View in EDS HTML Full Text PDF Full Text

Improving Struggling Fifth-Grade Students' Understanding of Fractions: A Randomized Controlled Trial of an Intervention That Stresses Both Concepts and Procedures

Saved in:
Bibliographic Details
Title: Improving Struggling Fifth-Grade Students' Understanding of Fractions: A Randomized Controlled Trial of an Intervention That Stresses Both Concepts and Procedures
Language: English
Authors: Jayanthi, Madhavi, Gersten, Russell, Schumacher, Robin F., Dimino, Joseph, Smolkowski, Keith, Spallone, Samantha
Source: Exceptional Children. Oct 2021 88(1):81-100.
Availability: SAGE Publications. 2455 Teller Road, Thousand Oaks, CA 91320. Tel: 800-818-7243; Tel: 805-499-9774; Fax: 800-583-2665; e-mail: journals@sagepub.com; Web site: http://sagepub.com
Peer Reviewed: Y
Page Count: 20
Publication Date: 2021
Sponsoring Agency: National Science Foundation (NSF)
Contract Number: DRL1535214
Document Type: Journal Articles
Reports - Research
Education Level: Elementary Education
Grade 5
Intermediate Grades
Middle Schools
Descriptors: Grade 5, Elementary School Students, Fractions, Mathematical Concepts, Concept Formation, Direct Instruction, Instructional Effectiveness, Intervention, Difficulty Level, Achievement Tests, Problem Solving
Assessment and Survey Identifiers: Wide Range Achievement Test
DOI: 10.1177/00144029211008851
ISSN: 0014-4029
Abstract: Using a randomized controlled trial, we examined the effect of a fractions intervention for students experiencing mathematical difficulties in Grade 5. Students who were eligible for the study (n = 205) were randomly assigned to intervention and comparison conditions, blocked by teacher. The intervention used systematic, explicit instruction and relied on linear representations (e.g., Cuisenaire Rods and number lines) to demonstrate key fractions concepts. Enhancing students' mathematical explanations was also a focus. Results indicated that intervention students significantly outperformed students from the comparison condition on measures of fractions proficiency and understanding (g = 0.66-0.78), number line estimation (g = 0.80-1.08), fractions procedures (g = 1.07), and explanation tasks (g = 0.68-1.23). Findings suggest that interventions designed to include explicit instruction, along with consistent use of the number line and opportunities to explain reasoning, can promote students' proficiency and understanding of fractions.
Abstractor: As Provided
Entry Date: 2022
Accession Number: EJ1320419
Database: ERIC
Full text is not displayed to guests.
FullText Links:
  – Type: pdflink
    Url: https://content.ebscohost.com/cds/retrieve?content=AQICAHj0k_4E0hTGH8RJwT4gCJyBsGNe_WN95AvKlDbXJGqwxwEyxrrX31PvgYG0JdJlAarlAAAA4zCB4AYJKoZIhvcNAQcGoIHSMIHPAgEAMIHJBgkqhkiG9w0BBwEwHgYJYIZIAWUDBAEuMBEEDGM9yKEE_-nrY-nQQwIBEICBm526AO33_2_kG8N-jQr85eG1WMo_pAc415CrYje9A0ef1dRJe9WcuPg6EQ7loguFLHZ6TLctYhMdFkps3SV-N3jwCP5vmUV9usWSKxWXubXcLkY4rhUFR55Md8E8cJMMuya4fT-byRxozZAp42SjnBRnFFSwFCP31jtrwx6HCJtQTHzpmmkKtgtrlZqw2EiU_Dnc21Rhq1CbscDN
Text:
  Availability: 1
  Value: <anid>AN0154098903;exc01oct.21;2021Dec15.01:41;v2.2.500</anid> <title id="AN0154098903-1">Improving Struggling Fifth-Grade Students' Understanding of Fractions: A Randomized Controlled Trial of an Intervention That Stresses Both Concepts and Procedures </title> <p>Using a randomized controlled trial, we examined the effect of a fractions intervention for students experiencing mathematical difficulties in Grade 5. Students who were eligible for the study (n = 205) were randomly assigned to intervention and comparison conditions, blocked by teacher. The intervention used systematic, explicit instruction and relied on linear representations (e.g., Cuisenaire Rods and number lines) to demonstrate key fractions concepts. Enhancing students' mathematical explanations was also a focus. Results indicated that intervention students significantly outperformed students from the comparison condition on measures of fractions proficiency and understanding (g = 0.66–0.78), number line estimation (g = 0.80–1.08), fractions procedures (g = 1.07), and explanation tasks (g = 0.68–1.23). Findings suggest that interventions designed to include explicit instruction, along with consistent use of the number line and opportunities to explain reasoning, can promote students' proficiency and understanding of fractions.</p> <p>A solid understanding of fractions is a key predictor of success in algebra, which is a critical gateway to higher education and many high-paying vocations (e.g., [<reflink idref="bib34" id="ref1">34</reflink>]; [<reflink idref="bib42" id="ref2">42</reflink>]). This predictive relationship was established through a longitudinal study of two large data sets from the United Kingdom and the United States that found that knowledge of fractions at age 10 (i.e., Grade 5) predicted performance in algebra and overall mathematics achievement in Grade 11, above and beyond the predictive power of general math ability, IQ, or socioeconomic status ([<reflink idref="bib42" id="ref3">42</reflink>]).</p> <p>Unfortunately, fractions present many challenges for students. Up until around Grade 3, the mathematics curriculum focuses primarily on whole numbers. However, in Grades 4 and 5, fractions become a key part of the curriculum, and students encounter a level of abstraction that is rarely present when working with whole numbers. For example, whole numbers have one unique successor (2 always follows 1; 11 always follows 10), and whole-number magnitude can be learned through counting discrete objects and place value. A fraction, however, represents magnitude through the relationship between the numerator and denominator. Fractions can represent quantities in many ways, such as a part of a whole, part of a set of objects, or length on a number line. Unlike whole numbers, a fraction can have an infinite number of equivalences (e.g., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>2</mn><mn>4</mn></mfrac></mrow></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>0</mn><mo>.</mo><mn>5</mn></mrow></math> </ephtml> ) that represent the same point on the number line. Many students fail to understand how fractions concepts differ from whole-number concepts and, in error, apply their whole-number understandings when solving fractions problems ([<reflink idref="bib40" id="ref4">40</reflink>]).</p> <p> <emph>Many students fail to understand how fractions concepts differ from whole-number concepts and, in error, apply their whole-number understandings when solving fractions problems</emph> </p> <p>For students who are already behind in mathematics, the introduction of fractions compounds their struggles (e.g., [<reflink idref="bib32" id="ref5">32</reflink>]). [<reflink idref="bib37" id="ref6">37</reflink>] found that many students failed to grow appreciably in their knowledge of fractions across Grades 4 through 6, with 42% of students in Grade 6 failing to understand fractions magnitude. Given how essential the topic of fractions is to future learning in mathematics, focusing intervention on fractions concepts across Grades 4 and 5 is worthwhile.</p> <p>In this introduction, we briefly describe prior relevant work that coalesced to guide the study's goals. Specifically, we examine intervention research on (a) using the number line as a key representation of fractions' magnitude, (b) enhancing students' explanations of fractions concepts and procedures, and (c) including foundational and grade-level content for students experiencing difficulties with fractions.</p> <hd id="AN0154098903-2">The Critical Role of the Number Line</hd> <p>The number line is a critical mathematical representation that can be used across elementary and middle grades to build understanding across whole numbers, fractions, decimals, and positive and negative integers ([<reflink idref="bib28" id="ref7">28</reflink>]). In the past decade, the number line has been included in intervention research when teaching fractions and decimals concepts to struggling learners (e.g., [<reflink idref="bib2" id="ref8">2</reflink>]; [<reflink idref="bib11" id="ref9">11</reflink>]; [<reflink idref="bib29" id="ref10">29</reflink>]), especially for teaching fractions magnitude. Some studies have shown that proficiency in estimating fractions magnitude on a number line also improved addition and subtraction of fractions (e.g., [<reflink idref="bib12" id="ref11">12</reflink>]; [<reflink idref="bib45" id="ref12">45</reflink>]).</p> <p>Estimating fractions' magnitude on a number line was also found to be predictive of how likely students were to learn strategies for solving linear equations in Algebra 1 ([<reflink idref="bib4" id="ref13">4</reflink>]). In addition, students who were taught to compare fractional magnitude using a number line were significantly more likely to transfer their understanding to new areas of fractions learning than those who were taught fractions primarily through discrete area models ([<reflink idref="bib17" id="ref14">17</reflink>]). Furthermore, in [<reflink idref="bib13" id="ref15">13</reflink>], students' ability to estimate fractions magnitude on the number line mediated effects on general fractions knowledge, which indicated a strong association between the number line and more general understandings of fractions.</p> <p>Given the positive benefits associated with using the number line to teach fractions magnitude concepts across Grades 4 through 6, our intervention used the number line with struggling learners in Grade 5 to teach magnitude and to facilitate students' learning of concepts underlying fractions operations. The number line was also one of the key visual representations used as a mechanism for developing students' explanations of fractions concepts.</p> <hd id="AN0154098903-3">Developing Students' Explanations</hd> <p>Virtually all contemporary mathematics state standards and assessments ask students to provide explanations of their mathematical reasoning when solving problems. Yet for students who are struggling to learn to solve problems correctly, providing a thoughtful and accurate explanation may not be possible without support from their teacher. Students are likely to benefit when taught how to explain through explicit modeling, practice, and feedback from the teacher.</p> <p>A key theme in earlier research was to build students' verbal explanations of the strategy they used to solve problems. When conducting a meta-analysis of mathematics interventions over 10 years ago, [<reflink idref="bib15" id="ref16">15</reflink>] identified the verbalization of mathematical ideas as one of six instructional components included across the set of studies. Having students verbalize their solution process or "think aloud" as they solve problems was identified in eight of the 42 studies that were reviewed. The approaches that were described in the studies focused on verbalization of a problem strategy. They did not assess how students used their knowledge of underlying mathematical concepts to solve problems, nor did they include instructional approaches to further enhance students' explanations.</p> <p>More recent intervention work has begun to examine ways to develop students' explanations of their mathematical reasoning beyond student verbalization of their solution process. For example, [<reflink idref="bib7" id="ref17">7</reflink>] asked students to verbalize their mathematical thinking while using manipulatives to represent problems during an intervention for kindergarteners. In [<reflink idref="bib11" id="ref18">11</reflink>], students were taught how to explain their solutions for problems focused on fractions magnitude using visual representations. Interventionists first modeled explanations by connecting the mathematical concept to a visual representation. Students then analyzed and practiced these explanations verbally and in writing while continuing to use the visual representations. Building on the literature on error analysis and worked examples, [<reflink idref="bib10" id="ref19">10</reflink>] taught students to analyze worked examples and explain whether the work was completed correctly or incorrectly and why. When the work was incorrect, students also provided an explanation of the correction of the error. Students had to justify their responses using precise mathematical language.