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Unique and Combined Effects of Small-Group Fraction Vocabulary and Arithmetic Interventions for Students with Mathematical Difficulties

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Title: Unique and Combined Effects of Small-Group Fraction Vocabulary and Arithmetic Interventions for Students with Mathematical Difficulties
Language: English
Authors: Xin Lin (ORCID 0000-0002-8077-5134), Juehang Zhang, Haorui Cui (ORCID 0009-0006-9332-1023), Xiuwen Song, Xiaonan Han
Source: Journal of Learning Disabilities. 2026 59(3):161-171.
Availability: SAGE Publications and Hammill Institute on Disabilities. 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: https://sagepub.com
Peer Reviewed: Y
Page Count: 11
Publication Date: 2026
Document Type: Journal Articles
Reports - Research
Education Level: Elementary Education
Grade 4
Intermediate Grades
Descriptors: Fractions, Mathematics Instruction, Grade 4, Elementary School Students, Vocabulary Development, Intervention, Arithmetic, Learning Problems, Program Effectiveness
DOI: 10.1177/00222194251342191
ISSN: 0022-2194
1538-4780
Abstract: The purpose of this study was to assess the effects of a fraction vocabulary intervention with fraction arithmetic components on fraction vocabulary knowledge and fraction arithmetic competencies among fourth-grade Chinese students with mathematics difficulties. We randomly assigned 70 students with mathematics difficulties to three conditions: fraction vocabulary only (n = 23), fraction vocabulary with an arithmetic component (n = 23), and a business-as-usual (BaU) condition (n = 24). The students in the fraction vocabulary intervention conditions participated in 10 sessions, occurring three times per week. Students within both intervention conditions showed significantly better performance in fraction vocabulary knowledge than those in the BaU condition. However, no notable distinctions were observed between the two intervention conditions in terms of fraction arithmetic. Only students who received the fraction vocabulary intervention with an arithmetic component exhibited enhanced performance in subtraction with like denominators compared to the BaU condition.
Abstractor: As Provided
Entry Date: 2026
Accession Number: EJ1502936
Database: ERIC
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  Value: <anid>AN0192937327;led01may.26;2026Apr14.06:50;v2.2.500</anid> <title id="AN0192937327-1">Unique and Combined Effects of Small-Group Fraction Vocabulary and Arithmetic Interventions for Students With Mathematical Difficulties </title> <p>The purpose of this study was to assess the effects of a fraction vocabulary intervention with fraction arithmetic components on fraction vocabulary knowledge and fraction arithmetic competencies among fourth-grade Chinese students with mathematics difficulties. We randomly assigned 70 students with mathematics difficulties to three conditions: fraction vocabulary only (n = 23), fraction vocabulary with an arithmetic component (n = 23), and a business-as-usual (BaU) condition (n = 24). The students in the fraction vocabulary intervention conditions participated in 10 sessions, occurring three times per week. Students within both intervention conditions showed significantly better performance in fraction vocabulary knowledge than those in the BaU condition. However, no notable distinctions were observed between the two intervention conditions in terms of fraction arithmetic. Only students who received the fraction vocabulary intervention with an arithmetic component exhibited enhanced performance in subtraction with like denominators compared to the BaU condition.</p> <p>Keywords: fractions; intervention; mathematics difficulties; mathematics vocabulary; China</p> <p>Recent research has emphasized the importance of understanding mathematics vocabulary for developing mathematical proficiency ([<reflink idref="bib14" id="ref1">14</reflink>]; [<reflink idref="bib22" id="ref2">22</reflink>]; [<reflink idref="bib24" id="ref3">24</reflink>]; [<reflink idref="bib26" id="ref4">26</reflink>]). According to U.S. mathematics standards ([<reflink idref="bib19" id="ref5">19</reflink>]), students are expected to effectively communicate their ideas, articulate problem-solving approaches, construct sound arguments, and evaluate mathematical reasoning using clear and precise vocabulary. In China, the 2022 edition of the mathematics curriculum standard also emphasizes mathematics vocabulary as a crucial bridge connecting abstract mathematical concepts to real-world applications ([<reflink idref="bib17" id="ref6">17</reflink>]). Without a solid understanding of mathematics vocabulary, students may struggle to comprehend word problems, interpret mathematical symbols, and effectively communicate their mathematical reasoning ([<reflink idref="bib12" id="ref7">12</reflink>]; [<reflink idref="bib30" id="ref8">30</reflink>]). Therefore, a strong grasp of mathematics vocabulary is vital for students' success in mathematics.</p> <hd id="AN0192937327-2">Importance and Difficulty of Fraction Vocabulary</hd> <p>However, it is important to acknowledge that many students, especially those with mathematical difficulties (MD), encounter difficulties in acquiring mathematics vocabulary ([<reflink idref="bib7" id="ref9">7</reflink>]; [<reflink idref="bib13" id="ref10">13</reflink>]). As students progress through grade levels, the complexity of mathematics knowledge increases, which in turn leads to increase in the complexity and specificity of mathematics vocabulary ([<reflink idref="bib15" id="ref11">15</reflink>]). The accumulation of mathematics vocabulary across grade levels further compounds the difficulties faced by students with MD. This highlights the need for targeted interventions and instructional approaches that address the specific challenges faced by students with MD in learning increasingly abstract mathematics concepts and corresponding mathematics vocabulary at later grades ([<reflink idref="bib15" id="ref12">15</reflink>]).</p> <p>One of the mathematics domains that can be particularly challenging for many students at late elementary grades is fractions, largely due to the inherent abstractness associated with fractions ([<reflink idref="bib18" id="ref13">18</reflink>]; [<reflink idref="bib28" id="ref14">28</reflink>]). However, understanding fractions is essential for learning more advanced mathematics and success in the workforce ([<reflink idref="bib3" id="ref15">3</reflink>]; [<reflink idref="bib20" id="ref16">20</reflink>]; [<reflink idref="bib29" id="ref17">29</reflink>]; [<reflink idref="bib33" id="ref18">33</reflink>]). More alarmingly, a limited understanding of fractions is especially prevalent among students with MD who also experience challenges with whole-number concepts and operations ([<reflink idref="bib1" id="ref19">1</reflink>]). Therefore, there is a clear need to teach mathematics vocabulary specific to fractions to support students with MD ([<reflink idref="bib14" id="ref20">14</reflink>]).</p> <hd id="AN0192937327-3">Conceptual and Procedural Vocabulary in Fractions May Require Distinct Instructional Approach...</hd> <p>Although fractions as a domain are inherently abstract ([<reflink idref="bib18" id="ref21">18</reflink>]), some vocabulary terms within this domain are even more abstract than others ([<reflink idref="bib15" id="ref22">15</reflink>]). In this study, we distinguished procedural and conceptual vocabulary in fraction learning. Procedural vocabulary can be considered more abstract than conceptual vocabulary because it often refers to specific steps, processes, or rules that require an understanding of the underlying logic of mathematical procedures, rather than just recognizing terms or ideas ([<reflink idref="bib4" id="ref23">4</reflink>]; [<reflink idref="bib27" id="ref24">27</reflink>]). Procedural terms like <emph>regrouping</emph> or <emph>least common denominator</emph> represent complex, multi-step processes that may not have tangible, everyday references and often rely on prior mathematical knowledge to understand fully ([<reflink idref="bib15" id="ref25">15</reflink>]). In contrast, conceptual vocabulary includes terms like <emph>fractions</emph> or <emph>variable</emph>, which are generally more straightforward in meaning and often easier to connect with visual models or real-world concepts ([<reflink idref="bib4" id="ref26">4</reflink>]; [<reflink idref="bib27" id="ref27">27</reflink>]).