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Proficiency with Number Concepts and Operations: Replicating the Efficacy of a First-Grade Mathematics Intervention

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Title: Proficiency with Number Concepts and Operations: Replicating the Efficacy of a First-Grade Mathematics Intervention
Language: English
Authors: Christian T. Doabler, Ben Clarke, Jessica E. Turtura, Marah Sutherland (ORCID 0000-0002-6108-8515), Jenna A. Gersib (ORCID 0000-0002-1270-1889), Taylor Lesner, Madison Cook, Georgia L. Kimmel (ORCID 0000-0001-7649-3073), Keith Smolkowski, Derek Kosty
Source: Grantee Submission. 2023.
Peer Reviewed: Y
Page Count: 32
Publication Date: 2023
Sponsoring Agency: Institute of Education Sciences (ED)
Contract Number: R324A090341
R324A160046
Document Type: Reports - Research
Education Level: Early Childhood Education
Elementary Education
Grade 1
Primary Education
Descriptors: Number Concepts, Mathematics Instruction, Grade 1, Elementary School Students, Intervention, Outcome Measures, Concept Formation, Teaching Methods, Small Group Instruction, Comparative Analysis, Mathematics Tests
Geographic Terms: Massachusetts (Boston)
DOI: 10.1177/00222194231209017
Abstract: Conceptual replications are part and parcel of education science. Methodologically rigorous conceptual replication studies permit researchers to test and strengthen the generalizability of a study's initial findings. The current conceptual replication sought to replicate the efficacy of a small-group, first-grade mathematics intervention with 240 first-grade students with mathematics difficulties in a new geographical region. Participating students were randomized into one of three conditions: (a) 2:1 mathematics intervention group, (b) 5:1 mathematics intervention group, or (c) business-as-usual instruction. Relative to the original study, findings from the replication varied. When comparing the treatment groups to the control, results suggested positive effects on all outcome measures, including a follow-up assessment administered one year later. However, differences between the two treatment groups based on group size were not found in the mathematics outcome measures. Both groups also received commensurate levels of observed instructional interactions. Implications for unpacking contextual differences between original research and their replications as well as using future research to explore the quantity and quality of instructional interactions as ways to explain variation in findings of group size are discussed. [This is the online first version of an article published in "Journal of Learning Disabilities."]
Abstractor: As Provided
IES Funded: Yes
Entry Date: 2024
Accession Number: ED641777
Database: ERIC
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  Value: <anid>AN0177928058;led01jul.24;2024Jun19.05:56;v2.2.500</anid> <title id="AN0177928058-1">Proficiency With Number Concepts and Operations: Replicating the Efficacy of a First-Grade Mathematics Intervention </title> <p>Conceptual replications are part and parcel of education science. Methodologically rigorous conceptual replication studies permit researchers to test and strengthen the generalizability of a study's initial findings. The current conceptual replication sought to replicate the efficacy of a small-group, first-grade mathematics intervention with 240 first-grade students with mathematics difficulties in a new geographical region. Participating students were randomized into one of three conditions: (a) 2:1 mathematics intervention group, (b) 5:1 mathematics intervention group, or (c) business-as-usual instruction. Relative to the original study, findings from the replication varied. When comparing the treatment groups to the control, results suggested positive effects on all outcome measures, including a follow-up assessment administered one year later. However, differences between the two treatment groups based on group size were not found in the mathematics outcome measures. Both groups also received commensurate levels of observed instructional interactions. Implications for unpacking contextual differences between original research and their replications as well as using future research to explore the quantity and quality of instructional interactions as ways to explain variation in findings of group size are discussed.</p> <p>Keywords: replication; explicit and systematic instruction; number sense; instructional interactions; mathematics difficulties</p> <p>In [<reflink idref="bib37" id="ref1">37</reflink>], Ioannidis aptly observed, "It is rigorous replication that guarantees science" (p. 2). Replication research, both successful and failed attempts ([<reflink idref="bib41" id="ref2">41</reflink>]), can quell false discoveries and advance a reliable scientific knowledge base ([<reflink idref="bib47" id="ref3">47</reflink>]). In the field of education, replicating causal effects of treatments or interventions across methodologically rigorous studies can help establish evidence-based practices and improve the outcomes of students with or at risk for learning disabilities ([<reflink idref="bib22" id="ref4">22</reflink>]).</p> <p>The complex and dynamic nature of schools and classrooms raises challenges for conducting direct replications. For example, reproducing contextual variables from the original study, such as the quantity and quality of mathematics instructional interactions facilitated in the counterfactual, makes exact duplication near impossible. Another challenge is intervening with a second sample of participants who are identified from the same population involved in the initial research ([<reflink idref="bib22" id="ref5">22</reflink>]). Given these difficulties, scholars tend to conduct conceptual replications ([<reflink idref="bib44" id="ref6">44</reflink>]). Such studies enable the field to determine the degree to which findings obtained in initial research generalize across settings, conditions, and participants. Conceptual replications typically differ from an initial study by only one or two contextual dimensions. These planned variations are important because they can promote corroboration and increase the generalizability of findings garnered by original research. Conceptual replications also hold important value even when results differ from initial studies. For example, researchers can gain information about variations in treatment response and the different factors considered critical for program implementation. The current study conducted a conceptual replication of a first-grade mathematics intervention in classrooms from a different geographical region than the original study ([<reflink idref="bib15" id="ref7">15</reflink>]).</p> <hd id="AN0177928058-2">State of Replication Research in the Field of Education</hd> <p>In recent years, major funding agencies have taken significant strides at mitigating the publication and funding bias that has plagued replication research in the field of education ([<reflink idref="bib10" id="ref8">10</reflink>]). In [<reflink idref="bib48" id="ref9">48</reflink>], the National Science Foundation (NSF) and the Institute of Education Sciences (IES) released a joint-led report that offered a set of guidelines for submitting proposals to conduct replication studies. In broad terms, the NSF-IES guidelines emphasized that proposals make plain how and to what extent proposed replication studies will (a) advance understanding of an intervention's prior treatment effects; (b) include any planned variations from the original research; (c) offer safeguards to ensure objectivity; and (d) document contextual factors of the proposed study to inform future replications, such as the intensity and quality of instruction delivered in the control condition.</p> <p>To illustrate the replication guidelines, the [<reflink idref="bib48" id="ref10">48</reflink>] report offered three high-quality examples of conceptual replications supported through NSF and IES awards. The report included these three studies because they represented models for designing, conducting, and reporting conceptual replication studies. Of the three studies, one focused on early reading instruction while the other two investigated early mathematics intervention. Here, we briefly review the two mathematics studies given their relevance to the current study.</p> <p>The first showcased study sought to replicate the impact of Building Blocks, a preschool mathematics intervention focused on supporting children's spatial thinking. In the original study, [<reflink idref="bib52" id="ref11">52</reflink>] reported a large effect (<emph>g</emph> = 0.85) when working with 65 preschool children. To understand the extent to which findings behind the Building Blocks program generalized to a more diverse sample of preschool children, [<reflink idref="bib19" id="ref12">19</reflink>] conducted a conceptual replication study. The replication differed from the initial study on two contextual factors. First, the replication study more than doubled the sample size from the initial study, intervening with over 200 preschool children. In addition to scaling up the sample size, [<reflink idref="bib19" id="ref13">19</reflink>] also conducted the replication study in 25 Head Start and state preschool classrooms in California and New York, reporting similar effects (<emph>g</emph> = 0.62) to the initial study ([<reflink idref="bib52" id="ref14">52</reflink>]).</p> <p>The second example highlighted in the [<reflink idref="bib48" id="ref15">48</reflink>] replication report focused on findings from the ROOTS efficacy project ([<reflink idref="bib13" id="ref16">13</reflink>]). The ROOTS program is a small-group (Tier 2) kindergarten mathematics intervention designed to support the development of early number sense among kindergarten students with or at risk for mathematics difficulties (MD). The ROOTS efficacy project defined students with or at risk for MD as performing at or below the 35th percentile on a standard mathematics assessment. Critical to the program of research behind the ROOTS intervention was a conceptual replication study that varied from the original research in terms of geographical location ([<reflink idref="bib26" id="ref17">26</reflink>]). The original study occurred in suburban districts in the northwestern region of the United States, while the replication study was conducted in urban districts from the northeast.</p> <p>To test the impact of the ROOTS intervention, the original study ([<reflink idref="bib17" id="ref18">17</reflink>]) and the replication ([<reflink idref="bib26" id="ref19">26</reflink>]) randomly assigned students within classrooms to one of two treatment conditions or a business-as-usual control condition. One treatment group provided ROOTS in a format with a 2:1 student–teacher ratio, while the other offered the intervention in a 5:1 student–teacher ratio. The original study and the replication involved 290 and 319 at-risk kindergarten students, respectively. When comparing differences between ROOTS groups and the control condition, significant effects on three measures were replicated in the latter study. These outcome measures included the ROOTS Assessment of Early Numeracy Skills ([<reflink idref="bib26" id="ref20">26</reflink>]), Assessing Student Proficiency in Early Number Sense (ASPENS; [<reflink idref="bib18" id="ref21">18</reflink>]), and the Test of Early Mathematics Ability–Edition 3 (TEMA-3; [<reflink idref="bib32" id="ref22">32</reflink>]). However, unlike the original research, the replication reported an impact on two distal outcome measures: Number Sense Brief ([<reflink idref="bib40" id="ref23">40</reflink>]) and the Stanford Early School Achievement Test ([<reflink idref="bib36" id="ref24">36</reflink>]). [<reflink idref="bib26" id="ref25">26</reflink>] conjectured that the stronger treatment effects found in the replication were likely due to intervention timing. The onset of the ROOTS intervention in the replication occurred nearly 2 months earlier in the school year than the original study.</p> <p>Collectively, the studies conducted by [<reflink idref="bib19" id="ref26">19</reflink>] and [<reflink idref="bib26" id="ref27">26</reflink>] represent high-quality examples of conceptual replications. Both studies employed rigorous research designs to examine whether the effects of their mathematics interventions were generalized across different settings and participating samples. While [<reflink idref="bib19" id="ref28">19</reflink>] and [<reflink idref="bib26" id="ref29">26</reflink>] successfully reproduced the findings of their original studies, it is important to note that both research teams replicated their own work. Research suggests the likelihood of findings being reproduced increases with the presence of author overlap between replications and initial studies ([<reflink idref="bib43" id="ref30">43</reflink>]).