View in EDS HTML Full Text PDF Full Text

Using Manipulative-Based Instructional Sequences to Increase the Understanding of Fractional Concepts of Students with Mathematical Learning Disabilities

Saved in:
Bibliographic Details
Title: Using Manipulative-Based Instructional Sequences to Increase the Understanding of Fractional Concepts of Students with Mathematical Learning Disabilities
Language: English
Authors: Kathryn Lavin Brave (ORCID 0000-0002-6806-4145), Izzy Berman, Debita Basu, Alexis Szkotak
Source: TEACHING Exceptional Children. 2025 57(5):368-378.
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: https://sagepub.com
Peer Reviewed: Y
Page Count: 11
Publication Date: 2025
Document Type: Journal Articles
Reports - Descriptive
Descriptors: Teaching Methods, Manipulative Materials, Fractions, Mathematical Concepts, Concept Formation, Students with Disabilities, Learning Disabilities, Mathematical Logic, Mathematics Instruction, Computation
DOI: 10.1177/00400599241231228
ISSN: 0040-0599
2163-5684
Abstract: Manipulative-based instructional sequences have proven to be successful with students with disabilities. However, instruction must not only support the acquisition of conceptual and procedural knowledge but also build on students' strengths. This article describes how teachers can use manipulative-based instructional sequences to support the fractional understanding of students who are diagnosed with mathematical learning disabilities. We begin by describing common misconceptions that may affect students' abilities to reason with and calculate fractions. We then explain how teachers can use a concrete-semi-concrete representational-abstract (CRA) model to plan and implement lessons that maximize learning and minimize frustration. Readers are also provided with detailed descriptions of how teachers can assess fraction knowledge, reinforce conceptual understanding, and promote procedural fluency.
Abstractor: As Provided
Entry Date: 2025
Accession Number: EJ1475234
Database: ERIC
Full text is not displayed to guests.
FullText Links:
  – Type: pdflink
    Url: https://content.ebscohost.com/cds/retrieve?content=AQICAHj0k_4E0hTGH8RJwT4gCJyBsGNe_WN95AvKlDbXJGqwxwF0aQX0f1XVOYV_HT75wbOKAAAA4jCB3wYJKoZIhvcNAQcGoIHRMIHOAgEAMIHIBgkqhkiG9w0BBwEwHgYJYIZIAWUDBAEuMBEEDFXgotk2gPp6ATbeSAIBEICBmmhQ6FEJb9XYDkJt3TrUgXA0z9kgTLYG0G8V8rpTXsuMNtFh6Rta49sizoDyYv-rt2zt3hrgVlhaS5g4Vy3YscHuPfMR97Lexls_z-OWU6oCNHCGkmPRowdp9LvZXQFw6DEhuHV7tOcOEo87WDVt9VKPYjOXu5DqYzAPx7Qyfj7UBLXuNmh1etQf2a_SVaT87OwE6EvyhYDDOas=
Text:
  Availability: 1
  Value: <anid>AN0185908517;tec01may.25;2025Jun16.03:26;v2.2.500</anid> <title id="AN0185908517-1">Using Manipulative-Based Instructional Sequences to Increase the Understanding of Fractional Concepts of Students With Mathematical Learning Disabilities </title> <p>Manipulative-based instructional sequences have proven to be successful with students with disabilities. However, instruction must not only support the acquisition of conceptual and procedural knowledge but also build on students' strengths. This article describes how teachers can use manipulative-based instructional sequences to support the fractional understanding of students who are diagnosed with mathematical learning disabilities. We begin by describing common misconceptions that may affect students' abilities to reason with and calculate fractions. We then explain how teachers can use a concrete-semi-concrete representational-abstract (CRA) model to plan and implement lessons that maximize learning and minimize frustration. Readers are also provided with detailed descriptions of how teachers can assess fraction knowledge, reinforce conceptual understanding, and promote procedural fluency.</p> <p>Graph</p> <p>The cultivation of mathematical talent remains a priority in the United States. However, since the 1970s, National Association of Educational Progress outcomes have revealed overwhelming inequities in mathematics education. These outcomes are particularly alarming when compared by disability status. In 2017, 86% of American fourth-grade students with disabilities scored below the proficiency level in mathematics. This was 30% more than those who were not identified with disabilities ([<reflink idref="bib24" id="ref1">24</reflink>]). Although gaps were found in all areas (number properties and operations, measurement, geometry, data analysis, and algebra), students with severe learning difficulties "were 32 times more likely than students with intact whole-number knowledge to experience difficulty with fractions" ([<reflink idref="bib23" id="ref2">23</reflink>], p. 36). This is disconcerting not only because the understanding of fractions is essential in everyday life (e.g., cooking, money, medicine) but also because it is linked to overall mathematical achievement. This achievement, in turn, is linked to college performance and future financial success ([<reflink idref="bib37" id="ref3">37</reflink>]).