</p> <p>As a result, students may find it easier to connect conceptual vocabulary to visual representations, making these terms less abstract than procedural vocabulary ([<reflink idref="bib27" id="ref28">27</reflink>]). Procedural vocabulary requires students to mentally reconstruct detailed steps and understand the underlying logic, often without visual or contextual support ([<reflink idref="bib4" id="ref29">4</reflink>]; [<reflink idref="bib16" id="ref30">16</reflink>]). This added complexity requires a higher level of abstraction, which can be challenging—especially if a single instructional method, like a vocabulary grid, is used to teach all fraction vocabulary terms ([<reflink idref="bib11" id="ref31">11</reflink>]). Although a vocabulary grid that systematically organizes fraction vocabulary with definitions, examples, non-examples, and practice problems provides students with a clear framework to support their learning ([<reflink idref="bib14" id="ref32">14</reflink>]; [<reflink idref="bib30" id="ref33">30</reflink>]), teaching procedural vocabulary may need additional instruction.</p> <hd id="AN0192937327-4">Teach Procedural Vocabulary in Fractions</hd> <p>If procedural vocabulary is not reinforced with the actual practice of related arithmetic operations, students may resort to rote memorization rather than developing true understanding ([<reflink idref="bib31" id="ref34">31</reflink>]). For example, when teaching the phrase—<emph>reduction to common denominator</emph>, explaining the step-by-step process of adding fractions with unlike denominators could ensure deeper understanding. Moreover, such step-by-step instruction could improve students' comprehension of various related fraction vocabulary. For instance, if we have to add <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>3</mn><mn>8</mn></mfrac><mo>,</mo></mrow></math> </ephtml> students need to understand the procedural vocabulary of <emph>least common denominator</emph> (in this case, 8), <emph>converting the fractions</emph> to have the same <emph>denominator</emph>, adding the <emph>numerators</emph>, and simplifying the fraction to <emph>the simplest form</emph> if necessary.</p> <p>We decided to teach three distinct arithmetic operations in fractions and to spread the instruction for these operations across three lessons when covering procedural vocabulary. In the lesson about <emph>like fractions</emph> and <emph>unlike fractions</emph>, we taught adding and subtracting like fractions while explaining the concept of <emph>like fractions</emph>. In the lesson that covered <emph>common denominators</emph> and <emph>reduction to common denominators</emph>, we taught adding and subtracting unlike fractions while discussing the process of finding <emph>common denominators</emph>. In the lesson on <emph>reciprocals</emph>, we taught multiplication and division of fractions. Although various fraction vocabulary terms may be involved in explaining the fraction arithmetic operations, we focused on these procedural vocabulary terms because we need to integrate the instruction of arithmetic operations into the fraction vocabulary instruction routine.</p> <p>Furthermore, teaching fraction vocabulary alongside step-by-step illustrations of relevant arithmetic operation procedures could directly enhance fraction arithmetic abilities. To date, no intervention with a primary focus on mathematics vocabulary has examined the combined effects of incorporating instruction on associated arithmetic operations within a mathematics vocabulary intervention. This study aims to investigate whether teaching procedural vocabulary alongside the corresponding arithmetic operations leads to greater improvements in both procedural vocabulary and arithmetic skills than teaching conceptual and procedural vocabulary using only a vocabulary grid.</p> <hd id="AN0192937327-5">Transfer to Fraction Arithmetic: Potential Difference Between the United States and China</hd> <p>Due to the differences in mathematics teaching between the United States and China, students' responses to fraction vocabulary intervention may vary. Specifically, previous research involving U.S. students suggested that a better understanding of mathematics vocabulary can lead to improved mathematical performance (e.g., [<reflink idref="bib14" id="ref35">14</reflink>]; [<reflink idref="bib25" id="ref36">25</reflink>], [<reflink idref="bib26" id="ref37">26</reflink>]). For example, [<reflink idref="bib14" id="ref38">14</reflink>] focused on U.S. fourth-grade students with MD and revealed that an improved understanding of fraction vocabulary may lead to enhanced skills in fraction addition and subtraction with the same denominator. However, the relationship may not apply to Chinese students. According to U.S. mathematics standards, the learning process is gradual, starting with basic fraction concepts in Grades 1 and 2, and introducing more complex operations by Grade 3 ([<reflink idref="bib19" id="ref39">19</reflink>]). U.S. students also learn to visualize concepts such as "<emph>like fraction</emph>s" as adding fractions with the same denominator, helping them understand how to apply these concepts to fraction arithmetic. As U.S. students build procedural skills through conceptual understanding, they are better able to translate their grasp of fraction vocabulary into mathematical operations involving fractions.</p> <p>In contrast, China's mathematics curriculum places greater emphasis on practical applications and contextual operations, typically introducing decimals and fractions in Grades 3 and 4 ([<reflink idref="bib17" id="ref40">17</reflink>]). Chinese students often acquire procedural skills through direct, step-by-step instruction and repetitive practice ([<reflink idref="bib5" id="ref41">5</reflink>]). For example, they learn mnemonic expressions such as "When adding or subtracting fractions with the same denominator, the numerators of the fractions are added while keeping the denominator unchanged." Since there is a lack of connection from conceptual understanding to procedural skills, improved knowledge of fraction vocabulary may not necessarily lead to enhanced fraction arithmetic skills among Chinese students. In this study, we implement a fraction vocabulary intervention with Chinese students to explore whether improved mathematics vocabulary can lead to enhanced mathematics performance.</p> <hd id="AN0192937327-6">The Present Study</hd> <p>The purpose of this study was to investigate the effect of a fraction vocabulary intervention, with versus without embedded arithmetic instruction, on enhancing fraction vocabulary and arithmetic among Chinese fourth-grade students with MD. Previous research has demonstrated the initial efficacy of a fraction vocabulary intervention for fourth-grade students experiencing MD in the United States ([<reflink idref="bib14" id="ref42">14</reflink>]). We sought and received permission from [<reflink idref="bib14" id="ref43">14</reflink>] to access and utilize their instructional materials. This study extended [<reflink idref="bib14" id="ref44">14</reflink>] quasi-experiment in several ways. First, the present study was a randomized controlled trial, rather than a quasi-experiment. Second, the present study introduced a new arithmetic instruction component to explicitly teach and provide repeated practice opportunities for fraction arithmetic operation procedures associated with the selected procedural vocabulary. Third, we modified the instructional activity to suit a small-group setting. We have replaced the previous vocabulary log review, where students needed to recall the terms and their definitions learned in the day's instruction, with keywords review activity.