Exploring the Relationship Between Initial Mathematics Skill and a Sixth-Grade Fractions Intervention

Abstract

This study explored whether initial skill moderated outcomes of Promoting Algebra Readiness, a Tier 2 sixth-grade mathematics intervention targeting conceptual and procedural knowledge of fractions. The study analyzed data from a quasi-experimental pilot study in which at-risk students (N = 198) from Oregon middle schools were assigned to the treatment or control condition at the school level. Proximal and distal measures of math proficiency were collected in the fall (pretest) and spring (post-test). Analyses examined initial student achievement as a moderator of mathematics outcomes. Results indicated that intervention outcomes were not moderated by initial skill. Implications for tiered mathematics instruction and future mathematics intervention research are discussed.

Keywords

at risk content-area instruction instructional strategies assessment observation research research design or utilization

Proficiency with fractions is foundational for success in algebra and other advanced mathematics domains, and therefore a gateway to careers in science, technology, engineering, and mathematics (STEM; Sadler & Tai, 2007; Siegler et al., 2012). In 2008, the National Mathematics Advisory Panel (NMAP) identified improving fractions performance as a national priority. The Common Core State Standards for Mathematics (CCSS-M, 2010) reflect this goal, with an emphasis on rational number content in the late elementary and middle school grades. Despite increased recognition of the importance of fractions learning, however, this critical content remains problematic for many students. Data from the National Assessment of Educational Progress indicate many eighth graders lack a basic understanding of fractions, despite repeated exposure to rational number content across several years of mathematics instruction. In recent assessments, only 49% of eighth graders correctly ordered the fractions 2/7, 1/2, and 5/9, while just 41% correctly answered a word problem requiring simple fraction computation (National Center for Education Statistics, 2021). Difficulties with fractions are especially pronounced for students with or at risk for math learning difficulties (MD; Jordan et al., 2017; Mazzocco & Devlin, 2008; Tian & Siegler, 2017).

Individual Differences in Fractions Knowledge

Proficiency with fractions requires both conceptual and procedural knowledge (Tian & Siegler, 2017). The literature to date suggests that conceptual understanding of fractions is foundational to procedural learning, including fraction computation and problem-solving (Jordan et al., 2017; Hecht & Vagi, 2010; National Mathematics Advisory Panel [NMAP], 2008). Conceptual knowledge involves developing a deep understanding of what fractions are, including that fractions represent a defined magnitude determined by the relationship between the numerator and denominator and are composed of unit fractional parts (NMAP, 2008). Students also must learn that salient properties of whole numbers do not necessarily apply to fractions, which have their own unique properties (e.g., density). For example, to reason about and compare fractions, students must understand the relationship between numerators and denominators, rather than treating them as separate whole numbers (Siegler & Pyke, 2013). Students who lack conceptual understanding of fractions, or rational number sense, often overgeneralize whole number concepts, leading to common errors working with fractions (Mazzocco & Devlin, 2008). In addition, to develop fractions proficiency, students must build understanding of multiple interpretations of fractions. Although fractions are most often interpreted as representing parts of a whole in the U.S. mathematics curriculum, they can also be understood as a point on a number line, division operation, quotient, or ratio, with each interpretation contributing to flexible understanding and use of fractions across a range of mathematical domains and problems (Siegler & Pyke, 2013; Wu, 2011).

The literature suggests that several aspects of fractions knowledge are particularly challenging for students with MD, including placing fractions on the number line, comparing and ordering fraction quantities, finding equivalent fractions, performing fraction calculations, and solving fraction word problems (Hecht & Vagi, 2010; Jordan et al., 2017; Mazzocco & Devlin, 2008; Siegler & Pyke, 2013). Fraction magnitude understanding appears to be a hallmark difficulty of students with MD as compared with peers without MD who experience difficulties learning fractions (Mazzocco & Devlin, 2008; Resnick et al., 2016; Siegler & Pyke, 2013). This is problematic because the understanding of fraction magnitudes is a key component of rational number sense and overall competence with fractions (Mazzocco & Devlin, 2008), critical to understanding fraction arithmetic, fraction problem solving, and other areas of advanced mathematics (Siegler & Pyke, 2013). It is therefore not surprising that conceptual understanding of fractions and magnitudes has emerged as a consistent, strong predictor of both later success with fractions and broader mathematics achievement (Jordan et al., 2013).

Research on fractions learning suggests that the late elementary and early middle school grades are a critical window for developing fraction understanding. Resnick et al. (2016) described distinct growth trajectories in fraction understanding across Grades 4 to 6, including a substantive group of students (42%) who benefited little from 3 years of formal fractions instruction and were far less likely to meet state mathematics standards at the end of sixth grade than their peers with greater fraction understanding. Other studies have found that students with MD enter middle school with lower fractions knowledge and skills and make slower progress than their peers across Grades 6 to 8, contributing to a substantial widening of achievement gaps between students with lower versus higher initial fractions knowledge during this period (Siegler & Pyke, 2013). Taken together, these findings suggest the importance of identifying and addressing gaps in foundational fraction knowledge before the end of sixth grade. Students who do not develop proficiency with fractions by early middle school will lack the prerequisite knowledge necessary to access grade-level mathematics content. These students are likely to experience ongoing and compounding difficulties in mathematics, since rational numbers are deeply embedded in advanced math content (Hwang et al., 2019).

