Abstract
Proficiency with fractions is one of the most significant predictors of later mathematics achievement. However, there are currently no meta-analyses that assess the literature base on fractions for students with or at risk for disabilities using quality indicators. We applied the 2014 Council for Exceptional Children Standards for Evidence-Based Practices in Special Education (CEC EBP) to 21 studies, both single-case and group designs, with instructionally based fraction interventions, published from 1986 to 2017. Ten of the included studies met all of the CEC EBP quality indicators, and effect sizes ranged from g = 0.42 to 11.51 across interventions. Publication bias was mixed but limited across the research base. Included studies examined the effects of anchored instruction, explicit instruction, graduated instruction, strategy instruction, and video modeling; explicit instruction was determined to be an evidence-based practice when applying the CEC EBP standards. We offer limitations and directions for future research in this area.
Proficiency in mathematics is a necessary aspect of adult independence (National Mathematics Advisory Panel [NMAP], 2008). However, scores on the national mathematics assessment have shown that the majority of students have consistently performed below proficiency for more than a decade (National Assessment of Educational Progress [NAEP], 2015). In 2015, 25% of 12th-grade students were at or above proficient in mathematics. In turn, 38% of students were below the basic achievement level. Additionally, 12th-grade students with disabilities scored in the basic range, an average of 37 points below the average of students without disabilities (NAEP, 2015).
Mirroring the inadequate achievement of students in later mathematics skills, American students consistently fall short in their ability to calculate fractions (NMAP, 2008). This is important to note, as Siegler and colleagues (2012) found the most significant predictor of later mathematics achievement (e.g., algebra) is a student’s knowledge of fractions, followed by knowledge of division. N. Jordan, Resnick, Rodrigues, Hansen, and Dyson (2017) conducted a longitudinal study to understand how fraction knowledge develops for third- through sixth-grade students with and without mathematics disabilities. Similar to the findings by Siegler et al., they found that slow growth in fraction knowledge predicted later failure in mathematics.
Siegler, Thompson, and Schneider (2011) presented a theory of numerical development that posits, “Numerical development is at its core a process of progressively broadening the class of numbers that are understood to possess magnitudes and of learning the functions that connect that increasingly broad and varied set of numbers to their magnitudes” (p. 274). This is conceptualized as a deepening of the understanding of whole numbers, existing on a number line, to include all rational numbers. The implications of this are that fraction knowledge should evolve with numerical development rather than as a secondary domain, as has been the case in other frameworks (Siegler et al., 2011).
fraction knowledge should evolve with numerical development rather than as a secondary domain
In 2010, a team of researchers led by Robert Siegler developed the What Works Clearinghouse practice guide on developing effective fractions instruction for kindergarten through eighth grade (Siegler et al., 2010). They identified five recommendations for promoting procedural fluency, or “the ability to apply procedures accurately, efficiently, and flexibly,” in fractions (National Council of Teachers of Mathematics, 2014). They include (a) building on students’ background knowledge of sharing and proportions to develop fraction concepts, (b) recognizing that fractions are numbers and they fall along a number line just like whole numbers, (c) understanding the validation for computations with fractions, (d) understanding various strategies for solving problems (e.g., rate, ratio, proportion), and (e) improving teachers’ knowledge of fractions and how best to teach them (Siegler et al., 2010). These approaches are supported by minimal (recommendations [a], [d], and [e]) to moderate (recommendations [b] and [c]) evidence from research, suggesting research in this area is still warranted.
Approaches to Fraction Instruction
Previous research has evaluated the effects of four main approaches to fraction instruction: anchored, explicit, graduated, and strategy instruction (Misquitta, 2011). Although there is overlap among these approaches (i.e., all use explicit instruction), each one has its own distinct differences.
Anchored Instruction
Anchored instruction for fractions involves the framing of mathematical problems within relevant and practical contexts to help facilitate problem solving and computation skills (Cognition and Technology Group at Vanderbilt, 1997). This instructional approach involves teachers presenting students with videos, vignettes, or building projects; teaching students how to identify relevant information; and solving problems using fractions computation (Bottge, 1999). In many recent investigations, Bottge and colleagues (e.g., Bottge, Rueda, Grant, Stephens, & Laroque, 2010) include a culminating activity solving problems anchored in the task of designing and building a hovercraft. The body of work on this approach has evolved over time with the most recent investigation labeled as enhanced anchored instruction. Authors note that enhanced anchored instruction provides more direct teaching of fractions and multiple representations of problems in an effort to avoid overloading working memory (Bottge et al., 2015). Although anchored instruction has a wealth of research to support its effectiveness, some of the results suggest only small to moderate effects (Misquitta, 2011).