</p> <p>Finally, we considered various fraction arithmetic competencies beyond just adding and subtracting fractions with the same denominator in [<reflink idref="bib14" id="ref45">14</reflink>]. Recognizing the unique procedures and understandings associated with each operation, we considered fraction addition and subtraction with like and unlike denominators, as well as multiplication and division. When adding or subtracting fractions with like denominators, students can simply add or subtract the numerators and keep the common denominator unchanged. However, when dealing with fractions that have unlike denominators, students need to find a common denominator before performing addition or subtraction ([<reflink idref="bib33" id="ref46">33</reflink>]).</p> <p>The following research questions guided this study:</p> <p></p> <ulist> <item> <bold> Research Question 1: </bold> What is the effect of a fraction vocabulary intervention alone (FV-only) compared to a business-as-usual (BaU) condition on fraction vocabulary and fraction arithmetic for Chinese fourth-grade students with MD?</item> <p></p> <item> <bold> Research Question 2: </bold> What is the effect of adding fraction arithmetic instruction to a fraction vocabulary intervention (FV+A) compared to FV-only and a BaU condition on fraction vocabulary and fraction arithmetic for fourth-grade Chinese students with MD?</item> </ulist> <hd id="AN0192937327-7">Method</hd> <p></p> <hd id="AN0192937327-8">Participants</hd> <p>Our research was conducted in participating schools with approval from the institutional review board (IRB) in our university and local elementary school. The school served over 2,000 students from 39 distinct classes in Southern China. Based on the teachers' report, students at this school come from a diverse range of socio-economic backgrounds, with most falling into the slightly below middle or middle categories. In addition, approximately 30% of the students come from low-income families, with many parents engaged in low-paying manual labor. During the 2023–2024 academic year, we selected participants in this research from seven fourth-grade classes at the school. Among the fourth-grade students, the school provides special education services for two students with Autism Spectrum Disorder and one student with Hyperactivity Disorder. Like the majority of schools in China, this school does not identify students with specific learning disabilities ([<reflink idref="bib34" id="ref47">34</reflink>]). Students who scored below the 25th percentile, a commonly used cutoff point in MD research, were identified as experiencing MD ([<reflink idref="bib21" id="ref48">21</reflink>]). This determination was made based on their performance on the calculation subtest of the <emph>Wide Range Achievement Test–4</emph> (WRAT-4, [<reflink idref="bib32" id="ref49">32</reflink>]) during the initial screening and their eligibility to participate. Out of the 364 participants in the initial screening, 90 students were identified as students with MD. Only 72 students (32 male and 40 female, with a mean age of 9.59 years) whose parents provided consent for participation in the research study were enrolled.</p> <p>We adopted an experimental study design and randomly assigned the 72 students with MD to three conditions. The intervention was conducted in small groups, with four students assigned to each group. However, we excluded two students who performed significantly better than their peers in the first three intervention lessons. As a result, we ultimately enrolled 70 students in the study. Ultimately, 23 students (12 male and 11 female, with a mean age of 9.56 years) remained in the FV-only condition, 23 students (10 male and 13 female, with a mean age of 9.74 years) remained in the FV+A intervention condition, and 24 students (9 male and 15 female with a mean age of 9.47 years) were assigned to the BaU condition. Students in the BaU condition did not receive any intervention related to the current study.</p> <hd id="AN0192937327-9">Measures</hd> <p>To identify students experiencing MD, we used the calculation subtest of the WRAT-4 ([<reflink idref="bib32" id="ref50">32</reflink>]). In this assessment, students had to solve 40 problems of increasing difficulty. They were given 15 minutes to complete the test using paper and pencil, and the highest achievable score was 40. Cronbach's α for this sample was.69.</p> <hd id="AN0192937327-10">Fraction Vocabulary</hd> <p>We utilized the Chinese version of the <emph>Fraction Vocabulary Measure</emph>, which was adapted from [<reflink idref="bib14" id="ref51">14</reflink>], to assess students' gains in fraction vocabulary ([<reflink idref="bib6" id="ref52">6</reflink>]). Of the 38 vocabulary terms included, five were designated as procedural vocabulary terms, for which we taught the corresponding arithmetic operations: <emph>like fractions</emph>, <emph>unlike fractions</emph>, <emph>least common denominators</emph>, <emph>reduction to common denominators</emph>, and <emph>reciprocals</emph>. The remaining 33 terms were classified as conceptual vocabulary. Although these terms do not strictly fall into procedural or conceptual vocabulary by definition, this classification reflects our specific approach in the fraction vocabulary intervention.</p> <p>For each vocabulary term, we developed three levels of questions based on [<reflink idref="bib9" id="ref53">9</reflink>] framework: recall, understanding, and use in tasks. Recall questions requested students to recall a term; comprehension questions requested students to respond to a prompt to demonstrate their understanding (e.g., circle the improper fraction); and use questions asked students to generate a fraction number words (e.g., write a composite number) based on the prompt. Please refer to Supplemental Appendix A for a comprehensive list of terms and questions featured in the Chinese <emph>Fraction Vocabulary Measure</emph>. The score was the number of questions answered correctly within 20 minutes. Cronbach's α was.87 for this sample.</p> <hd id="AN0192937327-11">Fraction Arithmetic</hd> <p>The <emph>Fraction Arithmetic</emph> included fraction addition, subtraction, multiplication, and division, with six problems each. There were four addition and subtraction problems with the same denominators, and two with different denominators. Supplemental Appendix B contains the problems. Students earned one point for each correct answer, and their score was determined by the number of correct responses within a 15-minute timeframe. Cronbach's α for this sample was.81.</p> <hd id="AN0192937327-12">Intervention</hd> <p>The fraction vocabulary intervention comprised nine lessons, including eight instruction lessons and a review lesson. Each instruction lesson introduced one to seven closely related vocabulary terms, followed by a review lesson reviewing all the vocabulary terms. This tutoring occurred outside the regular classroom in a quiet room (e.g., Science and Technology Room, Psychological Activity Room) during the students' after-school hours. In these sessions, the teacher guided the students in discussions, practice, and review mathematics problems.</p> <p>The fraction vocabulary intervention consisted of 40 terms, including 12 briefly mentioned terms. Most of these terms overlapped with those in [<reflink idref="bib14" id="ref54">14</reflink>]. Table 1 provides a detailed overview of the fraction vocabulary covered in each lesson for the FV-only condition and the FV+A condition. The sequence of vocabulary terms in the two conditions was adjusted to ensure a balanced allocation of instructional time between the two conditions due to the specific teaching approach for the procedural vocabulary terms and their associated arithmetic operations (such as <emph>like fractions</emph>, <emph>reduction to common denominator</emph>, <emph>common denominators</emph>, and <emph>reciprocal</emph>).