Research on Fraction Interventions

Increased awareness of the importance of fractions knowledge for later mathematics success and poor national math achievement in recent years has sparked an emerging body of research on interventions targeting fractions content. Early efforts to improve fractions knowledge have yielded promising results, with studies finding significant positive gains across a range of fraction concepts and skills including computation (Braithwaite & Siegler, 2021; L. S. Fuchs et al., 2013; Jayanthi et al., 2021), equivalency (Hunt, 2014), and magnitude understanding (Braithwaite & Siegler, 2021; Dyson et al., 2020; L. S. Fuchs et al., 2013; Jayanthi et al., 2021). However, recent reviews highlight the gaps in the fractions intervention literature that must be addressed to improve fractions learning outcomes for all students (e.g., Hwang et al., 2019; Roesslein & Codding, 2019; Shin & Bryant, 2015). Specifically, interventions with a dual focus on conceptual understanding and procedural knowledge are needed to support struggling learners in developing fractions proficiency (Hwang et al., 2019; Roesslein & Codding, 2019). Given that middle school students with MD likely have more complex, multifaceted mathematics difficulties, comprehensive intervention approaches that connect key fractions content to earlier mathematics learning and prepare students for algebra may be necessary to adequately address skill gaps and support improved outcomes (Shin & Bryant, 2015).

Recognizing this need, we developed Promoting Algebra Readiness (PAR; Clarke et al., 2012), a Tier 2 sixth-grade fractions intervention. Pilot study results examining the overall impact of PAR supported its promise for improving fractions outcomes for sixth-grade students with MD (Clarke et al., 2020). Students who received the PAR intervention demonstrated significant, positive gains over at-risk peers who did not receive PAR on two proximal measures (Hedges’s g = 0.68–0.84), with non-significant positive outcomes on all remaining proximal and distal math achievement measures (Hedges’s g = 0.15–0.44).

Understanding Response Variation

In recent years, there have been calls in the field for researchers to pursue more nuanced understanding of the effectiveness of academic interventions, looking beyond main effects to better understand what programs work for whom and under what conditions (D. Fuchs & Fuchs, 2019; Miller et al., 2014). In response, there has been increased interest in conducting follow-up analyses to examine student-level characteristics that moderate intervention response, and better understand how these factors may be linked to differential and/or inadequate response to generally effective intervention programs. Identifying cognitive, behavioral, and academic student characteristics that moderate response to intervention is of particular interest, given the importance of student-level moderators within the context of multi-tiered systems of support (MTSS). These models are predicated upon understanding student response such that students are assigned to tiers of instruction, as well as specific interventions and supports within tiers, that are best aligned with their needs. A greater understanding of student-level factors associated with differential response to evidence-based intervention may ultimately inform more effective and efficient screening and instructional placement decisions based on the characteristics of individual at-risk students (Miller et al., 2014). For example, universal screening practices might be adjusted such that students at higher risk are placed directly into more intensive intervention (Al Otaiba et al., 2014; Lam & McMaster, 2014).

One student-level moderator that may be critical in understanding response variation is pre-intervention academic skill (D. Fuchs & Fuchs, 2019). To date, the evidence supporting initial mathematics skill as a predictor of differential response to mathematics intervention has been mixed, with some studies finding comparable treatment effects across the range of initial skill level (L. S. Fuchs et al., 2016, 2019); and others finding differential effects favoring students with lower (Clarke et al., 2019) or higher (Toll & Van Luit, 2013) initial skill. Toll and Van Luit (2013) found that a kindergarten math intervention was effective for students with higher early numeracy skills (between the 25th and 50th percentiles) prior to intervention, with no significant positive effects for students who entered intervention with early numeracy skills below the 25th percentile. In contrast, Clarke et al. (2019) found that students with lower pre-intervention mathematics skill benefited more from a kindergarten mathematics intervention than their peers with higher initial skill. L. S. Fuchs et al. (2019), however, found that initial mathematics skill did not moderate response to a first-grade mathematics computation intervention, with at-risk students across the range of initial skill demonstrating comparable gains.