Explicit Instruction
Explicit instruction involves providing clear explanations, modeling steps or procedures, opportunities for supported and independent practice, ongoing feedback, and summative assessment (Fuchs et al., 2013). Researchers have also examined the effects of direct instruction, a form of explicit instruction that also involves designing an instructional format, including teacher wording, learning tasks, and correction procedures (Scarlato & Burr, 2002). Researchers evaluating the effects of explicit instruction have focused on both fraction procedures (e.g., adding and multiplying fractions) and concepts (e.g., part-whole conceptualization; Fuchs et al., 2013). Explicit instruction for fractions has produced overall positive effects (e.g., Shin & Bryant, 2015). Further, all other approaches (anchored, graduated, and strategy) involve components of explicit instruction.
Graduated Instruction
Another approach to mathematics instruction that has been applied to fraction skills is the use of graduated instruction, or graduated sequencing, which entails presenting concrete, representational, and abstract (CRA) examples and problems (Butler, Miller, Crehan, Babbitt, & Pierce, 2003). This approach involves the use of manipulatives (e.g., fraction tiles), pictures (e.g., circle cut into equal pieces), and fraction mathematics problems to facilitate each graduated condition of CRA, respectively. Researchers have demonstrated that the CRA approach for graduated instruction is more effective than relying on abstract instruction alone (e.g., L. Jordan, Miller, & Mercer, 1998).
Strategy Instruction
Strategy instruction is another approach used to teach fractions that has demonstrated efficacy for students with learning disabilities and other populations (Graham & Harris, 2003). A strategy is a goal-directed process for completing a task (Keene & Zimmerman, 2007). Researchers examining the effects of teaching strategies for fraction skills have taken varied approaches, including the use of mnemonics (Test & Ellis, 2005) and cue cards (Joseph & Hunter, 2001). Zhang, Stecker, Huckabee, and Miller (2016) used a variety of strategies to help support students, including cross-multiplication, number line, and visual representation. In her previous review of the fraction literature, Misquitta (2011) found positive results across studies using strategy instruction.
Reviews of Fraction Instruction
There have been only two recent reviews of the literature on fraction interventions for students with or at risk for disabilities (Misquitta, 2011; Shin & Bryant, 2015). Misquitta (2011) found 10 studies that specifically looked at interventions to promote understanding of fractions and found overall positive results for three of the intervention types (strategy instruction, graduated sequencing, and direct [explicit] instruction). Investigations that utilized anchored instruction resulted in mixed results (e.g., Bottge, Heinrichs, Mehta, & Hung, 2002). In sum, Misquitta found explicit instruction to be a necessary component for improving the mathematics proficiency of students with disabilities. However, the review was narrative in nature and did not calculate an omnibus effect size (meta-analysis), nor did it attempt to address study bias by assessing the included studies for methodological rigor. Shin and Bryant (2015) conducted a more recent review that included dissertations and aligned with the Common Core State Standards in Mathematics (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010). They examined 17 studies reporting overall positive effects for studies that utilized systematic (using multistep strategies to solve problems) and concrete (using manipulatives to solve problems) instruction, with mixed results once again seen for studies of anchored instruction. However, the methods of this review have limitations. First, the search criteria were vague and not easily replicable. Second, although the authors calculated effect sizes, they did not attempt to meta-analyze the findings, using the effect sizes as simply descriptive measures. Finally, neither review attempted to address study bias in the included studies using accepted standards for studies (e.g., the Council for Exceptional Children’s [CEC; 2014] Standards for Evidence-Based Practices in Special Education [CEC EBP]). Bias in the literature can be a result of two main constructs, bias in the studies due to faulty methods (Cook, 2014) or publication bias, which is a result of journals having a tendency to publish only those studies with positive results (Maag & Losinski, 2015). Publication bias is especially problematic in the social sciences, as the tendency to publish only positive results may influence the extent to which researchers may have confidence in an intervention’s effectiveness. The CEC EBP is a means of addressing bias within the methods of the studies; statistical tests are often conducted to assess publication bias.
Purpose
Thus, our purpose in conducting the current meta-analysis is to expand upon the reviews by Misquitta (2011) and Shin and Bryant (2015) through an updated and systematic search of the experimental literature on fractions achievement of students with disabilities, evaluating the research base for quality indicators (QIs), utilizing meta-analytic techniques, and addressing bias in the included studies. The current research is predicated on the following research questions: (a) What is the quality of the current literature on fraction interventions for students with or at risk for disabilities when studies are assessed using the CEC EBP? (b) Are any of the practices used to increase fraction knowledge for students with or at risk for disabilities an evidence-based practice as measured by the CEC EBP? (c) How effective are interventions addressing fraction skills for students with or at risk for disabilities? (4) To what extent is bias present in the literature base on fraction skills for students with or at risk for disabilities?
Method
To answer the research questions, we conducted a systematic search of the research on fraction problem solving with students with or at risk for disabilities. First, we conducted a database search on March 9, 2018, using the following databases: ERIC, Education Full Text, Academic Search Premiere, Medline, and PsycINFO. The Boolean phrase was as broad as possible (fraction* AND disabilit*) to ensure we captured all relevant titles. Including the word disabilit* also allowed us to capture studies that included students at risk for disabilities in the search. Both authors simultaneously worked as a team to screen articles for inclusion. Any disagreements were discussed together to reach a final decision.