</p> <hd id="AN0177928058-3">Replication and Treatment Intensity</hd> <p>In addition to bolstering evidence of treatment effects, replication research also offers additional opportunities to ascertain the treatment intensity of interventions. Researchers often operationalize treatment intensity as an alterable variable that can be purposefully manipulated to maximize student learning and obtain an optimal level of instruction ([<reflink idref="bib54" id="ref31">54</reflink>]). The current study manipulated group size to increase the treatment intensity for at-risk learners. Our program of research has focused on group size because the instructional format in which interventions are delivered is key to differentiating instruction and thus meeting the needs of students with MD. Findings from research conducted in the elementary grades suggest that reduced group size represents an opportune venue for facilitating high-quality instructional interactions between teachers and students, and among students ([<reflink idref="bib27" id="ref32">27</reflink>]; [<reflink idref="bib30" id="ref33">30</reflink>]; [<reflink idref="bib31" id="ref34">31</reflink>]). For example, in a small-group setting, teachers can offer frequent opportunities for individual students and the group at large to verbalize or use concrete models to demonstrate their mathematical understanding and thinking. Despite the theoretical value of group size, the field's understanding of group size in mathematics intervention research is still in its infancy. Therefore, the current replication was conducted to continue a broader investigation of group size and its role in mathematics interventions for students with or at risk for MD.</p> <p>Two recent meta-analyses have synthesized the differential effects of group size on student mathematics outcomes. [<reflink idref="bib49" id="ref35">49</reflink>] analyzed 50 early numeracy intervention studies targeting preschool, kindergarten, and first-grade students, including students with or at risk for MD. Interventions were coded on several study characteristics, including (a) instructional arrangement (i.e., group size), (b) intensity, (c) instructional approach, (d) numeracy content, (e) intervention agent, and (f) methodological quality. Of note, [<reflink idref="bib49" id="ref36">49</reflink>] found group size was the only study characteristic that accounted for a significant proportion of variance in the treatment effects (<emph>R</emph><sups>2</sups> = 0.37). When examining group size specifically, findings suggested treatments that offered instruction in small groups of three to five students had moderate to large effects (<emph>g</emph> = 0.76). The effect of interventions delivered in one-on-one formats was moderate (<emph>g</emph> = 0.65).</p> <p>More recently, [<reflink idref="bib39" id="ref37">39</reflink>] conducted a meta-analysis of 39 Tier 2 mathematics intervention studies involving students in prekindergarten through twelfth grade with or at risk for MD. The bulk of the examined sample was comprised of studies conducted in the elementary grades. In addition to investigating the overall treatment effects of the 39 studies, Jitendra and colleagues also explored the extent to which different study characteristics, including intervention duration and type, group size, implementer, and type of outcome measure, predicted students' differential response. Regarding group size, Jitendra and colleagues found the use of intervention formats with two or three students had positive and significant moderating effects, with an effect increase of 0.29 (Hedges' <emph>g</emph>) above the average adjusted effect of 0.46. However, unlike [<reflink idref="bib49" id="ref38">49</reflink>], the results in [<reflink idref="bib39" id="ref39">39</reflink>] indicated that one-to-one intervention formats did not moderate intervention effects. According to [<reflink idref="bib39" id="ref40">39</reflink>], the moderating effects of small-group instruction with two or three students may have been attributable to the fact that, relative to one-to-one formats, these small-group interventions foster more frequent instructional interactions between teachers and students, and among students.</p> <p>Given the mixed findings of [<reflink idref="bib49" id="ref41">49</reflink>] and [<reflink idref="bib39" id="ref42">39</reflink>], we investigated group size within a broader efficacy trial to further understand the treatment effects of a first-grade mathematics intervention on student mathematics outcomes. The current study also manipulated group size to explore the extent to which it impacts the quantity and quality of the instructional interactions facilitated during the early mathematics intervention. Prevailing research views instructional interactions as a critical element of academic development ([<reflink idref="bib50" id="ref43">50</reflink>]) and a defining mechanism to intensify instruction ([<reflink idref="bib30" id="ref44">30</reflink>]). Here, we operationally define instructional interactions as a dynamic interplay of a teacher offering clear examples and demonstrations of new mathematics content; students working independently and with their peers; and a teacher providing students with timely, specific academic feedback. Instructional interactions, when facilitated well, offer students opportunities to interact with their teacher and peers around foundational mathematics content, such as numbers and operations ([<reflink idref="bib24" id="ref45">24</reflink>]).</p> <hd id="AN0177928058-4">Purpose of the Study</hd> <p>The purpose of the current research was to conduct a conceptual replication study within a larger federally funded efficacy project ([<reflink idref="bib14" id="ref46">14</reflink>]) that tested the treatment effects of the Fusion mathematics intervention across four cohorts of first-grade students with or at risk for MD. Fusion is a 60-lesson, Tier 2 first-grade mathematics intervention designed to build mathematical proficiency. [<reflink idref="bib15" id="ref47">15</reflink>] conducted the initial Fusion study with the first two cohorts of the larger efficacy trial. The first cohort participated during the 2016 to 2017 school year, whereas the second cohort was from 2017 to 2018.</p> <p>Employing a partially nested randomized controlled trial, [<reflink idref="bib15" id="ref48">15</reflink>] randomly assigned 459 first-grade students within 53 first-grade classrooms to one of three conditions: (a) Fusion intervention group with a 2:1 student–teacher ratio (<emph>n</emph> = 91), (b) Fusion intervention group with a 5:1 student–teacher ratio (<emph>n</emph> = 230), or (c) a no-treatment control condition (<emph>n</emph> = 138). Students assigned to either treatment condition received Fusion in 30-min sessions, five days per week, for approximately 12 weeks. Because Fusion is designed as a supplemental intervention, treatment students received the intervention in addition to their core mathematics instruction. Students assigned to the control condition received core mathematics instruction as well as mathematics intervention services provided by their school. When such services were available, control students received the <emph>i-Ready Math</emph> curriculum ([<reflink idref="bib23" id="ref49">23</reflink>]), an individualized, online curriculum designed for 90 minutes per week.</p> <p>Findings from the initial study indicated significant, positive effects favoring the Fusion intervention over control on two of four mathematics outcome measures (i.e., ProFusion, [<reflink idref="bib11" id="ref50">11</reflink>]; ASPENS, [<reflink idref="bib18" id="ref51">18</reflink>]), with effect sizes (Hedges' <emph>g</emph>) ranging from 0.20 to 0.77 ([<reflink idref="bib15" id="ref52">15</reflink>]). Non-significant positive impacts were observed for two distal measures at posttest: (TEMA-3; [<reflink idref="bib32" id="ref53">32</reflink>]; first-grade easyCBM Math, [<reflink idref="bib3" id="ref54">3</reflink>]). The initial study also reported students in the 2:1 Fusion groups made significantly greater gains than those in the 5:1 Fusion groups on the TEMA-3 and first-grade easyCBM Math, with effect sizes (Hedges' <emph>g</emph>) ranging from 0.21 to 0.29. Follow-up results collected in second grade indicated students in the 2:1 Fusion groups maintained these differential gains on the second-grade easyCBM Math ([<reflink idref="bib3" id="ref55">3</reflink>]; [<reflink idref="bib9" id="ref56">9</reflink>] = 0.34). Finally, compared to the larger Fusion groups, treatment students in the 2:1 intervention groups received significantly more individual practice opportunities.</p> <p>Because the larger Fusion efficacy project was situated within a framework of systematic replication, we sought to bolster the generalizability of findings behind the Fusion intervention by conducting the current study in a different geographical area than the original Fusion work. This change in location, while holding constant the research design, analytics procedures, and the Fusion intervention, including dosage and implementation support, resulted in the replication study working with a new cohort of first-grade teachers and students in classrooms that used different core mathematics programs than those reported in the original work. This conceptual replication addressed three research questions:</p> <p></p> <ulist> <item> What was the overall impact of the Fusion intervention as compared to a business-as-usual control condition?</item> <p></p> <item> Was there a differential impact on student mathematics outcomes between the 2:1 Fusion and the 5:1 Fusion conditions?</item> <p></p> <item> Was there a differential impact on the observed quantity or quality of explicit instructional interactions between the 2:1 Fusion and 5:1 Fusion conditions?</item> </ulist> <hd id="AN0177928058-5">Method</hd> <p></p> <hd id="AN0177928058-6">Research Design and Context</hd> <p>This study analyzed data collected from the third cohort of a multi-cohort, federally funded efficacy project involving Fusion, a Tier 2 first-grade mathematics intervention. The third cohort participated in the larger efficacy project during the 2017 to 2018 school year. The Fusion Efficacy Trial ([<reflink idref="bib14" id="ref57">14</reflink>]) employed a partially nested randomized controlled trial ([<reflink idref="bib6" id="ref58">6</reflink>]). Figure 1 illustrates the partially nested study design, including the random assignment process, assessment schedule, and participant flow through follow-up. Blocking on classrooms, 240 first-grade students were randomly assigned within their classrooms to one of three conditions: (a) Fusion groups with 2:1 student–teacher ratios, (b) Fusion groups with 5:1 student–teacher ratios, and (c) a no-treatment control condition (i.e., business-as-usual). In all, 48 students were assigned to the 2:1 Fusion group, 120 to the 5:1 Fusion group, and 72 to the control condition. Students in the two Fusion groups received the intervention in addition to district-approved core mathematics instruction. In total, 43 Fusion groups were conducted in the third cohort of the Fusion Efficacy Trial (21 = 2:1 Fusion, 22 = 5:1 Fusion). Five Fusion groups (three 2:1 Fusion and two 5:1 Fusion) did not form due to staffing shortages, but the corresponding students were assessed and included in the analysis as randomized. Finally, it is important to note that an independent evaluator conducted the study's randomization process and all data analyses.</p> <p>DIAGRAM: Figure 1. Participant Flow Diagram for the Fusion Efficacy Project (Cohort 3). Note. The analysis sample varied by outcome measure. For outcomes collected at pretest and posttest, we used a Mixed Time × Condition model that included all cases with valid data at either pretest or posttest. For outcomes collected only at posttest or follow-up, we used a mixed-model analysis of covariance (ANCOVA) that only includes complete cases.aThe project did not have intervention staff to deliver Fusion to students. Although students in the four unstaffed classrooms were assigned to condition, they were never notified nor treated differentially in any way. Hence, we treated them as if they were never assigned.