</p> <p>Table 1. Intervention Lesson Overview.</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="char" char="." /><col align="char" char="." /></colgroup><thead><tr><th align="center">Lesson</th><th align="center">FV-only</th><th align="center">FV+A</th></tr></thead><tbody><tr><td>1</td><td><italic>Whole</italic>, <italic>One Unit</italic>, <italic>Fraction</italic>, <italic>Numerator</italic>, <italic>Denominator</italic>, <italic>Fraction Bar</italic>, <italic>Unit Fraction</italic></td><td><italic>Whole</italic>, <italic>One Unit</italic>, <italic>Fraction</italic>, <italic>Numerator</italic>, <italic>Denominator</italic>, <italic>Fraction Bar</italic>, <italic>Unit Fraction</italic></td></tr><tr><td>2</td><td><italic>Proper Fraction</italic>, <italic>Improper Fraction</italic>, <italic>Mixed Number</italic></td><td><italic>Proper Fraction</italic>, <italic>Improper Fraction</italic>, <italic>Mixed Number</italic></td></tr><tr><td>3</td><td><italic>Factors</italic>, <italic>Multiple</italic></td><td><italic>Factors</italic>, <italic>Multiple</italic></td></tr><tr><td>4</td><td><italic>Prime Number</italic>, <italic>Composite Number</italic>, <italic>Common Factor</italic>, <italic>Greatest Common Factor</italic></td><td><italic>Common Factor</italic>, <italic>Greatest Common Factor</italic>, <italic>Common Multiple</italic>, <italic>Least Common Multiple</italic></td></tr><tr><td>5</td><td><italic>Reduction of Fraction</italic> (<italic>Equivalent Fraction</italic>), <italic>Fraction in Simplest Form</italic></td><td><italic>Prime Number</italic>, <italic>Composite Number</italic>, <italic>Reduction of Fraction</italic> (<italic>Equivalent Fraction</italic>), <italic>Fraction in Simplest Form</italic></td></tr><tr><td>6</td><td><italic>Unlike Fractions</italic>, <italic>Like Fractions</italic></td><td><italic>Unlike Fractions</italic>, <italic>Like Fractions</italic></td></tr><tr><td>7</td><td><italic>Reciprocal, Common Multiple</italic>, <italic>Least Common Multiple</italic></td><td><italic>Reduction to Common Denominator</italic>, <italic>Common Denominators</italic> (<italic>Least Common Denominator</italic>)</td></tr><tr><td>8</td><td><italic>Reduction to Common Denominator</italic>, <italic>Common Denominators</italic> (<italic>Least Common Denominator</italic>)</td><td><italic>Reciprocal</italic></td></tr></tbody></table> </ephtml> </p> <p>1 <emph>Note</emph>. Lesson 9 reviewed all the fraction vocabulary terms introduced from Lesson 1 to Lesson 8. FV-only = fraction vocabulary intervention; FV + A = fraction vocabulary intervention with arithmetic instruction.</p> <p>During the final review lesson, students completed 15 practice problems under the guidance of tutors, each of which is designed to assess students' understanding of one to four vocabulary terms. Before solving each problem, the tutors first lead the students to recall the definitions of the related vocabularies, and then allowed them to solve it independently.</p> <hd id="AN0192937327-13">Description of Intervention Activities</hd> <p>We implemented the fraction vocabulary instruction routine outlined by [<reflink idref="bib14" id="ref55">14</reflink>]. FV-only and FV+A students participated in three activities in each session: (<reflink idref="bib1" id="ref56">1</reflink>) Read and Match, (<reflink idref="bib2" id="ref57">2</reflink>) Vocabulary Map Instruction, and (<reflink idref="bib3" id="ref58">3</reflink>) Keyword Fill-in Review. The activities were identical for both groups, except that FV+A students learned and practiced fraction operations for procedural vocabulary, whereas FV-only students participated in a Word Cloud activity to adjust for time differences. See Supplemental Appendix C for an example of student worksheet.</p> <hd id="AN0192937327-14">Read and Match</hd> <p>The first activity asked students to spend 2 minutes reviewing the fraction vocabulary terms introduced previously, where they matched the term on the left side with its corresponding definition on the right side. The tutor provided immediate, corrective feedback on any noted errors.</p> <hd id="AN0192937327-15">Vocabulary Map Instruction</hd> <p>The vocabulary map consisted of four sections: the term, definition, example and nonexample, and problem-based practice. Please see Figure 1 for an example student worksheet.</p> <p>MAP: Figure 1. Sample Vocabulary Map for FV-only and FV+A Conditions. Note. The worksheet on the left is for the FV-only condition, whereas the worksheet on the right is for the FV+A condition. FV-only = fraction vocabulary intervention; FV + A = fraction vocabulary intervention with arithmetic instruction.</p> <hd id="AN0192937327-16">Step 1—Rate the Term</hd> <p>Before introducing new terms, the tutor asked the students to rate each term based on their knowledge on a 1–4 scale within 2 minutes. The scale ranged as follows: 1 = "I do not recognize this term at all," 2 = "I have seen or heard this term before," 3 = "I have some idea of the term's meaning," 4 = "I am very familiar with this term."</p> <hd id="AN0192937327-17">Step 2—Define the Term</hd> <p>After the students rated the terms, the tutor introduced the definitions of the terms using specific examples. Once students had developed an understanding of the term, the tutors elaborated on the definition of such term and emphasized the keywords within it. For instance, to help students learn <emph>factor</emph>, tutors presented factors using finding the factors for a specific number to introduce the definition (e.g., the <emph>factors</emph> of the number 6 are 1, 2, 3, and 6). Following this, tutors proceeded to explain common factors and greatest common factor using concrete examples. For the procedural vocabulary, students in the FV+A group not only learned the definition of the term but also learned the arithmetic operations associated with that vocabulary to gain a deeper understanding of these terms.</p> <hd id="AN0192937327-18">Step 3—Reinforce Meaning With Examples and Non-examples</hd> <p>Once students had grasped the definition, they proceeded to fill in worksheets to provide examples and non-examples. This step was designed to enhance their understanding by prompting them to apply their knowledge in various situations and explain the reasons for their choices.</p> <hd id="AN0192937327-19">Step 4—Problem-Based Practices</hd> <p>During this 2-minute activity, students were presented with one practice problem related to the target vocabulary, enabling them to apply their acquired vocabulary knowledge. For the procedural vocabulary, FV+A students solved a simple fraction operation problem. On the other hand, students in the FV-only group engaged in three activities using a Word Cloud. They started by identifying the vocabulary introduced in the lesson, then utilized specific vocabulary terms to construct sentences explaining the definitions, and finally classified the fraction numbers in the Word Cloud according to their understanding of the definitions (See Figure 1 for example Word Cloud and three practices included in this step).</p> <hd id="AN0192937327-20">Keyword Fill-in Review</hd> <p>In the final activity, students participated in a brief 2-minute Keyword Fill-in task where they inserted keywords into blanks to complete sentence definitions. Following their individual problem-solving attempts, students receive affirmative and corrective feedback to correct any errors.</p> <hd id="AN0192937327-21">Intervention Training</hd> <p>Six graduate research assistants were in charge of conducting all pretesting, tutoring, and posttesting. Each tutor was pursuing a master's or doctoral degree in an education-related field and had prior experience working as a primary school mathematics teacher. In early October of the school year, the tutors underwent a 3-hour training session conducted by the first author to familiarize themselves with the scripted lessons and the administration of assessments. At the beginning of the training, each tutor's knowledge of fraction vocabulary was also evaluated. On average, the tutors answered 99.6% of the fraction vocabulary items correctly. Before their initial tutoring session, the tutors underwent reliability checks through mock tutoring sessions with the first author and received feedback for improvement. The first author regularly checked in with them virtually each week to review progress and address any student behavior concerns. Each tutoring session lasted around 20 minutes.