</p> <p>"Research suggests many students with MLDs exhibit misconceptions in representing and reasoning with fractions.</p> <p>"Manipulative-based instructional sequences can be used to provide students with accessible and effective opportunities for learning fractions.</p> <p> <emph>Dr. Brave teaches mathematics in a fifth-grade inclusive classroom. Six of the students in the class are identified with mathematical learning disabilities (MLDs). MLDs are identified by a team of qualified individuals who evaluate students' achievement and aptitude levels in mathematics calculations, problem solving, and visual-spatial reasoning. Levels are then compared across age groups to determine whether data support the identification of MLDs.</emph> </p> <p> <emph>Students interact in small-group settings to work toward achieving individualized education program goals and accessing grade-level curriculum. Dr. Brave is currently working in collaboration with the general education teacher to support the students in becoming fluent with fraction operations, specifically 5.NFA.1 (use equivalent fractions as a strategy to add and subtract fractions). Realizing that progress depends on the students' abilities to build procedural fluency from conceptual understanding, she searches for evidence-based practices that support the understanding of fractions.</emph> </p> <p>The following article aims to provide educators like Dr. Brave with strategies for promoting the acquisition of fractional concepts in the mathematics classroom. We draw on prior research ([<reflink idref="bib1" id="ref4">1</reflink>]; [<reflink idref="bib5" id="ref5">5</reflink>]; [<reflink idref="bib14" id="ref6">14</reflink>]) to present recommendations for assessing fraction knowledge and identifying common misconceptions among students with MLDs. We also expand on the work of [<reflink idref="bib2" id="ref7">2</reflink>] and [<reflink idref="bib37" id="ref8">37</reflink>] to describe how manipulative-based instructional sequences can be used to provide students with accessible and effective opportunities for learning fractions (see Figure 1).</p> <p>Graph: Figure 1 Framework for planning and implementing manipulative-based instructional sequences</p> <hd id="AN0185908517-2">Assessing Student Fraction Knowledge</hd> <p>According to the [<reflink idref="bib8" id="ref9">8</reflink>], fractions are formally introduced in first grade when students are expected to "partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of." Students spend the next few years developing a deeper understanding of fractions as numbers, and in Grades 4 and 5, they explore magnitude, apply equivalence, and perform calculations. These concepts are not only used in our everyday lives (e.g., measurement, cooking, etc.), but they also serve as the foundation for understanding ratios, proportions, and eventually algebra ([<reflink idref="bib26" id="ref10">26</reflink>]).</p> <p>Despite the importance of fractions in and out of school, research suggests many students with MLDs exhibit misconceptions in representing and reasoning with fractions ([<reflink idref="bib16" id="ref11">16</reflink>]; [<reflink idref="bib22" id="ref12">22</reflink>]; [<reflink idref="bib33" id="ref13">33</reflink>]). Because these misconceptions may be rooted in deeper mathematical misunderstandings, it is important that teachers use formative assessments to evaluate thinking strategies and plan for instructional sequences that place students at the center of learning. Although there are several means through which understanding can be assessed, the most common task involves a comparison of two fractions (e.g., 2/6 and 2/12; [<reflink idref="bib1" id="ref14">1</reflink>]; [<reflink idref="bib14" id="ref15">14</reflink>]). As students complete the task, teachers ask questions related to (a) the role of equal parts in fractionality, (b) the relationships between numerators and denominators, (c) the relationship between the whole and the fraction, and (d) the role of context in describing the fractions relation to an object, shape, or quantity ([<reflink idref="bib1" id="ref16">1</reflink>]; [<reflink idref="bib28" id="ref17">28</reflink>]). Answers will allow teachers to determine not only whether misconceptions exist but also whether students fully comprehend "fractions as <emph>numbers</emph> with <emph>magnitude</emph>" ([<reflink idref="bib14" id="ref18">14</reflink>], p. 123).</p> <p>"Misunderstandings related to the procedural and conceptual understanding of fractions are not uncommon.