), and 16 on adolescents aged 13 years and older (seventh grade and above). In one study [
We found a variety of primary and secondary outcomes [
With respect to the primary outcomes, seven of the studies in the sample targeted precursor skills (e.g., counting and number comparison in [
Instructional delivery varied in terms of the length of instructional units, number of lessons, and the length of each lesson. Not all 38 studies reported data on the length of the interventions, however. The data reported here represent only the studies that contained sufficient information for an analysis of instructional delivery. For 15 studies, the length of the instructional units ranged from three days (e.g., [
Finally, 36 studies in the sample included information about instructional contexts and settings. The results indicated considerable variability here as well; nineteen studies delivered targeted mathematics interventions to individual students outside the classroom, four offered interventions to students in small groups outside class, and 13 offered wholeclass instruction.
Information on the type of manipulative used across the samples studies is in the rightmost column of Table
In this section, we report on the quality ratings of the group studies and singlecase studies using the criteria laid out by Gersten et al. [
According to Gersten et al.’s [
Immediate performance was measured in all 16 group studies. Of the 16 studies, immediate learning was the only outcome measure targeted in seven of these, maintenance was additionally assessed in six, transfer and learning were assessed in an additional three, and none of the 16 group studies assessed both maintenance and transfer. In terms of the studies based on singlecase designs, all 22 assessed immediate performance as part of the experimental assessment. Of these, eight studies focused on immediate learning only, five assessed maintenance in addition to learning, two assessed transfer and learning, and seven of the 22 studies based on singlecase design assessed both maintenance and transfer.
All 38 studies demonstrated immediate student learning, either statistically or descriptively. The 16 group studies demonstrating learning effects provide some evidence of positive change for interventions involving manipulatives, but the benefits of manipulatives must be tempered because we judged only three of these as being of high quality and one as acceptable. Only the three highquality group studies reported effect sizes (between 0.245 and 2.50). The designs in these three studies did not all permit conclusions about the differential effects of interventions with manipulatives relative to interventions without. (In the study of Fuchs et al. [
With respect to singlecase studies in the sample (
Maintenance of gains was measured using delayed tests in 18 of the 38 studies in the sample, consisting of six group and 12 singlecase studies. Among these, maintenance ranged from a few days (e.g., [
Of the 12 singlecase studies that assessed maintenance, we found four of high quality and seven of acceptable quality by seven research teams in seven locations (i.e., seven states in the US) for 45 children with MLD. The 11 singlecase studies of high and acceptable quality were all based on multiplebaseline designs, which are considered suitable for establishing a functional relation between the manipulation of the intervention and the dependent variable [
Transfer was measured less often than immediate learning and even less often than maintenance. Twelve studies in the sample claimed to demonstrate transfer: three were group studies and nine were singlecase studies. Of the three group studies, one was of high quality with an effect size of 1.06 (i.e., [
Of the nine singlecase studies that assessed transfer, we classified two as high quality and seven as acceptable. Our analysis also revealed that these singlecase studies were conducted by five different research teams in five geographic locations (i.e., five states in the US) for 38 children with MLD. As was the case with maintenance, however, a closer look at how transfer was assessed in these studies reduces the confidence one can place in the effects claimed by the authors. Transfer was measured by administering performance on tasks that are to a greater or lesser extent different from those in the intervention. Desired performance on transfer tasks can be attributed to the student applying what was learned to contexts beyond the confines of the intervention. In seven of the nine singlecase studies, however, the authors administered the transfer measures either immediately following the intervention or after a period of time upon its completion. Such data are an important component of assessing the impacts of an instructional intervention, but because they were not part of the experimental assessments, we are unable to conclude from these descriptive data alone that the interventions in these nine studies are evidencebased. In the remaining two studies [
In sum, only the singlecase research in our sample allows us to conclude that interventions with manipulatives can be considered evidencebased for children with mathematics difficulties and this only for immediate learning. Considerable methodological weaknesses prevent a comparable conclusion to be drawn from the group studies for all three outcomes targeted in this review. We also note that the wide variability in the outcome measures targeted in the singlecase studies hinders our ability to draw conclusions about the effects of manipulatives for the learning of specific topics or learning outcomes.
To determine the value added by manipulatives themselves, one must assess the difference between the targeted intervention with manipulatives and the same intervention without them. Suitable comparison groups (or phases in the case of singlecase studies) are thus required—treatmentasusual comparisons leave various alternative explanations open regarding the reasons for the effects, whereas comparisons of identical interventions without the independent variable of interest (in our case, manipulatives) would serve to isolate the effects of manipulatives alone [
Among the 38 studies, only five studies were designed to establish the value added by manipulatives. Three of the five studies were group studies and two were singlecase. All studies assessed immediate learning, but only one was judged as high quality, one as acceptable, and three were judged as not acceptable (see Table
Quality ratings and research design to establish value added by manipulatives and causal effects of interventions.
Authors  Quality rating  Study design to establish value added by manipulatives  Study design to establish causal effect of intervention  Conclusion 





