</p> <hd id="AN0192937327-22">Fidelity of Implementation</hd> <p>The lessons were scripted to ensure fidelity of implementation. Tutors audio-recorded every session. We randomly sampled >20% of sessions for each tutor and completed a checklist to identify whether the essential components were implemented as intended. For each lesson, a checklist of around 50 items needed to be marked off. See Supplemental Appendix D for an example fidelity checklist. Adherence averaged 96.4%, with a range of 89.7%–100.0%. In addition, we included three items to roughly assess the quality of delivery. On average, the tutors performed well in implementing activities and monitoring students, with scores of 2.9 and 2.89 out of 3, respectively. Tutors also tracked the number of sessions each student received in fraction vocabulary intervention. On average, FV-only students completed 8.83 days of intervention (range 8–9; <emph>SD</emph> = 0.38), and FV+A students completed 8.6 days of intervention (range 7–9; <emph>SD</emph> = 0.60). The average lesson duration was around 28.66 minutes for the FV-only condition and 29.35 minutes for the FV+A condition, with no statistically significant difference (<emph>p</emph> =.33).</p> <hd id="AN0192937327-23">BaU Comparison</hd> <p>Students in the BaU condition did not receive any intervention from the research team. Instead, they participated in after-school math classes where the teacher led discussions, practice sessions, and math problem reviews. To understand the typical Grade 4 math class experience for students, we surveyed teachers to gather detailed information on the frequency of fraction vocabulary instruction they provided. The survey asked teachers to specify the frequency of this instruction during after-school math lessons, with response options including: every day, a few times a week, once a week, a few times a month, once a month, and never. On average, teachers indicated that they assessed students' understanding of fraction vocabulary before instruction and used concept maps to help students grasp fraction vocabulary several times per week. Once a week, they taught fraction vocabulary using both examples and non-examples. However, they rarely utilized problem-based practice to improve students' grasp of fraction vocabulary, seldom explained important fraction vocabulary encountered during problem-solving, or prompted students to discuss and communicate using fraction vocabulary, doing so less than once a week. Teachers also noted using Frayer Models to teach fraction vocabulary terms once a month.</p> <hd id="AN0192937327-24">Data Analysis</hd> <p>In order to compare gains among the three groups, we used analysis of variance (ANOVA) and Kruskal-Wallis tests based on the normality of the outcome variables ([<reflink idref="bib10" id="ref59">10</reflink>]). For the overall fraction vocabulary, conceptual vocabulary in fractions, and overall fraction arithmetic, the outcome variables exhibited normal distribution, allowing us to proceed with ANOVA for comparison. However, for other outcome variables like fraction addition and subtraction, as well as procedural vocabulary in fractions, deviations from normal distribution were observed, leading us to use Kruskal-Wallis tests for comparison.</p> <p>In our analysis, the between-subjects factor was the intervention status (FV-only vs. FV+A vs. BaU), whereas the within-subjects factors included the fraction vocabulary and arithmetic outcomes. Univariate follow-up tests were conducted only in cases where a statistical difference between groups was identified, allowing for a comparison between each pair of conditions.</p> <hd id="AN0192937327-25">Results</hd> <p>Table 2 presents unadjusted pretest and posttest means, along with the adjusted posttest means, for students in the FV-only, FV+A, and BaU conditions. At pretest, we identified no significant variances in fraction vocabulary and fraction arithmetic. For the subdimensions within fraction arithmetic, please refer to Supplemental Appendix E. In other words, FV-only, FV+A, and BaU students demonstrated comparable performance on preintervention fraction vocabulary and fraction arithmetic.</p> <p>Table 2. Performance by Condition: Intervention Only, Intervention + Instruction, No Intervention.</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /></colgroup><thead><tr><th /><th align="center" colspan="2">FV-only(<italic>n</italic> = 23)</th><th align="center" colspan="2">FV+A(<italic>n</italic> = 23)</th><th align="center" colspan="2">Control(<italic>n</italic> = 24)</th></tr><tr><th align="center">Variable</th><th align="center"><italic>M</italic></th><th align="center">SD/SE<xref ref-type="table-fn" rid="tfn3">a</xref></th><th align="center"><italic>M</italic></th><th align="center">SD/SE<xref ref-type="table-fn" rid="tfn3">a</xref></th><th align="center"><italic>M</italic></th><th align="center">SD/SE<xref ref-type="table-fn" rid="tfn3">a</xref></th></tr></thead><tbody><tr><td colspan="7">Fraction vocabulary</td></tr><tr><td>Pretest</td><td>8.91</td><td>3.64</td><td>9.83</td><td>4.27</td><td>8.25</td><td>4.36</td></tr><tr><td>Posttest</td><td>18.65</td><td>6.41</td><td>19.17</td><td>5.98</td><td>13.08</td><td>5.79</td></tr><tr><td>Adjusted posttest</td><td>18.68</td><td>1.22</td><td>18.81</td><td>1.23</td><td>13.40</td><td>1.20</td></tr><tr><td>Procedural vocabulary</td><td /><td /><td /><td /><td /><td /></tr><tr><td>Pretest</td><td>0.38</td><td>0.71</td><td>0.65</td><td>0.71</td><td>0.42</td><td>0.65</td></tr><tr><td>Posttest</td><td>2.30</td><td>1.40</td><td>2.43</td><td>1.53</td><td>1.04</td><td>1.08</td></tr><tr><td>Adjusted posttest</td><td>2.30</td><td>0.28</td><td>2.45</td><td>0.29</td><td>1.04</td><td>0.28</td></tr><tr><td>Conceptual vocabulary</td><td /><td /><td /><td /><td /><td /></tr><tr><td>Pretest</td><td>8.57</td><td>0.79</td><td>9.17</td><td>0.79</td><td>7.83</td><td>0.77</td></tr><tr><td>Posttest</td><td>16.35</td><td>5.38</td><td>16.74</td><td>4.97</td><td>12.04</td><td>4.89</td></tr><tr><td>Adjusted posttest</td><td>16.33</td><td>1.01</td><td>16.45</td><td>1.02</td><td>12.34</td><td>1.00</td></tr><tr><td colspan="7">Fraction arithmetic</td></tr><tr><td>Pretest</td><td>6.61</td><td>3.86</td><td>7.35</td><td>3.58</td><td>6.33</td><td>3.00</td></tr><tr><td>Posttest</td><td>18.65</td><td>6.41</td><td>19.17</td><td>5.97</td><td>13.08</td><td>5.79</td></tr><tr><td>Adjusted posttest</td><td>18.68</td><td>1.22</td><td>18.81</td><td>1.23</td><td>13.40</td><td>1.20</td></tr></tbody></table> </ephtml> </p> <ulist> <item>2 <emph>Note.</emph> Adjusted posttest is posttest score adjusted for pretest score. FV-only = fraction vocabulary intervention; FV + A = fraction vocabulary intervention with arithmetic instruction.</item> <item>3 For pretest and posttest, <emph>SD</emph>; for adjusted posttest, <emph>SE.</emph></item> </ulist> <p>To compare the means of overall fraction vocabulary, conceptual vocabulary, and fraction arithmetic across three groups, we conducted an ANOVA given the normal distribution of the outcome variables. Please refer to Table 3 for comparisons of gains on fraction vocabulary and arithmetic. The ANOVA revealed significant large group differences on both fraction vocabulary gains, <emph>F</emph> (<reflink idref="bib2" id="ref60">2</reflink>, 70) = 4.51, <emph>p</emph> <.05, and conceptual vocabulary gains, <emph>F</emph> (<reflink idref="bib2" id="ref61">2</reflink>, 70) = 3.43, <emph>p</emph> <.05. Post-hoc tests revealed significantly higher gains for students in the FV+A and FV-only than the BaU, while no significant difference was found between FV+A and FV-only for both overall fraction vocabulary gains and conceptual vocabulary gains. For overall fraction arithmetic, the ANOVA revealed nonsignificant group differences among the three groups.