The role of initial skill in predicting response to mathematics intervention in the later grades is even less clear. In the domain of rational numbers, a single study has empirically evaluated initial mathematics skill as a moderator of response to intervention. L. S. Fuchs et al. (2016) examined whether response to a fourth-grade fractions intervention was associated with pre-intervention whole number calculation skill. The authors found that, across the range of initial skill, at-risk students made comparable growth in fraction knowledge and computation and were equally likely to close achievement gaps with peers who were not at risk. Further research is needed to better understand the relation between initial skill and response to mathematics intervention, particularly in the later grades. Given the importance of fractions content and the necessity of remediating difficulties in a timely manner to prevent widening skill gaps, it is critical to understand student-level moderators of response to evidence-based fractions programs. Therefore, the purpose of this study was to examine initial mathematics skill as a moderator of response to PAR.

Importance to Pilot Work

Addressing student-level moderators of intervention response at the pilot stage of intervention development confers a unique advantage in that findings, while exploratory in nature, can inform research design decisions in more formal efficacy trials. Gaining insight, at this early stage, regarding the robustness and generalizability of an intervention may help to more clearly define the intervention’s intensity and target audience, allowing for better-informed decisions about specific populations of students that may be more likely to benefit from intervention. For example, if moderation analyses suggest greater benefit for students with lower initial skill, the intervention may be best suited for students with substantial knowledge and skill gaps (who may need more intensive, Tier 3 supports), whereas students with higher initial skill may be better served by an alternate program. Researchers and curriculum developers might also consider modifications to intervention content or delivery to increase the robustness of the program (e.g., incorporating in-program assessment to guide flexible instructional pacing). However, if gains are comparable for at-risk students across a wide range of initial skill, this lends support to the program’s use as a Tier 2 support appropriate for many at-risk students.

Purpose and Research Questions

More research on effective fractions intervention is needed to inform efforts to implement MTSS service delivery models in mathematics in the upper elementary and middle school grades. To address the need for evidence-based, comprehensive intervention programs targeting critical fractions content, we developed PAR, a Tier 2 sixth-grade intervention intended to increase conceptual understanding of and procedural fluency with fractions. Pilot study results supported the promise of PAR for improving fractions outcomes for sixth-grade students with MD. The authors found condition differences favoring PAR over the control condition on seven mathematics measures: EasyCBM (g = 0.44), Algebra Readiness Progress Monitoring (ARPM; g = 0.15), AIMSweb (g = 0.15), and PAR mastery tests for Strands 1 to 4 (g = 0.25, 0.84, 0.68, and 0.20, respectively), with statistically significant outcomes for the PAR Strand 2 and 3 mastery tests. To equip schools to meet the needs of all students, however, it is critical to better understand how student-level factors such as initial skill moderate intervention response. The present study sought to address this need through secondary analyses of the data from Clarke et al. (2020), with a focus on the following research question:

Research Question 1: Did students’ initial mathematics skill moderate response to the PAR intervention?

Method

This study analyzed pilot study data collected during the federally funded Project PAR development initiative (Clarke et al., 2012). A quasi-experimental design was used to evaluate the promise of the PAR intervention for improving math outcomes for sixth-grade students with or at risk for MD. Full methods for this study were previously published (Clarke et al., 2020).

Participants

Schools

Seven middle schools from four Oregon school districts participated in the study. Assignment of schools to the treatment (n = 4) or control (n = 3) conditions was non-random and based on schools’ ability to implement a daily, 45-min mathematics intervention with existing school personnel. Condition assignment at the school level was intended to minimize treatment diffusion across conditions.

Teachers

In schools assigned to the PAR condition, six certified teachers delivered PAR during an intervention class period, with some teaching multiple classes. Teachers were employed by the school districts and trained by the research team in PAR implementation. In control condition schools, students received business-as-usual instruction from three teachers. Two schools used an intervention class format similar to treatment group schools; one provided individual instruction to students at various times throughout the day. Treatment condition teachers, on average, had 9 years of teaching experience (including 7.5 years of experience teaching middle school mathematics and 1.5 years providing math intervention), whereas control group teachers averaged 10.67 years teaching experience (including 7.67 years of experience teaching middle school mathematics and 5 years providing math intervention). All teachers reported having taken graduate-level coursework in both mathematics and mathematics instruction.

Students

Within participating schools, sixth-grade students were considered at risk for MD, and therefore eligible for study participation, if they had scored below the 40th percentile on the state mathematics assessment in fifth grade the previous school year. This cut score was selected to ensure participating students represented a range of initial skill levels and were likely to benefit from supplemental intervention in preparation for algebra. After obtaining parental consent, the final sample included 110 students in the treatment group and 84 students in the control group. Over the course of the study, one student moved into each condition from classes or schools not participating in the study. In addition, eight students in the control group and 11 in the treatment group moved to non-participating schools or classes. Across conditions, participating students were predominantly White (85% of PAR students and 92% of control students), and 30% of PAR students and 31% of control students identified as Hispanic. Among students assigned to the PAR intervention, 8% were English language learners and 8% received special education services. In the control group, 12% were English language learners and 27% received special education services. Descriptive statistics for demographic variables by condition are reported in Table 1.