</p> <p>In this study, we approached explanations by explicitly teaching students to explain their mathematical reasoning. We taught students to use precise mathematical vocabulary, compared examples of complete and accurate explanations with ones that were not adequate, explicitly taught and modeled the process for explaining mathematical tasks using a general heuristic, and integrated high-leverage practices (e.g., strategic prompting) for eliciting more information from students while they explain their mathematical reasoning (e.g., [<reflink idref="bib9" id="ref20">9</reflink>]; [<reflink idref="bib41" id="ref21">41</reflink>]). In using this approach, we guided students through the process of explaining their thinking while emphasizing the connection of mathematical concepts to procedures.</p> <hd id="AN0154098903-4">Inclusion of Challenging Grade-Level Material and Relevant Foundational Material</hd> <p>To maximize the effect of a small-group intervention, especially when addressing topics that are essential for future success in mathematics, such as fractions, it is critical that the intervention focus on challenging material from the students' current grade level. It is equally important for interventions to focus on foundational material by reviewing, reteaching, and reinforcing content from previous grades that is essential for understanding current grade-level material. This approach—focus on both grade-level and foundational material—is clearly supported by the [<reflink idref="bib21" id="ref22">21</reflink>], with its goal of providing meaningful access to the general curriculum to students with disabilities.</p> <p>Several prior, efficacious research studies spanning Grades 1 through 4 ([<reflink idref="bib5" id="ref23">5</reflink>]; [<reflink idref="bib11" id="ref24">11</reflink>]; [<reflink idref="bib16" id="ref25">16</reflink>]; [<reflink idref="bib26" id="ref26">26</reflink>]; [<reflink idref="bib29" id="ref27">29</reflink>]) have followed this approach and focused on both grade-level and foundational material. Although this dual focus is much easier to implement when providing intervention to Grade 1 students because they have been in school for less than 2 years, it has been done with equally effective results in higher grades, where the focus is on more difficult content, like fractions and decimals (e.g., [<reflink idref="bib29" id="ref28">29</reflink>]) or ratio and proportion word problems (e.g., [<reflink idref="bib25" id="ref29">25</reflink>]).</p> <p>Prior fractions intervention research has focused on Grades 4 through 6, likely due to the importance of learning fractions within those grade levels. In this study, we addressed fractions learning in Grade 5 so that intervention can tackle missed foundational fractions understandings from Grades 3 and 4 (e.g., magnitude, equivalence) and also confront Grade 5 fractions material (e.g., the four operations with fractions). In focusing on students in Grade 5, intervention supports students in learning fractions concepts before starting middle school.</p> <hd id="AN0154098903-5">Purpose of the Present Study</hd> <p>The goal of this efficacy study was to conduct a rigorous, randomized controlled trial to evaluate the effect of a small-group fractions intervention. The Adapted TransMath (ATM) Fractions Intervention (adapted from Level 2; [<reflink idref="bib49" id="ref30">49</reflink>]) centralizes linear representations for teaching fractions concepts, supports students' explanations using correct mathematical language, and includes explicit, systematic instruction on challenging fifth-grade material and essential foundational material from early grades. We hypothesized that this multicomponent approach would increase struggling students' knowledge of fractions, arguably the most challenging elementary mathematics content. A secondary goal was to look at the relationship between estimating fractions magnitude on the number line and fractions achievement through mediation analysis. We also explored potential moderators, such as initial mathematics knowledge, district enrollment, gender, and free or reduced lunch.</p> <hd id="AN0154098903-6">Method</hd> <p></p> <hd id="AN0154098903-7">Procedure</hd> <p>Thirty-four Grade 5 teachers from 14 elementary schools across three districts (two urban districts on the West Coast and one urban-adjacent district in the Southeast) volunteered to participate in the study. Parent consent forms were sent home with the students at the start of the school year. In September, screening measures were administered to all Grade 5 students with parent consent from participating teachers' classes. Eligible Grade 5 students with consent were administered pretests and then randomly assigned to intervention or business-as-usual conditions, blocked by the teacher to control for the quality of mathematics instruction in the general classroom. Twenty-one intervention groups, composed of four or five students, were formed. For logistical reasons, each small group was composed of students across classrooms who were available for intervention at the same time.</p> <p>The implementation of the ATM Fractions Intervention began in late October and was delivered three times a week to nine intervention groups and four times a week to 12 intervention groups based on local needs. Intervention was provided outside the classroom in a separate room. Intervention was completed in March for 15 groups, in February for two groups, and in April for four groups. Completion times varied due to school holidays and events. Student explanation tasks were administered at five time points during the intervention period. Data were collected for all intervention students and for a randomly selected subsample of 34 comparison students (see section on subsample in Participants).</p> <p>Posttests were administered at each school to treatment and comparison students as soon as intervention ended for each group and were completed within 2 weeks of groups receiving their last session. Assessments were administered in small groups except for the Number Line Estimation, which was administered individually.</p> <hd id="AN0154098903-8">Participants</hd> <p>A total of 1,123 Grade 5 students, with parent consent, were screened for participation in the study using the Test for Understanding of Fractions, Fourth Grade (TUF-4; [<reflink idref="bib22" id="ref31">22</reflink>]). Students who scored between the 15th and 37th percentiles, based on a large norming sample (see [<reflink idref="bib24" id="ref32">24</reflink>]), were eligible to participate. This range is often used in mathematics intervention research to capture students who are likely to receive Tier 2 intervention services ([<reflink idref="bib36" id="ref33">36</reflink>]). Students who scored in this range (<emph>n</emph> = 326) were eligible to participate in the study. Of these eligible students, 58 students were excluded before random assignment by the classroom teacher for the following reasons: conflicting schedules due to receiving other school-based services (<emph>n</emph> = 37), parents' rescinding consent (<emph>n</emph> = 10), and other (e.g., student health, student mobility, English language learner; <emph>n</emph> = 10). In addition, another 63 were randomly dropped before randomization, because according to power estimates, a sample of 205 students was needed to have sufficient power to detect effects. After these exclusions, the remaining 205 Grade 5 students were randomly assigned to intervention (<emph>n</emph> = 102) and business-as-usual (<emph>n</emph> = 103) conditions.</p> <p>The sample with complete pre- and posttest data (<emph>n</emph> = 186 students; <emph>n</emph> = 87 in intervention, <emph>n</emph> = 99 in comparison) was used in the analysis. Student characteristics for this sample are summarized in Table 1. The sample included students who varied along racial and ethnic lines and also in terms of economic status. As data on disability status were not available from one school district, information on individualized education program or 504 plan status is presented only for 50% of the sample. Chi-square analysis and <emph>t</emph> tests revealed no statistically significant differences between conditions on any demographic variable or pretest measure.</p> <p>Graph</p> <p>Table 1. Baseline Characteristics of the Student Analytic Sample.</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 /><th align="center">Intervention(<italic>n</italic> = 87)</th><th align="center">Comparison(<italic>n</italic> = 99)</th><th /><th /></tr><tr><th align="center">Variable</th><th align="center">%</th><th align="center">%</th><th align="center">χ<sup>2</sup> or <italic>t</italic> (<italic>df</italic>)</th><th align="center"><italic>p</italic></th></tr></thead><tbody><tr><td>Gender</td><td /><td /><td>0.49 (1)</td><td>.485</td></tr><tr><td> Female</td><td>49.43</td><td>54.55</td><td /><td /></tr><tr><td>Race-ethnicity</td><td /><td /><td>3.41 (5)</td><td>.636</td></tr><tr><td> African American or Black</td><td>14.94</td><td>15.15</td><td /><td /></tr><tr><td> Asian</td><td>5.75</td><td>6.06</td><td /><td /></tr><tr><td> Hispanic or Latino</td><td>18.39</td><td>16.16</td><td /><td /></tr><tr><td> White</td><td>42.53</td><td>34.34</td><td /><td /></tr><tr><td> Multiracial</td><td>18.39</td><td>27.27</td><td /><td /></tr><tr><td> Missing</td><td>0.00</td><td>1.01</td><td /><td /></tr><tr><td>Free or reduced lunch</td><td /><td /><td>0.19 (2)</td><td>.909</td></tr><tr><td> Yes</td><td>57.47</td><td>54.55</td><td /><td /></tr><tr><td> No</td><td>27.59</td><td>30.30</td><td /><td /></tr><tr><td> Missing</td><td>14.94</td><td>15.15</td><td /><td /></tr><tr><td>IEP or 504 plan</td><td /><td /><td>0.10 (2)</td><td>.950</td></tr><tr><td> Yes</td><td>6.90</td><td>8.08</td><td /><td /></tr><tr><td> No</td><td>43.68</td><td>42.42</td><td /><td /></tr><tr><td> Missing<xref ref-type="table-fn" rid="tfn2">a</xref></td><td>49.43</td><td>49.49</td><td /><td /></tr><tr><td>Pretest measure, mean raw score (<italic>SD</italic>)</td><td /><td /><td /><td /></tr><tr><td> TUF-4</td><td>11.49 (1.44)</td><td>11.45 (1.55)</td><td>0.18 (184)</td><td>.857</td></tr><tr><td> WRAT-4</td><td>97.91 (10.46)</td><td>96.18 (11.21)</td><td>1.08 (184)</td><td>.281</td></tr><tr><td> Procedures (addition and subtraction)</td><td>5.69 (4.03)</td><td>5.13 (4.12)</td><td>0.93 (184)</td><td>.353</td></tr><tr><td> NLE 0–1</td><td>72.50 (10.54)</td><td>74.11 (12.14)</td><td>0.96 (184)</td><td>.339</td></tr></tbody></table> </ephtml> </p> <p>1 <emph>Note</emph>. Total analytic sample, <emph>N</emph> = 186 students (intervention, <emph>n</emph> = 87; comparison, <emph>n</emph> = 99). IEP = individualized education program; TUF-4 = Test for Understanding of Fractions, Fourth Grade; WRAT-4 = Wide Range Achievement Test 4; NLE = Number Line Estimation.</p> <ulist> <item>2 IEP or 504 status was unavailable from one school district.</item> <item>3 WRAT-4 scores are standard scores. WRAT-4 standard score of 97.91 = 44.45th percentile. WRAT-4 standard score of 96.18 = 39.95th percentile.</item> </ulist> <p>Overall attrition was 9.27%; differential attrition between the two conditions was 10.82%. These rates are acceptable by What Works Clearinghouse (WWC) standards ([<reflink idref="bib46" id="ref34">46</reflink>]). An attrition analysis was conducted to assess attrition issues related to each condition and for the overall sample. There were no statistically significant differences between the overall pretest means for the randomized sample and the analytic sample with complete posttest data for any of the four pretest measures (<emph>t</emph> = −0.07 to −0.39; <emph>p</emph> =.70 to.95). The results of an ANOVA indicate similar nonsignificant differences between conditions for the randomized and analytic samples on the four pretest measures (<emph>t</emph> = 0.19 to 0.50; <emph>p</emph> =.52 to.85).</p> <hd id="AN0154098903-9">Subsample</hd> <p>To assess the effect of the intervention on student explanations, based on power estimates, 35 intervention students and 34 comparison students were randomly selected from the overall sample of 102 treatment and 103 comparison students.</p> <hd id="AN0154098903-10">Interventionists</hd> <p>Ten interventionists (all female) were hired by the research team. Of the 10 interventionists, six were credentialed teachers. Their mean experience for teaching elementary mathematics was 7.7 years (<emph>SD</emph> = 8.92) and for teaching Grade 5 mathematics was 2.4 years (<emph>SD</emph> = 4.86). The four interventionists who were not credentialed teachers had worked with students in schools or other tutoring settings. Five interventionists had a master's degree in education or a related field. Interventionists were assigned to intervention groups based on their availability. The caseload ranged from one to three groups per interventionist. One interventionist taught one group, seven interventionists taught two groups, and two interventionists taught three groups.