</p> <p>In assessing student understanding, it is important to note that misunderstandings related to the procedural and conceptual understanding of fractions are not uncommon. According to [<reflink idref="bib35" id="ref19">35</reflink>], conceptual understanding "refers to <emph>connected</emph> knowledge" (p. 19) that explains why and how a mathematical solution is correct. On the other hand, procedural understanding involves the strategic application of appropriate processes and procedures ([<reflink idref="bib35" id="ref20">35</reflink>]). However, most often, students with MLDs exhibit conceptual misunderstandings resulting from a reliance on natural numbers, or "whole number bias" ([<reflink idref="bib5" id="ref21">5</reflink>]). Students exhibiting whole number bias view fractions as two independent numbers divided by a hyphen (or bar) that serves as nothing more than a separator between natural numbers ([<reflink idref="bib5" id="ref22">5</reflink>]; [<reflink idref="bib33" id="ref23">33</reflink>]). Subsequently, when comparing or ordering fractions, magnitude is determined by whole number values (e.g., 2/12 > 2/6; 3/9 > 1/2) rather than numerator/denominator relationships ([<reflink idref="bib5" id="ref24">5</reflink>]). Furthermore, when adding or subtracting fractions, students with whole number bias may also treat numerators and denominators as independent of one another, calculating each separately. For example, a student exhibiting whole number bias may identify the sum of 1/2 and 1/5 as 2/7 or the difference between 1/7 and 1/2 as 0/5. These errors are commonly attributed to a conceptual-change theory, during which students are expected to shift their conceptual understanding of whole numbers to fractions ([<reflink idref="bib36" id="ref25">36</reflink>]). As educators, it is important to note that these changes are often gradual and may reflect deeper misconceptions (see Table 1). Understanding how to identify these misconceptions will benefit teachers not only when assessing student learning but also when planning for specially designed instruction.</p> <p>Table 1 Students With Mathematical Learning Disabilities: Common Fraction Misconceptions</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="char" char="." /></colgroup><thead><tr><th align="left">Misconception</th><th align="center">Example</th></tr></thead><tbody><tr><td>The student exhibits whole number bias (identifying fractions as two distinct whole numbers).</td><td>When ordering fractions with like numerators, the student orders fractions according to the denominators (e.g., 2/12 > 2/8 > 2/3 > 2/1).</td></tr><tr><td>The student lacks attention to equal parts (<xref ref-type="bibr" rid="bibr1">Alkhateeb, 2019</xref>).</td><td>When identifying fractional parts, the student claims the circle below is divided into thirds.<graphic href="10.1177_00400599241231228-img3.tif" content-type="Graph" /></td></tr><tr><td>The student ignores the size of the referent whole when comparing fractions.</td><td>When comparing magnitude, the student does not consider the size of the referent whole (e.g., 1/2 of a car = 1/2 of a hot dog).</td></tr><tr><td>The student misrepresents fractions that are greater than 1 (e.g., 12/6, 8/5, 9/2).</td><td>When adding 7/8 of a cake and 5/8 of a cake, the student states that altogether, there is 12/8 of a cake.</td></tr><tr><td>The student exhibits frustration shifting fractional thinking between models (area, length, set).</td><td>When asked to show ways to represent 3/4, the student draws a circle that is divided into fourths and shades three parts. However, when asked to show another way to represent 3/4, the student is unable to do so (<xref ref-type="bibr" rid="bibr7">Clark et al., 2008</xref>).</td></tr></tbody></table> </ephtml> </p> <p> <emph>After Dr. Brave and the fifth-grade teachers reviewed the preassessment data, they realized that all students identified with MLDs exhibited some form of whole number bias. This was evident in either fraction compari</emph>s<emph>on tasks or fraction computation tasks. However, supporting students' learning would require a deeper understanding of students' strengths and needs.</emph></p> <p> <emph>To identify the root causes of misunderstandings, Dr. Brave and the fifth-grade teachers followed up with face-to-face conversations. Conversations revealed that although all students displayed unique mathematical strengths (from drawing to cooking), students' needs varied. Some students showed they needed to continue to examine the idea of fractions as numbers (3.NF). Others showed they were ready to apply the ideas of equivalence and ordering (4.NF). The teachers agreed these skills were foundational to adding and subtracting fractions with like and unlike denominators (4.NF and 5.NF).