Table 1.

PAR Study Characteristics of Student Participants.

Student characteristics	PAR (n = 110) (%)	Comparison (n = 84) (%)
Male	41	52
Race
White	85	92
Black	5	2
Hawaiian/Pacific Islander	1	0
American Indian/Alaskan Native	1	1
Asian	3	1
Multiracial	4	4
Hispanic ethnicity	30	31
LEP	8	12
SPED	8	27

Note. PAR = Promoting Algebra Readiness; LEP = limited English proficiency, SPED = eligible for special education services.

Measures

Measures of mathematics proficiency were administered to students in the treatment and control groups during the fall (pretest) and spring (post-test) of sixth grade. Measures included a researcher-developed proximal assessment of skills taught during PAR (see Clarke et al., 2020), a distal measure of skills necessary for algebra success, a curriculum-based measure of computation skills, and a broad standardized measure of mathematics proficiency. Students’ scores on the statewide mathematics assessment, OAKS (Oregon Department of Education, n.d.), were reported to the research team as part of the screening process. Student demographic data were obtained from participating schools’ student databases. The full measurement net is presented in Clarke et al. (2020). The following measures were used in the present study:

AIMSweb Mathematics Computation

AIMSweb Mathematics Computation (M-COMP) is a brief, standardized, curriculum-based measure of fluency with basic facts; computation; and conversion of whole and rational numbers, integers, and exponents for students in Grades 1 to 8. The Grade 6 fall and spring benchmark forms were used to assess proficiency with grade-level mathematics content. Based on the design and purpose of the AIMSweb measures for use in screening for individual risk, the fall benchmark was used as a measure of initial skill in the present study. Fisher’s z-transformed reliability coefficients for M-COMP range from .82 to .90, split-half reliability ranges from .85 to .93, and coefficient alpha ranges from .82 to .91 (Pearson Education, 2010). Concurrent validity with the Group Mathematics Assessment and Diagnostic Evaluation is .76 at Grade 8 and .73 at Grade 3.

Algebra Readiness Progress Monitoring

The ARPM (Ketterlin-Geller et al., 2015) is a set of three brief, standardized, group-administered curriculum-based measures of knowledge and skills essential for success in algebra, including knowledge of number properties, quantity discrimination and comparison, and proportional reasoning. Students are given 3 min per measure to answer as many items as possible. Each measure consists of 20 to 25 multiple selection items. The ARPM has high internal consistency (ranging from .92 to .97 across measures). Rasch model fit statistics and item–total correlations are appropriate for measuring individual student progress (Ketterlin-Geller et al., 2015). ARPM was an outcome measure in the present study.

EasyCBM

EasyCBM Mathematics (Alonzo et al., 2016) is a standardized, computer-administered measure of mathematics proficiency for students in Grades K to 8. Items for each grade level are mapped to the CCSS-M domains. The measure is untimed and typically takes between 18 and 30 min to administer. EasyCBM Mathematics has high internal consistency, ranging from .92 to .95 in the middle school grades. Concurrent validity with the SAT-10 is .82 (Anderson & Donchik, 2016). EasyCBM was an outcome measure in the present study.

Procedures

Promoting Algebra Readiness

Promoting Algebra Readiness is a 93-lesson, Tier 2 intervention for students in upper elementary and early middle school grades targeting rational number concepts and skills spanning four content strands: (a) multiplication and division of whole numbers, (b) fractions as numbers, (c) addition and subtraction of fractions, and (d) multiplication and division of fractions. PAR is aligned with CCSS-M (2010) Grade 6 objectives within the Number System domain focused on rational number content. PAR also targets prerequisite knowledge and skills that are necessary to develop mastery of Grade 6 rational number objectives by integrating content aligned with earlier grade-level standards (e.g., those focused on rational number knowledge or conceptually related whole number knowledge, such as multiplication and division; Clarke et al., 2020) throughout. This approach was intended to ensure that PAR meets the needs of students with MD, who may lack the foundational knowledge to demonstrate mastery of grade-level rational number content. The PAR instructional approach is based on principles of explicit and systematic instruction aligned with best practice guidelines for teaching students struggling in mathematics (L. S. Fuchs et al., 2021). Key features include explicitly stated instructional objectives, activities to activate relevant background knowledge (i.e., entry tasks to link daily objectives to earlier learning), teacher modeling of new skills and concepts, guided practice with instructional scaffolding faded over time, and independent practice opportunities with guidelines for differentiation based on student mastery. PAR utilizes a range of visual representations aligned with the concrete, semi-concrete, abstract instructional sequence (L. S. Fuchs et al., 2021) to support development of deep conceptual understanding. For example, to help students understand that all fractions are composed of unit fractional parts, students initially work with linking cubes representing copies of the unit fraction, then number line models showing unit fraction iteration, before using this knowledge to solve problems involving addition and subtraction of fractions with like denominators at the abstract level.