</p> <hd id="AN0177928058-7">Participants</hd> <p></p> <hd id="AN0177928058-8">Schools</hd> <p>Nine elementary schools from two urban school districts located in the metropolitan area of Boston, MA participated in the study. The districts, respectively, reported their student populations comprising 15.3% and 18.4% of students with disabilities and 3.6% and 37.3% as English learners. Also, 1.3% and 5.9% of students in the districts identified as Asian, 5.1% and 6.2% as Black, 8.8% and 86.2% as Hispanic, 0.1% and 0.2% of students in each district identified as Native American, 0.0% and 0.1% identified as Native Hawaiian/Pacific Islander, 6.4% and 75.0% identified as White, and 0.8% and 3.7% identified as multiple races. Free or reduced lunch rates for the districts were 28.1% and 66.3%, respectively.</p> <hd id="AN0177928058-9">Classrooms</hd> <p>The study took place in 24 first-grade classrooms, each taught by a certified teacher. Participating teachers had an average of 13.0 (<emph>SD</emph> = 7.3) years of teaching experience and 9.5 (<emph>SD</emph> = 5.9) years of experience teaching first grade; 63% had a master's degree in education; and 33% of teachers had completed an algebra course at the college level. Of the 24 teachers, 100% identified as female, 88% as White, 4% as Hispanic, and 8% declined to respond. Classrooms had an average of 22.4 students (<emph>SD</emph> = 4.1). Teachers reported providing mathematics instruction in English via a variety of commercially available mathematics curricula, with the vast majority (71%) using the <emph>enVision</emph> mathematics program and some (8%) using the <emph>i-Ready Math</emph> curriculum ([<reflink idref="bib23" id="ref59">23</reflink>]). Approximately half of the teachers also reported using teacher-developed mathematics materials. Teachers also primarily provided whole group instruction (88%) and utilized mathematics centers (83%) in their daily instructional routines.</p> <hd id="AN0177928058-10">Students and Inclusion Criteria</hd> <p>The current study applied the same student eligibility process as the original Fusion study. First, in each participating classroom, all students with parental consent were screened in the late fall of their first-grade year. The screening process comprised the three measures of the first-grade ASPENS battery ([<reflink idref="bib18" id="ref60">18</reflink>]). These measures included Magnitude Comparison, Missing Number, and Base-10/Basic Facts. To determine risk status, the ASPENS uses a composite score derived from the concatenation of student performance on the three measures. Student composite scores are then placed in a performance category: Benchmark, Strategic, or Intensive. A Benchmark performance suggests a student is on track for developing mathematical proficiency with whole numbers. The Strategic performance category represents students with scores below the Benchmark cut score and above the Intensive performance category. Composite scores in the Strategic category suggest students have a 50% chance of completing first grade at grade level and scoring at or above the 35th percentile on the spring TerraNova 3 ([<reflink idref="bib18" id="ref61">18</reflink>]). The Intensive performance category represents students at significant risk for MD. Students scoring in the Intensive category have approximately a 40% chance of achieving at or above the 35th percentile on the TerraNova 3 in the spring of first grade ([<reflink idref="bib18" id="ref62">18</reflink>]).</p> <p>In the current study, students were eligible and thus considered at risk for MD if they had an ASPENS' first-grade composite score at or below 26. This cut score placed students in the ASPENS' Strategic and Intensive performance categories based on first-grade winter benchmarks. Students with ASPENS composite scores in the Strategic or Intensive categories were rank-ordered in each participating classroom by an independent evaluator. The rank-ordering process involved approximately 10 students in each classroom. Within each classroom, the independent evaluator then randomly assigned two of the 10 students with the lowest ASPENS composite scores to a 2:1 Fusion intervention group, five students to a 5:1 Fusion intervention group, and three students to a control (i.e., business-as-usual) condition. A total of 614 first-grade students were screened for Fusion eligibility. Of these students, 240 met eligibility criteria and were randomly assigned within each of the 24 classrooms to 2:1 Fusion (<emph>n</emph> = 48), 5:1 Fusion (<emph>n</emph> = 120), or the control condition (<emph>n</emph> = 72). Demographic data for these students indicated that 15% of students had disabilities, 20% were emergent bilingual / English learners, and 57% of participants were female. Students identified as Hispanic (42%), White (41%), Black (9%), Asian (4%), or multiple races (4%).</p> <hd id="AN0177928058-11">Interventionists</hd> <p>Thirteen interventionists, hired specifically for the project, taught the 43 Fusion groups. Seventy-five percent of interventionists were identified as female. The majority, 67%, of interventionists identified as White, 17% as Hispanic, and 17% as Black. Most interventionists (78%) had previous experience teaching small groups, 92% held a bachelor's degree or higher, and 33% held a current teaching certification. All 13 interventionists had taken a college-level algebra course. On average, interventionists had 1.3 (<emph>SD</emph> = 0.5) years of teaching experience.</p> <hd id="AN0177928058-12">Procedures</hd> <p></p> <hd id="AN0177928058-13">Fusion Intervention</hd> <p>Fusion is a small-group, first-grade mathematics intervention aimed at building mathematical proficiency with foundational whole-number concepts and operations. The intervention contains 60 lessons (30 minutes each) that target the Operations and Algebraic Thinking, and Number and Operations in Base-10 strands in the first-grade Common Core State Standards for Mathematics ([<reflink idref="bib21" id="ref63">21</reflink>]). Fusion's scope and sequence use a systematic instructional approach to introduce new concepts and skills through a progression of mathematical development for first grade. As lessons progress, students encounter increasingly complex content to expand on skills taught earlier in the intervention. Such an approach also permits distributed practice across the program so that students can maintain their understanding of previously taught mathematical content.</p> <p>The first half of the Fusion intervention (Lessons 1–30) builds proficiency with numbers up to 100 through identifying, modeling, writing, and sequencing numbers. For example, a primary focus of the first 30 lessons is place value understanding of two-digit numbers. Students are also explicitly taught strategies to fluently recall addition and subtraction number combinations (i.e., math facts) within 10. The second half of the Fusion (Lessons 31–60) teaches students to solve two-digit addition and subtraction problems and compare two-digit numbers using "greater than, "less than," and "equal to" terminology. Additionally, in Lessons 31–60, students expand their repertoire of number combinations (i.e., math facts), learning doubles facts and common number families (e.g., 3, 4, and 7). Lessons 31–60 also build a deep understanding of the underlying structures of the word problem types identified in the first-grade Common Core State Standards for Mathematics ([<reflink idref="bib21" id="ref64">21</reflink>]). In that respect, students learn to represent and solve <emph>add to, take from, put together</emph>, and <emph>take apart</emph> problems.</p> <p>To build students' conceptual understanding, the intervention uses a host of physical mathematical models including number lines, number families, layered place value cards, base-10 blocks, and a hundreds chart. Fusion provides students with concurrent exposure to visual representations and mathematical symbols. For example, when learning the place value of two-digit numbers, students use base-10 blocks and unit cubes to model the tens and ones, and then write the targeted numbers in numerical form.</p> <p>Based on its systematic, explicit infrastructure, the Fusion intervention offers scripted lessons to support teachers in (a) delivering clear demonstrations and explanations of targeted mathematics content, (b) facilitating frequent student practice opportunities, and (c) offering timely academic feedback. The lesson scripting enables teachers' use of precise and consistent mathematical language and capacity to promote high-quality instructional interactions centered on whole number concepts and skills. These interactions are intended to facilitate deep mathematical thinking and reasoning, through individual and group-level mathematics verbalizations ([<reflink idref="bib27" id="ref65">27</reflink>]; [<reflink idref="bib30" id="ref66">30</reflink>]). For example, when teaching the commutative law of addition, the interventionist writes two problems on the board (e.g., 3 + 1 = 1 + 3) and asks students to discuss with their partner how the two problems are alike. In the latter part of the lesson, students discuss with a partner their understanding of the commutative law.</p> <p>Similar to the original study ([<reflink idref="bib15" id="ref67">15</reflink>]), the current research investigated the impact of Fusion when delivered in small-group formats (i.e., two or five students per interventionist), 30-min per day, five days per week for approximately 12 weeks. This 12-week window of instruction began in early winter and ended in the spring. Since Fusion serves as supplemental mathematics instruction, interventionists delivered the intervention at times that did not conflict with students' core Tier 1 mathematics instruction. Finally, it is important to note that 50 (31%) treatment students received mathematics intervention support in addition to Fusion. Teachers reported this support occurred three days per week, 45 minutes per day.</p> <hd id="AN0177928058-14">Control Condition</hd> <p>Students assigned to the business-as-usual control condition received core mathematics instruction as well as any standard mathematics intervention services provided by their school. Features of the intervention services received by control students were documented by teacher surveys. Teachers reported that 26 students (38%) in the control condition received mathematics intervention support beyond core mathematics instruction. Interestingly, these surveys did not identify use of specific commercially available mathematics intervention programs. Instead, the most widely reported intervention materials appeared to be teacher-developed interventions. On average, this support was delivered three days per week (<emph>SD</emph> = 1.3) for 37.9 minutes per day (<emph>SD</emph> = 11.2). The most common setting for intervention delivery was small groups (88%), followed by whole class (12%) and 1:1 instruction (4%). The majority of intervention support was provided by classroom teachers (77%). Of the 26 control students who received intervention support, 10 (38%) students received it during core mathematics instruction, four (15%) at the same time as Fusion was delivered, and 13 (50%) at another time of the school day.</p> <hd id="AN0177928058-15">Professional Development and Fidelity of Implementation</hd> <p>To implement the Fusion intervention, all interventionists participated in two 4-hour professional development workshops. The first workshop occurred prior to the start of Fusion, and the second midway through the intervention. Both workshops were delivered in person and focused on evidence-based practices in early mathematics instruction and strategies for small-group instruction. Interventionists each received, on average, two coaching visits to promote implementation fidelity. During such visits, trained Fusion coaches observed the intervention implementation and gave feedback on instructional quality and fidelity.