</emph> </p> <hd id="AN0185908517-3">Using a Manipulative-Based instructional Sequence to Scaffold Instruction</hd> <p>The understanding of fractions as numbers is crucial in determining future success ([<reflink idref="bib22" id="ref26">22</reflink>]). Because students with MLDs face challenges in learning fractional concepts, the National Council of Teachers of Mathematics promotes instruction that reinforces the connections between mathematical concepts and procedures. Therefore, researchers ([<reflink idref="bib2" id="ref27">2</reflink>]; [<reflink idref="bib30" id="ref28">30</reflink>]) have advocated for the implementation of manipulative-based instructional sequences to support the development of mathematical reasoning. According to [<reflink idref="bib30" id="ref29">30</reflink>], manipulative-based instructional models are "graduated sequence(s) of instruction to first teach students to use manipulatives (concrete or virtual), then representations (e.g., drawings), and final abstract methods (e.g., algorithms, math facts) to gain conceptual and procedural understanding of math" (p. 38)." In other words, students gain an understanding of mathematical concepts by acting on physical materials, creating semi-concrete representations, and communicating ideas through numerals and abstract symbols. These sequences have proved successful not only in working with students with MLDs but also in increasing students' sense of empowerment ([<reflink idref="bib2" id="ref30">2</reflink>]; [<reflink idref="bib30" id="ref31">30</reflink>]; [<reflink idref="bib37" id="ref32">37</reflink>]).</p> <p>The following examples illustrate how teachers like Dr. Brave can utilize manipulative-based instructional sequences in the classroom to support the understanding of fractional concepts and procedural fluency. Examples are presented according to students' progressions of learning (Common Core State Standards Numbers and Operations – Fractions) and the stages of the manipulative-based instructional sequence. Examples also include guidelines for planning and recommendations for effective implementation.</p> <hd id="AN0185908517-4">Planning for the Concrete Phase</hd> <p> <emph>During preassessments, students were asked to share their mathematical reasoning. As students explained how they completed tasks related to comparisons and computation, they were encouraged to use mathematical tools and/or semi-concrete representations. Many concrete manipulatives (e.g., fraction circles, number lines, and fractions strips) were available for the students. However, Dr. Brave noticed that although students attempted to use the tools, the tools were either incompatible with the students' strategies (e.g., fraction strips representing sevenths and ninths were not included in the set, number lines only represented whole numbers) or incongruent with the students' object manipulation skills (e.g., frustration when folding paper into equal parts, difficulty "lining up" fraction strips). She also noticed that despite the use of tools, students continued to rely on whole number knowledge to compare and compute fractions. This led to even greater misconceptions and as a result, decreased students' mathematical agency.</emph> </p> <p>In planning for the concrete phases of the instructional sequence, begin by choosing manipulatives or "concrete representations that encourage learning through movement or action" ([<reflink idref="bib35" id="ref33">35</reflink>], p. 108). In the aforementioned case, the learning goal ("understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b"; [<reflink idref="bib9" id="ref34">9</reflink>]) is essential in determining what types of mathematical manipulatives best support conceptual understanding ([<reflink idref="bib2" id="ref35">2</reflink>]). To understand the relationship between the numerator and the denominator, students must not only recognize parts as equal shares of a whole but also describe how the whole relates to the size of its parts. Research has shown students with MLDs are more likely to experience challenges connecting these part-whole concepts, particularly when presented symbolically ([<reflink idref="bib22" id="ref36">22</reflink>]). On an instructional level, this means teachers like Dr. Brave must begin by planning for the learning of fractions on a conceptual level. This may be done simultaneously to or prior to introducing them symbolically. Furthermore, teachers may also consult the Universal Design for Learning (UDL) framework in planning for multiple means of representation, action, and expression ([<reflink idref="bib6" id="ref37">6</reflink>]). This will provide a means for decreasing barriers and optimizing learning prior to implementing instructional strategies.</p> <p>An assortment of concrete tools may be used to support conceptual understanding (e.g., fraction strips, fraction circles, paper strips, Cuisenaire rods, pattern blocks, counters). In the aforementioned case, the goal is for students to gain an understanding of fractional concepts rather than aimlessly move manipulatives according to their understandings of whole numbers ([<reflink idref="bib34" id="ref38">34</reflink>]). Therefore, teachers may choose to use manipulatives that do not display numerals or symbols prior to discussing the abstract. Because students with MLDs may need additional support connecting concepts to symbols ([<reflink idref="bib27" id="ref39">27</reflink>]), this may be beneficial in laying the foundation for future understanding.