The PAR intervention was delivered during math intervention class periods; groups ranged in size from 10 to 24 students (m = 16). All but two intervention groups completed the entire PAR intervention; the remaining groups did not complete Strand 4 due to scheduling constraints. Intervention supports were provided in addition to core mathematics instruction. Core mathematics curricula implemented in treatment schools included Common Core/Eureka Mathematics, College Preparatory Mathematics, and teacher-created instructional materials.

Control

The control group received mathematics intervention supports according to typical school practices, in addition to core math instruction. In two control schools, intervention supports were delivered during math intervention class periods, whereas the remaining control school provided supplemental math instruction to individual students at various times throughout the school day. Intervention programs delivered to control group students included Core Focus, Engage New York, and teacher-created materials. Core mathematics curricula implemented in control schools included College Preparatory Mathematics, Envisions, and Engage New York. Control group teachers reported regular use of various effective instructional practices, including concrete and semi-concrete representations of mathematics concepts (e.g., number lines, place value models), other visual aids (e.g., graphic organizers), teacher verbalizations of mathematical reasoning (or think alouds), opportunities to verbalize mathematical reasoning, opportunities for peer interaction, and guided practice.

Data collection

Prior to PAR implementation, measures of mathematics proficiency (including EasyCBM Grade 6 fall benchmark, AIMSweb M-COMP Grade 6 fall benchmark, and ARPM) were administered to students in both the treatment and control conditions by trained project staff. Administration order was counterbalanced to address potential practice effects. A make-up testing session was offered for students absent on the initial pretesting day. At the end of Strand 4, the EasyCBM Grade 6 spring benchmark, AIMSweb M-COMP Grade 6 spring benchmark, and ARPM were administered. Again, administration order was counterbalanced, and a make-up testing session was offered to students who were absent on the day of post-testing. Prior to administering assessments, research staff received training to ensure reliable assessment administration. Assessors were required to complete fidelity checklists when administering assessments to ensure standardized administration procedures were followed. Refresher training was provided prior to post-test data collection.

Professional development and coaching

Teachers in the treatment condition were trained to implement the PAR intervention by project staff. Prior to PAR implementation in the fall, teachers received 6 hr of professional development focused on Strands 1 and 2. In the winter, teachers received an additional 6 hr of training focused on Strands 3 and 4. PAR trainers were educators experienced in design and delivery of mathematics interventions who had been involved in developing the PAR intervention. Training sessions focused on the content and delivery of PAR. Trainers modeled lessons and activities and provided opportunities for teachers to practice and receive feedback on delivery of lesson components. Trainers also provided coaching support throughout the study, visiting each intervention class twice per strand to support high-quality implementation of PAR. During these visits, coaches provided interventionists with feedback on implementation of key components of the PAR intervention and modeled skills interventionists did not implement fully or correctly. In addition, trainers answered interventionist questions over email between visits.

Fidelity of implementation

Research staff with extensive knowledge of intervention design and content assessed the extent to which PAR was implemented as intended via direct observation. Each PAR class was observed twice per strand, or approximately once per month, for a total of approximately 9% of sessions observed (8 out of approximately 93 sessions). Observers used a researcher-developed scale to rate completion and quality of key intervention components and activities (e.g., entry task, guided practice, etc.). Completion of each component was rated on a 3-point scale (1 = not performed to 3 = complete); whereas quality was rated on a 4-point scale (1 = low quality to 4 = high quality). Total scores for level and quality of completion were calculated as the mean across individual item ratings for each scale. Observation data indicated that PAR was implemented with a high degree of fidelity. PAR interventionists generally completed most planned activities for the session (M = 2.5, SD = 0.3) and implemented activities with moderate to high quality (M = 2.8, SD = 1.0) during observations. Intraclass correlation coefficients (ICCs) indicated substantial agreement across observers, with ICCs exceeding .80 for total scores for level and quality of completion.

Statistical Analysis

We conducted analyses designed to address our research question about differential response to PAR based on initial AIMSweb scores. Clarke et al. (2020) examined overall effects of the PAR intervention on mathematics achievement with mixed-model Time × Condition analysis (Murray, 1998) designed to account for the intraclass correlation associated with students nested within schools, the level of assignment to study condition. The analysis tested for differences between conditions on change in outcomes from pretest (T₁) to post-test (T₂), with gains for individual students clustered within schools. The model included time, condition, and the Time × Condition interaction, with time coded 0 at T₁ and 1 at T₂ and condition coded 0 for control and 1 for PAR. In this study, we examined whether initial mathematics skill based on MCOMP scores predicted differential response to the PAR intervention. Therefore, we expanded the statistical model to include pretest MCOMP as a student-level predictor variable and its interaction with the condition, time, and the Time × Condition terms.