Bouck et al. [ 
Not acceptable  TI group  Not applicable  No valueadded 
No causal effect  














Fuchs et al. [ 
Not acceptable  TI group (on the measurement interpretation of fractions, with manipulatives); control group (on the partwhole interpretation of fractions, with manipulatives)  Random assignment to groups  No valueadded 
Causal effect^{a}  
Yang et al. [ 
Not acceptable  TI group; TI group without difficulties; TI group with lowSES children  Nonrandom assignment to groups  No valueadded 
No causal effect  
Fuchs et al. [ 
High  TI group (on the measurement interpretation of fractions, with manipulatives); TI group with no difficulties; control group (on the partwhole interpretation of fractions, with manipulatives)  Random assignment to groups  No valueadded 
Causal effect^{a}  
Watt [ 
High  TI group with targeted intervention; NI group  Random assignment to groups  No valueadded 
Causal effect  





















Miller and Kaffar [ 
Acceptable  TI group; TAU group  Nonrandom assignment to groups  No valueadded 
No causal effect  







Powell and Fuchs [ 
High  TI group (on word problem solving and equalsign, with manipulatives); TI group with no difficulties; control group (intervention on word problem solving, with manipulatives)  Nonrandom assignment to groups  No valueadded 
No causal effect  














Smith and Montani [ 
Not acceptable  TI group  Not applicable  No valueadded 
No causal effect  
Tournaki et al. [ 
Not acceptable  TI group; TAU group; control group  Random assignment to groups  Valueadded 
Causal effect  
Witzel [ 
Not acceptable  TI group (CRA); control group (RA)  Nonrandom assignment to groups  Valueadded 
No causal effect  
Butler et al. [ 
Not acceptable  TI group (CRA); control group (RA)  Nonrandom assignment to groups  Valueadded 
No causal effect  







Witzel et al. [ 
Not acceptable  TI group (CRA); TAU group  Nonrandom assignment to groups  No valueadded 
No causal effect  





















Jordan et al. [ 
Not acceptable  TI group (CRA); TAU group  Nonrandom assignment to groups  No valueadded 
No causal effect  
Miller et al. [ 
Not acceptable  TI group  Not applicable  No valueadded 
No causal effect  





















Mercer and Miller [ 
Not acceptable  TI group  Not applicable  No valueadded 
No causal effect  





