Next, beginning with 2005, we conducted a hand search of journals containing more than one article identified in the database search: Exceptional Children, Journal of Educational Psychology, and Journal of Learning Disabilities. Last, we conducted an ancestral search of the citations of all included articles and previous reviews. We concluded the search process on April 10, 2018.
Inclusion Criteria
Both authors screened the titles and abstracts of all identified articles to determine inclusion in the synthesis. For a study to be included in the current synthesis, a study had to (a) present findings from an experiment (randomized control trial, quasiexperimental with a control group, single-case design, or regression discontinuity design), (b) have an instructionally based fraction intervention as an independent variable, (c) have a dependent variable that measured fractions in isolation (e.g., in Cieslar, McLaughlin, and Derby [2008], we could not disaggregate fractions as a distinct dependent variable), and (d) involve students with or at risk for disabilities in a K–12 setting. This inclusion criteria differs from previous reviews in that we included neither studies without a control group (e.g., Bottge, Rueda, & Skivington, 2006) nor studies that reanalyzed existing data (e.g., Bottge, Cohen, & Choi, 2018).
Coding Procedures
Both authors then coded all studies that met the inclusion criteria for study characteristics and methodological indicators using the CEC EBP (CEC, 2014). The CEC EBP is a guide for assessing the quality of a study’s reporting of eight factors related to the methodology of a study. Subsequent to coding, the researchers met to compare results, with initial interrater agreement between them determined by comparing the respective coding sheets and dividing the number of agreements by the number of opportunities for agreement. We calculated interrater agreement using point-by-point correspondence (Cooper, Heron, & Heward, 2007) for each study by dividing the sum of agreements by the number of QI components (i.e., 22 for single case, 24 for group), multiplied by 100. We also calculated agreement for each QI component across studies by dividing the sum of agreements by 21 (total number of studies), multiplied by 100. Interrater agreement was 100% for 23 of the 28 indicators; 95.2% for Elements 3.1, 3.2, and 6.3; 93.3% for 8.3; and 85.7% for 5.1. Mean interrater agreement was 98.7% by QI and 98.6% by studies. When disagreements occurred, the researchers discussed the issue until a consensus was reached, resulting in 100% agreement between coders.
The variables coded from each study included the following: (a) number of participants, (b) grade, (c) disability type, (d) gender, (e) dependent variable, and (f) design (see Table 1). To understand the independent variables used, we also coded intervention characteristics, including (a) intervention construct, (b) intervention, (c) intervention agent, (d) group size, (e) intervention duration, and (f) session length. The intervention constructs, or groupings, were based on previous reviews of fractions and included strategy instruction, graduated instruction, anchored instruction, and explicit instruction (Misquitta, 2011).
Study Characteristics.
Note. ADHD = attention deficit hyperactivity disorder; AP = applied (word) problems; AR = at risk; ASD = autism spectrum disorder; B = block; C = cluster; CO = comparison problems; CRA = concrete-representational-abstract; CU = computation problems; E = equivalency; GE = general education; GT = gifted; HI = hearing impairment; I = inclusion; ID = identification problems; LD = learning disability; M = mean, NL = number line; NS = not specified; OHI = other health impairment; Quasi = quasiexperimental; R = ratio; RCT = randomized control trial; RS = resource; SC = self-contained; SCD = single-case design; VI = visual impairment.
QIs
The methodological quality of the study was assessed using the CEC EBP (CEC, 2014). These standards were used because they assess study bias resulting from faulty methods, have been used extensively in the field, and correspond to other accepted standards (e.g., Gersten et al. 2005; Horner et al., 2005). To meet each standard, the study had to list the information required explicitly (or provide enough information that it can be reasonably inferred). For example, if a study did not report the grade (or age) of the participants, it would not meet QI 2.1. The CEC EBP includes eight standards, with many having substandards, for a total of 24 (group design) or 22 (single-case design) elements to evaluate.
For a study to contribute to the evidence base as measured by the CEC EBP, studies must meet 100% of the quality indicators (CEC, 2014). To be considered evidence based, there must be (a) two group-design studies with randomized assignment (60+ participants), (b) four group-design studies without randomized assignment (60+ participants), (c) five single-case-design studies (20+ participants), (d) one group-design study with randomized assignment (30+ participants) and three single-case-design studies (10+ participants), or (e) two group-design studies without randomized assignment (60+ participants) and three single-case-design studies (10+ participants). Furthermore, no studies can have negative effects, and the ratio of positive to neutral effects must be 3:1 or greater (CEC, 2014).
