Abstract
In this synthesis, we analyzed 10 prealgebraic reasoning interventions for students with mathematics difficulty (MD) in Grades 6 through 8. All interventions focused on one or more prealgebraic concepts including integer operations, algebraic expressions and equations, and functions. Of the 10 intervention studies, six employed single-case design methodology and four employed group design. We synthesized intervention effects and identified instructional practices utilized within interventions. Results indicated positive proximal student performance across studies, generally positive maintenance effects, and mixed transfer effects. Manipulative-based instruction and explicit instruction were the two most used instructional practices within interventions, supporting their use in improving targeted prealgebraic reasoning skills among students with MD. Additional implications for research and practice are discussed.
Proficiency in mathematics is increasingly prioritized in the workforce and postsecondary education (Lee, 2012; Wei et al., 2013; Witzel, 2016). However, data from the National Assessment of Educational Progress (NAEP) indicate that over half of school-aged students struggle to attain mathematics proficiency (National Center for Education Statistics [NCES], 2022). One common area of difficulty for students is prealgebraic reasoning, a skillset that is emphasized in mathematics coursework in Grades 6 through 8 (National Governors Association Center for Best Practices & Council of Chief State School Officers [NGA & CCSSO], 2010). In this synthesis, prealgebraic reasoning is defined as the integration of arithmetic knowledge, relational understanding of the equal sign (i.e., equivalence of values on both sides), and abstract problem-solving (Ketterlin-Geller & Chard, 2011; McMullen et al., 2017; Pillay et al., 1998; Powell, 2012). For instance, solving for x in the problem
Prealgebraic reasoning ability is essential to success in secondary and postsecondary mathematics, because advanced mathematics coursework such as algebra and calculus requires the integration of arithmetic knowledge, relational understanding of the equal sign, and abstract problem-solving skills (Ketterlin-Geller & Chard, 2011). Prealgebraic reasoning is also useful in numerous workforce settings, such as when modeling costs that increase at a steady rate. Because of the importance of prealgebraic reasoning, there is an instructional emphasis on this area in Grades 6 through 8 (NGA & CCSSO, 2010). Nonetheless, many students encounter difficulty developing prealgebraic reasoning skills (Ketterlin-Geller & Chard, 2011; NCES, 2022; Witzel, 2016). This is particularly the case among students with mathematics difficulty (MD; Ketterlin-Geller & Chard, 2011; Powell et al., 2021). Such students can benefit from targeted intervention, setting them up for future success in mathematics (Powell et al., 2021). Evaluating the current research base in this area may lead to the identification of effective approaches to prealgebraic reasoning intervention.
Mathematics difficulty
Many students experience substantial difficulty acquiring critical mathematics skills such as prealgebraic reasoning (NCES, 2022). Some of these students have a specific learning disability in mathematics (SLD-M). Students with SLD-M include students who demonstrate (a) limited response to increasingly intensive intervention, (b) a discrepancy between cognitive functioning and mathematics performance, or (c) patterns of strengths and weaknesses (Elias, 2021; Maki et al., 2015). (Please note, there has been considerable criticism of the discrepancy model for SLD-M identification [Taylor et al., 2017], although some states still use this model for identification purposes [Maki et al., 2015].)
Geary (2004) approximated that 5–8% of the student population has SLD-M. Such students may struggle with fact retrieval, computation, and representation of visuospatial mathematics content. Mathematics language may also present challenges for students with SLD-M, including comprehending word problems, developing mathematics vocabulary, and attributing meaning to mathematics symbols (Bone, Bouck, & Witmer, 2021; Geary, 2004; Powell et al., 2019; Witzel, 2016). Students integrate a number of these skillsets as they acquire prealgebraic reasoning skills. Because those with SLD-M are likely to have underdeveloped foundational skills, they are at increased risk of experiencing difficulty with prealgebraic reasoning (Geary, 2004; Witzel, 2016).
Although more than 5% of students may have an identified SLD-M (Geary, 2004), a far greater number exhibit below-expected mathematics performance (NCES, 2022). For example, on the 2022 NAEP, only 27% of eighth-grade students achieved proficiency in mathematics. This low proficiency rate is reflective of students’ difficulties acquiring grade-level skills in mathematics, many of which center on prealgebraic reasoning in Grades 6 through 8 (NGA & CCSSO, 2010). This means there are students who experience MD without an official identification of an SLD-M.
Given the low rates of mathematics proficiency across the United States (NCES, 2022), some researchers have characterized students with MD as both those with SLD-M and those without an identified disability but who exhibit below-expected mathematics performance (Jitendra et al., 2021; Nelson & Powell, 2018; Stevens et al., 2018). Researchers commonly identify below-expected mathematics performance through screener assessments (Nelson & Powell, 2018). There is no standardized cutoff score to use for identification, although across studies percentile cutoffs have typically ranged from the 10th percentile up to the 40th percentile (Jitendra et al., 2021; Nelson & Powell, 2018). Other researchers have identified below-expected mathematics performance through teacher nomination or placement in a remedial mathematics class (Bottge et al., 2001; Nelson & Powell, 2018).