</p> <hd id="AN0154098903-11">Intervention Condition</hd> <p>The ATM Fractions Intervention includes 52 35-min fractions lessons that were adapted from the TransMath curriculum (Level 2; [<reflink idref="bib49" id="ref35">49</reflink>]). The TransMath curriculum is designed for use in large-group settings for students struggling in mathematics in Grades 4 to 8 and includes the full range of the mathematics content covered in those grades. The adaptation process started by identifying the fractions lessons from TransMath that addressed fourth- and fifth-grade-level content. The fractions lessons from TransMath were reorganized by the research team to create 35-min lessons that focused only on fractions and were appropriate for a small-group intervention setting. This reorganization entailed dividing some lessons into two or three lessons, with each lesson structured to include review, explicit instruction of concepts, supported student practice, and independent student practice. Also, material that did not focus on fractions (e.g., geometry) was eliminated from the lessons. These adapted lessons were piloted and revised prior to implementation in this study ([<reflink idref="bib39" id="ref36">39</reflink>]). Details on how lessons were adapted are described in the pilot study.</p> <hd id="AN0154098903-12">Lesson structure</hd> <p>Each 35-min lesson included four parts. Lessons began with a 5-min review of foundational material relevant for that lesson. For example, reviewing factors was included when the lesson pertained to finding greatest common denominators. During the next 10-min block, interventionists provided strategic, explicit instruction and modeled key mathematical concepts and procedures using representations. They also taught explicit mathematical vocabulary words that were pertinent and central to understanding the fractions content or reviewed recently taught words. Interventionists used think-alouds to explicitly represent concepts and demonstrate procedures. Even though interventionists were leading the instruction, students were engaged in their learning by using concrete representations concurrently, writing equations, and answering questions posed by the interventionists to check for understanding. Additional supported practice followed, also for 10 min, where students solved problems as a group or with a partner and verbally explained their solutions. Interventionists provided corrective feedback to address any misunderstandings. There was a focus on verbal explanations in all lessons. During the last 10 min, students solved problems independently, and every three or four lessons, starting with Lesson 11, students were asked to provide written explanations of their work. Lessons that included written explanations were predetermined by the research team and consistent across all groups. All 52 lessons were systematically designed to provide sufficient time for student learning of complex fractions ([<reflink idref="bib15" id="ref37">15</reflink>]).</p> <p> <emph>Even though interventionists were leading the instruction, students were engaged in their learning by using concrete representations concurrently, writing equations, and answering questions</emph> </p> <hd id="AN0154098903-13">Fractions content</hd> <p>The intervention lessons covered grade-level (i.e., Grade 5) content in fractions as well as foundational material on fractions from Grade 4 identified in contemporary mathematics standards (e.g., [<reflink idref="bib6" id="ref38">6</reflink>]). In general, Grade 4 standards focus on foundational fractions concepts, such as equivalence and ordering, and understanding unit fractions. They also address addition and subtraction with like denominators and multiplication of a fraction with a whole number. Grade 5 standards extend fractions concepts to addition and subtraction with unlike denominators, multiplication of a fraction by a fraction, and division of a whole number by a unit fraction.</p> <p>Lessons 1 to 18 addressed foundational understanding. These lessons emphasized understanding what a fraction is, magnitude of fractions, equivalent fractions, developing understanding by comparing two fractions, ordering fractions from least to greatest, and estimating fraction placement on the number line. Instruction focused on how to reason about fractions magnitude by helping students understand the part–whole relationship of the numerator and denominator when evaluating a fraction's relative magnitude and by using benchmark fractions to evaluate magnitude when placing fractions on a number line. Lessons 19 to 28 focused on addition and subtraction of fractions, and Lessons 29 to 42 addressed multiplication and division of fractions. For instance, the lessons focused on the underlying concepts and procedures for (a) addition and subtraction of fractions with like denominators ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math> </ephtml> ) and unlike denominators ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></math> </ephtml> ), (b) multiplication of a whole number by a fraction ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>2</mn><mo>×</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></math> </ephtml> ) and a fraction times a fraction ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></math> </ephtml> ), and (c) division of a whole number by a unit fraction ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>2</mn><mo>÷</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></math> </ephtml> ) and a unit fraction divided by a whole number ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>÷</mo><mn>2</mn></mrow></math> </ephtml> ). The lessons also focused on critical fractions concepts related to computational procedures. Lessons explored, for instance, why fractions with unlike denominators (or fractions representing different units) cannot be added or subtracted before the problem is modified to include like denominators and how the multiplication of two fractions involves finding a fraction of a fraction (e.g., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mfrac><mn>4</mn><mn>5</mn></mfrac></mrow></math> </ephtml> is the same as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math> </ephtml> of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>4</mn><mn>5</mn></mfrac></mrow></math> </ephtml> ). These procedures were covered only after the fraction concepts were taught using representations.</p> <p>The final set of lessons, Lessons 43 to 52, included material on adding and subtracting fractions greater than 1 (e.g., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>4</mn><mn>3</mn></mfrac></mrow></math> </ephtml> or <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></math> </ephtml> ). Thus, lessons focused on fractions less than 1 (e.g., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></math> </ephtml> ), fractions greater than or equal to 1 (e.g., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>4</mn><mn>3</mn></mfrac></mrow></math> </ephtml> or <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>5</mn><mn>5</mn></mfrac></mrow></math> </ephtml> ), and fractions that are represented with a whole number and a fraction (e.g., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></math> </ephtml> ). Throughout, other requisite skills with whole numbers were included (e.g., multiples and factors) to support solving fractions computation problems.</p> <p>Word problems were included throughout the intervention as fractions skills were taught. When solving word problems, instruction focused on understanding the story to determine which operation was appropriate for solving the problem. Students were then taught to set up an equation and solve it to find the answer. The scope and sequence of the ATM Fractions Intervention is presented in Supplemental Table 1 online.</p> <hd id="AN0154098903-14">Supports for building explanations</hd> <p>A total of 31 mathematical vocabulary words were taught explicitly during the intervention. Words were introduced and defined during instruction, and students were shown representations of each new word. For example, "relative size" and "magnitude" were introduced when Lesson 9 focused on using number lines to compare fractions. Students and interventionists developed vocabulary word walls as new words were introduced.</p> <p>Additional supports were provided for building explanations. The interventionist provided verbal prompts and encouraged students to refer to the vocabulary word wall as they worked on providing verbal and written explanations. Additionally, for written explanations, a strategic prompt card was used to help students maintain focus and write a thorough and mathematically accurate explanation of their work. The first two prompts reminded students to think about what the problem was asking and to reflect on how they approached solving the problem. The next two prompts asked students to write a description of their problem solving with accurate vocabulary and to justify their approach based on their mathematical understanding.</p> <hd id="AN0154098903-15">Representations</hd> <p>Number lines and Cuisenaire Rods, both linear representations, were used primarily to facilitate understanding of fractions concepts and procedures. Cuisenaire Rods (a concrete representation) were used to support students as they learned to represent fractions using the number line (a semiconcrete representation). These representations were presented concurrently with mathematical notation. Area models were used occasionally to clarify or demonstrate specific fractions concepts (e.g., part–whole relationship, multiplication of fractions). Overall, these representations were introduced, used iteratively, and faded over time. The goal was for students to solve problems without relying on the representations, but they were available to students at all times, if needed.</p> <hd id="AN0154098903-16">Interventionist training and supports</hd> <p>The 2-day training (8 hr per day) on the ATM Fractions Intervention was provided by one of the authors. The training on Day 1 covered the scope and sequence of the fractions content in the intervention, the recurring structure of each lesson, and how to navigate the lesson plans. Key representations in the intervention—Cuisenaire Rods and number lines—used to represent fractions concepts and procedures were covered thoroughly. Training also focused on using accurate mathematical language during instruction. Additionally, the training focused on techniques for prompting verbal and written explanations from students. A large portion of training (Day 2) was devoted to hands-on training. Practice activities focused on the often-challenging tasks for most interventionists: representing fractions accurately, explaining concepts using representations and in mathematically accurate language, and prompting and questioning students to guide their explanations rather than filling in their thinking. Interventionists role-played portions of lessons during these practice opportunities.</p> <p>Ongoing training included weekly meetings via conference calls for the first 3 weeks of the intervention. As the intervention progressed, group calls were held with interventionists once every 3 weeks, and individual calls were conducted on an as-needed basis. During these calls, lesson goals were reviewed, and interventionists posed questions, raised concerns, and discussed their experiences so that the research team could provide support. Audio recordings of intervention sessions were reviewed on an ongoing basis, and feedback was provided to interventionists as needed.