</p> <p>Table 3. Comparisons of Gains on the Fraction Vocabulary and Fraction Arithmetic Outcome Variables.</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /></colgroup><thead><tr><th /><th align="center">BaU vs. FV vs. FV+A</th><th align="center">FV vs. BaU</th><th align="center">FV+A vs. BAU</th><th align="center">FV+A vs. FV</th><th /></tr><tr><th align="center">Variable</th><th align="center">F/H<xref ref-type="table-fn" rid="tfn5">a</xref></th><th align="center" colspan="3">MD/Z<xref ref-type="table-fn" rid="tfn5">b</xref></th><th align="center">Statistical outcomes</th></tr></thead><tbody><tr><td>Gains</td><td /><td /><td /><td /><td /></tr><tr><td>Fraction vocabulary</td><td>4.51<xref ref-type="table-fn" rid="tfn6">*</xref></td><td>4.91<xref ref-type="table-fn" rid="tfn6">**</xref></td><td>4.51<xref ref-type="table-fn" rid="tfn6">*</xref></td><td>−0.39</td><td>(FV+A = FV) > BaU</td></tr><tr><td>Procedural vocabulary</td><td>10.58<xref ref-type="table-fn" rid="tfn6">**</xref></td><td>2.50<xref ref-type="table-fn" rid="tfn6">**</xref></td><td>3.04<xref ref-type="table-fn" rid="tfn6">*</xref></td><td>0.52</td><td>(FV+A = FV) > BaU</td></tr><tr><td>Conceptual vocabulary</td><td>3.43<xref ref-type="table-fn" rid="tfn6">*</xref></td><td>3.57<xref ref-type="table-fn" rid="tfn6">*</xref></td><td>3.57<xref ref-type="table-fn" rid="tfn6">*</xref></td><td>−0.22</td><td>(FV+A = FV) > BaU</td></tr><tr><td>Fraction arithmetic</td><td>1.51</td><td colspan="3">NS</td><td>FV+A = FV = BaU</td></tr><tr><td>Add/sub like fractions</td><td>0.63</td><td colspan="3" rowspan="4">NS</td><td rowspan="4">FV+A = FV = BaU</td></tr><tr><td>Add/sub unlike fractions</td><td>0.62</td></tr><tr><td>Multiplication</td><td>2.88</td></tr><tr><td>Division</td><td>1.38</td></tr></tbody></table> </ephtml> </p> <ulist> <item>4 <emph>Note.</emph> FV = FV-only; NS = not significant. BaU = business as usual; FV + A = fraction vocabulary intervention with arithmetic instruction.</item> <item>5 For variables that follow normal distribution <emph>F</emph>; for variables that do not follow normal distribution <emph>H</emph>. <sups>b</sups> For variables that follow normal distribution <emph>MD</emph>; for variables that do not follow normal distribution <emph>Z</emph>.</item> <item>6 <emph>p</emph> <.05; **<emph>p</emph> <.01; significant differences in means or median as determined by Bonferroni.</item> </ulist> <p>For the remaining outcome variables that were not normally distributed, we employed the Kruskal-Wallis test to assess the differences among the three groups. Specifically, when examining procedural vocabulary in fractions, the results indicated a significant difference among the groups, <emph>H</emph> (<reflink idref="bib2" id="ref62">2</reflink>) = 10.58, <emph>p</emph> <.01. Post-hoc analyses revealed that students in the FV+A and FV-only conditions exhibited significantly greater improvements compared to those in the BaU condition, although there was no notable difference between the FV+A and FV-only groups in terms of procedural vocabulary gains. In the remaining fraction arithmetic outcomes, the Kruskal-Wallis tests revealed no significant differences among the three groups.</p> <hd id="AN0192937327-26">Discussion</hd> <p>In this study, we explored the effect of a fraction vocabulary intervention with versus without embedded arithmetic instruction. We extended [<reflink idref="bib14" id="ref63">14</reflink>] quasi-experiment by including a fraction arithmetic instruction component and used a randomized controlled trial. Our findings revealed both intervention groups showed significantly better performance in fraction vocabulary knowledge compared to the BaU group. However, no notable distinctions were observed between the two intervention conditions in terms of fraction vocabulary and fraction arithmetic. Only students within the FV+A condition exhibited enhanced performance in subtraction with like denominators compared to the BaU condition.</p> <hd id="AN0192937327-27">Fraction Vocabulary Intervention Effects</hd> <p>The findings of the present study revealed superior fraction vocabulary knowledge for FV-only and FV+A students over BaU students, with effect sizes of 0.85 and 0.69. Moreover, the difference in fraction vocabulary gains did not differ between the two conditions. We attribute superior fraction vocabulary performance to the extensive instruction students in both conditions received. Our fraction vocabulary intervention primarily incorporated the use of a vocabulary map to introduce and teach fraction vocabulary terms. We also utilized other instructional components, such as read and match, word cloud, and keyword fill-ins to review knowledge of fraction vocabulary terms. This finding is similar to that of the previous study involving U.S. students ([<reflink idref="bib14" id="ref64">14</reflink>]).</p> <p>It is important to note that the current intervention was administered in a small group setting, with four students per group, contrasting with the previous study by [<reflink idref="bib14" id="ref65">14</reflink>] which utilized one-on-one instruction. The results of the present research indicated that this fraction vocabulary intervention delivered primarily through vocabulary map can indeed be effective when implemented in a small group format. Considering the financial and time constraints faced by numerous schools ([<reflink idref="bib2" id="ref66">2</reflink>]), we suggest implementing mathematics vocabulary instruction routine in small groups whenever a group of students experiencing MD requires supplemental and targeted support in understanding mathematics vocabulary.</p> <hd id="AN0192937327-28">Fraction Arithmetic</hd> <p>Overall, our findings demonstrated that there were no significant differences between the two intervention conditions and the students in the BaU condition regarding fraction arithmetic. Even though students in the FV+A condition had some exposure to learning and practicing fraction arithmetic operations scattered throughout the 10 sessions, it is important to highlight that due to the limited instruction on the selected procedural vocabulary and corresponding fraction arithmetic (only three lessons), students in the FV+A condition had very limited exposure to learn and practice fraction arithmetic operations. For example, we only allocated one session to teach the term—<emph>reciprocal</emph> and corresponding arithmetic operation—fraction multiplication and division. This created a potential issue of information overload for FV+A students within the brief learning period.</p> <p>Furthermore, students with MD already possess restricted cognitive resources ([<reflink idref="bib13" id="ref67">13</reflink>]). The complexity of fraction arithmetic, which involves multiple steps, combined with their difficulties in performing basic whole-number operations, presents an even greater challenge for them. As a result, exposing these students to complex tasks like fraction arithmetic may increase their cognitive load, potentially hindering their performance. However, it is noteworthy that even though the students in the FV+A condition did not exhibit significantly greater improvements compared to other two conditions, their performance did not decline despite the potential for cognitive overload. In fact, their scores demonstrated an increase, suggesting that while trying to cover more content in a limited timeframe can be challenging and may lead to confusion, it does not necessarily result in a decrease in performance.