Peterson et al. [ 
Not acceptable  TI group (CRA); control group (A)  Nonrandom assignment to groups  No valueadded 
No causal effect 
To establish causal effects of interventions with manipulatives, group studies must incorporate random assignment of the unit of analysis to conditions (e.g., [
Although these analyses provide evidence of causal effects of interventions with manipulatives for all three outcome measures, some of these studies were nevertheless of poor quality (i.e., not acceptable) according to our previous analysis (i.e., the findings to Research Question 2). In other words, in five studies (three assessing immediate learning, one assessing maintenance, and one transfer), the experimental design for establishing cause was present, but too many quality indicators were absent for us to classify the studies as evidencebased. This information should be taken into consideration when interpreting the findings on causal effects in this section.
Our aim in this chapter was to conduct a review of the literature to evaluate the impact of using manipulatives, either physical or virtual, on the mathematics learning, maintenance, and transfer in children with MLD. We used the frameworks established by Gersten et al. [
Our first research question addressed the instructional contexts in which the interventions were delivered, namely, the skills that were targeted, the characteristics of the interventions themselves, and the types of manipulatives used. The interventions varied considerably in total duration, the length of each session, and the number of sessions; the size of the groups receiving the intervention (onetoone, small group, whole class) also varied from study to study. In addition, the types of manipulatives varied greatly as well, with some interventions using concrete materials, pictorial representations (as in the case of CRA), or virtual manipulatives. Interventions involving manipulatives also differed with respect to the mathematical domain targeted (i.e., precursor skills, arithmetic computation, word problem solving, and advanced mathematical skills).
Our second research question focused on whether interventions that include manipulatives can be considered evidencebased for children with MLD in terms of immediate learning, maintenance, and transfer. A quick glance at the findings in our sample may suggest that, overall, mathematics interventions with manipulatives are effective for children with MLD. Among the 38 studies, all showed immediate improvement, reported either statistically or descriptively. Applying the criteria established by Gersten et al. [
In contrast, the studies using singlecase designs credibly demonstrated immediate learning of such mathematical outcomes as arithmetic computation, word problem solving, and advanced mathematical skills. We note, however, that skills such as transcoding (e.g., the ability to read and write numerals) and the development of counting principles, such as cardinality and onetoone correspondence, were not examined in the studies we reviewed. Given that such that competencies are important predictors for school success in mathematics [
We were also interested in examining the effects of interventions with manipulatives on students’ maintenance and transfer. Learning that lasts over time has obvious benefits for both teachers and students and has surfaced as a particular challenge in responding to the needs of children with MLD. In addition, the goal of mathematics education goes beyond simply reproducing the material taught during instruction; the ultimate goal is for students to meaningfully apply (i.e., transfer) new knowledge to other tasks and contexts. The studies we reviewed, however, did not reveal credible effects of maintenance or transfer. A maintenance effect was found in only one highquality group study, and only a small handful of group studies measured transfer, with just one of them judged as high quality. Relative to group studies, a larger number of studies based on singlecase designs claimed maintenance and transfer effects, but our analysis of their methodologies produced disappointing conclusions. The maintenance and transfer outcomes assessed in the singlecase studies were not incorporated into the betweenseries and multiplebaseline designs in ways that established functional relations between manipulations of the instructional interventions and the dependent variables.
Despite the promising findings with respect to immediate learning, we found too many instructional variations across the sample to develop prescriptive models for how to use manipulatives with MLD children. A larger number of controlled studies that isolate specific instructional features, such as length of instruction and type of manipulative used for specific outcome measures, are required to draw more definitive conclusions. In a study that manipulated the length of instruction, for example, Kroesbergen and Van Luit [
Along the same lines, we observed positive effects of interventions involving manipulatives for students with a wide variety of individual differences, such as age (e.g., 6 to 17 years old, first to twelfth grade) and the type of difficulty described (e.g., children with MLD, children with mathematics difficulties at school but without an official or known diagnosis, and children at risk of developing MLD). Again, however, positive effects were found regardless of students’ age or specific learning challenges, which makes specific prescriptions for uses of manipulatives with the MLD population elusive. Furthermore, student characteristics related to general cognitive ability or executive functioning skills were not directly addressed or tested in any of the studies reviewed. This is a glaring omission, as relatively recent work has identified general cognitive factors as moderators of intervention effects with atrisk elementary students. Fuchs and her colleagues (i.e., [
Prior knowledge is another student characteristic that can impact the conclusions drawn about the effects of mathematics interventions involving manipulatives. Peterson and McNeil (2013), for example, demonstrated that children’s counting performance was compromised if they had what the authors called “established knowledge” of the manipulatives they were counting. They speculated that the children were distracted by what they knew about the objects represented by the counters (e.g., their knowledge of zebra when they were counting with manipulatives that looked like zebras); in contrast, the students performed significantly better with objects that were unfamiliar to them about which they had no prior knowledge. In another study also with typically developing students, Osana et al. [