</p> <hd id="AN0154098903-17">Comparison Condition</hd> <p>The comparison condition was business as usual (i.e., what the school typically provides in the way of additional instruction, informal support, or alternative intervention). Information on the comparison students' activities was obtained by surveying classroom mathematics teachers. The majority of students in the comparison condition received only general mathematics instruction with some informal support; they did not receive additional structured intervention in mathematics. Yet, a small percentage of teachers (17.24%) from one school district reported that students in the comparison group received either mathematics or reading intervention, with reading prioritized over mathematics, during an intervention period that was part of their response-to-intervention framework. Students moved into intervention based on testing data. Intervention was provided in large groups (approximately 15 students) and focused on reteaching core content. Intervention did not use a structured program.</p> <hd id="AN0154098903-18">General Classroom Mathematics Instruction</hd> <p>General classroom teachers were asked to complete a survey in fall and spring about the nature of their fractions instruction. Most teachers (89.66%) indicated that they used the district-adopted textbook (i.e., <emph>My Math</emph>, <emph>California Math</emph>, and <emph>GO Math!</emph>). Of those, 80.77% reported that they supplemented the textbook with materials that were either obtained online or developed by the district to address state standards. General fractions instruction began at different times across the schools and varied within districts. Over half of the teachers (51.72%) started teaching fractions in the fall (September to December). Close to a quarter of the teachers taught fractions beginning in the spring (24.14% started in January or February; 24.14 % started in March or April).</p> <p>Teachers reported using concrete manipulatives (75.86%), number lines (96.55%), and other visual representations (96.55%) when teaching fractions concepts and operations. Most of the teachers reported using number lines to teach fraction equivalence (96.55%). In contrast, fewer teachers used number lines to teach fraction magnitude (51.72%), fraction addition and subtraction (65.52%), and fraction multiplication and division (31.03%). Teachers also reported teaching students to solve fraction word problems by drawing a picture (100%), thinking about problem types (82.76%), making a table (65.52%), writing an equation (93.10%), and focusing on keywords (which is an outdated and ineffective method for teaching word problems; 100%).</p> <p>Very few teachers indicated using prompt cards as a support for solving problems and generating explanations of solutions (less than 7%). However, 93.1% of teachers did report that they require students to explain their solutions using mathematically correct language.</p> <hd id="AN0154098903-19">Measures</hd> <p></p> <hd id="AN0154098903-20">Wide Range Achievement Test 4 (WRAT-4) Math Computation subtest</hd> <p>The 40-item Math Computation subtest measures general mathematics achievement. It was administered at pretest only. Median reliabilities range from.83 to.87 ([<reflink idref="bib47" id="ref39">47</reflink>]).</p> <hd id="AN0154098903-21">TUF-4</hd> <p>The TUF-4 ([<reflink idref="bib22" id="ref40">22</reflink>]) was administered at pretest and posttest. It also served as the screener. The measure consists of 26 multiple-choice fractions items selected from publicly available National Assessment of Educational Progress (NAEP) measures ([<reflink idref="bib33" id="ref41">33</reflink>]), [<reflink idref="bib20" id="ref42">20</reflink>], and measures used by the Institute of Education Sciences Center for Improving Learning of Fractions. The items were reviewed by two research mathematicians involved in mathematics education for comprehensiveness and alignment with Grade 4 (and some Grade 3) Common Core State Standards for Mathematical Practice (CCSS-M) standards, the precision of mathematical language used, and the extent to which understanding of key mathematical ideas was addressed. The measure demonstrated a coefficient alpha reliability of.80 in the current study and.86 in a previous large-scale study ([<reflink idref="bib24" id="ref43">24</reflink>]).</p> <hd id="AN0154098903-22">Test for Understanding of Fractions, Fifth Grade (TUF-5)</hd> <p>The TUF-5 ([<reflink idref="bib23" id="ref44">23</reflink>]), used at posttest only, includes 18 items derived from NAEP and Partnership for Assessment of Readiness for College and Careers assessments ([<reflink idref="bib35" id="ref45">35</reflink>]). It was reviewed by two mathematics educators to ensure comprehensiveness and alignment with CCSS-M Grade 5 standards and precision of mathematical language. The internal consistency for the measure is.76 for the current at-risk sample and.85 for a sample that included both at-risk and not-at-risk students.</p> <hd id="AN0154098903-23">Test of Fraction Procedures</hd> <p>This 24-item measure, used as a posttest, was adapted from the measure developed by [<reflink idref="bib27" id="ref46">27</reflink>]. It assesses students' skill with the four arithmetic operations involving fractions. The adaptation focused on having the same number of items for each of the four operations and on revising some multiplication and division items so that the difficulty level of fractions items was at Grade 5. The measure has an internal consistency of.89. An abbreviated, 12-item version of this test, containing only the addition and subtraction items, was used as a pretest. The internal consistency for the pretest measure is.86.</p> <hd id="AN0154098903-24">Number Line Estimation (NLE)</hd> <p>Two measures, NLE 0–1 and NLE 0–2 ([<reflink idref="bib18" id="ref47">18</reflink>], adapted from [<reflink idref="bib43" id="ref48">43</reflink>]), were used to assess students' ability to place fractions on number lines with endpoints from 0 to 1 and from 0 to 2, respectively. NLE 0–1 was used as both a pretest and a posttest and included nine items, whereas the NLE 0–2 was used only as a posttest and included 19 items. Test-retest reliability for these measures is.80 ([<reflink idref="bib29" id="ref49">29</reflink>]).</p> <hd id="AN0154098903-25">Student explanation tasks</hd> <p>Five explanation tasks were administered at five points in time during the study. For these tasks, students had to solve a problem and provide a written explanation for their solution (i.e., their rationale). The five explanation tasks were on the following topics: estimating a fraction on a number line, ordering fractions on a number line, fraction addition word problems, fraction multiplication word problems, and fraction subtraction word problems. The research team developed a unique rubric to score each problem. Each rubric assessed the accuracy of the answer and the quality of the written explanation. The scoring rubrics for tasks involving a word problem included an additional section for scoring the selection of the correct operation to solve the problem. The maximum points that could be awarded varied across the five explanation tasks depending on the complexity of the problem being solved and the concepts that students would explain. The range in total possible points across the five explanation tasks is 5 to 9 points. Full points were awarded for a correct and simplified answer, and partial points were given for a correct answer that was not simplified.</p> <p>The research team considered what critical understandings the students should have to solve the problem (e.g., the rationale for solving the problem, what operation to use and why, and how students would execute the operation). For example, the scoring rubric for the written explanations for the fifth performance assessment (i.e., "Manny is training for a marathon. He ran <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>10</mn><mfrac><mn>2</mn><mn>5</mn></mfrac></mrow></math> </ephtml> miles on Saturday. On Sunday, he ran <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>7</mn><mfrac><mn>9</mn><mrow><mn>10</mn></mrow></mfrac></mrow></math> </ephtml> miles. How many more miles did Manny run on Saturday than Sunday?") allowed for a total of 5 points, 1 point for each of the following elements: (a) a student writes about changing <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>2</mn><mn>5</mn></mfrac></mrow></math> </ephtml> to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>4</mn><mrow><mn>10</mn></mrow></mfrac></mrow></math> </ephtml> or about common denominators, (b) a student presents a rationale for subtracting, (c) a student mentions not being able to subtract without regrouping, (d) a student explains the regrouping process, and (e) a student mentions simplifying the answer. Three of the scoring rubrics for tasks that involved a word problem included an additional section for selecting the correct operation to solve the problem.</p> <p>Two members of the research team used the rubrics to score all student responses. Given the moderate level of inference necessary for scoring the assessments, linear weighted kappa (κ) was calculated to assess interrater agreement between the two scorers ([<reflink idref="bib44" id="ref50">44</reflink>]). The mean interrater reliability was 98.45% for correct answer (κ = 0.94), 93.12% for appropriate written explanation (κ = 0.70), and 98.54% for correct operation (κ = 0.97).</p> <hd id="AN0154098903-26">Fidelity of Implementation</hd> <p>Fidelity for each intervention group was assessed using audio recordings of eight lessons that cover all phases of the intervention period (i.e., Lessons 4, 11, 20, 27, 31, 35, 43, and 47). These lessons were selected because they cover critical intervention topics and instructional approaches, including using representations and supporting explanations.</p> <p>Fidelity was assessed in two ways: procedural fidelity and quality of implementation. Procedural fidelity was assessed using checklists that were developed by the research team on the lessons' curricular content and included procedures and activities detailed in the lesson plans provided to the interventionists. See Supplemental Table 2 online for sample items.</p> <p>Fidelity checklists on average included 55 items (range = 34 to 82 items). Lessons that were dense or addressed a complex topic included more items than those lessons that covered less or relatively easier content material. For instance, Lesson 31, which covered the concept of fraction multiplication using area models, included 82 items.</p> <p>Each item was rated as observed or not observed. As the intervention was not scripted and interventionists were encouraged to adjust the lesson according to students' needs and understanding of concepts taught, our expectation was that interventionists would implement on average 80% of the activities and procedures in each lesson. Procedural fidelity was calculated as percentage of activities implemented (number of activities observed ÷ total number of activities [observed and not observed] × 100). On average, the interventionists completed 80.37% of the activities (median = 83.67%; range = 73.40%–91.40%). Often, lack of time to complete the lesson—especially complex, content-intense lessons—or consolidating an activity to create time for the next activity was the reason for unobserved fidelity items. Another potential reason was the use of a fine-grained checklist with an unscripted intervention, where interventionists followed the lesson plans rather than a scripted curriculum.</p> <p>Quality of implementation was assessed by rating the interventionists on a 5-point Likert scale (1 = <emph>low</emph>, 3 = <emph>moderate</emph>, and 5 = <emph>high</emph>) on qualities associated with engaging mathematics instruction: maintaining positive rapport with students, using precise mathematical language, supporting students' explanations, assessing students' grasp of material, pacing, and providing math-oriented feedback. The mean quality rating was 3.99 (median = 4.00, <emph>SD</emph> = 0.83), indicating that the quality of instruction provided by the interventionists was well above average. See Supplemental Table 3 online for means and standard deviations on each item.</p> <p>Interrater reliability was assessed on 13.10% of the sessions (<emph>n</emph> = 22) by having two raters independently assess for the procedural fidelity using exact agreement and for instructional quality using agreement within 1 point due to the level of inference required for those items. The mean interrater reliability for procedural fidelity was 81.57% (median = 81.03%). The mean interrater reliability for quality of implementation ratings was 75.97% (median = 85.71%).