Differential response to intervention implies that the condition difference depends on a moderator (e.g., pretest MCOMP scores). For a statistically significant moderator, the difference between the two conditions will be larger (or smaller) at higher (or lower) levels of the moderator. To detect these differences, we estimated the difference between conditions and confidence bounds at multiple points along the moderator (Jaccard & Turrisi, 2003). We used these estimates to compute the regions of statistical significance based on the confidence intervals and graphed the results with the method recommended by Preacher et al. (2006) for interpretation. The graphs depict the condition effect and its 95% confidence intervals across the range of moderator scores. Regions in which both confidence bounds exclude zero value for condition differences are interpreted as regions of significant intervention effects.

We fit statistical models to our data using SAS PROC MIXED version 14.2 (SAS Institute, 2016) with maximum likelihood estimation. Maximum likelihood estimation with all available data produces potentially unbiased results even in the face of substantial missing data, provided the missing data were missing at random (Schafer & Graham, 2002). In the present study, missing data do not likely represent a meaningful departure from the missing at random assumption, meaning that missing data did not likely depend on unobserved determinants of the outcomes of interest (Little & Rubin, 2002) or negatively affect internal validity. Missing data (10%) is explained by student-level attrition, which did not significantly differ by condition. The effect of attrition on outcomes also did not vary by condition (Clarke et al., 2020).

The models assume independent and normally distributed dependent variables. We addressed the first, more important assumption (van Belle, 2008) by explicitly modeling the multilevel nature of the data. The data also did not markedly deviate from univariate normality; skewness and kurtosis fell within ±0.9 for the outcomes evaluated in this study.

Results

The potential for differential attrition and the main effects for the PAR intervention were presented in Clarke et al. (2020).

Clarke et al. (2020) found non-significant positive main effects of PAR on pretest to post-test gains in EasyCBM scores (Hedges’s g = 0.44) and ARPM scores (g = 0.15). Table 2 presents the tests of differential response to PAR on these outcomes as a function of pretest MCOMP scores. These tests of moderation suggest that intervention differences were not significantly moderated by pretest MCOMP scores for either outcome: EasyCBM (t₃₄₀ = −0.42, p = .6747) or ARPM (t₃₅₀ = 0.80, p = .4259).

Table 2.

Moderation Results From Mixed Time × Condition Analysis of Change in Student Outcomes.

Effect or statistic	EasyCBM	ARPM
Fixed effects
Intercept	24.5****(1.5)	33.6****(1.4)
Time	−0.3 (1.2)	2.8 (1.8)
Condition	1.4 (2.0)	0.2 (1.9)
Time × Condition	3.2* (1.5)	1.2 (2.3)
Pretest	0.7****(0.1)	1.0****(0.3)
Pretest × Condition	−0.2 (0.2)	−0.2 (0.4)
Pretest × Time	0.1 (0.1)	0.1 (0.4)
Pretest × Time × Condition	−0.1 (0.2)	0.4 (0.5)
Variances
School-level intercept	4.3 (3.2)	0.0 (.)
School-level gains	1.3 (1.1)	0.0 (.)
Individual-level intercept	13.2****(2.3)	42.9****(12.8)
Residual (error)	13.4****(1.5)	118.9****(12.8)
Pretest × Time × Condition effects
p value	.6747	.4259
Degrees of freedom	340	350

Note. Table entries show parameter estimates with standard errors in parentheses except for p values and degrees of freedom. Pretest AIMSweb scores were centered at the mean. Bounded = variances were constrained to zero to achieve model fit. ARPM = Algebra Readiness Progress Monitoring.

p < .10. **p < .05. ***p < .01. ****p < .001.

Figure 1 presents the condition differences in outcome across the range of pretest MCOMP scores. The graphs show the estimated difference between conditions (dark line) with the 95% confidence intervals (light lines) across the range of pretest MCOMP scores (Preacher et al., 2006). Zero on the vertical axis represents no difference between conditions. The vertical lines within the graph represent sample percentiles (5th, 25th, 50th, 75th, and 95th).

Figure 1.

Differential effects of PAR on the (A) EasyCBM and (B) ARPM scores based on pretest AIMSweb scores.

Graph A shows results for the EasyCBM outcome. The vertical lines show that about 50% of the students had pretest MCOMP scores below 11.0, 25% below 8.0, and 5% below 4.0. The confidence bounds exclude zero at values of 4.7 to 12.7, or for students scoring between the 9th and 65th percentile for the sample on the pretest MCOMP assessment. This suggests that students scoring in this range (corresponding to the 7th–31st percentile for the MCOMP national normative sample) achieved greater gains on the EasyCBM when exposed to PAR than students in the control condition schools. In Graph B, for the ARPM outcome, the confidence bounds included zero across the full range of pretest scores. This suggests that the effect of PAR was not statistically significant across the range of pretest MCOMP scores.