Given that many students with and without SLD-M exhibit persistent difficulty in mathematics, in this synthesis students with MD included those with a documented learning disability as well as those without a disability who exhibited below-expected mathematics performance (Jitendra et al., 2021; Stevens et al., 2018). To comprehensively identify and include relevant studies, we considered below-expected mathematics performance to be identified through a screener assessment score below the 40th percentile, teacher nomination, or placement into a remedial mathematics class (Bottge et al., 2001; Nelson & Powell, 2018).
Mathematics instruction in Grades 6 through 8
Across many states, school districts use the Common Core State Standards domains to guide their mathematics scope and sequence of instruction. In Grades 6 through 8, Common Core State Standards domains include geometry, ratios and proportional relationships, statistics and probability, expressions and equations, functions, and the number system (NGA & CCSSO, 2010). Within the last three domains, prealgebraic reasoning is a primary focus of instruction at these grade levels. In the Expressions and Equations domain, students in Grades 6 and 7 use algebraic equations to model real-world situations and determine unknown quantities. Within the Functions domain, students in Grade 8 model relationships with functions and represent functions in different ways (e.g., equations, graphs, and tables). The skills addressed in The Number System domain directly support algebra readiness by introducing students to a wider range of number types, such as negative integers, that are commonplace in Algebra 1 coursework. Students in Grades 6 through 8 also learn content that is less directly applicable to algebra, such as how to find probabilities of events, measure angles, and calculate surface area.
Although students with MD encounter challenges with mathematics across grade levels, these difficulties can intensify as students transition from elementary to middle school (i.e., Grades 6 through 8; NCES, 2022; Powell et al., 2021; Witzel, 2016). Students with MD in Grades 6 through 8 can struggle with both fundamental skills and newly introduced concepts, creating a complex set of potential challenges (Cirino et al., 2016). For instance, misconceptions about place value and lack of fluency with whole-number operations can exacerbate students’ difficulties with middle-school content (Moser Opitz et al., 2017). Even for students whose foundational skills are relatively intact, middle-school content may present difficulties due to its increased abstractness, novel mathematics language, and multistep procedures (Geary, 2004; Witzel, 2016). Many students with MD in Grades 6 through 8 require targeted intervention to support their successful transition to middle-school mathematics (Lee, 2012).
Prealgebraic reasoning
Prealgebraic reasoning centers on integrating arithmetic knowledge, relational understanding of the equal sign, and abstract problem-solving (McMullen et al., 2017; Pillay et al., 1998; Witzel, 2016). Although prealgebraic reasoning is a primary focus in Grades 6 through 8, students begin to develop prealgebraic reasoning skills as early as elementary school (Powell, 2012). For instance, solving nonstandard equations such as 10 = __ + 7 requires prealgebraic reasoning (Bone et al., 2021b; Powell, 2012; Powell et al., 2020). First, students must understand the equal sign as relational (i.e., that 10 is the same value as some number added to seven). Second, students must have the arithmetic skill to compute the value that belongs in the blank space. Finally, this problem is devoid of context and thus completely abstract. In these ways, solving nonstandard equations integrates arithmetic skill, relational understanding of the equal sign, and abstract problem-solving.
Prealgebraic reasoning becomes a focal point of instruction in Grades 6 through 8 as students acquire additional skills that integrate their arithmetic and algebraic knowledge (Pillay et al., 1998; Witzel, 2016). Students in these grade levels learn to represent unknown quantities with letters and solve for these quantities by balancing algebraic equations (e.g., solving for x in the equation 9 = 5x – 6; NGA & CCSSO, 2010). Students also expand their knowledge of the number system to include negative integers and use this knowledge to represent relationships algebraically. By Grade 8, students apply their knowledge of algebraic equations to recognize and model functions. For example, consider the following scenario: Naomi pays $100 to buy a lawn mower and charges her neighbors $15 each to mow their lawns. Students learn to create a linear model for this type of situation, such as y = −100 + 15x, where y represents total earnings and x represents the number of lawns mowed. Creating this algebraic equation relies on knowledge of negative integers, understanding of arithmetic operations, relational understanding of the equal sign, and functions. Integration of these skills is at the core of prealgebraic reasoning, and it equips students to translate real-life scenarios to abstract models.