</p> <hd id="AN0154098903-27">Data Analysis</hd> <p></p> <hd id="AN0154098903-28">Primary analysis</hd> <p>For all outcomes, an analysis of covariance (ANCOVA) for partially nested data ([<reflink idref="bib1" id="ref51">1</reflink>]) was conducted, as intervention students were nested within small instructional groups, unlike the comparison-group students. The analytic models accounted for this potential heterogeneity of variance across conditions. All analytic models included the WRAT-4 and NLE 0–1 pretests as covariates. The statistical models were fit to the data using SAS PROC MIXED Version 14.2 ([<reflink idref="bib38" id="ref52">38</reflink>]) and restricted maximum-likelihood estimation. To control for false discovery, [<reflink idref="bib3" id="ref53">3</reflink>] procedures were used; the <emph>p</emph> values were adjusted for the seven tests of main effects. Sensitivity analyses were conducted and are described in the supplemental section online.</p> <hd id="AN0154098903-29">Mediation analysis</hd> <p>This analysis examined the potential mediating effect of improved number-line-estimating skills (as measured by the NLE 0–1 measure) on fractions achievement (as measured by TUF-4) with a correlated gain-score mediation model. The NLE 0–1 was identified as a potential mediator as it is a widely used and accepted measure of students' magnitude understanding of fractions (e.g., [<reflink idref="bib43" id="ref54">43</reflink>]) and has been shown to mediate fractions achievement ([<reflink idref="bib12" id="ref55">12</reflink>]; [<reflink idref="bib17" id="ref56">17</reflink>]). The analysis focused on the mediating effect on TUF-4, because it measured fractions knowledge at pre- and posttest and allowed for the calculation of a gain score. The gain score approximates growth ([<reflink idref="bib48" id="ref57">48</reflink>]), and the analysis of gains decreases the likelihood of bias associated with traditional mediation tests ([<reflink idref="bib30" id="ref58">30</reflink>]). The mediation analysis was conducted in Mplus ([<reflink idref="bib31" id="ref59">31</reflink>]) with bias-corrected bootstrapped confidence intervals based on 5,000 samples ([<reflink idref="bib19" id="ref60">19</reflink>]) for the estimate of the indirect path.</p> <hd id="AN0154098903-30">Moderation analysis</hd> <p>We tested for the following potential moderator variables: initial math knowledge (all pretest measures), district, gender, and free- and reduced-price-lunch status. We expanded the model to include each moderator and its interaction with the condition.</p> <hd id="AN0154098903-31">Results</hd> <p></p> <hd id="AN0154098903-32">Fractions Outcome Measures</hd> <p>Posttest means and standard deviations are presented in Table 2. Results of analyses appear in Table 3. Effect sizes (Hedges's <emph>g</emph>) ranged from 0.66 to 1.08, and <emph>p</emph> values were all less than.001 with the Benjamini-Hochberg correction, indicating that students who received the intervention scored significantly better than comparison students on all outcome measures.</p> <p>Graph</p> <p>Table 2. Posttest Means and Standard Deviations.</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="." /></colgroup><thead><tr><th /><th align="center" colspan="3">Intervention(<italic>n</italic> = 87)</th><th align="center" colspan="3">Comparison(<italic>n</italic> = 99)</th></tr><tr><th align="center">Posttest measure</th><th align="center">Unadjusted mean</th><th align="center">Adjusted mean</th><th align="center">Unadjusted <italic>SD</italic></th><th align="center">Unadjusted mean</th><th align="center">Adjusted mean</th><th align="center">Unadjusted <italic>SD</italic></th></tr></thead><tbody><tr><td>TUF-4</td><td>16.84</td><td>16.74</td><td>4.16</td><td>13.46</td><td>13.49</td><td>4.13</td></tr><tr><td>TUF-5</td><td>8.36</td><td>8.28</td><td>3.92</td><td>5.86</td><td>5.92</td><td>3.25</td></tr><tr><td>Test of Fraction Procedures</td><td>26.42</td><td>26.33</td><td>10.79</td><td>15.40</td><td>15.54</td><td>9.46</td></tr><tr><td>NLE 0–1</td><td>90.65</td><td>90.76</td><td>7.53</td><td>80.03</td><td>79.97</td><td>11.67</td></tr><tr><td>NLE 0–2</td><td>87.24</td><td>87.31</td><td>8.33</td><td>80.82</td><td>80.76</td><td>8.15</td></tr></tbody></table> </ephtml> </p> <p>4 <emph>Note</emph>. Total sample size, <emph>N</emph> = 186 students (intervention, <emph>n</emph> = 87; comparison, <emph>n</emph> = 99). Sample size for TUF-5, Test of Fraction Procedures (Full), NLE 0–1, and NLE 0–2 posttests is 185 students (intervention, <emph>n</emph> = 86; comparison, <emph>n</emph> = 99). TUF-4 = Test for Understanding of Fractions, Fourth Grade; TUF-5 = Test for Understanding of Fractions, Fifth Grade; NLE = Number Line Estimation.</p> <p>Graph</p> <p>Table 3. Results of a Partially Nested Mixed-Model Analysis of Covariance on Students' Posttests.</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="." /></colgroup><thead><tr><th align="left" colspan="2">Effect or Statistic</th><th align="center">TUF-4</th><th align="center">TUF-5</th><th align="center">Test of Fraction Procedures</th><th align="center">NLE 0–1</th><th align="center">NLE 0–2</th></tr></thead><tbody><tr><td rowspan="4">Fixed effects</td><td>Intercept</td><td>1.46(2.48)</td><td>−5.50<xref ref-type="table-fn" rid="tfn6">*</xref>(2.34)</td><td>−14.92<xref ref-type="table-fn" rid="tfn6">*</xref>(6.15)</td><td>63.77<xref ref-type="table-fn" rid="tfn6">**</xref>(6.22)</td><td>67.48<xref ref-type="table-fn" rid="tfn6">**</xref>(4.98)</td></tr><tr><td>Condition (intervention)</td><td>3.25<xref ref-type="table-fn" rid="tfn6">**</xref>(0.56)</td><td>2.36<xref ref-type="table-fn" rid="tfn6">**</xref>(0.46)</td><td>10.79<xref ref-type="table-fn" rid="tfn6">**</xref>(1.37)</td><td>10.79<xref ref-type="table-fn" rid="tfn6">**</xref>(1.20)</td><td>6.56<xref ref-type="table-fn" rid="tfn6">**</xref>(0.95)</td></tr><tr><td>WRAT-4 pretest</td><td>0.16<xref ref-type="table-fn" rid="tfn6">**</xref>(0.02)</td><td>0.14<xref ref-type="table-fn" rid="tfn6">**</xref>(0.02)</td><td>0.40<xref ref-type="table-fn" rid="tfn6">**</xref>(0.06)</td><td>0.27<xref ref-type="table-fn" rid="tfn6">**</xref>(0.06)</td><td>0.23<xref ref-type="table-fn" rid="tfn6">**</xref>(0.05)</td></tr><tr><td>NLE 0–1 pretest</td><td>0.14<xref ref-type="table-fn" rid="tfn6">**</xref>(0.02)</td><td>0.09<xref ref-type="table-fn" rid="tfn6">**</xref>(0.02)</td><td>0.31<xref ref-type="table-fn" rid="tfn6">**</xref>(0.05)</td><td>−0.37<xref ref-type="table-fn" rid="tfn6">**</xref>(0.05)</td><td>−0.35<xref ref-type="table-fn" rid="tfn6">**</xref>(0.04)</td></tr><tr><td rowspan="4">Variances</td><td>Intervention group intercept</td><td>2.87<xref ref-type="table-fn" rid="tfn6">*</xref>(1.31)</td><td>1.00(0.77)</td><td>16.95<xref ref-type="table-fn" rid="tfn6">*</xref>(7.13)</td><td>4.88(4.91)</td><td>0.75(3.77)</td></tr><tr><td>Residual</td><td>6.44<xref ref-type="table-fn" rid="tfn6">**</xref>(1.11)</td><td>6.90<xref ref-type="table-fn" rid="tfn6">**</xref>(.96)</td><td>39.33<xref ref-type="table-fn" rid="tfn6">**</xref>(6.33)</td><td /><td /></tr><tr><td>Intervention residual</td><td /><td /><td /><td>38.21<xref ref-type="table-fn" rid="tfn6">**</xref>(6.68)</td><td>44.57<xref ref-type="table-fn" rid="tfn6">**</xref>(7.69)</td></tr><tr><td>Comparison residual</td><td /><td /><td /><td>68.03<xref ref-type="table-fn" rid="tfn6">**</xref>(12.03)</td><td>31.24<xref ref-type="table-fn" rid="tfn6">**</xref>(5.94)</td></tr><tr><td>ICC</td><td>Intervention groups</td><td>.31</td><td>.13</td><td>.30</td><td>.11</td><td>.02</td></tr><tr><td>Hedges' <italic>g</italic></td><td>Condition</td><td>0.784</td><td>0.660</td><td>1.068</td><td>1.083</td><td>0.796</td></tr><tr><td><italic>p</italic> values</td><td>Condition</td><td><.0001</td><td><.0001</td><td><.0001</td><td><.0001</td><td><.0001</td></tr><tr><td>BH <italic>p</italic> values</td><td>Condition</td><td><.0001</td><td><.0001</td><td><.0001</td><td><.0001</td><td><.0001</td></tr><tr><td><italic>df</italic></td><td>Condition</td><td>47</td><td>61</td><td>55</td><td>66</td><td>49</td></tr><tr><td colspan="2">Likelihood ratio χ2</td><td>0.34</td><td>0.30</td><td>0.01</td><td>5.08</td><td>2.74</td></tr><tr><td colspan="2"><italic>p</italic> values</td><td>.558</td><td>.585</td><td>.919</td><td>.024</td><td>.098</td></tr></tbody></table> </ephtml> </p> <ulist> <item>5 <emph>Note</emph>. Total sample size, <emph>N</emph> = 186 students (intervention, <emph>n</emph> = 87; comparison, <emph>n</emph> = 99). Sample size for TUF-5, Test of Fraction Procedures, NLE 0–1, and NLE 0–2 is 185 students (intervention, <emph>n</emph> = 86; comparison, <emph>n</emph> = 99). Fixed effects and variances shown as parameter estimates with standard errors in parentheses. The models nested only intervention students within groups; comparison students were unclustered. ICCs estimated only for intervention students nested within instructional groups. The degrees of freedom for tests of condition effects were based on the Satterthwaite approximation. Likelihood ratio tests, at bottom, compared homoscedastic residuals with heteroscedastic residuals with a criterion α of.20 and 1 degree of freedom. TUF-4 = Test for Understanding of Fractions, Fourth Grade; TUF-5 = Test for Understanding of Fractions, Fifth Grade; NLE = Number Line Estimation; ICC = intraclass correlation; BH = Benjamini-Hochberg.</item> <item>6 <emph>p</emph> =.05. **<emph>p</emph> =.0001.</item> </ulist> <p>The effects on TUF-4, which covered essential foundational knowledge, and TUF-5, which covered grade-level fractions content, were 0.78 and 0.66, respectively. Results were even stronger on the Test of Fraction Procedures (<emph>g</emph> = 1.07). The effects on the two number-line-estimation assessments, NLE 0–1 and NLE 0–2, were 1.08 and 0.80, respectively.</p> <hd id="AN0154098903-33">Student Explanation Tasks</hd> <p>The randomly selected subsample of 35 intervention students outperformed the randomly selected subsample of 34 comparison students on all five explanation tasks (<emph>g</emph> = 0.68–1.23). The findings were statistically significant using the [<reflink idref="bib3" id="ref61">3</reflink>] correction for multiple comparisons (<emph>p</emph> =.000–.006). Explanations given by intervention students were rated as more thorough than those from comparison students. In addition, intervention students provided more coherent rationales for their solution methods and more details demonstrating their understanding of the procedures they employed. Intervention students used on average 2.02 relevant math vocabulary words (e.g., "common denominator," "simplify," "unit fraction"), compared with an average of 0.60 words used by comparison students. For example, when explaining a word problem that required addition of three addends (e.g., "On Saturday, Jessie walked <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math> </ephtml> of a mile to the park. Then, from the park she walked <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>4</mn><mrow><mn>10</mn></mrow></mfrac></mrow></math> </ephtml> of a mile to her friend Emily's house. Next, Jessie and Emily walked <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>7</mn><mrow><mn>10</mn></mrow></mfrac></mrow></math> </ephtml> of a mile. How far did Jessie walk on Saturday?"), students in the intervention group used correct mathematical terminology (e.g., "numerator," "equivalent"). In contrast, comparison students used pronouns (e.g., "it" and "that") that did not accurately show their understanding. Additionally, students in the intervention group were more likely to say why the denominators needed to be from the same unit before adding, whereas comparison students did not provide that as part of their explanations. See Figure 1 for a sample student explanation. For an additional example of student explanation, see Supplemental Figure 1 online.</p> <p> <emph>Intervention students provided more coherent rationales for their solution methods and more details demonstrating their understanding of the procedures they employed</emph> </p> <p>Graph: Figure 1. Sample student responses to an explanation task on ordering fractions.</p> <hd id="AN0154098903-34">Mediation Analysis</hd> <p>We tested whether the effects of condition on gains on TUF-4 were potentially mediated by gains on NLE 0–1. The direct effect of treatment condition on gains in TUF-4 was statistically significant: 3.34 [2.13, 4.55]. The indirect effect, which addresses whether NLE 0–1 mediated the effect of condition on gains in TUF-4 dropped to 0.84 and was no longer statistically significant (confidence bounds exclude zero). This mediation analysis suggests that after accounting for the indirect effects of condition on gains in fraction understanding through gains on the number-line-estimation measure, the direct effects of condition on gains in fraction understanding become statistically nonsignificant. The result is consistent with the hypothesis of mediation by number line improvement.