</p> <p>However, as opposed to [<reflink idref="bib14" id="ref68">14</reflink>], our finding did not reveal that FV-only students performed better than BaU students in either fraction addition or subtraction with the same denominator. One possible explanation for this difference lies in the varied approaches used in teaching procedural skills and introducing fraction arithmetic operations between the United States and China. Chinese students often learn procedural skills through explicit, step-by-step instruction and repetitive practice. Conversely, the U.S. education system emphasizes the development of conceptual understanding as the foundation for procedural skills. For example, Chinese textbooks typically use whole-number multiplication and division to demonstrate the concept of fraction multiplication and division, highlighting the relationship between fraction division and multiplication as inverses. Conversely, U.S. textbooks often explain division concepts using "partitive" interpretations. For instance, they may illustrate division by showing how to use rubber bands to create <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></math> </ephtml> of a rectangle because there are four <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></math> </ephtml> sections in 1, 1÷ <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></math> </ephtml> = 4. As a consequence, U.S. students are more likely to utilize their conceptual understanding to cultivate procedural skills, facilitating the transfer of their fraction vocabulary knowledge, a form of conceptual understanding, to fraction arithmetic. For example, when U.S. students learn the term "like denominators," they may visualize adding fractions with like denominators as combining parts of a circle that has been divided into equal sections. However, Chinese students may not generate this kind of visualization; instead, they may focus on adding numerators and adding denominators separately (treating them as whole numbers).</p> <p>These mixed findings suggested that the transfer from improved mathematics vocabulary to enhanced mathematics performance is not necessarily direct or automatic, but rather contingent on other factors, such as the presence of a pathway from conceptual understanding to procedural skills in students' learning patterns.</p> <hd id="AN0192937327-29">Limitations and Future Research</hd> <p>Despite the promising results, we note several limitations. First, we did not teach all the fraction vocabulary in the same sequence for the two intervention conditions. This was done to ensure a balance in instructional time between the two conditions, despite our efforts to arrange and sequence the fraction vocabulary logically. Future research could investigate if a particular sequence of teaching mathematics vocabulary could lead to better learning outcomes for students compared to other sequences.</p> <p>Second, we provided students in the FV+A group a substantial amount of information on fraction vocabulary and its corresponding arithmetic operation procedures within a constrained instructional timeframe. One potential issue of presenting students with an excessive amount of information in a limited time frame is cognitive overload, particularly as mathematics vocabulary and knowledge become more intricate in higher elementary grade levels ([<reflink idref="bib23" id="ref69">23</reflink>]).</p> <p>In addition to the risk of cognitive overload, the absence of a significant increase in fraction arithmetic for students in FV+A condition may be linked to the limited number of instructional sessions and the insufficient opportunities for repetition and practice. The current fraction arithmetic operation instruction was provided over three sessions, plus one review session. In contrast, a typical fraction intervention (e.g., Fraction Face-Off; [<reflink idref="bib8" id="ref70">8</reflink>]) consists of more than 30 sessions, allowing students to practice fraction operations consistently throughout the program. Future research may incorporate adequate repetition and practice to assess the potential benefits of integrating fraction arithmetic within fraction vocabulary intervention for upper elementary students with MD.</p> <p>Future research could delve deeper into investigating the circumstances in which enhanced comprehension of mathematics vocabulary translates into improved performance in mathematics. The inconsistent finding on the transfer of fraction vocabulary understanding to improved fraction performance between the present study on Chinese students and prior study on U.S. students suggested that the transition may not always follow a straightforward trajectory. Instead, this process seems to be influenced by additional factors, such as the existence of a clear pathway that bridges the gap between conceptual understanding and procedural proficiency. Further exploration is warranted to better grasp the complex interplay between mathematical vocabulary acquisition and mathematical performance.</p> <hd id="AN0192937327-30">Implications</hd> <p>The present study has important practical implications. First, the present study demonstrated that teaching mathematics vocabulary using a vocabulary map in a small group setting can yield results comparable to one-on-one instruction. This finding is crucial considering the constraints in time and resources when it comes to teach students experiencing MD. The fact that the current study, conducted in a small group setting, and a previous study, conducted through one-on-one instruction, both led to comparable enhancements in the understanding of mathematics vocabulary suggests that future interventions for mathematics vocabulary could teach a group of students with MD within a small group setup. Second, our findings indicated comparable improvements in fraction vocabulary understanding across both intervention conditions. This suggests that the specific sequence in which we structure our fraction vocabulary instruction may not have a substantial influence on the effectiveness of the intervention, as long as the mathematics vocabulary terms are organized in a logical sequence.</p> <p>In conclusion, this study extends the research of [<reflink idref="bib14" id="ref71">14</reflink>] by examining the effects of a fraction vocabulary intervention that included embedded arithmetic instruction. The findings demonstrated small-group instruction can be as effective as one-on-one instruction for students with MD. In addition, findings from the study indicated that the specific sequence of vocabulary instruction may not be as crucial as previously believed. It also advised caution against combining complex vocabulary instruction with corresponding arithmetic operations, as this may overwhelm students. Overall, these insights underscore the importance of designing carefully structured vocabulary interventions that address the unique challenges faced by students.</p> <hd id="AN0192937327-31">Supplemental Material</hd> <p>Graph: Supplemental material, sj-docx-1-ldx-10.1177_00222194251342191 for Unique and Combined Effects of Small-Group Fraction Vocabulary and Arithmetic Interventions for Students With Mathematical Difficulties by Xin Lin, Juehang Zhang, Haorui Cui, Xiuwen Song and Xiaonan Han in Journal of Learning Disabilities</p> <p>We would like to express our sincere gratitude to Zhuhai Xiangzhou District No. 16 Primary School for their valuable support and participation in this study.</p> <ref id="AN0192937327-32"> <title> References </title> <blist> <bibl id="bib1" idref="ref19" type="bt">1</bibl> <bibtext> Barbieri C. A., Rodrigues J., Dyson N., Jordan N. C. (2020). 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Chinese Journal of Special Education, 9, 57–63.</bibtext> </blist> </ref> <ref id="AN0192937327-33"> <title> Footnotes </title> <blist> <bibtext> The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.</bibtext> </blist> <blist> <bibtext> This research was supported by Grant SRG2023-00002-FED from the Faculty of Education in the University of Macau, named "Develop and implement Mathematics Vocabulary Intervention for students Experiencing Mathematics Learning Difficulties."