Discussion

The purpose of this study was to explore whether initial mathematics skill predicted differential response to a Tier 2 sixth-grade fractions intervention. Secondary analyses yielded no statistical evidence for moderation by initial skill, suggesting that the impact of the PAR intervention was comparable for at-risk students with a wide range of pre-intervention mathematics skill profiles. Given the mixed results of other studies examining moderation of response to mathematics intervention by initial skill, we did not have a strong hypothesis as to whether response to the PAR intervention would vary by student initial skill level. However, results of the present analyses do align with those reported by L. S. Fuchs et al. (2016), who found that initial skill did not moderate response to a Tier 2 fourth-grade fractions intervention. The present findings are based on an underpowered pilot study, which yielded nonsignificant positive outcomes on all measures used in these analyses, and should thus be interpreted with caution. The remainder of this paper discusses the results within the context of implications for designing and evaluating fraction interventions, current study limitations, and areas for future research.

These findings raise several interesting points to consider as the field aims to better understand response variation in intervention work, particularly in more advanced mathematical domains. Clear patterns regarding the role of initial skill in predicting response to mathematics intervention have not yet emerged, with some studies finding differential gains favoring students with lower (Clarke et al., 2019) or higher (Toll & Van Luit, 2013) initial skill, and others, like the present study, finding no evidence of moderation (L. S. Fuchs et al., 2016, 2019). These divergent findings, in conjunction with the relative paucity of research examining student-level moderators of response to mathematics intervention, preclude clear conclusions regarding the role of initial skill in mathematics. Discrepant results across studies may be partially accounted for by key differences in the targeted content and/or instructional design of the interventions themselves. The present study, like that by L. S. Fuchs et al. (2016), found no evidence that students’ broad initial mathematics skill moderated response to fractions intervention. Given that fractions are difficult content for many students to master, it is likely that at-risk students included in both studies were in need of additional fractions support, regardless of initial skill in mathematics more broadly. However, evidence of moderation by initial skill in prior studies may reflect a poorer match between student need and intervention content for students on either the lower or higher end of the pre-intervention mathematics skill spectrum. The scope of PAR is also relatively comprehensive, addressing a wide range of prerequisite skills that students struggling in mathematics may not have mastered by sixth grade. This in-depth, comprehensive content coverage may have helped to compensate for difficulties associated with lower initial skill, mitigating potential greater benefit of PAR for students with higher initial skill.

Disparate findings regarding the role of initial skill in predicting response to mathematics intervention may also reflect differences in intensity of intervention. Implementation of PAR in the present pilot study was generally aligned with conceptualizations of what Tier 2 supports should look like, in that a standard protocol intervention was provided to groups of at-risk students who had not responded adequately to core mathematics instruction. However, evidence of moderation by initial skill in prior studies may reflect variations in instructional intensity that resulted in a better match between student need and intervention design for students with lower (or higher) initial mathematics skill. For example, Clarke et al. (2019) suggested that the greater gains observed for at-risk students with lower initial skill may reflect the intensive design of their kindergarten mathematics intervention, which may have been a better match for students with more significant skill gaps. Future intervention research examining student-level moderation may clarify how initial skill and intervention intensity interact to predict response variability. For example, a more formal evaluation of the PAR intervention may compare large group implementation like that of the present study with the more intensive, small-group format more typical of the mathematics intervention literature to date.

Alternatively, divergent findings regarding the role of initial skill in predicting response to mathematics intervention may reflect differences in how pre-intervention skill level was operationalized across studies. In this study, initial mathematics skill was defined as performance on a measure of general mathematics competency that is typically used as a screener for risk for MD and is distal to the content of the PAR intervention. Similarly, in the only other study examining initial skill as a moderator of fractions intervention response to date, L. S. Fuchs et al. (2016) defined initial skill as performance on a measure of whole number calculation skill that was distal to content taught in their fourth-grade fractions intervention. Operationalizing initial skill with a more proximal measure of fraction knowledge may have yielded a very different pattern of results. It is plausible that individual students’ competence with fractions content taught in fourth and fifth grade may be a stronger predictor of response to PAR than general mathematics performance, such that students on the higher (or lower) end of the pre-intervention skill distribution might have benefited more from receiving PAR. Future research in this area is needed to better understand whether differential operationalization of initial mathematics skill partially accounts for the discrepant findings to date.

Limitations and Future Directions

In addition to the limitations noted above related to study power, one significant limitation is that the present study utilized a business-as-usual control condition without documenting key components of the counterfactual, such as the instructional architecture, content focus, dosage, and fidelity of implementation of typical district mathematics intervention activities (especially in the control school that did not deliver intervention during a dedicated class period). The lack of a consistent pattern across studies examining initial skill as a moderator of mathematics outcomes to date highlights the need to thoroughly document the counterfactual to enhance understanding of intervention effects (Lemons et al., 2014). Doing so can help to clarify differences between conditions (e.g., intervention dosage, instructional quality, and design features). Although control group teachers were asked to self-report their use of effective mathematics instructional practices, including formal observations to document the presence of key instructional design features (e.g., teacher models, guided practice, frequent opportunities to respond) and the content and quality of instruction in control classrooms would have helped to contextualize both main effects of PAR and the present findings. Future research should include efforts to systematically document instruction provided to the control group and/or control for variables such as intervention dosage and fidelity of implementation that may confound results.