Students with MD frequently encounter difficulties integrating arithmetic and algebraic knowledge, leading to challenges developing prealgebraic reasoning skills (Witzel, 2016). Several factors contribute to these difficulties. First, students may see the equal sign as an operational rather than relational symbol (Powell, 2012; Powell et al., 2020). For instance, students may misinterpret the equal sign as an indication of where to put an answer, rather than an indication of equivalence. Such misinterpretations can cause challenges in accessing the conceptual underpinnings of manipulating algebraic equations. Some students also encounter difficulty assigning meaning to letters that represent unknown quantities (i.e., variables; Bone, Bouck, & Witmer, 2021; Witzel, 2016). Targeted instruction can support prealgebraic reasoning development among students with MD; it is, therefore, critical that practitioners provide such support to facilitate students’ algebra readiness and future mathematics success (Strickland, 2022; Watt et al., 2016).
Previous syntheses, systematic reviews, and meta-analyses
Students with MD require mathematics intervention to succeed across domains and grade levels, including prealgebraic reasoning in Grades 6 through 8 (Powell et al., 2021). In this synthesis, we used Stevens et al.'s (2018) definition of mathematics intervention as specialized mathematics instruction provided outside of the general education whole-class setting. We focused on intervention to examine efforts beyond general education classroom mathematics instruction that teachers or interventionists would use to provide targeted support specifically for students with MD. This support might be provided within multitiered systems of support (e.g., Tier-2 or Tier-3 tutoring) or intensive intervention delivered in small groups or individually to students with MD. To the authors’ knowledge, no synthesis, systematic review, or meta-analysis currently exists with a unique focus on prealgebraic reasoning intervention. However, researchers have conducted related syntheses, systematic reviews, and meta-analyses on middle-school mathematics intervention and algebra intervention (e.g., Hughes et al., 2014; Powell et al., 2021). We discuss several of these efforts in the following sections.
Secondary-level mathematics intervention
Powell et al. (2021) conducted the only systematic review to date that focused exclusively on mathematics intervention for students with MD in Grades 6 through 8. In their review, Powell et al. determined that 82% of included studies yielded significant positive results. They identified six instructional practices that researchers incorporated into effective interventions: explicit instruction, multiple representations, problem-solving instruction, mathematics-language instruction, use of mnemonics, and use of graphic organizers.
Researchers have also conducted meta-analyses on mathematics intervention in Grades 4 through 12 (Stevens et al., 2018) and Grades 6 through 12 (Jitendra et al., 2018; Myers et al., 2021). By focusing on secondary students within these meta-analyses, researchers addressed challenges students with MD encounter with increasingly complex mathematics content. Within all three meta-analyses, authors determined that mathematics intervention had significant, moderate effects. Effects were typically weaker for students in later grades than earlier grades, although these differences were not always significant (Jitendra et al., 2018; Stevens et al., 2018). This decreasing trend in effects across grades illustrates the challenge of supporting students in acquiring complex mathematical skills (Stevens et al., 2018). As such, all three author teams highlighted the need for more research that supports secondary students with MD (Jitendra et al., 2018; Myers et al., 2021; Stevens et al., 2018).
Algebra intervention
Researchers have investigated the effects of algebra intervention in one meta-analysis (Hughes et al., 2014) and several systematic reviews (Bone, Bouck, & Witmer, 2021; Hwang et al., 2019; Lee et al., 2020; Watt et al., 2016). In their meta-analysis, Hughes et al. (2014) determined that algebra intervention across grade levels for students with MD had a moderate weighted mean effect size of 0.62. Additionally, they identified that cognitive and model-based strategies were associated with the largest effects. Significant moderate effects were identified among interventions incorporating the concrete-representational-abstract (CRA) framework, which includes utilizing a combination of hands-on, visual, and symbolic representations. Researchers who conducted systematic reviews of algebra intervention also determined positive effects associated with use of cognitive strategies and CRA (Bone, Bouck, & Witmer, 2021; Hwang et al., 2019; Lee et al., 2020; Watt et al., 2016). Other effective algebra interventions incorporated explicit instruction, schema-based instruction, enhanced anchor instruction, peer-assisted learning strategies, and manipulatives outside of the CRA framework.
Although a multitude of instructional practices yielded positive effects on the algebra performance of students with MD, Bone, Bouck, and Witmer (2021) cautioned that there are limited high-quality studies focused on algebra intervention. They therefore determined that many of these strategies do not yet fully meet criteria as evidence-based practices within the domain of algebra intervention. They called for additional, rigorous algebra intervention research to address these limitations. Given that there is even less research on prealgebraic reasoning specifically, interventions on these skills clearly warrant examination.