</p> <hd id="AN0154098903-35">Moderation Analysis</hd> <p>The pretest score on WRAT-4 significantly moderated intervention effects on TUF-5 (<emph>p</emph> =.0264) and NLE 0–1 (<emph>p</emph> =.0132) posttest scores. Also, NLE 0–1 pretest performance significantly moderated the intervention effect on NLE 0–1 posttest (<emph>p</emph> =.0204). Higher WRAT-4 pretest scores were associated with a larger difference between intervention and comparison students' scores on the TUF-5 posttest. In contrast, lower pretest performance on WRAT-4 and NLE 0–1 was associated with a larger difference between intervention and comparison students' scores on the NLE 0–1 posttest. Free and reduced-price lunch significantly moderated intervention effects on the TUF-4 (<emph>p</emph> =.0374) and NLE 0–1 (<emph>p</emph> =.0400) posttests. The difference between treatment and comparison students' scores on the TUF-4 posttest was more pronounced among students who were not receiving free and reduced-price lunch; in contrast, on the NLE 0–1 posttest, the larger condition difference was due to students who were receiving free or reduced-priced lunch. Districts also significantly moderated intervention effects on student posttest performance on NLE 0–1. The difference between treatment and comparison students' NLE 0–1 posttest scores varied as a result of district membership (e.g., District 3 vs. 1 and 2; <emph>p</emph> =.0213). None of the other potential moderators significantly interacted with the intervention condition.</p> <hd id="AN0154098903-36">Correlation Between Fidelity and Student Outcomes</hd> <p>Both procedural fidelity and quality of implementation ratings were moderately and significantly correlated with outcomes related to knowledge of fractions, TUF-4, TUF-5, and the Test of Fraction Procedures (<emph>r</emph> =.38–.45, <emph>p</emph> <.001). Correlations between fidelity of implementation and NLE 0–2 were somewhat smaller but also statistically significant at <emph>p</emph> <.05 (<emph>r</emph> =.24 and.25 for procedural and quality fidelity of implementation, respectively).</p> <hd id="AN0154098903-37">Discussion</hd> <p>This randomized controlled trial examined the effect of a small-group fractions intervention on Grade 5 students with gaps in foundational fractions understandings. The intervention emphasized consistent use of the number line and activities to support students' explanations of mathematical ideas. Results indicated that the intervention students significantly outperformed students from the business-as-usual comparison condition on all measures of mathematics knowledge and proficiency (<emph>p</emph> <.001). Effect sizes (Hedges's <emph>g</emph>) were 0.78 and 0.66 on two measures assessing Grade 4 and Grade 5 general fractions knowledge and 1.07 on a measure of fractions operations. Thus, the fractions intervention had a significant effect on Grade 5 students' knowledge of challenging grade-level fractions material and crucial foundational material from Grade 4 on fractions as well as on their ability to perform arithmetic operations.</p> <hd id="AN0154098903-38">Crucial Role of the Number Line</hd> <p>Intervention students performed significantly better on number-line-estimation tasks (<emph>g</emph> = 1.08 for NLE 0–1; <emph>g</emph> = 0.80 for NLE 0–2). These measures assessed an aspect of fractions instruction stressed in virtually all contemporary state standards. Given the research demonstrating number line estimation as a key predictor of success in middle school mathematics ([<reflink idref="bib37" id="ref62">37</reflink>]) and also algebra ([<reflink idref="bib14" id="ref63">14</reflink>]; [<reflink idref="bib42" id="ref64">42</reflink>]), these positive effects indicated that there is at least a possibility that these students might experience some level of success as they advance to middle school. Also, as the intervention students were able to evaluate the relative magnitude of fractions with fairly high accuracy on the more difficult number line with 0 and 2 as endpoints, it is likely that their mental or abstract number line understanding has improved. This development of a sophisticated mental number line also suggests that the intervention students are more likely than the comparison students to be successful in advanced mathematics, such as algebra ([<reflink idref="bib42" id="ref65">42</reflink>]).</p> <p>The gains on TUF-4, the measure of foundational fractional knowledge, were mediated by gains in NLE 0–1, which replicated findings from [<reflink idref="bib13" id="ref66">13</reflink>]. This association is consistent with findings from previous studies suggesting that estimating magnitude on a number line is linked to better performance on fractions computation and word problems ([<reflink idref="bib12" id="ref67">12</reflink>]; [<reflink idref="bib45" id="ref68">45</reflink>]). Thus, increased time on estimating fractions magnitude on the number line may lead to increased understanding of general fractions knowledge.</p> <hd id="AN0154098903-39">Supporting Development of Students' Mathematical Explanations in Interventions</hd> <p>We hypothesized that the approach to written explanations—explicitly teaching students to generate explanations, using a strategic prompt to support and focus students on their problem-solving process, and emphasizing mathematically precise vocabulary—would lead to superior written explanations for students in the intervention group. Interventionists supported the development of written explanations by leading a discussion with each intervention group, after the written explanation activity concluded, to discuss quality aspects of each student's writing. The structured written explanation activities, in conjunction with the frequent questioning and opportunities for verbalizing their thinking and reasoning that occurred throughout the intervention, resulted in more thorough and higher-quality explanations for students participating in the intervention on the five explanation tasks (<emph>g</emph> = 0.68–1.23; <emph>p</emph> <.01). Explanations provided by intervention students included more mathematics vocabulary words and more details that demonstrated their understanding of their solution.</p> <hd id="AN0154098903-40">Moderators</hd> <p>Moderation analyses are only associations and thus cannot lead to causal inferences. However, they do suggest a potentially interesting pattern for the two major outcome measures. For TUF-5, a measure of fractions knowledge, higher student pretest scores on WRAT-4, a measure of general mathematics knowledge, were associated with larger condition differences favoring the intervention. In contrast, for NLE 0–1, a measure assessing relative fractions magnitude, lower student WRAT-4 pretest scores were associated with larger condition differences. One reason for this somewhat unusual pattern of findings could be that the NLE 0–1 is narrow in focus and assesses only understanding of relative fractions magnitude, whereas TUF-5 measures a full array of fractions topics, including computations and word problems.</p> <hd id="AN0154098903-41">Practical Implications</hd> <p>A vexing decision that interventionists and other service providers often have to make involves determining what content to teach during the limited intervention time in mathematics. Findings from this study indicate that intervention in Grade 5 can and should focus on challenging grade-level fractions content while also concurrently addressing the missing foundational material that is absolutely necessary for filling in gaps in learning.</p> <p>Focus on challenging grade-level fractions content while also concurrently addressing the missing foundational material that is absolutely necessary for filling in gaps in learning.</p> <p>Most curricula include a wide array of visual representations for teaching fractions, which can be overwhelming to teachers in choosing how to represent fractions concepts during instruction. Results from the study suggest that relying on a small set of continuous, linear representations (Cuisenaire Rods and number lines) when teaching fractions leads to improved student learning. The number line is of particular importance for teaching fractions concepts—a finding that is supported not only by this study but also by a long history of intervention research incorporating the number line to teach fractions, especially fractions magnitude (e.g., [<reflink idref="bib2" id="ref69">2</reflink>]; [<reflink idref="bib11" id="ref70">11</reflink>]; [<reflink idref="bib29" id="ref71">29</reflink>]). Students are likely to benefit from the consistent use of the number line to teach foundational fractions concepts as well as the four operations to solve computational problems.</p> <p>Virtually all contemporary standards ask students to explain their mathematical reasoning; yet, supporting the mathematical explanations of students who are experiencing difficulties can be especially challenging if they struggle with language in addition to mathematics. Furthermore, facilitating student explanations is seen as a time-consuming task that is often minimized to focus on competing demands and needs. Findings from this study suggest that teaching students how to explain and reason can be efficiently and successfully incorporated into the intervention routine by (a) offering frequent opportunities for students to provide both verbal and written explanations with support and feedback and (b) developing mathematical vocabulary visuals, like a word wall, to serve as a reminder of which words will enhance the precision of mathematical explanations.</p> <hd id="AN0154098903-42">Future Research</hd> <p>Given this study's consistent statistically significant findings, a logical next step would be to conduct conceptual replications (e.g., [<reflink idref="bib8" id="ref72">8</reflink>]). Several options come to mind. One option is to conduct the study with a variation in sample (e.g., focusing on students who were below the 15th percentile). This would help determine whether the intervention's instructional approaches, such as the strategies for eliciting student explanations, are effective for this population or whether further modifications are needed. Another option is to examine different instructional approaches for eliciting and supporting students' explanations within fractions intervention, given the potential for student growth in that area. Other intervention research may focus on expanding the use of the number line with other rational number topics (e.g., positive and negative integers) important for learning algebra or focus on mathematics topics that are emphasized less often in intervention, like geometry in middle school.</p> <hd id="AN0154098903-43">Limitations of the Study</hd> <p>The three districts in this study varied widely in family income, ethnicity, and school achievement. They also varied in their provision of mathematics intervention services. One district had a dedicated intervention block, where all students received additional intervention in either reading or mathematics, with priority given to reading. The other two districts provided informal supports during general mathematics instruction by differentiating instruction across learners. Thus, in only one of the three districts did students in the comparison condition receive consistent intervention provided by their schools, and it was not always in mathematics. Consequently, a limitation of the study is that the comparison students received a divergent range of formal and informal supports provided by their school. Thus, it is difficult to ascertain whether the positive effects were due to the instruction provided in the ATM Fractions Intervention or whether they were the result of receiving specialized small-group instruction in mathematics. Finally, as data on disability status were not available from one district, it is difficult to fully ascertain generalizability of results.