</p> <p>After you have narrowed choices to match the learning goals, consider the following questions:</p> <p></p> <ulist> <item> Which choices best support the transformation from a concrete stage to a semi-concrete representational stage or in some cases, a concrete stage to an abstract stage?</item> <p></p> <item> Will students benefit from creating their own fraction manipulatives? If not, will manipulatives that are divided equally prior to introducing fractional concepts (prepartitioned) best support learning? If so, will students be more successful with manipulatives of varying shapes (e.g., pattern blocks) or similar shapes (e.g., Cuisenaire rods or precut paper strips)?</item> <p></p> <item> Which choices allow for a variety of ways to partition (separate) the whole and iterate (repeatedly copy) the parts?</item> <p></p> <item> Will students have access to virtual or digital representations of concrete manipulatives? If so, do students' strengths and/or needs (e.g., fine motor skills, motivation levels, technological skills) support the use of technology, and are technological supports (e.g., base-10 grouping mechanisms, fraction number lines, etc.) built in to enhance learning and bolster independence ([<reflink idref="bib3" id="ref40">3</reflink>])?</item> <p></p> <item> Which manipulatives can be used in a variety of contexts (see Table 2)?</item> </ulist> <p>Table 2 Manipulatives/Tools for Modeling Fraction Concepts</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /></colgroup><thead><tr><th align="left">Concept model</th><th align="center">Definition</th><th align="center">Manipulatives/tools</th><th align="center">Considerations</th></tr></thead><tbody><tr><td>Area</td><td>An area or region is subdivided into equal parts (e.g., circle divided to show pie pieces, square divided to show brownie pieces)</td><td>Fraction circlesFraction tilesFraction squaresGeoboardsPattern blocksPaper foldingLego</td><td>*Equal partitioning may be difficult when working with odd denominators. *Models are more difficult to partition as denominators increase. *The area model is a good place to start exploring the act of "equal sharing and partitioning" (<xref ref-type="bibr" rid="bibr35">Van de Walle et al., 2019</xref>, p. 341).</td></tr><tr><td>Length</td><td>A unit of length is subdivided into equal parts (e.g., a ruler divided to show eighths of an inch, a mile divided into equal parts)</td><td>RulersPaper stripsCuisenaire rodsNumber lines</td><td>*Often, "model switching" occurs between the length model and the area model. Therefore, be explicit in linking the model to the context.</td></tr><tr><td>Set</td><td>A set or collection of items is divided equally into subsets (e.g., fraction of boys in a class, fraction of oranges/apples/bananas in a bowl of fruit)</td><td>Two-color countersCounting bearsCountable items that can be joined to create a set (e.g., buttons, balls, markers, crayons)</td><td>*Viewing the set as a whole may present challenges for some students (<xref ref-type="bibr" rid="bibr21">Lee & Lee, 2023</xref>).*Common misconceptions include the counting of the size of a subset rather than the number of equal subsets to identify the numerator (<xref ref-type="bibr" rid="bibr35">Van de Walle et al., 2019</xref>).</td></tr></tbody></table> </ephtml> </p> <p> <emph>After careful deliberation, Dr. Brave chooses to use prepartitioned paper strips to reintroduce fractional concepts. Although she is familiar with the pattern blocks, she realizes that because of their varying shapes (hexagons, rhombi, triangles, trapezoids), they may not be the best starting point for concept development. She is also familiar with the Cuisenaire rods; however, she highlights the fact that some of her students struggle with fine motor skills and may experience frustration with their small sizes. So, she uses colored 8 × 11 construction paper to create strips that represent wholes, halves, thirds, fourths, fifths, sixths, sevenths, eighths, 10ths, and 12ths. Each student is provided with a set of strips and matching parts. From here, Dr. Brave is explicit in her instruction and deliberate in her questioning, drawing attention to the part-whole relationships and the use of authentic contexts. After students master the concept, she makes an explicit connection to unit fractions as a number (e.g., 1/2, 1/4) and encourages students to label their paper strips accordingly. See Table 3 for an example of implementation.</emph> </p> <p>Table 3 Implementation of a Manipulative-Based Instructional Sequence</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /></colgroup><tbody><tr><td><bold>Concrete phase</bold></td></tr><tr><td><bold><italic>Dr. Brave</italic>:</bold> (Shows the photo of a snail) Good news everyone! I have adopted a pet snail, and I have signed her up for a race.Dr. Brave shows the students the orange strip of paper that represents the whole.<bold><italic>Dr. Brave</italic>:</bold> This is the length of the race that she will be running this week.Dr. Brave then instructs the students to take out their orange strips.