Purpose and research questions
Mathematics proficiency is critical for students to succeed in postsecondary settings, including the workforce and college (Lee, 2012; Wei et al., 2013). However, many students with MD struggle with both prealgebraic reasoning and algebra content, impeding their ability to graduate high school with adequate mathematics knowledge (Witzel, 2016). To better prepare students with MD for success in algebra, researchers must identify interventions that support students’ prealgebraic reasoning skills in Grades 6 through 8. Although researchers have conducted numerous reviews on algebra intervention for students with learning disabilities and MD (e.g., Hughes et al., 2014; Lee et al., 2020), none focused specifically on prealgebraic reasoning intervention. Moreover, only one systematic review currently exists with a focus on students with MD in Grades 6 through 8 (Powell et al., 2021). This synthesis will thus fill a gap in mathematics intervention literature by uniquely addressing prealgebraic reasoning intervention for students with MD in Grades 6 through 8. In this synthesis, we addressed the following research questions:
What are the effects of prealgebraic reasoning interventions for students with MD in Grades 6 through 8? What instructional practices are included in prealgebraic reasoning interventions for students with MD in Grades 6 through 8?
Method
Search procedures
Figure 1 displays a PRISMA diagram that details search procedures to identify relevant articles for this synthesis. The search process began with a systematic database search conducted in November of 2021 of four academic databases: Academic Search Complete, Education Source, Education Resources Information Center (ERIC), and PsycINFO. Two lines of search terms were used, the first to capture prealgebraic content and the second focused on students with MD. The following terms were used in the first line: algebra* OR equation OR linear OR integer OR “negative number*.” On the second line, the following terms were used: “math* disabilit*” OR “math* difficult*” OR “learning disabilit*” OR “learning difficult*” OR dyscalculia OR “low achiev*” OR “low perform*.”
PRISMA diagram.
The systematic database search initially yielded 5441 results. After removing duplicates, 3882 articles remained to screen for potential eligibility. Of the remaining articles, 3804 were excluded based on information provided in their titles and abstracts. We then sought out the remaining 78 articles for retrieval, 76 of which were successfully retrieved. The full text of each retrieved article was assessed for eligibility, and 68 of them were excluded because they did not meet full inclusion criteria. Ultimately, eight articles were identified for inclusion through the systematic database search.
To identify additional relevant articles, we supplemented our database search with 10-year table of contents searches of the following journals: Exceptional Children, Journal of Learning Disabilities, Learning Disability Quarterly, Learning Disabilities Research & Practice, and Remedial and Special Education. These journals were chosen because they frequently publish articles on mathematics intervention, including studies identified through our systematic database search. No additional articles were retrieved through the table of contents search. Additionally, we conducted forward and backward searches of all included articles. Two more articles were retrieved through forward searching. One of these articles was originally published online in 2021; however, its most recent citation is used throughout the article (Bone et al., 2022). Overall, a total of 10 studies were included in this synthesis.
Inclusion criteria
Studies were eligible for inclusion in the synthesis if they met the following criteria:
Authors employed a single-case design, experimental group design, or quasi-experimental group design in the study. Studies were published in peer-reviewed journals. Studies were available in English. Studies included at least one quantitative measurement of student outcomes on a mathematics measure. Studies included a mathematics intervention, defined as supplemental mathematics instruction outside of the general education whole-class setting (Stevens et al., 2018). The focus of the mathematics intervention was prealgebraic reasoning, specifically skills within the following Grades 6 through 8 content domains: Expressions and Equations, Functions, and integers within The Number System (NGA & CCSSO, 2010). Studies were eligible for inclusion if they disaggregated intervention content and outcomes by unit and at least 50% of units focused on prealgebraic reasoning content. In these cases, effects were analyzed only on the prealgebraic reasoning units. Study participants were students with MD. Specifically, students had to either have learning disabilities or exhibit below-expected performance in mathematics despite no disability diagnosis (Jitendra et al., 2021; Stevens et al., 2018). Below-expected mathematics performance could be determined by a screener assessment score below the 40th percentile, teacher nomination, or placement into a remedial mathematics class (Bottge et al., 2001; Jitendra et al., 2021; Nelson & Powell, 2018). If a study included students without MD, it was included if at least 50% of participants had MD or if authors provided disaggregated outcome data on participants with MD. Studies targeting students with intellectual disabilities or autism were excluded. Participants had to be in Grades 6 through 8 throughout the duration of the intervention, due to the curricular focus on prealgebraic reasoning at these grade levels (NGA & CCSSO, 2010). Just as with the previous criterion, studies were eligible for inclusion if at least 50% of participants were in Grades 6 through 8 or if they provided disaggregated data for students in these grades.
To capture the greatest number of studies that met all inclusion criteria, there were no limitations on the publication date of studies.