</p> <p>In addition to coaching visits, project staff observed Fusion implementation and conducted fidelity checks approximately three times per group. Trained observers used a four-point scale (coded as 1 = "None" to 4 = "All") to rate the extent to which the interventionist (a) met the lesson's instructional objectives, (b) followed the lesson scripting, and (c) employed the lesson's mathematical models. Observers also documented the number of activities taught in each observed lesson. Overall, interventionists met the instructional objectives (<emph>M</emph> = 3.3, <emph>SD</emph> = 0.5), followed the lesson scripting (<emph>M</emph> = 3.2, <emph>SD</emph> = 0.5), and used the prescribed models of mathematics (<emph>M</emph> = 3.5, <emph>SD</emph> = 0.5). In addition, the majority of the prescribed activities were delivered during the observed lessons (<emph>M</emph> = 5.1 of 6 activities per lesson, <emph>SD</emph> = 0.5).</p> <hd id="AN0177928058-16">Student Mathematics Outcome Measures</hd> <p>At pretest and posttest, trained research staff administered three mathematics outcome measures to assess students' understanding of whole number concepts and skills: ProFusion, ASPENS, and TEMA-3. These measures were individually administered. To assess far transfer of treatment effects, students completed the first-grade easyCBM Math assessment at posttest in first grade. The second-grade easyCBM Math assessment was administered in the winter of students' second grade to assess far transfer as well as persistence of treatment effects. Research staff were trained in administering each measure and obtained interscorer reliability greater than 0.85 for all assessments.</p> <hd id="AN0177928058-17">ProFusion</hd> <p>ProFusion ([<reflink idref="bib11" id="ref68">11</reflink>]) is an untimed, researcher-developed assessment that contains four subtests focused on (a) place value, (b) addition and subtraction facts, (c) multi-digit addition and subtraction, and (d) word problems. Anchored to the first-grade [<reflink idref="bib21" id="ref69">21</reflink>], ProFusion is individually administered and considered proximal to the Fusion intervention. Across the 36-item assessment, students write two-digit numerals from dictation, fill in missing numerals within strings of three numerals (e.g., 19 ____ 21), write numerals that match base-10 pictures, identify the number of tens and ones in two-digit numerals, and verbally identify a set of numerals. Students also complete four multi-digit addition problems and four multi-digit subtraction problems. The assessment concludes with two word problems, where students identify the problem type, represent the problem with a schematic diagram, and use a numerical equation to solve the problem. Intercorrelations between the ProFusion subtests ranged from.31 to.56 in the current sample, with an average intercorrelation of.41. The correlation between pretest and posttest ProFusion composite scores was.50. The composite score used for analysis demonstrates concurrent criterion validity with similar mathematics measures, such as the SAT-10, with correlations ranging from.56 to.68 ([<reflink idref="bib12" id="ref70">12</reflink>]).</p> <hd id="AN0177928058-18">Assessing Student Proficiency in Early Number Sense</hd> <p>Assessing Student Proficiency in Early Number Sense ([<reflink idref="bib18" id="ref71">18</reflink>]) consists of three curriculum-based measures: (a) Magnitude Comparison, (b) Missing Number, and (c) Base-10/Basic Facts. All three measures are timed, individually administered, and considered distal in the current study. For the Magnitude Comparison and Missing Number measures, students complete as many items as they can in 1 minute. The Base-10/Basic Facts measure contains 10 addition (e.g., 10 + 10) and 10 subtraction (e.g., 12 – 8) items and students complete as many problems as possible in 2 minutes. Authors report test–retest reliability in the moderate to high range (.74 to.85), and adequate concurrent validity with the TerraNova 3 (.51 to.63).</p> <hd id="AN0177928058-19">TEMA-3</hd> <p>The TEMA-3 assesses formal and informal mathematical understanding for children ages 3 to 8 ([<reflink idref="bib32" id="ref72">32</reflink>]). This distal assessment is a standardized, norm-referenced, individually administered measure that examines students' conceptual and procedural understanding, including in calculations and counting. The TEMA-3 reports alternate-form reliability at.97 and test–retest reliability of.93. In terms of concurrent validity, the TEMA-3 reported coefficients ranging from.53 to.91. Raw scores were used for analysis.</p> <hd id="AN0177928058-20">easyCBM Math</hd> <p>The easyCBM Math ([<reflink idref="bib3" id="ref73">3</reflink>]) is an online, group-administered, multiple-choice assessment system for kindergarten to eighth grade. Students independently complete easyCBM Math on a computer or tablet. In the current study, the first-grade easyCBM Math was administered at posttest only, whereas the second-grade easyCBM Math served as the follow-up assessment in the winter of students' second grade. Both measures were considered distal outcomes and intended to assess far transfer. The first- and second-grade easyCBM measures are 35-item assessments that align with the [<reflink idref="bib21" id="ref74">21</reflink>]. In first grade, mathematical content assessed includes representing and solving addition and subtraction problems, applying properties of operations and the relationship between addition and subtraction, adding and subtracting within 20, extending the counting sequence, using place value understanding, telling and writing time, representing and interpreting data, and reasoning with shapes and their attributes. The second-grade easyCBM Math assessment measures students' ability to represent and solve addition and subtraction problems involving addition and subtraction, add and subtract within 20, work with equal groups of objects to gain foundations for multiplication, use place value understanding and properties of operations to add and subtract, measure and estimate lengths in standard units, relate addition and subtraction to length, work with time and money, and reason with shapes and their attributes. The reliability and validity of easyCBM Math measures are well established. Internal reliabilities of first-grade easyCBM are high (.81 to.84). Concurrent validity of easyCBM Math scores with the SAT-10 range from.75 to.82.</p> <hd id="AN0177928058-21">Direct Observations of Fusion Instruction</hd> <p>Each Fusion group was observed approximately three times over the course of the study, in intervals of three weeks between each observation occasion. In total, 393 observations were conducted, with 94 (24%) including two observers to document interobserver agreement. The average observation lasted 25.2 minutes (<emph>SD</emph> = 2.7). The observers, masked to the research hypotheses, documented the quantity and quality of instructional interactions.</p> <hd id="AN0177928058-22">Classroom Observations of Student–Teacher Interactions–Mathematics</hd> <p>The COSTI-M ([<reflink idref="bib24" id="ref75">24</reflink>]) is an empirically validated instrument that documents the occurrences of (a) teacher models, (b) group student practice opportunities, (c) individual student practice opportunities, (d) teacher-provided feedback, and (e) student mistakes. Teacher modeling represents a teacher's verbalizations of thought processes (think-alouds) and their physical demonstrations of mathematical content. Academic feedback is operationalized as a teacher's verbal reply or physical demonstration to correct or affirm a student's response. Group practice opportunities include verbalizations by two or more students in unison. Individual practice opportunities are coded whenever a single student verbalizes or demonstrates their mathematical thinking, such as a student being called on to represent the numeral 45 with base-10 blocks. [<reflink idref="bib24" id="ref76">24</reflink>] reported predictive validity of the COSTI-M with the TEMA-3 (<emph>p</emph> =.004, Pseudo-<emph>R</emph><sups>2</sups> =.08) and the EN-CBM (<emph>p</emph> =.017, Pseudo-<emph>R</emph><sups>2</sups> =.05).</p> <hd id="AN0177928058-23">Quality of Explicit Mathematics Instruction (QEMI)</hd> <p>The QEMI ([<reflink idref="bib25" id="ref77">25</reflink>]) includes seven items that document the quality of (a) group practice opportunities, (b) individual practice opportunities, (c) student participation, (d) teacher modeling, (e) academic feedback, (f) efficiency of instructional delivery, and (g) instructional scaffolding. Observers used a four-point scale to gauge the quality of each item (1 = "Not Present" to 4 = "Highly Present"). Internal consistency of the QEMI is high (.93).</p> <hd id="AN0177928058-24">Observation Training</hd> <p>Trained observers conducted all direct observations. Observers received 10 hours of in-person training: six hours in initial training and a four-hour follow-up to recalibrate and minimize observer drift. The training focused on using the COSTI-M and QEMI instruments, direct observation practices, first-grade mathematics instruction, and implementation fidelity of the Fusion intervention. Observers completed two reliability checks and obtained an interobserver agreement of.85 or higher prior to independently observing classrooms.</p> <hd id="AN0177928058-25">Interobserver Agreement and Stability Intraclass Correlations Coefficients</hd> <p>Intraclass correlation coefficients <bold>(</bold>ICCs<bold>)</bold> were calculated to estimate interobserver agreement in the COSTI-M and QEMI measures. In total, 94 paired reliability observations were conducted. Agreement ranged from.63 to.99, representing substantial to a nearly perfect agreement ([<reflink idref="bib42" id="ref78">42</reflink>]). To estimate stability, ICCs were calculated across all three observations within each Fusion group. Stability ICCs for the COSTI-M were.11 for teacher demonstrations,.09 for individual practice,.37 for group practice,.33 for student mistakes, and.45 for academic feedback. The QEMI ICC was.54. Reliability ranged from.23 (individual practice) to.78 (QEMI scores).</p> <hd id="AN0177928058-26">Statistical Analysis</hd> <p>We addressed our research questions using the same analysis approach as the original study ([<reflink idref="bib15" id="ref79">15</reflink>]). First, we assessed overall Fusion intervention effects, with 2:1 and 5:1 Fusion groups as the intervention condition, on student outcomes using a mixed model (multilevel) Time × Condition analysis ([<reflink idref="bib45" id="ref80">45</reflink>]) designed to account for students partially nested within small groups ([<reflink idref="bib6" id="ref81">6</reflink>]; [<reflink idref="bib7" id="ref82">7</reflink>]). The study design called for the randomization of individual students to receive Fusion, nested within 2:1 or 5:1 Fusion groups, or a non-nested comparison condition, and the analytic model must account for the potential heterogeneity among variances across conditions ([<reflink idref="bib51" id="ref83">51</reflink>]). In particular, the Fusion groups required a group-level variance, while the unclustered controls did not. Furthermore, because the residual variances may have differed among conditions, we tested the assumption of homoscedasticity of residuals. The analysis tested for differences among conditions on gains in outcomes from the fall (T<subs>1</subs>) to spring (T<subs>2</subs>) of first grade and is described in detail by [<reflink idref="bib17" id="ref84">17</reflink>] and [<reflink idref="bib26" id="ref85">26</reflink>]. The statistical model included effects for time, coded 0 at T<subs>1</subs> and 1 at T<subs>2</subs>; condition, coded 0 for control and 1 for Fusion; and the interaction between time and condition. The model tests for net differences between conditions ([<reflink idref="bib45" id="ref86">45</reflink>]), which provides an unbiased and straightforward interpretation of the results ([<reflink idref="bib1" id="ref87">1</reflink>]; [<reflink idref="bib38" id="ref88">38</reflink>]). For two outcomes not collected at pretest—first-grade easyCBM and second-grade easyCBM—we used the analysis of covariance approach described by [<reflink idref="bib7" id="ref89">7</reflink>] and [<reflink idref="bib6" id="ref90">6</reflink>].</p> <p>Second, we examined the effects of the 2:1 versus the 5:1 Fusion group size on student outcomes using a fully nested mixed model (multilevel) Time × Group Size analysis ([<reflink idref="bib45" id="ref91">45</reflink>]) to account for the intraclass correlation associated with students nested within Fusion groups. Similar to the first set of analyses, the model included effects for time, coded 0 at T<subs>1</subs> and 1 at T<subs>2</subs>; group size, coded 0 for 5:1 Fusion and 1 for 2:1 Fusion conditions; and the interaction between time and group size. Mixed analysis of covariance models was used for the first-grade and second-grade easyCBM scores, both tested with ASPENS and TEMA-3 scores as pretest covariates. Third, we tested whether 2:1 and 5:1 Fusion groups experienced differential rates of observed instructional interactions using independent-sample <emph>t-</emph>tests.