Study Effects
We identified studies that used both group and single-case designs. We calculated effect sizes with equivalent metrics so that effects could be compared across group and single-case designs, an application noted by Shadish, Hedges, and Pustejovsky (2014). We calculated Hedges’ g for group designs, which controls for small sample sizes. Hedges’ g was calculated in Comprehensive Meta-analysis (Version 3.0) using the means and standard deviations from the posttest scores of the respective treatment and control groups. Because many different outcome measures and constructs (e.g., number line, computation problems) were used across and within studies, and it was assumed that the outcomes were correlated, we combined the effect sizes from each study into one average effect size as discussed by Shadish and Haddock (2009). We could not calculate an effect size for two of the included group-design studies (L. Jordan et al., 1999; Scarlato & Burr, 2002) due to insufficient information being reported in the studies.
We calculated between-case standard mean difference (BC-SMD; Shadish et al., 2014) for single-case designs, which is in the same metric as g, lending itself to comparability to the group designs (e.g., Losinski, Cuenca-Carlino, Zablocki, & Teagarden, 2014). For single-case designs, we decided to analyze only the complete treatment package, not component parts, against baseline data. For example, if a study investigated two or three different interventions before choosing a final intervention, we used only the final intervention chosen. For example, Kim, Wang, and Michaels (2015) delivered a CRA sequence. We took data only on the final (representational) stage as this was the culmination of the strategy. Finally, we extracted data from each study using WebPlotDigitizer (Version 3.11; Rohatgi, 2017) for entry into the online BC-SMD calculator (Pustejovsky, 2016). We then calculated omnibus and moderator analyses in R (Version 3.4.1) using computed scores (g and BC-SMD) for each study. Interpretation of g and BC-SMD follow the recommendations for the standard mean difference (d) by Cohen (1988), where d < 0.20 indicates a small effect and d > 0.80 indicates a large effect.
Publication bias
We used statistical tests to assess potential publication bias (Maag & Losinski, 2015). Two analyses were conducted using R (Version 3.4.1): Egger’s regression-of-the-intercepts test (Egger, Davey Smith, Schneider, & Minder, 1997) and Duval and Tweedie’s (2000) trim-and-fill method. Egger’s regression of the intercepts predicts the effect size under the assumption that all studies are present in the analysis. If Egger’s regression of the intercepts is zero, publication bias is likely not present; however, if it surpasses zero, it is likely publication bias exists within the literature obtained. Duval and Tweedie’s trim-and-fill method uses a funnel plot to impute included studies. For example, the funnel plot will be symmetrical in the event there is no bias. Conversely, a nonsymmetrical funnel plot indicates bias may be present, in which case hypothetical study effects are added until symmetry is achieved, and the effect size is recalculated.
Results
The database search yielded 1,774 articles. After duplicates were removed, a total of 1,200 articles remained, and the titles and abstracts were screened for inclusion (see Figure 1). After screening of the titles and abstracts, 22 articles remained and were read in their entirety by two researchers; then 10 articles were removed because they did not meet the inclusion criteria, leaving 12 articles. After the hand and ancestral searches were completed, an additional nine articles were added, for a total of 21 articles included in the synthesis.
Flow diagram of search procedures.
Study Characteristics
A total of 1,804 participants were included in the 21 studies with a median of 49% males (range, 33% to 100%) and the median grade being seventh (range, fourth grade to 11th grade). All studies included students with or at risk for disabilities and included participants in general education (n = 4 studies), at risk for math mathematics difficulties (n = 7), with disabilities (n = 15), and who were English language learners (n = 1). Of the studies with students receiving special education services, the following eligibility categories were represented: learning disability (n = 14), intellectual disability (n = 5), multiple disabilities (n = 4), autism spectrum disorder (n = 3), emotional or behavioral disorder (n = 3), other health impairment (n = 2), gifted (n = 1), speech language impairment (n = 1), visual impairment (n = 1), and not specified (n = 1). See Table 1 for more detailed study participant characteristics.
In most of the studies in this analysis (n = 14), researchers collected multiple dependent variables. However, Bouck et al. (2017); Joseph and Hunter (2001); Kim et al. (2015); Scarlato and Burr (2002); Test and Ellis (2005); Yakubova, Hughes, and Hornberger (2015); and Zhang et al. (2016) each measured only one. The dependent variables assessed included applied (word) problems (n = 9), comparison (n = 6), computation (n = 16), equivalency (n = 5), identification (n = 3), and number line (n = 4) or were not specified (Scarlato & Burr, 2002).
Intervention group size ranged from one-to-one instruction to small group (3–6 per group) with many of the studies (n = 9) not reporting the size of the groups. The most used design was a randomized control trial (n = 12), followed by single-case designs (n = 6) and quasi-experiments (n = 3). The intervention components coded included explicit instruction (n = 7), anchored instruction (n = 5), graduated instruction (n = 3), strategy instruction (n = 3), and video modeling (n = 1). Teachers were slightly more prevalent as the intervention agent (n = 11) compared to researchers (n = 10). The number of days of intervention ranged from 9 (Bottge, 1999) to 140 (Scarlato & Burr, 2002). Sessions lasted between 30 minutes (Test & Ellis, 2005) and 45 to 75 minutes (Bottge et al., 2015). See Table 2 for more information on intervention characteristics.