</bibtext> </blist> <blist> <bibtext> Xin Lin</bibtext> </blist> <blist> <bibtext>Graph</bibtext> </blist> <blist> <bibtext>https://orcid.org/0000-0002-8077-5134 Haorui Cui</bibtext> </blist> <blist> <bibtext>Graph https://orcid.org/0009-0006-9332-1023</bibtext> </blist> <blist> <bibtext> Supplemental material for this article is available at https://doi.org/10.1177/00222194251342191</bibtext> </blist> </ref> <aug> <p>By Xin Lin; Juehang Zhang; Haorui Cui; Xiuwen Song and Xiaonan Han</p> <p>Reported by Author; Author; Author; Author; Author</p> </aug> <nolink nlid="nl1" bibid="bib14" firstref="ref1"></nolink> <nolink nlid="nl2" bibid="bib22" firstref="ref2"></nolink> <nolink nlid="nl3" bibid="bib24" firstref="ref3"></nolink> <nolink nlid="nl4" bibid="bib26" firstref="ref4"></nolink> <nolink nlid="nl5" bibid="bib19" firstref="ref5"></nolink> <nolink nlid="nl6" bibid="bib17" firstref="ref6"></nolink> <nolink nlid="nl7" bibid="bib12" firstref="ref7"></nolink> <nolink nlid="nl8" bibid="bib30" firstref="ref8"></nolink> <nolink nlid="nl9" bibid="bib13" firstref="ref10"></nolink> <nolink nlid="nl10" bibid="bib15" firstref="ref11"></nolink> <nolink nlid="nl11" bibid="bib18" firstref="ref13"></nolink> <nolink nlid="nl12" bibid="bib28" firstref="ref14"></nolink> <nolink nlid="nl13" bibid="bib20" firstref="ref16"></nolink> <nolink nlid="nl14" bibid="bib29" firstref="ref17"></nolink> <nolink nlid="nl15" bibid="bib33" firstref="ref18"></nolink> <nolink nlid="nl16" bibid="bib27" firstref="ref24"></nolink> <nolink nlid="nl17" bibid="bib16" firstref="ref30"></nolink> <nolink nlid="nl18" bibid="bib11" firstref="ref31"></nolink> <nolink nlid="nl19" bibid="bib31" firstref="ref34"></nolink> <nolink nlid="nl20" bibid="bib25" firstref="ref36"></nolink> <nolink nlid="nl21" bibid="bib34" firstref="ref47"></nolink> <nolink nlid="nl22" bibid="bib21" firstref="ref48"></nolink> <nolink nlid="nl23" bibid="bib32" firstref="ref49"></nolink> <nolink nlid="nl24" bibid="bib10" firstref="ref59"></nolink> <nolink nlid="nl25" bibid="bib23" firstref="ref69"></nolink>
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Items – Name: Title
  Label: Title
  Group: Ti
  Data: Unique and Combined Effects of Small-Group Fraction Vocabulary and Arithmetic Interventions for Students with Mathematical Difficulties
– Name: Language
  Label: Language
  Group: Lang
  Data: English
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Xin+Lin%22">Xin Lin</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-8077-5134">0000-0002-8077-5134</externalLink>)<br /><searchLink fieldCode="AR" term="%22Juehang+Zhang%22">Juehang Zhang</searchLink><br /><searchLink fieldCode="AR" term="%22Haorui+Cui%22">Haorui Cui</searchLink> (ORCID <externalLink term="https://orcid.org/0009-0006-9332-1023">0009-0006-9332-1023</externalLink>)<br /><searchLink fieldCode="AR" term="%22Xiuwen+Song%22">Xiuwen Song</searchLink><br /><searchLink fieldCode="AR" term="%22Xiaonan+Han%22">Xiaonan Han</searchLink>
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="SO" term="%22Journal+of+Learning+Disabilities%22"><i>Journal of Learning Disabilities</i></searchLink>. 2026 59(3):161-171.
– Name: Avail
  Label: Availability
  Group: Avail
  Data: SAGE Publications and Hammill Institute on Disabilities. 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: https://sagepub.com
– Name: PeerReviewed
  Label: Peer Reviewed
  Group: SrcInfo
  Data: Y
– Name: Pages
  Label: Page Count
  Group: Src
  Data: 11
– Name: DatePubCY
  Label: Publication Date
  Group: Date
  Data: 2026
– 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+4%22">Grade 4</searchLink><br /><searchLink fieldCode="EL" term="%22Intermediate+Grades%22">Intermediate Grades</searchLink>
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Fractions%22">Fractions</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Instruction%22">Mathematics Instruction</searchLink><br /><searchLink fieldCode="DE" term="%22Grade+4%22">Grade 4</searchLink><br /><searchLink fieldCode="DE" term="%22Elementary+School+Students%22">Elementary School Students</searchLink><br /><searchLink fieldCode="DE" term="%22Vocabulary+Development%22">Vocabulary Development</searchLink><br /><searchLink fieldCode="DE" term="%22Intervention%22">Intervention</searchLink><br /><searchLink fieldCode="DE" term="%22Arithmetic%22">Arithmetic</searchLink><br /><searchLink fieldCode="DE" term="%22Learning+Problems%22">Learning Problems</searchLink><br /><searchLink fieldCode="DE" term="%22Program+Effectiveness%22">Program Effectiveness</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1177/00222194251342191
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 0022-2194<br />1538-4780
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: The purpose of this study was to assess the effects of a fraction vocabulary intervention with fraction arithmetic components on fraction vocabulary knowledge and fraction arithmetic competencies among fourth-grade Chinese students with mathematics difficulties. We randomly assigned 70 students with mathematics difficulties to three conditions: fraction vocabulary only (n = 23), fraction vocabulary with an arithmetic component (n = 23), and a business-as-usual (BaU) condition (n = 24). The students in the fraction vocabulary intervention conditions participated in 10 sessions, occurring three times per week. Students within both intervention conditions showed significantly better performance in fraction vocabulary knowledge than those in the BaU condition. However, no notable distinctions were observed between the two intervention conditions in terms of fraction arithmetic. Only students who received the fraction vocabulary intervention with an arithmetic component exhibited enhanced performance in subtraction with like denominators compared to the BaU condition.
– Name: AbstractInfo
  Label: Abstractor
  Group: Ab
  Data: As Provided
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  Label: Entry Date
  Group: Date
  Data: 2026
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  Label: Accession Number
  Group: ID
  Data: EJ1502936
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1502936
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1177/00222194251342191
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 11
        StartPage: 161
    Subjects:
      – SubjectFull: Fractions
        Type: general
      – SubjectFull: Mathematics Instruction
        Type: general
      – SubjectFull: Grade 4
        Type: general
      – SubjectFull: Elementary School Students
        Type: general
      – SubjectFull: Vocabulary Development
        Type: general
      – SubjectFull: Intervention
        Type: general
      – SubjectFull: Arithmetic
        Type: general
      – SubjectFull: Learning Problems
        Type: general
      – SubjectFull: Program Effectiveness
        Type: general
    Titles:
      – TitleFull: Unique and Combined Effects of Small-Group Fraction Vocabulary and Arithmetic Interventions for Students with Mathematical Difficulties
        Type: main
  BibRelationships:
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      – PersonEntity:
          Name:
            NameFull: Xin Lin
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            NameFull: Juehang Zhang
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            NameFull: Haorui Cui
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            NameFull: Xiuwen Song
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            NameFull: Xiaonan Han
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          Dates:
            – D: 01
              M: 05
              Type: published
              Y: 2026
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              Value: 1538-4780
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              Value: 59
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              Value: 3
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            – TitleFull: Journal of Learning Disabilities
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