<bold><italic>Dr. Brave</italic>:</bold> (Referring to the orange strip) As I shared, this is the length of the whole race. However, my snail is always tired, so she has decided to break the race into two parts. What color strips could we use to show how she can run the whole race in two parts?Students take their strips out of their bags and examine the strips to find two halves.<bold><italic>Mason</italic>:</bold> If I put the red strips next to each other, they are as long as the orange strip.<bold><italic>Maria</italic>:</bold> She can run the red part one day and the next red part the next day.<bold><italic>Dr. Brave</italic>:</bold> I agree. What else do you notice about the red strips?<bold><italic>Students</italic>:</bold> There are two strips.The strips are the same length.Dr. Brave stresses the relationship between the two equal parts and the whole.<bold><italic>Dr. Brave</italic>:</bold> Do you think that there is a colored strip that could represent how she could run the whole race in three parts?The following image shows the whole strip and the thirds (whole in three parts):<graphic href="10.1177_00400599241231228-img4.tif" content-type="Graph" />The process continues with fourths, fifths, sixths, and so on. Dr. Brave reinforces the fact that the strips are equal sizes and can be "iterated" or repeatedly copied to make a whole. She also draws attention to the changes in the sizes of the parts (decreases) as the number of parts increases.Once students have mastered the concept, Dr. Brave introduces the "unit fraction (1/3, 1/4, 1/5, etc.)." She then connects to the abstract by encouraging students to label their strips accordingly.</td></tr></tbody></table> </ephtml> </p> <p>1 <emph>Note.</emph> Teachers can also change the length of the referent whole to reinforce the fact that fraction comparisons depend on the size of the whole. This can be done with paper strips or Cuisenaire rods.</p> <p>It is critical that we understand the research as educators, and research shows that multiple methods of representation support the learning of fractions ([<reflink idref="bib30" id="ref41">30</reflink>]; [<reflink idref="bib35" id="ref42">35</reflink>]). Although teachers like Dr. Brave may begin with a linear model, it is important that students develop an understanding of fractional concepts using all three types of models (see Table 2). This is particularly important for students with MLDs because it supports the transfer of knowledge from familiar contexts to unfamiliar contexts ([<reflink idref="bib19" id="ref43">19</reflink>]).</p> <hd id="AN0185908517-5">Planning for the Semi-Concrete Representational Phase</hd> <p>The semi-concrete representational stage bridges the gap between the concrete and abstract stages. This phase is commonly referred to as the "seeing" phase of the manipulative-based instructional sequence because it typically involves the creation of drawings that represent mental models. Mental models are then stored in the students' brains and used to connect previously learned information to novel concepts, such as decimals and ratios ([<reflink idref="bib12" id="ref44">12</reflink>]).</p> <p>Drawings are commonly associated with the semi-concrete representational phase of an instructional sequence; however, it is important to remember that representations must be explicitly chosen to support student learning. Consider the following scenario.</p> <p> <emph>Maria was diagnosed with an MLD. Prior conversations showed she recognized fractions as parts of a whole and identified numerators and denominators as working in unison to describe magnitude; however, when asked to identify a fraction that was "the same as (or equivalent to) 1/2," she became discouraged. Not only was she frustrated when asked to draw and partition a circle (see Figure 2), but she was uneasy when "lining up" physical manipulatives to model equivalent fractions (see Figure 2). For this reason, she experienced a heightened sense of frustration when asked to identify equivalent fractions.</emph> </p> <p>Graph: Figure 2 Maria attempted to partition a circle to find fractions that are equivalent to 1/2 and then lined up the fraction strips (tiles) to test for equivalence</p> <p>In the aforementioned case, Maria attempted to illustrate the concepts of magnitude and equivalence (3. NF.A.3: explain equivalence of fractions in special cases and compare fractions by reasoning about their size). However, her fine motor skills affected her ability to represent fractions through drawings or concrete manipulatives. This is not uncommon; research has shown that students with MLDs are more likely to struggle with tasks that require visual perception and fine motor skills ([<reflink idref="bib29" id="ref45">29</reflink>]). Because these issues may result in frustration when lining up physical fraction strips, comparing pattern blocks, or copying shapes from the board, teachers may choose to omit the semi-concrete representational stage or present alternative means of representation ([<reflink idref="bib2" id="ref46">2</reflink>]).