Intervention Characteristics.
Note. CRA = concrete, representational, abstract; LAP = look at the sign denominator, ask yourself the question, pick your fraction type; NS = not specified; VRA = virtual, representational, abstract.
QIs
A summary of QIs that were met is presented by study in Figure 2. Ten studies met 100% of all elements, and an additional nine studies met 80% or more. Results by indicator are presented next.
Ten studies met 100% of all elements, and an additional nine studies met 80% or more.
Council for Exceptional Children (CEC) quality indicator (QI) matrix. Left y-axis displays the components of CEC (2014) QIs. Shaded cells = component was met; white cells = component was not met; diagonal lines = component was not applicable to the study. The right y-axis shows the number of absolute QIs met for each study (triangle marker).
1.0: Context and setting
All articles included enough information about the setting to determine if it was appropriate for inclusion in this analysis. Investigations took place in Grades 4 through 11, with investigations occurring most frequently at the middle school level.
2.0: Participants
All articles met QIs 2.1 and 2.2 by clearly describing the participants and their risk status. For example, Fuchs et al. (2013, 2014, 2016, 2017) defined risk using a cut score of performance below the 25th percentile on a standardized test; they further specified risk by looking at students in two groups: below the 15th percentile and between the 15th and 34th percentiles. Zhang and colleagues (2016) conducted a preliminary study to determine students’ strategic development levels prior to conducting their intervention.
3.0: Intervention agent
Similarly, all studies except one met QI 3.1 by reporting details about the intervention agent to determine if he or she was a researcher or classroom teacher as well as additional characteristics (e.g., years of teaching experience, gender). However, only 76.2% (n = 17) of studies met QI 3.2, as some studies did not provide details about what qualified the intervention agent to implement the intervention or how the intervention agents were trained. For example, Scarlato and Burr (2002) reported the years of teaching experience for both the intervention- and control-group teachers. Further, they specified that the intervention teacher had taken a graduate course in direct instruction whereas the control teacher had no training in direct instruction. Bottge and colleagues (2014, 2015) conducted a 2-day, 14-hr workshop to train both math and special education teacher participants to use enhanced anchored instruction.
4.0: Description of the practice
All studies met QIs 4.1 and 4.2 by providing descriptions of the interventions and procedures so as to allow for replication. Several studies included graphics or figures to illustrate the procedures used in the intervention (e.g., Bottge et al., 2002, 2010; Joseph & Hunter, 2001). Hunter (2014) included a figure depicting both the intervention and control conditions to illustrate their differences.
5.0: Implementation fidelity
Most (90.5%; n = 19) studies met QIs 5.1 and 5.2 by detailing procedures for collecting and results of implementation fidelity related to adherence to the intervention procedures and providing information regarding the dosage of the intervention. However, only 80.0% (n = 16; this indicator was not applicable for one study because it did not meet QI 5.1 or 5.2; Butler et al., 2003) met QI 5.3 by articulating that implementation data were collected throughout the intervention. The most common approach to monitoring adherence fidelity was through a checklist of required lesson elements (e.g., Bouck et al., 2017; Test & Ellis, 2005). Kelly, Gersten, and Carnine (1990) used a checklist and also held weekly meetings with teachers to discuss any omitted items. Fuchs et al. (2013, 2014, 2016, 2017) audiotaped all sessions and then used a checklist to evaluate fidelity of implementation of a predetermined percentage of randomly selected sessions.
6.0: Internal validity
A majority of the studies were group designs (n = 15), and six studies used single-case designs. Across all studies, 85.7% (n = 18) met QI 6.1 by systematically manipulating the independent variable, and 81.0% (n = 17) met QIs 6.2 and 6.3 by describing baseline or control conditions and limiting exposure to any intervention elements during baseline or control conditions. Of the 15 studies conducted using group designs, 93.3% (n = 14) met QI 6.4 by clearly describing assignment to groups, 73.3% (n = 11) met QI 6.8 by reporting attrition at a rate of 30% or lower of the total participants (those studies that did not meet this indicator failed to either report attrition or state that no attrition was present), and only 60.0% (n = 9) met QI 6.9 by reporting differential attrition at a rate of 10% or lower within groups. For the six single-case-design studies, 100% met QIs 6.5, 6.6, and 6.7 by providing at least three demonstrations of effect, including at least three data points during baseline phases, and using conventional designs that controlled for threats to internal validity.
7.0: Outcome measures or dependent variables
For QIs 7.1, 7.2, 7.3, and 7.4, 100% of studies met these indicators by targeting socially important outcomes, describing the outcome variables, reporting results for all variables, and administering measures with appropriate frequency and timing. Only 85.7% (n = 18) met QI 7.5 by reporting reliability of outcome measures. For example, Bottge (1999) assessed reliability of 25% of protocols with a mean of 98.0% agreement. Similarly, Kim et al. (2015) assessed reliability of 20% of protocols with 100% agreement. QI 7.6 applies only to group-design studies. Of the 15 group-design studies, 100% provided adequate evidence of validity.