</p> <hd id="AN0177928058-27">Model Estimation</hd> <p>Models specified with SAS PROC MIXED version 9.4 ([<reflink idref="bib53" id="ref92">53</reflink>]) used full-information maximum likelihood (ML) methods. ML estimation uses all available data, reducing potential bias—even in the face of substantial attrition—provided data are missing at random ([<reflink idref="bib33" id="ref93">33</reflink>]). Compared to complete-case analyses, ML relies on relatively benign assumptions and does not introduce bias ([<reflink idref="bib2" id="ref94">2</reflink>]; [<reflink idref="bib20" id="ref95">20</reflink>]). Student attrition was due primarily to absenteeism and school transfers.</p> <hd id="AN0177928058-28">Interpretation of Results</hd> <p>To interpret results, we focus on Hedges' <emph>g</emph> effect sizes, their 95% confidence intervals (CI), and model probabilities for hypothesis tests. As recommended by the American Statistical Association ([<reflink idref="bib55" id="ref96">55</reflink>]), we abstained from using bright-line rules such as claims of "statistical significance" when <emph>p</emph> <.05. <emph>P</emph> values measure the incompatibility between the observed data and all assumptions of the statistical model, including the null hypothesis, H<subs>0</subs> ([<reflink idref="bib35" id="ref97">35</reflink>]). This awkward definition determines neither which assumptions are incorrect nor the importance of the association. To complement <emph>p</emph> values, we report effect sizes, <emph>g</emph>, and <emph>model probabilities</emph>, <emph>w</emph>. The model probabilities indicate the strength of evidence for one model when compared with others, given the data at hand. Based on the Akaike Information Criterion, [<reflink idref="bib9" id="ref98">9</reflink>] describe <emph>w</emph> as the probability of selecting the same model with a "replicate data set from the same system" (p. 30) and "allow statements such as "the probability of [H<subs>A</subs>] is 0.78" (p. 26). Model probabilities better characterize the chance of a replicated result than <emph>p</emph> values. In this study, we compared models for two hypotheses: a model with the intervention effect (H<subs>A</subs>) and one without (H<subs>0</subs>). We reported the model probability for the model with the condition effect (H<subs>A</subs>), and with only two models, the model probability for H<subs>0</subs> is 1—<emph>w</emph>.</p> <hd id="AN0177928058-29">Results</hd> <p>Table 1 presents means, standard deviations, and sample sizes for the five dependent variables by assessment time and condition. The average pretest TEMA-3 raw score for the study sample was 34.4 (<emph>SD</emph> = 7.2), corresponding to the 32nd norm-referenced percentile rank. We found no differences between Fusion and control conditions at baseline on outcome measures (|Hedges' <emph>g</emph>| ≤.10). In what follows, we present results from tests of bias due to attrition, efficacy effects for Fusion (Research Question 1), effects of the 2:1 versus 5:1 Fusion group size on student outcomes (Research Question 2), and differential rates of instructional interactions between the 2:1 and 5:1 Fusion conditions (Research Question 3).</p> <p>Graph</p> <p>Table 1. Descriptive Statistics for Mathematics Outcome Measures by Condition and Assessment Time.</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /></colgroup><thead><tr><th align="left" rowspan="2">Measure</th><th align="center" colspan="3">Fall of Grade 1 (T<sub>1</sub>)</th><th align="center" colspan="4">Spring of Grade 1 (T<sub>2</sub>)</th><th align="center" colspan="2">Grade 2 (T<sub>3</sub>)</th></tr><tr><th align="center">2:1 Fusion</th><th align="center">5:1 Fusion</th><th align="center">Control</th><th align="center">2:1 Fusion</th><th align="center">5:1 Fusion</th><th align="center">Control</th><th align="center">2:1 Fusion</th><th align="center">5:1 Fusion</th><th align="center">Control</th></tr></thead><tbody><tr><td colspan="10">ProFusion</td></tr><tr><td><italic>M</italic></td><td>27.2</td><td>25.3</td><td>25.4</td><td>48.5</td><td>45.9</td><td>37.4</td><td /><td /><td /></tr><tr><td> (<italic>SD</italic>)</td><td>(8.3)</td><td>(8.8)</td><td>(9.1)</td><td>(10.8)</td><td>(10.6)</td><td>(9.1)</td><td /><td /><td /></tr><tr><td><italic>n</italic></td><td>46</td><td>118</td><td>69</td><td>41</td><td>110</td><td>62</td><td /><td /><td /></tr><tr><td /><td /><td /><td /><td>ASPENS</td><td /><td /><td /><td /><td /></tr><tr><td><italic>M</italic></td><td>18.8</td><td>18.3</td><td>20.0</td><td>46.6</td><td>44.1</td><td>43.0</td><td /><td /><td /></tr><tr><td> (<italic>SD</italic>)</td><td>(11.3)</td><td>(10.9)</td><td>(11.5)</td><td>(16.6)</td><td>(15.4)</td><td>(14.4)</td><td /><td /><td /></tr><tr><td><italic>n</italic></td><td>48</td><td>120</td><td>72</td><td>42</td><td>110</td><td>64</td><td /><td /><td /></tr><tr><td /><td /><td /><td /><td>TEMA-3</td><td /><td /><td /><td /><td /></tr><tr><td><italic>M</italic></td><td>34.3</td><td>34.2</td><td>34.9</td><td>44.1</td><td>42.8</td><td>41.2</td><td /><td /><td /></tr><tr><td> (<italic>SD</italic>)</td><td>(6.8)</td><td>(7.7)</td><td>(6.4)</td><td>(7.7)</td><td>(7.2)</td><td>(6.7)</td><td /><td /><td /></tr><tr><td><italic>n</italic></td><td>45</td><td>118</td><td>66</td><td>41</td><td>106</td><td>64</td><td /><td /><td /></tr><tr><td colspan="10">Grade 1 easyCBM</td></tr><tr><td><italic>M</italic></td><td /><td /><td /><td>25.2</td><td>24.1</td><td>24.0</td><td /><td /><td /></tr><tr><td> (<italic>SD</italic>)</td><td /><td /><td /><td>(4.9)</td><td>(5.6)</td><td>(5.4)</td><td /><td /><td /></tr><tr><td><italic>n</italic></td><td /><td /><td /><td>43</td><td>111</td><td>64</td><td /><td /><td /></tr><tr><td colspan="10">Grade 2 easyCBM</td></tr><tr><td><italic>M</italic></td><td /><td /><td /><td /><td /><td /><td>24.8</td><td>24.4</td><td>23.3</td></tr><tr><td> (<italic>SD</italic>)</td><td /><td /><td /><td /><td /><td /><td>(5.6)</td><td>(6.0)</td><td>(5.5)</td></tr><tr><td><italic>n</italic></td><td /><td /><td /><td /><td /><td /><td>35</td><td>100</td><td>60</td></tr></tbody></table> </ephtml> </p> <p>1 <emph>Note.</emph> The sample sizes represent students with a particular measure at each assessment period. The complete sample included 48 students in the 2:1 Fusion, 120 students in the 5:1 Fusion, and 72 students in the control condition. ASPENS = Assessing Student Proficiency in Early Number Sense ([<reflink idref="bib18" id="ref99">18</reflink>]); TEMA-3 = Test of Early Mathematics Ability–Third Edition ([<reflink idref="bib32" id="ref100">32</reflink>]).</p> <hd id="AN0177928058-30">Attrition</hd> <p>Among students with pretest data, the overall attrition rate at posttest was 10% in both study conditions. On the ASPENS screening measure, 10% of students without posttest data differed between conditions (Hedges' <emph>g</emph> = −0.65) more than students with posttest data (<emph>g</emph> = −0.09), but the interaction between condition and missingness at posttest implied minimal bias (interaction = 7.2, 95% CI [−2.7, 17.1], <emph>t</emph><subs>119</subs> = 1.44, <emph>p</emph> =.1524, <emph>w</emph> =.49). In this model, <emph>w</emph> represented the probability of the hypothesis that included the missingness-by-condition interaction compared to a hypothesis without the interaction.</p> <p>The overall attrition rate at second-grade follow-up was 19%, with a differential rate of 3%: 17% for control and 20% for Fusion. The 19% of students without follow-up data differed between conditions on ASPENS at screening (<emph>g</emph> = −0.41) more than students with follow-up data (<emph>g</emph> = −0.07), but the interaction between condition and missingness at follow-up also implied minimal bias (interaction = 4.2, 95% CI [−3.8, 12.3], <emph>t</emph><subs>119</subs> = 1.04, <emph>p</emph> =.3001, <emph>w</emph> =.37).</p> <p>The interpretation of attrition results is not straightforward. Differential rates of attrition offer little information ([<reflink idref="bib29" id="ref101">29</reflink>]) and baseline condition differences for students missing posttest data only rely on small samples. Attrition bias may be best conveyed by the missingness-by-condition interaction, although the approach has limits ([<reflink idref="bib34" id="ref102">34</reflink>]). We therefore chose ML estimation with all available data to balance the effects of non-response and to minimize bias ([<reflink idref="bib20" id="ref103">20</reflink>]; [<reflink idref="bib33" id="ref104">33</reflink>]).</p> <hd id="AN0177928058-31">Effects of Fusion Versus Control on Student Outcomes</hd> <p>Table 2 presents the results of the partially nested statistical models that estimate the difference between conditions at pretest (condition effect), gains across time for the control condition (time effect), and differential gains for the Fusion condition (Time × Condition interaction). The table presents the results of the homoscedastic model for each outcome because it was deemed equivalent to the more complicated heteroscedastic model. The bottom two rows of the table show the likelihood ratio test results that compared homoscedastic residuals to heteroscedastic residuals.</p> <p>Graph</p> <p>Table 2. Results of Partially Nested Time × Condition Analyses That Compared Fall-to-Spring Gains in Math Scores Between Fusion Students Nested Within Groups and Unclustered Control Students.</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /></colgroup><thead><tr><th align="center">Effect or statistic</th><th align="center">ProFusion</th><th align="center">ASPENS</th><th align="center">TEMA-3</th></tr></thead><tbody><tr><td>Model probability (<italic>w</italic>)</td><td>.99</td><td>.61</td><td>.84</td></tr><tr><td colspan="4">Fixed effects</td></tr><tr><td> Intercept</td><td>25.4<xref ref-type="table-fn" rid="tfn4">***</xref>(1.1)</td><td>20.0<xref ref-type="table-fn" rid="tfn4">***</xref>(1.6)</td><td>34.9<xref ref-type="table-fn" rid="tfn4">***</xref>(0.9)</td></tr><tr><td> Time</td><td>12.0<xref ref-type="table-fn" rid="tfn4">***</xref>(1.1)</td><td>22.8<xref ref-type="table-fn" rid="tfn4">***</xref>(1.5)</td><td>6.3<xref ref-type="table-fn" rid="tfn4">***</xref>(0.7)</td></tr><tr><td> Condition (Fusion)</td><td>0.5(1.4)</td><td>−1.6(1.9)</td><td>−0.6(1.1)</td></tr><tr><td> Time × Condition</td><td>8.5<xref ref-type="table-fn" rid="tfn4">***</xref>(1.4)</td><td>3.1†(1.8)</td><td>2.0<xref ref-type="table-fn" rid="tfn4">*</xref>(0.9)</td></tr><tr><td colspan="4">Variances</td></tr><tr><td> Group-level intercept</td><td>0.9(5.1)</td><td>0.4(9.3)</td><td>0.9(2.9)</td></tr><tr><td> Group-level gains</td><td>4.7(3.1)</td><td>2.8(5.0)</td><td>2.1(1.2)</td></tr><tr><td> Student-level pre–post covariance</td><td>50.3<xref ref-type="table-fn" rid="tfn4">***</xref>(8.1)</td><td>102.3<xref ref-type="table-fn" rid="tfn4">***</xref>(16.2)</td><td>37.0<xref ref-type="table-fn" rid="tfn4">***</xref>(4.9)</td></tr><tr><td> Residual</td><td>34.2<xref ref-type="table-fn" rid="tfn4">***</xref>(4.1)</td><td>70.0<xref ref-type="table-fn" rid="tfn4">***</xref>(8.2)</td><td>12.2<xref ref-type="table-fn" rid="tfn4">***</xref>(1.5)</td></tr><tr><td colspan="4">Hedges' <italic>g</italic> 95% CI</td></tr><tr><td> Time × Condition</td><td>0.82[0.56, 1.09]</td><td>0.20[−0.03, 0.43]</td><td>0.28[0.04, 0.52]</td></tr><tr><td colspan="4"><italic>p</italic>-value</td></tr><tr><td> Time × Condition</td><td><.0001</td><td>.0879</td><td>.0212</td></tr><tr><td colspan="4">BH <italic>p</italic>-value</td></tr><tr><td> Time × Condition</td><td>.0005</td><td>.1465</td><td>.0530</td></tr><tr><td colspan="4"><italic>df</italic></td></tr><tr><td> Time × Condition</td><td>138</td><td>159</td><td>130</td></tr><tr><td colspan="4">ICC</td></tr><tr><td> Fusion groups</td><td>.122</td><td>.039</td><td>.144</td></tr><tr><td>Likelihood ratio,<xref ref-type="table-fn" rid="tfn3">a</xref> χ2</td><td>3.08</td><td>2.59</td><td>2.33</td></tr><tr><td><italic>p</italic>-values</td><td>.3114</td><td>.2737</td><td>.2144</td></tr></tbody></table> </ephtml> </p> <ulist> <item>2 <emph>Note.