Coding and analysis procedures
We designed a coding manual to extract information from each included study. The manual included variables related to general study characteristics and quality, intervention effects, and instructional practices used within interventions. Study characteristics included details on research design, sample size, participant grade levels, and intervention content foci. Study quality codes were aligned with the Council for Exceptional Children (CEC)'s best-practice quality indicators for evidence in special education (Cook et al., 2015). These quality indicators address whether authors report information in multiple domains, including descriptions of context and setting, participants, intervention agents, instructional practices, implementation fidelity, internal validity, outcome measures, and data analysis. After coding each study's presence of quality indicators, we determined the study's quality-indicator score by dividing the number of quality indicators addressed by the total number of potential quality indicators. Coding general study characteristics and quality information provided a foundation for interpreting intervention effects and instructional practices used within interventions.
To address quantitative effects, we extracted information related to outcome measures and participant performance. We first coded whether each measure was standardized or researcher-developed, whether it was proximal or distal to the intervention content, and whether it was administered as an immediate measure of intervention growth or a maintenance measure. For group-design studies, p-values and effect sizes were extracted for each measure to assess the significance and magnitude of intervention effects. For single-case design studies, we calculated the percentage of nonoverlapping data (PND) for each participant's intervention performance as compared to their highest baseline performance. If single-case design studies also reported maintenance or transfer data, PNDs were included on these measures as well. Given the difference in data extraction based on research design, group-design quantitative effects were analyzed separately from single-case-design effects within this synthesis.
We then coded information related to instructional practices based on authors’ intervention descriptions. Specifically, our manual included codes for explicit instruction, manipulatives, word-problem attack strategies, word-problem schema instruction, vocabulary instruction, self-regulation strategies, behavioral reinforcement, and use of technology.
Coding reliability
The first author, a doctoral student studying special education, coded all 10 articles included in the synthesis. The first author then provided a 1-hr training on coding procedures to the third author, who is also a doctoral student studying special education. The third author double-coded five randomly selected articles (i.e., 50% of the included articles). The first author then calculated initial reliability to be 96%. Both authors met to resolve all discrepancies through discussion until 100% agreement was reached.
Results
Table 1 displays key characteristics of all 10 studies included in this synthesis. The studies ranged in publication year from 2000–2022. Four of the 10 studies employed a group design, two of which were randomized control trials and two of which were quasi-experimental studies. The remaining six studies involved single-case designs, five of which were multiple probe and one of which was an alternating-treatment design. Sample sizes ranged from three to 908 participants; single-case studies included between three and 14 participants, while group-design studies included between 44 and 908 participants.
Summary of study characteristics.
Note. CCSSM = Common Core State Standards in Mathematics; CM = concrete manipulatives; VM = virtual manipulatives; EAI = enhanced anchored instruction; CRA = concrete-representational-abstract framework.
Authors provided disaggregated data on the performance of students with mathematics difficulty (n = 14).
The majority of participants (76%) were in Grades 6–8.
Two out of four units within this intervention met inclusion criteria for prealgebraic content. Only data from these two units are included in this synthesis.
STAR is a word-problem-solving mnemonic.
Nine out of the 10 studies included only participants with MD. Participants in six of the 10 studies had identification of a learning disability, whereas participants in three studies did not have identified disabilities but exhibited below-expected performance in mathematics. In one study, students with both types of MD were included (Cuenca-Carlino et al., 2016). Four studies included eighth-grade student participants, three studies included seventh-grade participants, one study included sixth-grade participants, and two studies included students at multiple grade levels. One of the studies that included students at multiple grade levels also included students in Grades 9 through 12, although 76% of participants were in Grades 6 through 8 (Bottge et al., 2007).
We calculated a quality-indicator proportion for each study based on presence of CEC's best-practice quality indicators (Cook et al., 2015). Overall study-quality proportions ranged from 0.54 to 1.00, with an average quality-indicator score of 0.85.
Intervention effects
We reviewed intervention effects from group-design studies separately from effects in single-case studies, due to differences in study design and effect measurement.
Group-design effects
Table 2 summarizes intervention effects within the group-design studies included in this synthesis. Of the four group-design studies, three studies included a combination of researcher-developed measures and standardized measures. The remaining study used only one measure, which was researcher-developed (Bryant et al., 2020). We extracted 20 effect sizes across studies and measures. All effect sizes reported by Bryant et al. (2020) as well as Namkung and Bricko (2021) represented treatment to control group comparisons, and all participants in both studies had MD identifications. Bottge et al. (2001) reported comparisons of treatment participants with MD to control participants without MD, as well as comparisons of treatment participants with MD to treatment participants without MD. We extracted effect sizes from both comparisons. Bottge et al. (2007) did not include a control group in their study. Instead, they implemented differently sequenced treatments with two treatment groups of participants with MD. Thus, effect sizes from Bottge et al. (2007)'s study represent a combination of pre- to posttest comparisons and treatment group comparisons at specific time points.
Effects within group-design studies.
Note. T = treatment; EAI = enhanced anchored instruction; MD = mathematics difficulty; C = control; BAU = business as usual; RD = researcher-developed.