8.0: Data analysis
QIs 8.1 and 8.3 apply to group-design studies. Of the 15 group-design studies, 100% met QI 8.1 by using appropriate data analysis methods (or justifying irregular analyses). However, 93.3% (n = 14) met QI 8.3, because one study did not report an effect size or metrics needed to calculate an effect size (L. Jordan et al., 1999). One hundred percent of all single-case-design studies met QI 8.2 by including a graph of the primary outcome variable.
Study Effects
The omnibus effect size for fractions interventions of g = 1.17 (var. = .13) using a random-effects model would be considered large using the guidelines established by Cohen (1988). Individual studies ranged from −0.42 (var. = 0.10; Bottge et al., 2002) to 11.51 (var. = 1.67; Yakubova et al., 2015). Table 3 displays effects sizes and a forest plot for the included studies grouped by intervention construct. The largest effects were found for video modeling with one study (Yakubova et al., 2015; g = 11.51, var. = 2.95), followed by strategy instruction (g = 1.48; var. = 0.12) with three studies, graduated instruction (g = 1.60, var. = 0.14) with four studies, explicit instruction (g = 1.25, var. = 0.18) with seven studies, and a moderate effect shown for anchored instruction (g = 0.35, var. = 0.20) with five studies. Results of a meta-regression showed single-case designs significantly predicted a larger effect size compared to group designs (b = 1.994, SE = 0.61, z = 3.29, p = .001).
Effect sizes and forest plot for interventions targeting fractions.
Regarding the determination of an evidence-based practice, only two intervention constructs had sufficient evidence for consideration. Explicit instruction had a total of five group-design studies using randomization and meeting 100% of QIs with positive outcomes for a total of 961 participants. Therefore, based on CEC EBP standards, explicit instruction is an evidence-based practice for improving the fraction performance of students with or at risk for disabilities. These studies were conducted by two different research groups (Fuchs et al., 2013, 2014, 2016, 2017; Kelly et al., 1990). Enhanced anchored instruction had a total of three group-design studies using randomization and meeting 100% of QIs with positive outcomes for a total 193 students. Therefore, based on CEC EBP standards, anchored instruction is considered to have mixed evidence for improving the fraction performance of students with or at risk for disabilities. These studies were all conducted by the same research groups (Bottge et al., 2002, 2010, 2014, 2015; Bottge, 1999). The remaining interventions—graduated instruction, strategy instruction, and video modeling—had insufficient evidence to make a determination of their evidence base at this time. This is due to the insufficient number of studies meeting the CEC EBP guidelines for establishing an intervention as evidence based.
Publication bias
Publication bias analyses were conducted using the g effect size or the overall random-effects model. Results of Duvall and Tweedie’s (2000) trim-and-fill method showed small possibility of publication bias with an adjusted value of g = 1.08, suggesting that the funnel plot was symmetrical. With respect to Egger’s regression-of-the-intercept test (Egger et al., 1997), publication bias was shown to be negligible (intercept = 0.96, p = .19). In sum, the results of these analyses suggest there is a small chance publication bias may affect the results. In other words, due to the amount of studies and the disbursement of effect sizes, there are likely no studies that would impact the effect sizes in a negative manner.
Discussion
The current study examined the extant literature on fraction interventions for students with disabilities, based on 21 studies. Overall, significant effects were noted for each of the intervention constructs used. Regarding effect sizes, video modeling was shown to be the most effective, though with only one study involving four students (Yakubova et al., 2015). The next highest effect was found when graduated instruction was implemented (e.g., concrete, representational, abstract; Bouck et al., 2017; Butler et al., 2003; L. Jordan et al., 1999; Kim et al., 2015), followed closely by strategy instruction (Joseph & Hunter, 2001; Test & Ellis, 2005; Zhang et al., 2016) and explicit instruction, which held the widest research base (Fuchs et al., 2013, 2014, 2016, 2017; Hunt, 2014; Kelly et al., 1990; Kelly, Carnine, Gersten, & Grossen, 1986; Scarlato & Burr, 2002). With respect to the quality of the included studies, 10 studies met 100% of the QIs. We will discuss these results with respect to the research questions posed, followed by limitations of the meta-analysis and implications for future research.
QIs
Across the studies, variability of adherence with QI ranged from three to eight of the eight included QIs. Of the 28 individual indicators, including both group and single-case designs, all studies met 14 indicators. This is encouraging given that eight of the 21 studies were published prior to 2005, which marked the publication of the special issue of Exceptional Children on scientific methods for special education research (e.g., Gersten et al. 2005; Horner et al., 2005), a defining point for the evaluation of QIs when determining a study’s contribution to the evidence base.