Finally, students appear to benefit when they acquire what Uttal, Liu, and DeLoache [
Our third research question concerned whether the effects of manipulatives alone (i.e., the value added by the manipulatives themselves) could be determined from the research we reviewed, and the extent to which causal effects of interventions could be established in the sample. Only two high or acceptable quality studies were designed to establish the value added by manipulatives for immediate learning and transfer, providing little support for the benefits of manipulatives over and above the effects of comparable interventions without manipulatives. Concerning causal effects, random assignment was present in one group study of high quality and a suitable baseline was established in 17 singlecase studies of high or acceptable quality. Together, these results allow us to conclude that there is some evidence to show that interventions using manipulatives can cause positive mathematical outcomes in students with MLD.
Overall, we found the methodological quality in the sample far from perfect, which limits the conclusions that can be drawn. For example, the lessthanresounding evidence for causal effects of interventions with manipulatives can be explained, in large part, by the lack of carefully designed experimental group studies. Even among those studies that were designed to establish experimental control (i.e., predominantly singlecase studies), not all were judged to be of high or acceptable quality. Methodological shortcomings also dilute the quality of evidence on maintenance and transfer and can account for the lack of data on the value added by the manipulatives themselves. Additionally, the required information that would allow for complete assessments to be made regarding our three research questions was not available in many of the published reports. Without information about key methodological and procedural aspects of the research, the interpretability of the data is compromised, as are the pedagogical implications that are derived from them. For instance, in many of the singlecase studies that assessed maintenance and transfer, little to no information on possible confounding variables were provided (i.e., what took place in the period of time between the experiment and when the followup or transfer data were collected). Also, key pieces of information about the instructional interventions, for example, such as the total duration of the instruction, the number of sessions, and the length of each session, were frequently omitted. Furthermore, critically important details about the teacher’s (or researcher’s) practices, such what he or she said and presented to the children at key moments during instruction, were absent from almost all the reports. Information about the students themselves was rarely reported; children’s domaingeneral and domainspecific cognitive abilities were not assessed in the vast majority of studies, and socioeconomic variables were also rarely considered. Finally, very little information was provided about the types of manipulatives used and how they were used, and in some studies, the location of the data collection was not specified.
That few of the studies reviewed met desirable scientific thresholds does not necessarily imply that none of the interventions is, in fact, effective for some outcome or another. In fact, we maintain that the collection of studies in this review provides a number of instructional resources for practitioners. Well represented in the sample is CRA, for example, which is an application of “concreteness fading,” an empirically supported theoretical framework for instruction in mathematics and science [
Given the designs of the studies in the sample, we were unable to pinpoint the specific aspects of instruction that were responsible for the improvements observed. In the case of CRA, for example, did the explicit explanations, or when and how they were delivered, predict improved performance? Did the teachers’ use of the manipulatives during explanation and practice account for learning? Were the tools used to alleviate cognitive load responsible for the effects observed? These are questions that cannot be answered at this time, but we argue that teachers can nevertheless use the ideas and approaches described to test whether they are useful for the students in their own classrooms. Practitioners are accustomed to testing a variety of approaches, particularly with students for whom “traditional” instruction is not effective. We maintain that the research reviewed here can be viewed by educators as a collection of resources that can inspire and motivate their practice.
The present study is, to our knowledge, the first systematic review on immediate learning, maintenance, and transfer effects of manipulatives in the context of instruction with the MLD population. Despite methodological limitations found across the sample, we can tentatively conclude that interventions with manipulatives show promise for children who struggle to learn mathematics. Our optimism must be tempered by the wide heterogeneity in methodological quality, the absence of instructional variables and student characteristics that are known to influence intervention effects, and insufficient consideration of possible confounding and moderating variables that have been shown to impact mathematics learning with manipulatives in typically developing populations. More systematic studies are needed to contribute to current theory on the instructional potential of manipulatives in the MLD population and to build instructional models that are pedagogically useful for special educators. We are also aware of the inherent publication bias (i.e., the tendency for studies that show statistically significant effects to be published over those that show null results; see [
The data used in this article were presented at the 2018 meeting of the Mathematical Cognition and Learning Society in Oxford, UK.
The authors have declared that no conflicts of interest exist.
This research was supported by a grant from the Social Sciences and Humanities Research Council of Canada (43520152002) and by Concordia University. We would like to thank Thomas Kratochwill for his generous assistance in our interpretations of singlecase research design.