</p> <hd id="AN0154098903-44">Concluding Remarks</hd> <p>Given the significant role of fractions in future learning of mathematics, providing intervention in fractions concepts in Grade 5 is critical. Study findings indicate that an explicit and systematic intervention focused on foundational and grade-level mathematics and that incorporates a small set of visual representations and supports students' mathematical explanations led to improved student outcomes in fractions.</p> <hd id="AN0154098903-45">Supplemental Material</hd> <p>sj-docx-1-ecx-10.1177_00144029211008851.docx</p> <p>sj-docx-1-ecx-10.1177_00144029211008851 – Supplemental material for Improving Struggling Fifth-Grade Students' Understanding of Fractions: A Randomized Controlled Trial of an Intervention That Stresses Both Concepts and Procedures</p> <p></p> <p>Supplemental material, sj-docx-1-ecx-10.1177_00144029211008851 for Improving Struggling Fifth-Grade Students' Understanding of Fractions: A Randomized Controlled Trial of an Intervention That Stresses Both Concepts and Procedures by Madhavi Jayanthi, Russell Gersten, Robin F. Schumacher, Joseph Dimino, Keith Smolkowski and Samantha Spallone in Exceptional Children</p> <p></p> <ref id="AN0154098903-46"> <title> References </title> <blist> <bibl id="bib1" idref="ref51" type="bt">1</bibl> <bibtext> Baldwin S. A., Bauer D. J., Stice E., Rohde P. (2011). Evaluating models for partially clustered designs. Psychological Methods, 16(2), 149–165. https://doi.org/10.1037/a0023464</bibtext> </blist> <blist> <bibl id="bib2" idref="ref8" type="bt">2</bibl> <bibtext> Barbieri C. A., Rodrigues J., Dyson N., Jordan N. C. (2020). Improving fraction understanding in sixth graders with mathematics difficulties: Effects of a number line approach combined with cognitive learning strategies. Journal of Educational Psychology, 112(3), 628–648. https://doi.org/10.1037/edu0000384</bibtext> </blist> <blist> <bibl id="bib3" idref="ref53" type="bt">3</bibl> <bibtext> Benjamini Y., Hochberg Y. (1995). Controlling the false discovery rate: A practical and powerful approach to multiple testing. Journal of the Royal Statistical Society–Series B (Methodological), 57(1), 289–300. <ulink href="http://www.jstor.org/stable/2346101">http://www.jstor.org/stable/2346101</ulink></bibtext> </blist> <blist> <bibl id="bib4" idref="ref13" type="bt">4</bibl> <bibtext> Booth J. L., Newton K. J., Twiss-Garrity L. K. (2014). The impact of fraction magnitude knowledge on algebra performance and learning. Journal of Experimental Child Psychology, 118, 110–118. https://doi.org/10.1016/j.jecp.2013.09.001</bibtext> </blist> <blist> <bibl id="bib5" idref="ref23" type="bt">5</bibl> <bibtext> Bottge B. A., Ma X., Gassaway L., Toland M. D., Butler M., Cho S. J. (2014). Effects of blended instructional models on math performance. Exceptional Children, 80(4), 423–437. https://doi.org/10.1177/0014402914527240</bibtext> </blist> <blist> <bibl id="bib6" idref="ref38" type="bt">6</bibl> <bibtext> California Department of Education. (2013). California Common Core State Standards: Mathematics. <ulink href="http://www.cde.ca.gov/be/st/ss/documents/ccssmathstandardaug2013.pdf">http://www.cde.ca.gov/be/st/ss/documents/ccssmathstandardaug2013.pdf</ulink></bibtext> </blist> <blist> <bibl id="bib7" idref="ref17" type="bt">7</bibl> <bibtext> Clarke B., Doabler C. T., Smolkowski K., Kurtz Nelson E., Fien H., Baker S. K., Kosty D. (2016). Testing the immediate and long-term efficacy of a Tier 2 kindergarten mathematics intervention. Journal of Research on Educational Effectiveness, 9(4), 607–634. https://doi.org/10.1080/19345747.2015.1116034</bibtext> </blist> <blist> <bibl id="bib8" idref="ref72" type="bt">8</bibl> <bibtext> Coyne M. D., Cook B. G., Therrien W. J. (2016). Recommendations for replication research in special education: A framework of systematic, conceptual replications. Remedial and Special Education, 37(4), 244–253. https://doi.org/10.1177/0741932516648463</bibtext> </blist> <blist> <bibl id="bib9" idref="ref20" type="bt">9</bibl> <bibtext> Dougherty B., Bryant D. P., Bryant B. R., Darrough R. L., Pfannenstiel K. H. (2015). Developing concepts and generalizations to build algebraic thinking: The reversibility, flexibility, and generalization approach. Intervention in School and Clinic, 50(5), 273–281. https://doi.org/10.1177/1053451214560892</bibtext> </blist> <blist> <bibtext> Fuchs L. S., Malone A. S., Preacher K. J., Fuchs D., Wang A. Y., Pachmayr R. (2019). Effects of fourth- and fifth-grade Super Solvers intervention on fraction magnitude understanding and calculation skill: A research report. Vanderbilt University.</bibtext> </blist> <blist> <bibtext> Fuchs L. S., Malone A. S., Schumacher R. F., Namkung J., Hamlett C. L., Jordan N. C., Siegler R. S., Gersten R., Changas P. (2016). Supported self-explaining during fraction intervention. Journal of Educational Psychology, 108(4), 493–508. https://doi.org/10.1037/edu0000073</bibtext> </blist> <blist> <bibtext> Fuchs L. S., Schumacher R. F., Long J., Namkung J., Hamlett C. L., Cirino P. T., Jordan N. C., Siegler R., Gersten R., Changas P. (2013). Improving at-risk learners' understanding of fractions. Journal of Educational Psychology, 105(3), 683–700. https://doi.org/10.1037/a0032446</bibtext> </blist> <blist> <bibtext> Fuchs L. S., Schumacher R. F., Sterba S. K., Long J., Namkung J., Malone A., Hamlett C. L., Jordan N. C., Gersten R., Siegler R. S., Changas P. (2014). Does working memory moderate the effects of fraction intervention? An aptitude-treatment interaction. Journal of Educational Psychology, 106(2), 499–514. https://doi.org/10.1037/a0034341</bibtext> </blist> <blist> <bibtext> Geary D. C., Hoard M. K., Nugent L., Bailey D. H. (2012). Mathematical cognition deficits in children with learning disabilities and persistent low achievement: A five-year prospective study. Journal of Educational Psychology, 104(1), 206–223. https://doi.org/10.1037/a0025398</bibtext> </blist> <blist> <bibtext> Gersten R., Chard D. J., Jayanthi M., Baker S. K., Morphy P., Flojo J. (2009). Mathematics instruction for students with learning disabilities: A meta-analysis of instructional components. Review of Educational Research, 79(3), 1202–1242. https://doi.org/10.3102/0034654309334431</bibtext> </blist> <blist> <bibtext> Gersten R., Rolfhus E., Clarke B., Decker L. E., Wilkins C., Dimino J. (2015). Intervention for first graders with limited number knowledge: Large-scale replication of a randomized controlled trial. American Educational Research Journal, 52(3), 516–546. https://doi.org/10.3102/0002831214565787</bibtext> </blist> <blist> <bibtext> Hamdan N., Gunderson E. A. (2017). The number line is a critical spatial-numerical representation: Evidence from a fraction intervention. Developmental Psychology, 53(3), 587–596. https://doi.org/10.1037/dev0000252</bibtext> </blist> <blist> <bibtext> Hamlett C. L., Schumacher R. S., Fuchs L. S. (2011). 2011 0–2 Number Line adapted from Siegler et al. (2011). Available from L. S. Fuchs, Vanderbilt University, Nashville, TN.</bibtext> </blist> <blist> <bibtext> Hayes A. F. (2009). Beyond Baron and Kenny: Statistical mediation analysis in the new millennium. Communication Monographs, 76(4), 408–420. https://doi.org/10.1080/03637750903310360</bibtext> </blist> <blist> <bibtext> Illustrative Mathematics. (2013). Illustrative mathematics. <ulink href="http://www.illustrativemathematics.org">http://www.illustrativemathematics.org</ulink></bibtext> </blist> <blist> <bibtext> Individuals With Disabilities Education Act, 20 U.S.C. §§ 1400 et seq. (2004).</bibtext> </blist> <blist> <bibtext> Instructional Research Group. (2014). Test for Understanding of Fractions, Fourth Grade (TUF-4).</bibtext> </blist> <blist> <bibtext> Instructional Research Group. (2015). Test for Understanding of Fractions, Fifth Grade (TUF-5).</bibtext> </blist> <blist> <bibtext> Jayanthi M., Gersten R., Taylor M. J., Smolkowski K., Dimino J. (2017). Impact of the Developing Mathematical Ideas professional development program on Grade 4 students' and teachers' understanding of fractions (REL 2017–256). U.S. Department of Education, Institute of Education Sciences, National Center for Education Evaluation and Regional Assistance, Regional Educational Laboratory Southeast. <ulink href="http://ies.ed.gov/ncee/edlabs">http://ies.ed.gov/ncee/edlabs</ulink></bibtext> </blist> <blist> <bibtext> Jitendra A. K., Dupuis D. N., Star J. R., Rodriguez M. C. (2016). The effects of schema-based instruction on the proportional thinking of students with mathematics difficulties with and without reading difficulties. Journal of Learning Disabilities, 49(4), 354–367. https://doi.org/10.1177/0022219414554228</bibtext> </blist> <blist> <bibtext> Jitendra A. K., Harwell M. R., Dupuis D. N., Karl S. R. (2017). A randomized trial of the effects of schema-based instruction on proportional problem-solving for students with mathematics problem-solving difficulties. Journal of Learning Disabilities, 50(3), 322–336. https://doi.org/10.1177/0022219416629646</bibtext> </blist> <blist> <bibtext> Jordan N. C., Hansen N., Fuchs L. S., Siegler R. S., Gersten R., Micklos D. (2013). Developmental predictors of fraction concepts and procedures. Journal of Experimental Child Psychology, 116(1), 45–58. https://doi.org/10.1016/j.jecp.2013.02.001</bibtext> </blist> <blist> <bibtext> Lahme B., McLeman C., Nakamaye M., Umland K. (2019). The number line: Unifying the evolving definition of number in K–12 mathematics. Notices of the American Mathematical Society, 66(9), 1465–1470. https://<ulink href="http://www.ams.org/journals/notices/201909/rnoti-p1465.pdf">www.ams.org/journals/notices/201909/rnoti-p1465.pdf</ulink></bibtext> </blist> <blist> <bibtext> Malone A. S., Fuchs L. S., Sterba S. K., Fuchs D., Foreman-Murray L. (2019). Does an integrated focus on fractions and decimals improve at-risk students' rational number magnitude performance? Contemporary Educational Psychology, 59. https://doi.org/10.1016/j.cedpsych.2019.101782</bibtext> </blist> <blist> <bibtext> Maxwell S. E., Cole D. A., Mitchell M. A. (2011). Bias in cross-sectional analyses of longitudinal mediation: Partial and complete mediation under an autoregressive model. Multivariate Behavioral Research, 46(5), 816–841. https://doi.org/10.1080/00273171.2011.606716</bibtext> </blist> <blist> <bibtext> Muthén L. K., Muthén B. O. (1998-2017). Mplus user's guide (8th ed.). https://<ulink href="http://www.statmodel.com/download/usersguide/MplusUserGuideVer%5f8.pdf">www.statmodel.com/download/usersguide/MplusUserGuideVer%5f8.pdf</ulink></bibtext> </blist> <blist> <bibtext> Namkung J. M., Fuchs L. S. (2016). Cognitive predictors of calculations and number line estimation with whole numbers and fractions among at-risk students. Journal of Educational Psychology, 108(2), 214–228. https://doi.org/10.1037/edu0000055</bibtext> </blist> <blist> <bibtext> National Center for Education Statistics. (2009). The nation's report card: Mathematics 2009 (NCES 2010–451). Department of Education, Institute of Education Sciences. <ulink href="http://nces.ed.gov/nationsreportcard/pdf/main2009/2010451.pdf">http://nces.ed.gov/nationsreportcard/pdf/main2009/2010451.pdf</ulink></bibtext> </blist> <blist> <bibtext> National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. U.S. Department of Education.</bibtext> </blist> <blist> <bibtext> Pearson Education. (2015). Mathematics practice tests. https://parcc.pearson.com/practice-tests/math/</bibtext> </blist> <blist> <bibtext> Powell S. R., Fuchs L. S. (2015). Intensive intervention in mathematics. Learning Disabilities Research & Practice, 30(4), 182–192. https://doi.org/10.1111/ldrp.12087</bibtext> </blist> <blist> <bibtext> Resnick I., Jordan N. C., Hansen N., Rajan V., Rodrigues J., Siegler R. S., Fuchs L. S. (2016). Developmental growth trajectories in understanding of fraction magnitude from fourth through sixth grade. Developmental Psychology, 52(5), 746–757. https://doi.org/10.1037/dev0000102</bibtext> </blist> <blist> <bibtext> SAS Institute. (2016). SAS/STAT (Version 14.2). <ulink href="http://support.sas.com/documentation/index.html">http://support.sas.com/documentation/index.html</ulink></bibtext> </blist> <blist> <bibtext> Schumacher R. F., Jayanthi M., Gersten R., Dimino J., Spallone S., Haymond K. (2018). Using the number line to promote understanding of fractions for struggling fifth graders: A formative pilot study. Learning Disabilities Research & Practice, 33(4), 192–206. https://doi.org/10.1111/ldrp.12169</bibtext> </blist> <blist> <bibtext> Schumacher R. F., Malone A. S. (2017). Error patterns with fraction calculations at fourth grade as a function of students' mathematics achievement status. Elementary School Journal, 118(1), 105–127. https://doi.org/10.1086/692914</bibtext> </blist> <blist> <bibtext> Shaughnessy M., Boerst T., Ball D. L. (2015). Simulating teaching: New possibilities for assessing teaching practice. In Bartell T. G., Bieda K. N., Putnam R. T., Bradfield K., Dominguez H. (Eds.), Proceedings of the 37th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 924–927). Michigan State University.