</emph> Fixed effects and variances are shown with standard errors in parentheses. The models nested only Fusion students within groups. <emph>p</emph>-values are also provided with the BH correction. Degrees of freedom (<emph>df</emph>) for tests of fixed effects based on the Satterthwaite approximation. ASPENS = Assessing Student Proficiency in Early Number Sense ([<reflink idref="bib18" id="ref105">18</reflink>]); TEMA-3 = Test of Early Mathematics Ability–Third Edition ([<reflink idref="bib32" id="ref106">32</reflink>]); CI = confidence interval; BH = [<reflink idref="bib8" id="ref107">8</reflink>]; ICC = Intraclass correlation coefficient.</item> <item>3 The likelihood ratio tests compared homoscedastic to heteroscedastic residuals (α =.10, <emph>df</emph> = 1).</item> <item>4 <emph>p</emph> <.05. **<emph>p</emph> <.01. ***<emph>p</emph> <.001.</item> </ulist> <p>Students in the Fusion condition achieved greater gains than control students on the ProFusion assessment (<emph>g</emph> = 0.82 [0.56, 1.09], <emph>t</emph><subs>138</subs> = 6.13, <emph>p</emph> <.0001, <emph>w</emph> >.99) and TEMA-3 (<emph>g</emph> [95% CI] = 0.28 [0.04, 0.52], <emph>t</emph><subs>130</subs> = 2.33, <emph>p</emph> =.0212, <emph>w</emph> =.84). Time × Condition model results for these outcomes suggested that the hypothesis of a difference between conditions, measured by the Time × Condition effect, fit the data; that is, models for both ProFusion and TEMA-3 scores which included the Time × Condition interaction had considerably higher probabilities (<emph>w</emph>'s ≥.84) than models without the condition difference (<emph>w</emph>'s ≤.16). We found limited support for condition differences on ASPENS gains (<emph>g</emph> = 0.20 [−0.03, 0.43], <emph>w</emph> =.61). Similarly, the mixed-model analysis of covariance (ANCOVA) for outcomes not available at pretest generated limited evidence of condition differences on posttest first-grade easyCBM Math scores (<emph>g</emph> = 0.13 [−0.14, 0.40], <emph>w</emph> =.36) and follow-up second-grade easyCBM Math scores (<emph>g</emph> = 0.19 [−0.08, 0.46], <emph>w</emph> =.48).</p> <hd id="AN0177928058-32">Effects of 2:1 Versus 5:1 Fusion Groups on Student Outcomes</hd> <p>Table 3 presents the results of the fully nested Time × Group Size models comparing gains between 2:1 and 5:1 Fusion groups. The models in Table 3 tested fixed effects for differences among group sizes at pretest (2:1 Fusion group effect), gains across time for the 5:1 Fusion condition (time effect), and differential gains for the 2:1 Fusion condition (Time × Group Size interaction). We found limited evidence of group size differences favoring the 2:1 Fusion groups on ProFusion gains (<emph>g</emph> = 0.11 [−0.24, 0.46], <emph>w</emph> =.30), ASPENS gains (<emph>g</emph> = 0.15 [−0.14, 0.45], <emph>w</emph> =.37), and TEMA-3 gains (<emph>g</emph> = 0.24 [−0.06, 0.54], <emph>w</emph> =.55). Similarly, we found limited evidence of group size differences on first-grade easyCBM Math scores (<emph>g</emph> = 0.23 [−0.05, 0.52], <emph>w</emph> =.55) and follow-up second-grade easyCBM Math scores (<emph>g</emph> = 0.05 [−0.29, 0.38], <emph>w</emph> =.26).</p> <p>Graph</p> <p>Table 3. Results of Fully Nested Time × Condition Analyses That Compared Fall-to-Spring Gains in Mathematics Scores Between 2:1 and 5:1 Fusion Groups.</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /></colgroup><thead><tr><th align="center">Effect or statistic</th><th align="center">ProFusion</th><th align="center">ASPENS</th><th align="center">TEMA-3</th></tr></thead><tbody><tr><td>Model probability (<italic>w</italic>)</td><td>.30</td><td>.37</td><td>.55</td></tr><tr><td colspan="4">Fixed effects</td></tr><tr><td> Intercept</td><td>25.3<xref ref-type="table-fn" rid="tfn6">***</xref>(1.0)</td><td>18.3<xref ref-type="table-fn" rid="tfn6">***</xref>(1.3)</td><td>34.2<xref ref-type="table-fn" rid="tfn6">***</xref>(0.8)</td></tr><tr><td> Time</td><td>20.1<xref ref-type="table-fn" rid="tfn6">***</xref>(1.1)</td><td>25.2<xref ref-type="table-fn" rid="tfn6">***</xref>(1.3)</td><td>7.7<xref ref-type="table-fn" rid="tfn6">***</xref>(0.6)</td></tr><tr><td> 2:1 Fusion</td><td>2.0(1.8)</td><td>0.5(2.3)</td><td>0.3(1.4)</td></tr><tr><td> Time × 2:1 Fusion</td><td>1.2(1.8)</td><td>2.4(2.3)</td><td>1.8(1.1)</td></tr><tr><td colspan="4">Variances</td></tr><tr><td> Group-level intercept</td><td>−0.2(5.3)</td><td>−0.2(9.3)</td><td>1.2(3.4)</td></tr><tr><td> Group-level gains</td><td>5.6(3.9)</td><td>2.8(5.9)</td><td>1.9(1.3)</td></tr><tr><td> Student-level pre–post covariance</td><td>51.0<xref ref-type="table-fn" rid="tfn6">***</xref>(9.4)</td><td>100.1<xref ref-type="table-fn" rid="tfn6">***</xref>(18.6)</td><td>40.9<xref ref-type="table-fn" rid="tfn6">***</xref>(6.2)</td></tr><tr><td> Residual</td><td>36.6<xref ref-type="table-fn" rid="tfn6">***</xref>(4.9)</td><td>76.1<xref ref-type="table-fn" rid="tfn6">***</xref>(10.2)</td><td>12.5<xref ref-type="table-fn" rid="tfn6">***</xref>(1.7)</td></tr><tr><td colspan="4">Hedges' <italic>g</italic> 95% CI</td></tr><tr><td> Time × 2:1 Fusion</td><td>0.11[−0.24, 0.46]</td><td>0.15[−0.14, 0.45]</td><td>0.24[−0.06, 0.54]</td></tr><tr><td colspan="4"><italic>p-</italic>value</td></tr><tr><td> Time × 2:1 Fusion</td><td>.5254</td><td>.3077</td><td>.1133</td></tr><tr><td colspan="4">BH <italic>p-</italic>value</td></tr><tr><td> Time × 2:1 Fusion</td><td /><td /><td /></tr><tr><td colspan="4"><italic>df</italic></td></tr><tr><td> Time × 2:1 Fusion</td><td>45</td><td>45</td><td>45</td></tr><tr><td colspan="4">ICC</td></tr><tr><td> Fusion groups</td><td>.133</td><td>.036</td><td>.132</td></tr></tbody></table> </ephtml> </p> <ulist> <item>5 <emph>Note.</emph> Fixed effects and variances shown with standard errors in parentheses. The models nested students within Fusion groups. <emph>P</emph>-values also provided with the BH correction. Degrees of freedom (<emph>df</emph>) for tests of fixed effects based on the Satterthwaite approximation. ASPENS = Assessing Student Proficiency in Early Number Sense ([<reflink idref="bib18" id="ref108">18</reflink>]); TEMA-3 = Test of Early Mathematics Ability–Third Edition ([<reflink idref="bib32" id="ref109">32</reflink>]); CI = confidence interval; BH = Benjamini-Hochberg; ICC = Intraclass correlation coefficient.</item> <item>6 <emph>p</emph> <.05. **<emph>p</emph> <.01. ***<emph>p</emph> <.001.</item> </ulist> <hd id="AN0177928058-33">Effects of Group Size on the Quantity and Quality of Explicit Instructional Interactions</hd> <p>Table 4 presents descriptive statistics for the quantity (rates per minute) and quality of explicit instructional interactions as well as results of independent-samples <emph>t</emph>-tests comparing these observation measures by Fusion group size. Modest differences were observed for the teacher model rate (<emph>g</emph> = -0.14 [-0.74, 0.46], <emph>w</emph> =.26; favoring 5:1 Fusion groups), individual practice (<emph>g</emph> = 0.39 [-0.21, 1.00], <emph>w</emph> =.43; favoring 2:1 Fusion groups), group practice rate (<emph>g</emph> = 0.24 [-0.36, 0.84], <emph>w</emph> =.30; favoring 2:1 Fusion groups), student error rate (<emph>g</emph> = -0.21 [-0.81, 0.40], <emph>w</emph> =.29; with more errors occurring in the 5:1 Fusion groups), academic feedback rate (<emph>g</emph> = 0.29 [-0.31, 0.89], <emph>w</emph> =.34; favoring 2:1 Fusion groups), and the overall instruction quality rating (<emph>g</emph> = 0.41 [-0.19, 1.01], <emph>w</emph> =.44; favoring 2:1 Fusion groups). Results suggested that the hypothesis of a difference between group sizes on these observation measures had similar probabilities (<emph>w</emph>'s ≤.44) than the hypothesis of no difference between group sizes (<emph>w</emph>'s ≥.56).</p> <p>Graph</p> <p>Table 4. Results of Independent-Samples t Tests that Compared the Quantity and Quality of Explicit Instructional Interactions Between 2:1 and 5:1 Fusion Groups.</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /></colgroup><thead><tr><th align="center">Measure</th><th align="center">2:1 Fusion, <italic>M</italic> (<italic>SD</italic>)</th><th align="center">5:1 Fusion, <italic>M</italic> (<italic>SD</italic>)</th><th align="center"><italic>t</italic></th><th align="center"><italic>p</italic></th><th align="center">Hedges' <italic>g</italic> [95% CI]</th></tr></thead><tbody><tr><td>Teacher model rate</td><td>0.13 (0.08)</td><td>0.14 (0.09)</td><td>−0.47</td><td>.6422</td><td>−0.14[−0.74, 0.46]</td></tr><tr><td>Individual practice rate</td><td>2.48 (0.61)</td><td>2.24 (0.60)</td><td>1.32</td><td>.1930</td><td>0.39[−0.21, 1.00]</td></tr><tr><td>Group practice rate</td><td>1.11 (0.58)</td><td>0.98 (0.48)</td><td>0.80</td><td>.4294</td><td>0.24[−0.36, 0.84]</td></tr><tr><td>Student error rate</td><td>0.24 (0.14)</td><td>0.28 (0.20)</td><td>−0.69</td><td>.4919</td><td>−0.21[−0.81, 0.40]</td></tr><tr><td>Academic feedback rate</td><td>2.13 (0.54)</td><td>1.97 (0.57)</td><td>0.98</td><td>.3321</td><td>0.29[−0.31, 0.89]</td></tr><tr><td>Quality of explicit math instruction</td><td>3.01 (0.26)</td><td>2.90 (0.28)</td><td>1.37</td><td>.1768</td><td>0.41[−0.19, 1.01]</td></tr></tbody></table> </ephtml> </p> <p>7 <emph>Note.</emph> Independent samples <emph>t</emph> tests were based on 21 2:1 Fusion groups and 22 5:1 Fusion groups with observation data (<emph>df</emph> = 41). CI = confidence interval.</p> <hd id="AN0177928058-34">Discussion</hd> <p>We conducted this conceptual replication to determine whether the existing evidence behind the Fusion intervention ([<reflink idref="bib15" id="ref110">15</reflink>]) held up in first-grade classrooms from a different geographical region. The current study tested Fusion in classrooms located in the metropolitan area of Boston, MA, whereas the original Fusion study took place in districts throughout the state of Oregon. As a result of this location change, the current study worked in classrooms that implemented different core mathematics programs and enrolled a more racially and ethnically diverse student sample than included in the original Fusion study.</p> <hd id="AN0177928058-35">Overall Impact of Fusion</hd> <p>We first examined the effects of Fusion relative to a control condition. As juxtaposed in Table 5, findings from the replication study were similar to [<reflink idref="bib15" id="ref111">15</reflink>] in that treatment students demonstrated greater gains than their control peers on ProFusion, a researcher-developed mathematics assessment. The effect of Fusion on this measure was large, <emph>g</emph> = 0.82, 95% CI [0.56, 1.09]. Contrary to the original study, we found a non-significant finding on the ASPENS, a battery of fluency-based measures focused on early number sense. While the current study's finding on the ASPENS was positive (<emph>g</emph> = 0.20), the non-significant differences between Fusion and control students may have been attributed to our sample size. The replication study included about half as many students and Fusion groups as the original study.</p> <p>Graph</p> <p>Table 5. Published and Replicated Descriptive Statistics and Effect Sizes for Research Questions 1 and 2.</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /></colgroup><thead><tr><th align="center">Variable</th><th align="center" colspan="2">Original fusion study</th><th align="center" colspan="2">Replication study</th></tr></thead><tbody><tr><td><italic>n</italic></td><td colspan="2">460</td><td colspan="2">240</td></tr><tr><td> 2:1</td><td colspan="2">92</td><td colspan="2">48</td></tr><tr><td> 5:1</td><td colspan="2">230</td><td colspan="2">120</td></tr><tr><td> BAU</td><td colspan="2">138</td><td colspan="2">72</td></tr><tr><td>Classrooms</td><td colspan="2">53</td><td colspan="2">24</td></tr><tr><td>SPED</td><td colspan="2">17%</td><td colspan="2">15%</td></tr><tr><td>ELs</td><td colspan="2">14%</td><td colspan="2">20%</td></tr><tr><td>Females</td><td colspan="2">54%</td><td colspan="2">57%</td></tr><tr><th align="center" colspan="5">Ethnicity</th></tr><tr><td> American Indian</td><td colspan="2">1%</td><td colspan="2">0%</td></tr><tr><td> Asian</td><td colspan="2">3%</td><td colspan="2">4%</td></tr><tr><td> Black</td><td colspan="2">2%</td><td colspan="2">9%</td></tr><tr><td> Hispanic</td><td colspan="2">21%</td><td colspan="2">42%</td></tr><tr><td> White</td><td colspan="2">65%</td><td colspan="2">41%</td></tr><tr><td> Multiple races</td><td colspan="2">8%</td><td colspan="2">4%</td></tr><tr><td>Region</td><td colspan="2">Northwest</td><td colspan="2">Northeast</td></tr><tr><td>Primary Core Math Curriculum</td><td colspan="2">Houghton Mifflin Mathematics</td><td colspan="2">enVision</td></tr><tr><th align="center">Outcome Measure g [95% CI]</th><th align="center">Fusion vs. BAU</th><th align="center">2:1 vs. 5:1</th><th align="center">Fusion vs. BAU</th><th align="center">2:1 vs. 5:1</th></tr><tr><td>ProFusion</td><td>0.77<xref ref-type="table-fn" rid="tfn9">*</xref>[0.60, 0.93]</td><td>0.16[−0.04, 0.37]</td><td>0.82<xref ref-type="table-fn" rid="tfn9">*</xref>[0.56, 1.09]</td><td>0.11[−0.24, 0.46]</td></tr><tr><td>ASPENS</td><td>0.20<xref ref-type="table-fn" rid="tfn9">*</xref>[0.04, 0.36]</td><td>0.17[−0.04, 0.37]</td><td>0.20[−0.03, 0.43]</td><td>0.15[−0.14, 0.45]</td></tr><tr><td>TEMA-3</td><td>0.07[−0.09, 0.23]</td><td>0.21<xref ref-type="table-fn" rid="tfn9">*</xref>[0.02, 0.39</td><td>0.28<xref ref-type="table-fn" rid="tfn9">*</xref>[0.04, 0.52]</td><td>0.24[−0.06, 0.54]</td></tr><tr><td>Grade 1 easyCBM</td><td>0.02[−0.16, 0.20]</td><td>0.29<xref ref-type="table-fn" rid="tfn9">*</xref>[0.05, 0.53]</td><td>0.13[−0.14, 0.40]</td><td>0.23[−0.05, 0.52]</td></tr><tr><td>Grade 2 easyCBM</td><td>−0.01[−0.20, 0.18]</td><td>0.34<xref ref-type="table-fn" rid="tfn9">*</xref>[0.09, 0.58]</td><td>0.19[−0.08, 0.46]</td><td>0.05[−0.29, 0.38]</td></tr></tbody></table> </ephtml> </p> <ulist> <item>8 <emph>Note.