Across the four studies, authors reported seven effect sizes that reflected students’ proximal performance on researcher-developed measures. Although authors reported different effect size metrics (e.g., Cohen's d, partial eta squared), 100% of these effect sizes reflected significant, positive effects.
Further, authors reported maintenance effects in two of the four studies, reflecting data collected between 10 days and 4 weeks after the interventions ended (Bottge et al., 2001, 2007). A total of five effect sizes across the two studies demonstrated students’ maintenance performance, all of which were measured through researcher-developed assessments. Bottge et al. (2001) determined that treatment participants with MD performed comparably to treatment and control participants without MD; this was a favorable outcome that reflected a narrowing of significant pretest differences. Bottge et al. (2007) identified significant, positive maintenance effects in three out of four comparisons. In aggregate, 80% (i.e., four out of five) of reported effect sizes reflected favorable maintenance performance.
Authors also assessed effects of transfer to other mathematics domains in three out of four studies, and they reported a total of eight effect sizes (Bottge et al., 2001, 2007; Namkung & Bricko, 2021). In all three studies, transfer measures were standardized assessments. Namkung and Bricko (2021) reported insignificant effects of transfer to fluency and algebra skills. Similarly, Bottge et al. (2001) learned that treatment students with MD did not transfer their skills to arithmetic. Bottge et al. (2007) reported positive transfer effects on measures of problem solving and data interpretation. In total, four of the eight reported effect sizes demonstrated participants’ successful transfer of skills to other mathematics domains, yielding a positivity rate of 50%.
Single-case effects
Table 3 provides a summary of intervention effects within the single-case design studies in this synthesis. Five of the six single-case author teams utilized only researcher-developed assessments to measure intervention effects. In the sixth study, Scheuermann et al. (2009) included a combination of researcher-developed and standardized measures. Across all six studies, PNDs comparing baseline to intervention performance ranged from 93–100%, demonstrating efficacy of the interventions (Scruggs & Mastropieri, 2013). In addition to providing baseline and intervention data on proximal probes, Scheuermann et al. administered another proximal measure at pre- and posttest. Student performance on this measure demonstrated strong effects (p = .001, Δ = 2.32).
Effects within single-case design studies.
Note. CM = concrete manipulatives; VM = virtual manipulatives; RD = researcher-developed; PND = percentage of nonoverlapping data.
Five of the six studies also included maintenance data, the timing of which ranged from one week after intervention to 11 weeks after intervention. The PNDs comparing maintenance to baseline performance ranged from 43–100%, and the majority of these PND statistics were 100%. Additionally, Scheuermann et al. (2009) provided one maintenance effect size using Glass’ delta (p = .001, Δ = 1.92), which demonstrated a strong effect.
Authors in two of the six studies assessed transfer effects to other mathematics domains (Maccini & Ruhl, 2000; Scheuermann et al., 2009). Maccini and Ruhl (2000) administered researcher-developed near-transfer and far-transfer word-problem assessments to participants. The near-transfer average PND was 100% and the far-transfer average PND was 33%. Scheuermann et al. (2009) assessed transfer effects on three measures. First, they administered researcher-developed probes of uninstructed problems that were more complex than those covered in the intervention. The average PND on these probes was 79%. Additionally, they administered an assessment of textbook problems as well as the KeyMath-Revised. Results indicated significant effects on textbook problems (p = .018, Δ = 0.67) and on the KeyMath-Revised (p = .027, Δ = 0.54). Both effect sizes represented moderate effects (Scheuermann et al., 2009). Across both studies that reported transfer data, students demonstrated successful transfer on 80% of measures.
Intervention content and instructional practices
The most common intervention content focus was solving algebraic equations, reflected in five of the 10 studies. Three of the 10 interventions had a focus on integer operations, and the remaining two focused on functions. Regarding instructional practices, there were multiple commonalities across interventions. All but one intervention incorporated some form of manipulative use (e.g., concrete manipulatives, virtual manipulatives, and/or the CRA framework). Aside from manipulatives, authors utilized explicit instruction (i.e., modeling, guided practice, and independent practice) in seven of the 10 interventions. Five of the 10 interventions included technology-based instruction. Of these five interventions, two relied on apps, two incorporated video modeling, and one included both tools. For details on instructional practices within specific studies, see Table 1.
Although manipulatives, explicit instruction, and technology were the most common instructional practices that author teams reported, several other practices were also present throughout many of the interventions. Four of the 10 author teams reported emphasizing applied problem-solving in their interventions, either through real-world problems or word-problem instruction. Three interventions included student discussion opportunities, and three interventions incorporated vocabulary instruction. Two author teams included behavioral reinforcement in their interventions, and two author teams utilized self-monitoring checklists for participants.