In addition, results are inherently limited to the specific intervention and population studied here. Additional research is needed to better understand the impact of PAR, and how it relates to students’ initial mathematics skill, across diverse geographic locations and samples. Finally, moderation analyses are just one approach to evaluate the relation between pre-intervention skill and student outcomes. Future research in this area should evaluate initial skill as a moderator of intervention response from multiple angles. In particular, it is recommended that studies of initial skill examine post-intervention achievement gaps between students with higher and lower pre-intervention skills to better contextualize moderation effects. L. S. Fuchs et al. (2016) found that at-risk students with lower initial math skill were less likely to perform comparably to not-at-risk peers following intervention than those who began intervention with higher initial skill, despite finding that initial skill was not a moderator of responsiveness. As this finding suggests, examining post-intervention achievement levels compared with not-at-risk peers may identify a larger proportion of students as demonstrating inadequate response to intervention and requiring ongoing or more intensive supports. Additional research with varied methodological approaches is needed to better understand the complex relation between initial mathematics skill and response to fractions intervention.

Practical Implications

Intervention research that explores student-level moderators of responsiveness has the potential to inform more effective and efficient educational decision-making by providing insight regarding how individual student characteristics may affect the “fit” of an intervention for a specific student (Miller et al., 2014). This understanding may ultimately allow educators to maximize the likelihood of positive response by choosing intervention programs and strategies that are likely to be a good fit based on the characteristics of at-risk students at the individual or group level. As future research clarifies the role of initial skill in predicting response to mathematics intervention, practitioners may use this knowledge in several key ways. Educators seeking to narrow achievement gaps in mathematics may wish to implement math intervention programs associated with greater gains for students with lower initial skill. Alternatively, practitioners might employ initial skill as an additional consideration when assigning at-risk students to various interventions, selecting students with initial skill levels within a range that the intervention is likely to be most effective for based on the literature. Students’ initial skill level might also be used to justify acceleration within the response to intervention framework, for example, by providing students who are less likely to respond to a standardized Tier 2 intervention with more intensive, individualized supports right away.

Exploring questions such as these at the pilot stage may provide invaluable insights to inform more formal efficacy research. In this study, the lack of moderation of responsiveness by initial skill suggests that targeting at-risk students with a wide range of initial mathematics skill levels is a reasonable design choice at the more formal efficacy trial stage. These findings also lend support to the authors’ conceptualization of PAR as a supplemental, Tier 2 intervention designed to meet the needs of many students at risk for mathematics difficulties. However, upon finding a lack of moderation by initial skill, it would also be reasonable to consider modifications to intervention content or delivery that may better meet the needs of, and improve outcomes for, students with lower initial math skill. For example, Clarke et al. (2020) suggest the possibility of placing students within intervention programs based on initial skill such that students complete the lessons that align with their pre-intervention needs. This may be especially relevant for broad intervention programs like PAR that include foundational content taught in earlier grades. Students who demonstrate high levels of mastery of this foundational content may benefit from skipping early review lessons to focus on content that better aligns with their current level. The shear breadth of fractions content may also make fractions interventions like PAR well-suited for an adaptive or smart research design (Almirall et al., 2018) at the efficacy stage. For example, students with higher initial skill might be grouped together and accelerated through foundational content, while those who struggle with targeted content may be grouped and provided additional practice and review opportunities.

The present findings suggest that PAR outcomes are comparable regardless of students’ initial skill. It is therefore likely that skill gaps between at-risk students with lower versus higher initial math skill may persist following implementation of PAR, despite its promise as a Tier 2 intervention. To maximize outcomes for all students with MD, it may be important to evaluate alternate models of fractions intervention. Standardized Tier 2 intervention programs are a sound approach to address low fractions performance in schools, but supplements to core instruction and class-wide fractions intervention provide reasonable alternatives to support the large proportion of students struggling with fractions content. In addition, methods for intensifying PAR to better meet the needs of students with low initial skill (e.g., smaller group size, flexible pacing, supplemental supports for students with the lowest math skills) are also important to consider. Future research in this vein may help to clarify how PAR and other fractions intervention approaches may operate within a broader service-delivery framework, advancing the field’s understanding of how best to support all students in acquiring critical fractions knowledge.

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 research reported here was supported by the Institute of Education Sciences, U.S. Department of Education, through Grant number R324A120115 to the Center on Teaching and Learning. The opinions expressed are those of the authors and do not represent the views of the Institute or the U.S. Department of Education.

ORCID iDs

Nancy Nelson

Keith Smolkowski

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