</emph> BAU = business-as-usual; SPED = students receiving special education services; ELs = English learners; CI = confidence interval; ASPENS = Assessing Student Proficiency in Early Number Sense ([<reflink idref="bib18" id="ref112">18</reflink>]); TEMA-3 = Test of Early Mathematics Ability–Third Edition ([<reflink idref="bib32" id="ref113">32</reflink>]).</item> <item>9 Significant at <emph>p</emph> <.05.</item> </ulist> <p>Notably, the replication's results on the TEMA-3 were statistically significant and the effect size (<emph>g</emph> = 0.28) exceeded that reported in the original study (<emph>g</emph> = 0.07). Taken together, results on the TEMA-3 from the two Fusion studies have an important implication for students struggling with early mathematics. Aligned with the converging knowledge base of effective mathematics instruction ([<reflink idref="bib30" id="ref114">30</reflink>]), our findings suggest systematic and explicit mathematics instruction can help support at-risk learners in developing proficiency in a vital area of mathematics. Arguably, a robust understanding of whole numbers and operations lays the foundation for later mathematical learning ([<reflink idref="bib46" id="ref115">46</reflink>]).</p> <p>In terms of the first-grade easyCBM, results from the current study mirrored those reported in the original study. Both studies observed positive, albeit non-significant, effects on this distal outcome measure. We also did not detect a statistically significant finding on the second-grade, follow-up assessment in the current study. However, relative to the original Fusion study, the current effect size was positive on the second-grade easyCBM and trended toward an educationally meaningful difference between treatment and control students (<emph>g</emph> = 0.19). While preliminary, the persistence of Fusion's impact one year later in the current research suggests schools may have offered subsequent learning environments that permitted treatment students to sustain and build upon the concepts and skills acquired in the intervention ([<reflink idref="bib4" id="ref116">4</reflink>]). This finding is important as a fade-out of intervention effects is widespread ([<reflink idref="bib5" id="ref117">5</reflink>]).</p> <hd id="AN0177928058-36">Differential Impact of Group Size</hd> <p>We also examined the differential impact on student mathematics outcomes between the 2:1 Fusion and the 5:1 Fusion conditions. Reducing an intervention's group size is thought to influence many outcomes, most markedly treatment intensity and the differentiation of instruction ([<reflink idref="bib16" id="ref118">16</reflink>]; [<reflink idref="bib31" id="ref119">31</reflink>]). Thus, when an intervention's group size is reduced, its level of instructional intensity and individualization is expected to increase.</p> <p>Our work with group size and its experimental manipulation within the context of the Fusion intervention research has produced mixed findings. Results from the original Fusion study suggest students in the 2:1 Fusion groups significantly outperformed their peers in the 5:1 Fusion groups on three distal mathematics outcome measures, including a follow-up assessment administered in second grade. However, those results did not replicate in the current study. No statistically significant differences between the two Fusion intervention groups were found on any of the replication study's outcome measures. It is plausible that our results on group size differed across the two studies based on the difficulty of replicating experimental conditions in educational research. As noted by the [<reflink idref="bib47" id="ref120">47</reflink>], true replication in educational environments contains a much greater margin of error than in laboratory settings, as schools include many contextual factors that cannot be fully controlled, such as school norms and quality of learning experiences. This is true even in conceptual replications. For example, changing the geographical landscape in the present replication inevitably led to working in new first-grade classrooms, with new norms and learning experiences outside the Fusion intervention.</p> <p>While the current study demonstrated mixed findings relative to the original Fusion study, we believe there is intellectual merit regarding future investigations of group size. Above all, such efforts can address the urgent need to unpack this important mechanism of treatment intensity. To date, the number of mathematics intervention studies that have experimentally manipulated group size is thin ([<reflink idref="bib39" id="ref121">39</reflink>]). Therefore, future research at different grade levels and in various areas of mathematics is required to pinpoint optimal instructional arrangements that offer the intensity required to accelerate learning for students with MD.</p> <hd id="AN0177928058-37">Differential Impact of Explicit Instructional Interactions</hd> <p>Finally, we explored the extent to which the Fusion conditions differed in the quantity and quality of explicit instructional interactions. Because instructional interactions are a cornerstone of mathematical learning ([<reflink idref="bib50" id="ref122">50</reflink>]), we used a validated multifaceted observation system to document their quantity and quality in the Fusion groups. Overall, the current study reported no statistical evidence that varying group size differentiated the quantity and quality of instructional interactions students received. However, two interesting findings emerged when juxtaposing the observation data collected across the two Fusion studies.</p> <p>First, the current study found that the 2:1 and 5:1 Fusion groups offered comparable individual practice opportunities. However, the original study reported students in the 2:1 Fusion groups received higher rates of individual practice than students in the 5:1 groups. Prior research suggests that more individual practice is associated with better mathematics outcomes ([<reflink idref="bib24" id="ref123">24</reflink>]). Therefore, given the original Fusion study where students in the 2:1 groups outperformed their peers in the 5:1 groups, it may be that individual practice opportunities facilitated in small group settings are a key to intensifying mathematics instruction and, in turn, increasing student mathematics achievement. However, future research is warranted.</p> <p>Another interesting finding was in regard to the rate of group practice opportunities. Although non-significant, the original study reported a negative effect size for group practice (<emph>g</emph> = −0.25, 95% CI [−0.67, 0.16]), whereas it was positive in the replication study (<emph>g</emph> = 0.24, 95% CI [−0.26, 0.84]). These results indicate that the original study observed more group practice opportunities in the 5:1 groups, while the replication documented more group practice in the 2:1 groups. When facilitated well, group responses can provide all students with opportunities to chorally verbalize their mathematical thinking and understanding with peers. Research is needed, however, to understand what might be driving these mixed results and to consider how to utilize group practice to intensify mathematics instruction.</p> <hd id="AN0177928058-38">Implications for Research and Practice</hd> <p>We contend the current study has several implications for the field. First, given conceptual replications have gained increased attention as mechanisms for bolstering the knowledge base of effective education programs ([<reflink idref="bib22" id="ref124">22</reflink>]; [<reflink idref="bib44" id="ref125">44</reflink>]), we deem it important for the field to examine and learn from the contextual differences between original studies and their subsequent replications. Such efforts may provide insight into replication failure and, as in the case of the current study, mixed findings. One consideration to understand why our two studies produced mixed results centers around how a change in the geographical region resulted in working in first-grade classrooms that used different core mathematics programs than those reported in the original Fusion study. In the elementary grades, core mathematics programs represent important building blocks for later mathematics learning. Despite this importance, research suggests the quality of mathematics programs varies widely ([<reflink idref="bib28" id="ref126">28</reflink>]). While implementation fidelity and curricular quality were not measured, it is plausible that differences in the reported mathematics curricula contributed to our mixed findings between the two Fusion studies. Thus, future research is warranted to explore contextual differences, such as the quality of enacted mathematics curricula, between original studies and their replications.</p> <p>In addition, we see value in researchers unpacking the instructional interactions that occur among students during small-group interventions. Smaller groups (e.g., two students) are theorized to provide greater individualization; however, it may be that groups with four to five students offer students more meaningful opportunities to observe and interact with their peers. When students verbalize their mathematical thinking and understanding, an intervention group with five students likely offers a more diverse range of student perspectives than a group of two.</p> <p>For teachers, we believe this replication work has meaningful implications for their classrooms. First, while we found variation across outcomes based on group size, the explicit, systematic framework that served as the backbone of Fusion produced significant results on the ProFusion measure across settings and group sizes. The ProFusion results, which assessed key concepts and skills students need in order to succeed in later mathematics, including place value, addition and subtraction, and problem-solving, signal that students receiving additional targeted support can make significant improvements when given additional learning opportunities. Thus, we encourage schools to identify and provide more intensive instruction for students with MD based on the resources allocated to the school. Second, in light of the variation across settings, two key elements emerged across settings: individual and group practice opportunities. Providing frequent practice opportunities for students with MD affords students the ability to (a) receive frequent and immediate feedback, (b) verbalize their mathematical thinking, and (c) generalize key mathematical concepts and skills ([<reflink idref="bib30" id="ref127">30</reflink>]; [<reflink idref="bib31" id="ref128">31</reflink>]). Thus, regardless of group size, we recommend teachers routinely provide ample opportunities for students to practice new mathematical concepts and skills both individually and with their classmates.