Discussion
The purpose of this synthesis was to examine prealgebraic reasoning interventions for students with MD in Grades 6 through 8. Specifically, we focused our investigation on intervention effects and instructional practices used within interventions. This synthesis adds to existing literature by uniquely addressing prealgebraic reasoning interventions, which to the authors’ knowledge has not been a focus of any previous synthesis. Synthesizing intervention research in this area provides valuable information to researchers and practitioners, because (a) prealgebraic reasoning represents a major component of mathematics instruction in Grades 6 through 8 and (b) prealgebraic reasoning ability sets a foundation for success in more advanced mathematics (Ketterlin-Geller & Chard, 2011; NGA & CCSSO, 2010). Given the importance of prealgebraic reasoning in students’ mathematics trajectories, it is critical to provide research-based, targeted support to students experiencing difficulty in this area (i.e., students with MD).
Effects
Across both group-design and single-case studies in this synthesis, authors reported strong proximal performance among participants. These results validate the efficacy of the prealgebraic reasoning interventions included in this synthesis. Moreover, the majority of studies included maintenance data and most of these results were positive. Such results reflect participants’ sustained improvement of skills, which is highly encouraging. However, readers should note that maintenance data were collected between one and 11 weeks post-intervention across studies, which represents a wide range of timeframes as well as limited understanding of longer-term maintenance. Therefore, participants’ maintenance performance must be interpreted in light of some fairly narrow gaps between the intervention's conclusion and collection of maintenance data.
Results were mixed regarding transfer effects, based on the five studies that included transfer data. Authors of single-case studies typically reported stronger transfer effects than authors of group-design studies. Across both study designs, authors reported positive transfer effects on standardized measures such as the Iowa Test of Basic Skills: Data Interpretation, Iowa Test of Basic Skills: Problem Solving, and KeyMath-Revised (Bottge et al., 2007; Scheuermann et al., 2009). However, transfer to arithmetic and fluency was not evident (Bottge et al., 2001; Namkung & Bricko, 2021). These results suggest that prealgebraic reasoning skill may bolster performance in data analysis and problem-solving more so than in more procedural domains, such as arithmetic. It may be that abstract reasoning, a key component of prealgebra (Ketterlin-Geller & Chard, 2011), plays a stronger role in the former domains than the latter. Additionally, it is possible that arithmetic skill must already be developed for students to be successful with prealgebraic reasoning skills involving equation-solving and functions. However, with such limited data it is difficult to conclusively make these determinations.
Instructional practices
The most common instructional practice among interventions included in this synthesis was manipulative-based instruction. This result complements existing literature on algebra interventions, which has also indicated a high prevalence of manipulative use (Bone, Bouck, & Witmer, 2021; Hughes et al., 2014; Hwang et al., 2019; Lee et al., 2020; Watt et al., 2016). Importantly, authors of studies in this synthesis investigated different types of manipulative modalities to determine how to most effectively incorporate manipulatives into prealgebraic reasoning interventions. For instance, Bone, Bouck, and Satsangi (2021) alternated concrete and virtual manipulative instruction in their intervention on two-step algebraic equations. They determined that both manipulative types supported student growth, although social validity data indicated that students preferred using virtual manipulatives. This result illustrates that virtual manipulatives may be a suitable alternative to concrete manipulatives for students with MD in Grades 6 through 8. Additionally, three author teams relied on a traditional CRA framework (Bryant et al., 2020; Maccini & Ruhl, 2000; Scheuermann et al., 2009), which supported student growth in all interventions. However, two other author teams also determined positive effects by removing the representational phase of this framework, indicating that this phase may not be needed for prealgebraic content in Grades 6 through 8 (Bone et al., 2022; Namkung & Bricko, 2021). Both virtual manipulatives and removal of representation phases have the potential to streamline future prealgebraic reasoning intervention.
Additionally, this synthesis provided support for use of explicit instruction in prealgebraic reasoning intervention. Seven of the 10 interventions incorporated explicit instruction, supporting positive effects on all proximal measures and some distal measures. Just like with manipulative usage, these results complement other synthesized mathematics intervention research supporting use of explicit instruction (Dennis et al., 2016; Gersten et al., 2009; Powell et al., 2021).
Author teams incorporated other instructional practices into their interventions, although each was present in five or fewer studies. These practices included technology use, applied problem-solving, student-discussion opportunities, vocabulary instruction, behavior rewards, use of mnemonics, and self-monitoring strategies. These practices have demonstrated promise in this synthesis as well as across other middle-school mathematics intervention (Powell et al., 2021). Additional studies could further enhance the evidence behind these practices within prealgebraic reasoning intervention.
Study scope and quality
With only 10 studies included in this synthesis, the literature base on prealgebraic reasoning interventions for students with MD is limited. More intervention research is needed in this area in general, but particularly with a content focus on functions. Only two of the 10 studies in this synthesis focused on functions content (Bottge et al., 2001; Bottge et al., 2007), both of which were published by the same lead author and employed similar interventions. It is important to expand this literature base, as introductory functions instruction provides a foundation for much of high-school mathematics content (NGA & CCSSO, 2010).