However, there were common QIs that were routinely omitted. Although 20 studies provided descriptions of the intervention agent to ascertain his or her role (e.g., classroom teacher, faculty, research assistant), only 16 defined why the intervention agent was qualified to implement the intervention or what training he or she completed in order to meet these qualifications. Future researchers should look for ways to make more explicit the training of the intervention agent, as it may greatly affect the outcomes (Cook, Tankersley, & Harjusola-Webb, 2008). Related to QI 5.0, treatment fidelity, 19 studies reported clear procedures and results for treatment fidelity related to adherence and dosage. Of those, 17 made explicitly clear that treatment fidelity data were collected throughout the intervention. Future researchers will want to be explicit if fidelity of the training process is measured as appropriate training and qualifications are essential to both the replication process and fidelity of implementation.
Study Effects
Findings from the current meta-analysis are consistent with previous reviews (Misquitta, 2011; Shin & Bryant, 2015) of interventions that have been found effective for improving the fraction knowledge of students with disabilities. Specifically, four of the five constructs of interventions showed large effects, with the one moderate effect coming from anchored instruction. However, these results should be taken with the caution that some of the constructs had few studies within them and thus may not be as reliable. For example, the highest-performing construct, video modeling, had an extreme effect size (g = 11.51), but that mean effect size was based on one study with four participants. Additionally, this study was a single-case design, the effect sizes of which tend to be inflated (Shadish et al., 2014), as corroborated by the meta-regression we conducted, which showed a significant ability to predict higher scores from single-case designs as opposed to group designs. The second highest effect sizes were noted for graduated instruction, which was used in three studies, two of which were single-case designs. The one group-design study had an effect size that was two standard deviations lower than the next lowest single-case design, thus illustrating the possible discrepancies between the two designs. There was an additional group-design study for graduated instruction (L. Jordan et al., 1999); however, it did not provide sufficient data to calculate an effect size.
four of the five constructs of interventions showed large effects, with the one moderate effect coming from anchored instruction
Explicit instruction possessed the widest literature base, with six studies utilizing the method, yielding large effects. All of the studies were group designs, and the lowest effect was still considered large (g = 0.92; Fuchs et al., 2016). This robust literature coincides with findings from the Misquitta (2011) and Shin and Bryant (2015) reviews, which noted direct (explicit) instruction as the most promising method for teaching these skills. With five group-design studies meeting all QIs, it has sufficient evidence to be considered an evidence-based practice (CEC, 2014).
Graduated instruction, which uses components of explicit instruction when moving from a concrete representation of a problem to an abstract one, showed promise and had the benefit of a group design to aid in the generalization of results. However, that group design (Butler et al., 2003) had only a moderate effect (g = 0.47), and the other studies were single-case design, thus limiting the interpretability of the results. Relatedly, strategy instruction, which uses explicit instruction to teach specific approaches, also had a large effect within three studies. The strategy instruction construct here included three approaches. Zhang et al. (2016) specifically taught a strategy to calculate fractions problems, Test and Ellis (2005) used a mnemonic device to teach the addition and subtraction of fractions with unlike denominators, and Joseph and Hunter (2001) used cue cards to help students negotiate the given strategy. Strategy instruction was shown to be one of the more promising methods, though the evidence was from a series of single-case-design studies.
Finally, anchored instruction, which entails a real-world problem-solving approach (e.g., building a hovercraft), had modest effects. Five group-design studies are sufficient to meet the criteria, but the average effect size of 0.47 is only moderate. Thus, the confidence in the results is high, though the effects were not.
Publication bias
Results of the analyses of possible publication bias showed little evidence publication bias exists with the results of both Duvall and Tweedie’s (2000) trim-and-fill method and Egger’s regression-of-the-intercept test (Egger et al., 1997). However, these measures are not infallible, so the possibility of bias may still exist in the literature.
Limitations
There are limitations of this meta-analysis that should be noted. First, although every effort was made to find all articles dealing with interventions to improve the fraction performance of students with or at risk for disabilities, it is possible some were missed. Additionally, we did not attempt to include “gray literature” (e.g., unpublished manuscripts); rather, we attempted to account for unpublished studies through statistical analyses. Next, our combining of group and single-case designs into one meta-analysis is a somewhat novel approach, so caution should be taken when interpreting the measures of effects. Results of the meta-regression on study design (e.g., single-case vs. group designs) confirm the suspicions that single-case designs tend to be inflated and not easily interpreted (Shadish et al., 2014). However, the single-case- and group-design measures follow recent trends in meta-analysis and thus lend themselves to comparisons with other like designs. Third, there is always a possibility for error and variations in reader interpretations for QI coding. For that reason, we conducted reliability on 100% of studies. The high reliability between coders suggests that, if present, coding errors were minimal; however, results should still be interpreted in light of this consideration. To facilitate accuracy, future researchers should continue to have multiple raters examine the QIs. Next, we were unable to control for baseline inequivalence between the groups; therefore it is possible that the effect estimates reported here may not reflect the actual effects if other analyses were conducted. Another limitation of this study is that we were not able to disaggregate outcomes (e.g., number line, computation, applied problems) and are not able to make conclusions as to the effects of the interventions on individual outcomes. Finally, we used an average effect size for studies with multiple dependent variables, which may have overestimated the standard errors when the effect sizes are independent (Moeyaert et al., 2017). Thus, results from the effect sizes should be taken with caution.