</p> <hd id="AN0177928058-39">Limitations</hd> <p>Findings from this replication study should be interpreted with caution in light of several limitations. The first limitation was author overlap with the original study. Because our research team conducted both Fusion efficacy trials, we employed an independent evaluator to manage and analyze all project data. We contend this helped alleviate any potential bias of our findings. Also, due to budgetary limitations, we were unable to directly observe the core mathematics instruction delivered in the participating first-grade classrooms. Consequently, this restricted us from gaining a richer understanding of the enacted core mathematics curriculum. Finally, the current study's sample size was approximately half of that of the original study. Logistical issues with conducting distance research limited our opportunity to include a second student cohort in the Boston, MA area.</p> <hd id="AN0177928058-40">Conclusion</hd> <p>The early elementary grades represent a critical window for students to develop a robust understanding of whole numbers and operations. One way to get students who struggle early with mathematics on track for mathematical success is to implement validated, small-group mathematics interventions that spark frequent and high-quality instructional interactions. Findings from the two Fusion efficacy trials suggest the intervention has the capacity to support students' understanding of whole numbers. However, we aim to further advance a framework of systematic replication to bolster the scientific credibility of the Fusion intervention.</p> <ref id="AN0177928058-41"> <title> References </title> <blist> <bibl id="bib1" idref="ref87" type="bt">1</bibl> <bibtext> Allison P. D. (1990). Change scores as dependent variables in regression analysis. Sociological Methodology, 20, 93–114. https://doi.org/10.2307/271083</bibtext> </blist> <blist> <bibl id="bib2" idref="ref94" type="bt">2</bibl> <bibtext> Allison P. D. (2009). Missing data. In Millsap R. E., Maydeu-Olivares A., Millsap R. E., Maydeu-Olivares A. 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Additionally, the terms of this arrangement have been reviewed and approved by The University of Texas at Austin in accordance with its policy on objectivity in research. An independent external evaluator and coauthor of this publication completed the research analysis described in the article.</bibtext> </blist> <blist> <bibtext> The research reported here was supported by the U.S. Department of Education, Institute of Education Sciences through Grants R324A090341 and R324A160046 to the Center on Teaching and Learning at the University of Oregon.</bibtext> </blist> <blist> <bibtext> Marah Sutherland</bibtext> </blist> <blist> <bibtext>Graph</bibtext> </blist> <blist> <bibtext>https://orcid.org/0000-0002-6108-8515 Jenna A. Gersib</bibtext> </blist> <blist> <bibtext>Graph</bibtext> </blist> <blist> <bibtext>https://orcid.org/0000-0002-1270-1889 Georgia L. Kimmel</bibtext> </blist> <blist> <bibtext>Graph https://orcid.org/0000-0001-7649-3073</bibtext> </blist> </ref> <aug> <p>By Christian T. Doabler; Ben Clarke; Jessica E. Turtura; Marah Sutherland; Jenna A. Gersib; Taylor Lesner; Madison Cook; Georgia L. Kimmel; Keith Smolkowski and Derek Kosty</p> <p>Reported by Author; Author; Author; Author; Author; Author; Author; Author; Author; Author</p> </aug> <nolink nlid="nl1" bibid="bib37" firstref="ref1"></nolink> <nolink nlid="nl2" bibid="bib41" firstref="ref2"></nolink> <nolink nlid="nl3" bibid="bib47" firstref="ref3"></nolink> <nolink nlid="nl4" bibid="bib22" firstref="ref4"></nolink> <nolink nlid="nl5" bibid="bib44" firstref="ref6"></nolink> <nolink nlid="nl6" bibid="bib15" firstref="ref7"></nolink> <nolink nlid="nl7" bibid="bib10" firstref="ref8"></nolink> <nolink nlid="nl8" bibid="bib48" firstref="ref9"></nolink> <nolink nlid="nl9" bibid="bib52" firstref="ref11"></nolink> <nolink nlid="nl10" bibid="bib19" firstref="ref12"></nolink> <nolink nlid="nl11" bibid="bib13" firstref="ref16"></nolink> <nolink nlid="nl12" bibid="bib26" firstref="ref17"></nolink> <nolink nlid="nl13" bibid="bib17" firstref="ref18"></nolink> <nolink nlid="nl14" bibid="bib18" firstref="ref21"></nolink> <nolink nlid="nl15" bibid="bib32" firstref="ref22"></nolink> <nolink nlid="nl16" bibid="bib40" firstref="ref23"></nolink> <nolink nlid="nl17" bibid="bib36" firstref="ref24"></nolink> <nolink nlid="nl18" bibid="bib43" firstref="ref30"></nolink> <nolink nlid="nl19" bibid="bib54" firstref="ref31"></nolink> <nolink nlid="nl20" bibid="bib27" firstref="ref32"></nolink> <nolink nlid="nl21" bibid="bib30" firstref="ref33"></nolink> <nolink nlid="nl22" bibid="bib31" firstref="ref34"></nolink> <nolink nlid="nl23" bibid="bib49" firstref="ref35"></nolink> <nolink nlid="nl24" bibid="bib39" firstref="ref37"></nolink> <nolink nlid="nl25" bibid="bib50" firstref="ref43"></nolink> <nolink nlid="nl26" bibid="bib24" firstref="ref45"></nolink> <nolink nlid="nl27" bibid="bib14" firstref="ref46"></nolink> <nolink nlid="nl28" bibid="bib23" firstref="ref49"></nolink> <nolink nlid="nl29" bibid="bib11" firstref="ref50"></nolink> <nolink nlid="nl30" bibid="bib21" firstref="ref63"></nolink> <nolink nlid="nl31" bibid="bib12" firstref="ref70"></nolink> <nolink nlid="nl32" bibid="bib25" firstref="ref77"></nolink> <nolink nlid="nl33" bibid="bib42" firstref="ref78"></nolink> <nolink nlid="nl34" bibid="bib45" firstref="ref80"></nolink> <nolink nlid="nl35" bibid="bib51" firstref="ref83"></nolink> <nolink nlid="nl36" bibid="bib38" firstref="ref88"></nolink> <nolink nlid="nl37" bibid="bib53" firstref="ref92"></nolink> <nolink nlid="nl38" bibid="bib33" firstref="ref93"></nolink> <nolink nlid="nl39" bibid="bib20" firstref="ref95"></nolink> <nolink nlid="nl40" bibid="bib55" firstref="ref96"></nolink> <nolink nlid="nl41" bibid="bib35" firstref="ref97"></nolink> <nolink nlid="nl42" bibid="bib29" firstref="ref101"></nolink> <nolink nlid="nl43" bibid="bib34" firstref="ref102"></nolink> <nolink nlid="nl44" bibid="bib46" firstref="ref115"></nolink> <nolink nlid="nl45" bibid="bib16" firstref="ref118"></nolink> <nolink nlid="nl46" bibid="bib28" firstref="ref126"></nolink>
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  Data: <searchLink fieldCode="AR" term="%22Christian+T%2E+Doabler%22">Christian T. Doabler</searchLink><br /><searchLink fieldCode="AR" term="%22Ben+Clarke%22">Ben Clarke</searchLink><br /><searchLink fieldCode="AR" term="%22Jessica+E%2E+Turtura%22">Jessica E. Turtura</searchLink><br /><searchLink fieldCode="AR" term="%22Marah+Sutherland%22">Marah Sutherland</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-6108-8515">0000-0002-6108-8515</externalLink>)<br /><searchLink fieldCode="AR" term="%22Jenna+A%2E+Gersib%22">Jenna A. Gersib</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-1270-1889">0000-0002-1270-1889</externalLink>)<br /><searchLink fieldCode="AR" term="%22Taylor+Lesner%22">Taylor Lesner</searchLink><br /><searchLink fieldCode="AR" term="%22Madison+Cook%22">Madison Cook</searchLink><br /><searchLink fieldCode="AR" term="%22Georgia+L%2E+Kimmel%22">Georgia L. Kimmel</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0001-7649-3073">0000-0001-7649-3073</externalLink>)<br /><searchLink fieldCode="AR" term="%22Keith+Smolkowski%22">Keith Smolkowski</searchLink><br /><searchLink fieldCode="AR" term="%22Derek+Kosty%22">Derek Kosty</searchLink>
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="SO" term="%22Grantee+Submission%22"><i>Grantee Submission</i></searchLink>. 2023.
– Name: PeerReviewed
  Label: Peer Reviewed
  Group: SrcInfo
  Data: Y
– Name: Pages
  Label: Page Count
  Group: Src
  Data: 32
– Name: DatePubCY
  Label: Publication Date
  Group: Date
  Data: 2023
– Name: SourceSuprt
  Label: Sponsoring Agency
  Group: SrcSuprt
  Data: Institute of Education Sciences (ED)
– Name: NumberContract
  Label: Contract Number
  Group: NumCntrct
  Data: R324A090341<br />R324A160046
– Name: TypeDocument
  Label: Document Type
  Group: TypDoc
  Data: Reports - Research
– Name: Audience
  Label: Education Level
  Group: Audnce
  Data: <searchLink fieldCode="EL" term="%22Early+Childhood+Education%22">Early Childhood Education</searchLink><br /><searchLink fieldCode="EL" term="%22Elementary+Education%22">Elementary Education</searchLink><br /><searchLink fieldCode="EL" term="%22Grade+1%22">Grade 1</searchLink><br /><searchLink fieldCode="EL" term="%22Primary+Education%22">Primary Education</searchLink>
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Number+Concepts%22">Number Concepts</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Instruction%22">Mathematics Instruction</searchLink><br /><searchLink fieldCode="DE" term="%22Grade+1%22">Grade 1</searchLink><br /><searchLink fieldCode="DE" term="%22Elementary+School+Students%22">Elementary School Students</searchLink><br /><searchLink fieldCode="DE" term="%22Intervention%22">Intervention</searchLink><br /><searchLink fieldCode="DE" term="%22Outcome+Measures%22">Outcome Measures</searchLink><br /><searchLink fieldCode="DE" term="%22Concept+Formation%22">Concept Formation</searchLink><br /><searchLink fieldCode="DE" term="%22Teaching+Methods%22">Teaching Methods</searchLink><br /><searchLink fieldCode="DE" term="%22Small+Group+Instruction%22">Small Group Instruction</searchLink><br /><searchLink fieldCode="DE" term="%22Comparative+Analysis%22">Comparative Analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Tests%22">Mathematics Tests</searchLink>
– Name: Subject
  Label: Geographic Terms
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Massachusetts+%28Boston%29%22">Massachusetts (Boston)</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1177/00222194231209017
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Conceptual replications are part and parcel of education science. Methodologically rigorous conceptual replication studies permit researchers to test and strengthen the generalizability of a study's initial findings. The current conceptual replication sought to replicate the efficacy of a small-group, first-grade mathematics intervention with 240 first-grade students with mathematics difficulties in a new geographical region. Participating students were randomized into one of three conditions: (a) 2:1 mathematics intervention group, (b) 5:1 mathematics intervention group, or (c) business-as-usual instruction. Relative to the original study, findings from the replication varied. When comparing the treatment groups to the control, results suggested positive effects on all outcome measures, including a follow-up assessment administered one year later. However, differences between the two treatment groups based on group size were not found in the mathematics outcome measures. Both groups also received commensurate levels of observed instructional interactions. Implications for unpacking contextual differences between original research and their replications as well as using future research to explore the quantity and quality of instructional interactions as ways to explain variation in findings of group size are discussed. [This is the online first version of an article published in "Journal of Learning Disabilities."]
– Name: AbstractInfo
  Label: Abstractor
  Group: Ab
  Data: As Provided
– Name: CodeSource
  Label: IES Funded
  Group: SrcInfo
  Data: Yes
– Name: DateEntry
  Label: Entry Date
  Group: Date
  Data: 2024
– Name: AN
  Label: Accession Number
  Group: ID
  Data: ED641777
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=ED641777
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1177/00222194231209017
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 32
    Subjects:
      – SubjectFull: Number Concepts
        Type: general
      – SubjectFull: Mathematics Instruction
        Type: general
      – SubjectFull: Grade 1
        Type: general
      – SubjectFull: Elementary School Students
        Type: general
      – SubjectFull: Intervention
        Type: general
      – SubjectFull: Outcome Measures
        Type: general
      – SubjectFull: Concept Formation
        Type: general
      – SubjectFull: Teaching Methods
        Type: general
      – SubjectFull: Small Group Instruction
        Type: general
      – SubjectFull: Comparative Analysis
        Type: general
      – SubjectFull: Mathematics Tests
        Type: general
      – SubjectFull: Massachusetts (Boston)
        Type: general
    Titles:
      – TitleFull: Proficiency with Number Concepts and Operations: Replicating the Efficacy of a First-Grade Mathematics Intervention
        Type: main
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            NameFull: Derek Kosty
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          Dates:
            – D: 28
              M: 11
              Type: published
              Y: 2023
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            – TitleFull: Grantee Submission
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