Despite the small number of studies included in this synthesis, overall quality among the included studies was relatively high. Author teams addressed most quality indicators (Cook et al., 2015), with an average quality-indicator proportion of 0.85. All but one study earned a quality-indicator score above 0.70, indicating an overall high level of quality among studies (Powell et al., 2021). The most common missing information related to technical features of measures (i.e., reliability and validity information). It is critical that authors report this information so that results can be effectively interpreted and compared across studies.
Limitations
There are several limitations to this synthesis. First, based on our operational definition of MD, we included only studies whose participants either had learning disabilities and/or exhibited below-expected performance in mathematics (Stevens et al., 2018). Studies whose participants had autism and intellectual disabilities were excluded. This limits the generalizability of the results of this synthesis, and it also limited the number of included studies.
Additionally, our inclusion criteria for prealgebraic reasoning interventions were restricted to content focused on algebraic expressions and equations, functions, and integers (NGA & CCSSO, 2010). Other researchers might have operationalized prealgebraic content slightly differently. For instance, some researchers might have focused solely on algebraic expressions and equations, while others might have been more inclusive than we were by adding in content on ratios and proportions. The results of this synthesis must therefore be interpreted in reference to the specific domains included in our operational definition of prealgebraic reasoning content. Finally, this synthesis included only peer-reviewed journal articles. Thus, prealgebraic reasoning studies published in other sources were excluded (e.g., dissertations).
Implications for research
The results of this synthesis lead to several implications for researchers. First, as noted, researchers should conduct more studies on prealgebraic reasoning interventions for students with MD. This is particularly the case in the domain of functions, where very few interventions are available for practitioners.
Regarding use of instructional practices, researchers have conducted innovative work on manipulative usage within interventions. Future research should extend this line of inquiry to determine the most effective and socially valid methods of incorporating manipulatives into prealgebraic reasoning instruction. Researchers may also want to investigate whether the representational phase of CRA is necessary for all students. Additionally, researchers should continue to investigate several promising practices in prealgebraic reasoning intervention: technology use, applied problem-solving, student-discussion opportunities, vocabulary instruction, behavioral reinforcement, mnemonics, and self-monitoring supports. One practice that seems particularly useful for promoting transfer is applied problem-solving instruction. Although transfer effects were mixed among studies, all author teams that reported positive transfer effects incorporated applied problem-solving into their interventions (Bottge et al., 2007; Maccini & Ruhl, 2000; Scheuermann et al., 2009).
Implications for practice
Practitioners can utilize the results of this synthesis in multiple ways. First, all interventions resulted in positive proximal growth among students. Manipulatives and explicit instruction were the most common instructional practices across interventions, lending support to their use among practitioners. Moreover, given encouraging results of both virtual and concrete manipulative usage, practitioners can feel positively about using both or either, depending on needs and preferences within their school setting. Finally, several other practices showed promising impacts on students’ prealgebraic reasoning; these practices included technology use, applied problem-solving, student-discussion opportunities, vocabulary instruction, behavioral reinforcement, mnemonics, and self-monitoring supports.
Practitioners should also emphasize instruction on transfer, given that transfer data among studies indicated mixed results. Specifically, practitioners should support students to make clear connections between prealgebraic reasoning skills and other mathematics domains. For instance, when studying how to graph linear functions, practitioners can model how this skill relates to displaying and analyzing real-world data (NGA & CCSSO, 2010). Additionally, to support students’ algebra readiness, practitioners should facilitate connections between prealgebraic reasoning and Algebra 1 content (e.g., applying an understanding of integers to modeling and graphing functions).
Conclusion
A primary purpose of this synthesis was to assess effects of prealgebraic reasoning interventions for students with MD in Grades 6 through 8. Across both group-design and single-case studies, intervention effects demonstrated strong proximal growth among participants. Moreover, most authors reported maintenance data, and the majority of these results indicated participants’ sustained growth. Transfer effects were mixed, although participants demonstrated successful transfer to other mathematics domains on multiple standardized measures. The second aim of this synthesis was to provide insight into instructional practices used within prealgebraic reasoning interventions for students with MD in Grades 6 through 8. Manipulative-based instruction was the most common instructional practice within the 10 intervention studies included in this synthesis, followed by explicit instruction. In sum, despite the limited number of studies that met inclusion criteria for this synthesis, results indicate that participants with MD respond positively to prealgebraic reasoning interventions. Manipulative-based instruction and explicit instruction hold particular promise for building students’ prealgebraic reasoning skills.
Footnotes
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author(s) received no financial support for the research, authorship, and/or publication of this article.