Implications for Practice and Future Directions
Given the need to improve fraction skills, in particular for students with disabilities (NAEP, 2015), it is encouraging to see such a wide array of strategies used to increase the fraction performance of students with disabilities. It is also heartening to see investigations occurring most frequently at the middle school level, as historically in the field of special education, more extensive research has been done at the elementary level (e.g., Losinski, Ennis, Sanders, & Nelson, 2018). However, as we seek to bolster students’ fraction skills to prepare them for algebra and other higher-order mathematics (Siegler et al., 2012), further investigations that address the introduction of fractions in fourth and fifth grades are warranted. We should seek to implement evidence-based practices when concepts are first taught (Tier 1), not just as a means of remediation (Tiers 2 and 3; e.g., Every Student Succeeds Act, 2015; Individuals With Disabilities Education Improvement Act, 2006). Another important finding of this meta-analysis is the fact that teachers were responsible for implementation of over half of the studies examined (i.e., 11 of 21). This is an important consideration as we seek to bridge the research-to-practice gap and empower teachers to implement evidence-based practices (Cook et al., 2008; Cook & Schirmer, 2006) and build capacity for sustained use (Odom et al., 2005).
Given the need to improve fraction skills, in particular for students with disabilities (NAEP, 2015), it is encouraging to see such a wide array of strategies used to increase the fraction performance of students with disabilities
There were 21 studies targeting fractions outcomes for students with disabilities; however, more research is needed in the area of fraction instruction. Researchers have demonstrated the importance of fraction knowledge for both real-life outcomes and predictors of higher-level mathematics performance (Booth, Newton, & Twiss-Garrity, 2014; Siegler et al., 2012). Of the five investigated intervention constructs, only explicit instruction is considered evidence based. Six of eight studies were implemented by a researcher rather than the classroom teacher (e.g., Fuchs et al., 2013). The large number of participants for which this intervention has demonstrated effectiveness is notably impressive, in particular given the extensive work done by Lynn Fuchs and colleagues; however, future researchers may want to examine the utility of explicit instruction for fractions when implemented by a classroom teacher to explore issues of feasibility and acceptability (Ennis, Lane, & Oakes; 2018; Odom et al., 2005).
Anchored instruction was determined to have mixed evidence, because although there was a large evidence base with a large number of participants, there were some studies with negative effects. In addition, studies were conducted by the same research team. Other researchers should consider replicating the work begun by Brian Bottge and colleagues, particularly the use of enhanced anchored instruction, which used explicit instruction with anchored instruction and achieved beneficial outcomes.
Graduated instruction, strategy instruction, and video modeling all have insufficient evidence to be determined an evidence-based practice. Graduated instruction has a wide research base for teaching other mathematics concepts (e.g., Agrawal & Morin, 2016). Future researchers should continue to replicate these findings to improve students’ fraction skills. Only three studies examined the utility of strategy instruction to improve fraction skills. Strategy instruction, in particular, self-regulated strategy development (SRSD), has been effective for teaching reading and writing (e.g., Ennis & Jolivette, 2014; Sanders et al., in press) to students with disabilities and may have promise in the area of mathematics (e.g., Case, Harris, & Graham, 1992; Cuenca-Carlino, Freeman-Green, Stephenson, & Hauth, 2016). Given the multistep procedures involved in understanding fractions, future researchers may want to explore the utility of SRSD for fractions. Regarding video modeling, not many conclusions can be drawn based on only one study; however, given the large effects found by Yakubova et al. (2015) and the utility of video modeling in social and behavioral contexts (Mason, Ganz, Parker, Burke, & Camargo, 2012), future researchers may also want to explore the utility of this strategy to teach fractions.
Finally, it is important to examine these findings in light of the What Works Clearinghouse recommendations for improving fraction performance (Siegler et al., 2010). Many of the approaches analyzed in this meta-analysis address Recommendations 1 through 4: (1) building on students’ background knowledge, (2) recognizing fractions as they fall along a number line, (3) understanding the validation for computations with fractions, and (4) understanding various strategies for solving problems (Siegler et al., 2010). Future researchers should consider these core components as they investigate ways to improve the fractions skills of students with disabilities. Regarding Recommendation 5 (improving teachers’ knowledge of fractions and how to teach them), conducting investigations that empower teachers to implement the strategy in their classroom is one way to begin this necessary element of professional development and bridge the research-to-practice gap.