</bibtext> </blist> <blist> <bibtext> Siegler R. S., Duncan G. J., Davis-Kean P. E., Duckworth K., Claessens A., Engel M., Susperreguy M. I., Chen M. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23(7), 691–697. https://doi.org/10.1177/0956797612440101</bibtext> </blist> <blist> <bibtext> Siegler R. S., Thompson C. A., Schneider M. (2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62(4), 273–296. https://doi.org/10.1016/j.cogpsych.2011.03.001</bibtext> </blist> <blist> <bibtext> Sim J., Wright C. C. (2005). The kappa statistic in reliability studies: Use, interpretation, and sample size requirements. Physical Therapy, 85(3), 257–268. https://doi.org/10.1093/ptj/85.3.257</bibtext> </blist> <blist> <bibtext> Tian J., Siegler R. S. (2017). Fractions learning in children with mathematics difficulties. Journal of Learning Disabilities, 50(6), 614–620. https://doi.org/10.1177/0022219416662032</bibtext> </blist> <blist> <bibtext> U.S. Department of Education, Institute of Education Sciences, & What Works Clearinghouse. (2020). What Works Clearinghouse: Standards handbook (Version 4.1). https://ies.ed.gov/ncee/wwc/Handbooks</bibtext> </blist> <blist> <bibtext> Wilkinson G. S., Robertson G. J. (2006). WRAT4: Wide Range Achievement Test. NCS Pearson.</bibtext> </blist> <blist> <bibtext> Willett J. B. (1988). Questions and answers in the measurement of change. Review of Research in Education, 15, 345–422. https://doi.org/10.2307/1167368</bibtext> </blist> <blist> <bibtext> Woodward J., Stroh M. (2015). TransMath: Fluency with rational numbers (3rd ed.). Voyager Sopris Learning.</bibtext> </blist> </ref> <ref id="AN0154098903-47"> <title> Footnotes </title> <blist> <bibtext> This project was funded by the National Science Foundation (Grant No. DRL-1535214). The authors would like to thank Karen Karp, who provided ongoing counsel throughout the project; Drew Bailey and Francis (Skip) Fennell for their fresh perspectives and guidance during the study; and Kristin Umland, Jim Lewis, and Kim Paulsen for their review of the outcome measures. In addition, the authors would like to thank Kelly Haymond, Pamela Foremski, and Christopher Tran for assisting with the research and preparation of the report.</bibtext> </blist> <blist> <bibtext> Supplemental material for this article is available online.</bibtext> </blist> </ref> <aug> <p>By Madhavi Jayanthi; Russell Gersten; Robin F. Schumacher; Joseph Dimino; Keith Smolkowski and Samantha Spallone</p> <p>Reported by Author; Author; Author; Author; Author; Author</p> </aug> <nolink nlid="nl1" bibid="bib34" firstref="ref1"></nolink> <nolink nlid="nl2" bibid="bib42" firstref="ref2"></nolink> <nolink nlid="nl3" bibid="bib40" firstref="ref4"></nolink> <nolink nlid="nl4" bibid="bib32" firstref="ref5"></nolink> <nolink nlid="nl5" bibid="bib37" firstref="ref6"></nolink> <nolink nlid="nl6" bibid="bib28" firstref="ref7"></nolink> <nolink nlid="nl7" bibid="bib11" firstref="ref9"></nolink> <nolink nlid="nl8" bibid="bib29" firstref="ref10"></nolink> <nolink nlid="nl9" bibid="bib12" firstref="ref11"></nolink> <nolink nlid="nl10" bibid="bib45" firstref="ref12"></nolink> <nolink nlid="nl11" bibid="bib17" firstref="ref14"></nolink> <nolink nlid="nl12" bibid="bib13" firstref="ref15"></nolink> <nolink nlid="nl13" bibid="bib15" firstref="ref16"></nolink> <nolink nlid="nl14" bibid="bib10" firstref="ref19"></nolink> <nolink nlid="nl15" bibid="bib41" firstref="ref21"></nolink> <nolink nlid="nl16" bibid="bib21" firstref="ref22"></nolink> <nolink nlid="nl17" bibid="bib16" firstref="ref25"></nolink> <nolink nlid="nl18" bibid="bib26" firstref="ref26"></nolink> <nolink nlid="nl19" bibid="bib25" firstref="ref29"></nolink> <nolink nlid="nl20" bibid="bib49" firstref="ref30"></nolink> <nolink nlid="nl21" bibid="bib22" firstref="ref31"></nolink> <nolink nlid="nl22" bibid="bib24" firstref="ref32"></nolink> <nolink nlid="nl23" bibid="bib36" firstref="ref33"></nolink> <nolink nlid="nl24" bibid="bib46" firstref="ref34"></nolink> <nolink nlid="nl25" bibid="bib39" firstref="ref36"></nolink> <nolink nlid="nl26" bibid="bib47" firstref="ref39"></nolink> <nolink nlid="nl27" bibid="bib33" firstref="ref41"></nolink> <nolink nlid="nl28" bibid="bib20" firstref="ref42"></nolink> <nolink nlid="nl29" bibid="bib23" firstref="ref44"></nolink> <nolink nlid="nl30" bibid="bib35" firstref="ref45"></nolink> <nolink nlid="nl31" bibid="bib27" firstref="ref46"></nolink> <nolink nlid="nl32" bibid="bib18" firstref="ref47"></nolink> <nolink nlid="nl33" bibid="bib43" firstref="ref48"></nolink> <nolink nlid="nl34" bibid="bib44" firstref="ref50"></nolink> <nolink nlid="nl35" bibid="bib38" firstref="ref52"></nolink> <nolink nlid="nl36" bibid="bib48" firstref="ref57"></nolink> <nolink nlid="nl37" bibid="bib30" firstref="ref58"></nolink> <nolink nlid="nl38" bibid="bib31" firstref="ref59"></nolink> <nolink nlid="nl39" bibid="bib19" firstref="ref60"></nolink> <nolink nlid="nl40" bibid="bib14" firstref="ref63"></nolink>
Header DbId: eric
DbLabel: ERIC
An: EJ1320419
AccessLevel: 3
PubType: Academic Journal
PubTypeId: academicJournal
PreciseRelevancyScore: 0
IllustrationInfo
Items – Name: Title
  Label: Title
  Group: Ti
  Data: Improving Struggling Fifth-Grade Students' Understanding of Fractions: A Randomized Controlled Trial of an Intervention That Stresses Both Concepts and Procedures
– Name: Language
  Label: Language
  Group: Lang
  Data: English
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Jayanthi%2C+Madhavi%22">Jayanthi, Madhavi</searchLink><br /><searchLink fieldCode="AR" term="%22Gersten%2C+Russell%22">Gersten, Russell</searchLink><br /><searchLink fieldCode="AR" term="%22Schumacher%2C+Robin+F%2E%22">Schumacher, Robin F.</searchLink><br /><searchLink fieldCode="AR" term="%22Dimino%2C+Joseph%22">Dimino, Joseph</searchLink><br /><searchLink fieldCode="AR" term="%22Smolkowski%2C+Keith%22">Smolkowski, Keith</searchLink><br /><searchLink fieldCode="AR" term="%22Spallone%2C+Samantha%22">Spallone, Samantha</searchLink>
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="SO" term="%22Exceptional+Children%22"><i>Exceptional Children</i></searchLink>. Oct 2021 88(1):81-100.
– Name: Avail
  Label: Availability
  Group: Avail
  Data: SAGE Publications. 2455 Teller Road, Thousand Oaks, CA 91320. Tel: 800-818-7243; Tel: 805-499-9774; Fax: 800-583-2665; e-mail: journals@sagepub.com; Web site: http://sagepub.com
– Name: PeerReviewed
  Label: Peer Reviewed
  Group: SrcInfo
  Data: Y
– Name: Pages
  Label: Page Count
  Group: Src
  Data: 20
– Name: DatePubCY
  Label: Publication Date
  Group: Date
  Data: 2021
– Name: SourceSuprt
  Label: Sponsoring Agency
  Group: SrcSuprt
  Data: National Science Foundation (NSF)
– Name: NumberContract
  Label: Contract Number
  Group: NumCntrct
  Data: DRL1535214
– Name: TypeDocument
  Label: Document Type
  Group: TypDoc
  Data: Journal Articles<br />Reports - Research
– Name: Audience
  Label: Education Level
  Group: Audnce
  Data: <searchLink fieldCode="EL" term="%22Elementary+Education%22">Elementary Education</searchLink><br /><searchLink fieldCode="EL" term="%22Grade+5%22">Grade 5</searchLink><br /><searchLink fieldCode="EL" term="%22Intermediate+Grades%22">Intermediate Grades</searchLink><br /><searchLink fieldCode="EL" term="%22Middle+Schools%22">Middle Schools</searchLink>
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Grade+5%22">Grade 5</searchLink><br /><searchLink fieldCode="DE" term="%22Elementary+School+Students%22">Elementary School Students</searchLink><br /><searchLink fieldCode="DE" term="%22Fractions%22">Fractions</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+Concepts%22">Mathematical Concepts</searchLink><br /><searchLink fieldCode="DE" term="%22Concept+Formation%22">Concept Formation</searchLink><br /><searchLink fieldCode="DE" term="%22Direct+Instruction%22">Direct Instruction</searchLink><br /><searchLink fieldCode="DE" term="%22Instructional+Effectiveness%22">Instructional Effectiveness</searchLink><br /><searchLink fieldCode="DE" term="%22Intervention%22">Intervention</searchLink><br /><searchLink fieldCode="DE" term="%22Difficulty+Level%22">Difficulty Level</searchLink><br /><searchLink fieldCode="DE" term="%22Achievement+Tests%22">Achievement Tests</searchLink><br /><searchLink fieldCode="DE" term="%22Problem+Solving%22">Problem Solving</searchLink>
– Name: SubjectThesaurus
  Label: Assessment and Survey Identifiers
  Group: Su
  Data: <searchLink fieldCode="SU" term="%22Wide+Range+Achievement+Test%22">Wide Range Achievement Test</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1177/00144029211008851
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 0014-4029
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Using a randomized controlled trial, we examined the effect of a fractions intervention for students experiencing mathematical difficulties in Grade 5. Students who were eligible for the study (n = 205) were randomly assigned to intervention and comparison conditions, blocked by teacher. The intervention used systematic, explicit instruction and relied on linear representations (e.g., Cuisenaire Rods and number lines) to demonstrate key fractions concepts. Enhancing students' mathematical explanations was also a focus. Results indicated that intervention students significantly outperformed students from the comparison condition on measures of fractions proficiency and understanding (g = 0.66-0.78), number line estimation (g = 0.80-1.08), fractions procedures (g = 1.07), and explanation tasks (g = 0.68-1.23). Findings suggest that interventions designed to include explicit instruction, along with consistent use of the number line and opportunities to explain reasoning, can promote students' proficiency and understanding of fractions.
– Name: AbstractInfo
  Label: Abstractor
  Group: Ab
  Data: As Provided
– Name: DateEntry
  Label: Entry Date
  Group: Date
  Data: 2022
– Name: AN
  Label: Accession Number
  Group: ID
  Data: EJ1320419
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1320419
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1177/00144029211008851
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 20
        StartPage: 81
    Subjects:
      – SubjectFull: Grade 5
        Type: general
      – SubjectFull: Elementary School Students
        Type: general
      – SubjectFull: Fractions
        Type: general
      – SubjectFull: Mathematical Concepts
        Type: general
      – SubjectFull: Concept Formation
        Type: general
      – SubjectFull: Direct Instruction
        Type: general
      – SubjectFull: Instructional Effectiveness
        Type: general
      – SubjectFull: Intervention
        Type: general
      – SubjectFull: Difficulty Level
        Type: general
      – SubjectFull: Achievement Tests
        Type: general
      – SubjectFull: Problem Solving
        Type: general
      – SubjectFull: Wide Range Achievement Test
        Type: general
    Titles:
      – TitleFull: Improving Struggling Fifth-Grade Students' Understanding of Fractions: A Randomized Controlled Trial of an Intervention That Stresses Both Concepts and Procedures
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Jayanthi, Madhavi
      – PersonEntity:
          Name:
            NameFull: Gersten, Russell
      – PersonEntity:
          Name:
            NameFull: Schumacher, Robin F.
      – PersonEntity:
          Name:
            NameFull: Dimino, Joseph
      – PersonEntity:
          Name:
            NameFull: Smolkowski, Keith
      – PersonEntity:
          Name:
            NameFull: Spallone, Samantha
    IsPartOfRelationships:
      – BibEntity:
          Dates:
            – D: 01
              M: 10
              Type: published
              Y: 2021
          Identifiers:
            – Type: issn-print
              Value: 0014-4029
          Numbering:
            – Type: volume
              Value: 88
            – Type: issue
              Value: 1
          Titles:
            – TitleFull: Exceptional Children
              Type: main
ResultId 1