</p> <p>Although several alternatives allow for exploration on a semi-concrete representational level, one practice that proves effective with both students with and without MLDs includes the use of number lines ([<reflink idref="bib30" id="ref47">30</reflink>]). Number lines not only reinforce multiple mathematics skills, but they also support the continuity of learning from elementary to secondary school. Additional research shows that number lines reinforce part-whole relationships and deepen students' understanding of linear models. Because students are required to think about the relationships between numerators and denominators to determine magnitude, number lines also support the reduction of students' whole number biases ([<reflink idref="bib31" id="ref48">31</reflink>]). Furthermore, number lines allow teachers to expand the understanding of fractions beyond the typical area model while also reinforcing the fact that an infinite number of fractions fall between two whole numbers ([<reflink idref="bib15" id="ref49">15</reflink>]).</p> <p>When planning for students with MLDs, prepartition, color code, or pair fraction tiles with number lines to support retention and memory mapping ([<reflink idref="bib30" id="ref50">30</reflink>]). Number lines may also be presented horizontally or vertically. Vertical number lines are useful in reinforcing magnitude and transitioning to the concept of negative and positive integers ([<reflink idref="bib4" id="ref51">4</reflink>]; [<reflink idref="bib32" id="ref52">32</reflink>]).</p> <p>Virtual number lines may also be used to support students with fine motor issues. Sites, such as Didax (https://<ulink href="http://www.didax.com/apps/fraction-number-line/),">www.didax.com/apps/fraction-number-line/),</ulink> allow for the exploration of fractions on number lines while also allowing users to match virtual fraction tiles to number lines. The Math Learning Center also features a fraction app (https://apps.mathlearningcenter.org/fractions/) that allows for the equal partitioning and iterating of linear models and area models (see Figure 3). Linear models can be connected to number lines via the writing tool. Students may then take a screenshot to create personalized semi-concrete representations.</p> <p>Graph: Figure 3 Students may use the Didax site (https://<ulink href="http://www.didax.com/apps/fraction-number-line/">www.didax.com/apps/fraction-number-line/</ulink>) to pair fraction tiles with the number lines or the Math Learning Center app (https://apps.mathlearningcenter.org/fractions/) to partition length models</p> <p>There are also several online visual aids that have been created to reinforce the concept of fractions. Unfortunately, however, these visuals are often used to reinforce memorization rather than the understanding of concepts ([<reflink idref="bib17" id="ref53">17</reflink>]). A common example of this is the prepartitioned fraction wall. Although several iterations of prepartitioned fraction walls can be found online, researchers ([<reflink idref="bib10" id="ref54">10</reflink>]) have found that that the static nature of the wall reinforces partitioning but not iterating and as a result, recommend that the fraction wall be used with supplemental models to allow for a deeper understanding of how the whole can be "broken apart" and "reconstructed."</p> <p> <emph>In planning for the semi-concrete representational phase of instruction, Dr. Brave reflected on successes from virtual learning. During virtual learning, the students used Didax fraction tiles and number lines to reinforce fractional concepts. These fraction tiles (strips) differed from the physical models because they clicked directly onto the canvas in rows. This would alleviate any frustrations associated with the lining up of manipulatives, allowing for precise comparisons (by stacking) and the act of addition (by joining). More importantly was the fact that the fraction tiles could be paired with number lines. Number lines could not only be used to explore magnitude and equivalence, but they could also be used in future lessons to calculate sums and differences (see Table 4 for an example of implementation).</emph> </p> <p>Table 4 Implementation of a Manipulative-Based Instructional Sequence</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /></colgroup><tbody><tr><td><bold>Semi-concrete representational phase</bold></td></tr><tr><td>Before introducing the students to equations with unlike denominators, Dr. Brave revisited the standard:CCSS.MATH.CONTENT.3.NF.A.3 (explain equivalence of fractions in special cases and compare fractions by reasoning about their size).<bold><italic>Dr. Brave</italic>:</bold><italic>The other day, we talked about my snail and the race that she was about to run. Well, the race is over, and I would like to show you how far she ran.</italic> Dr. Brave shows the following image:<graphic href="10.1177_00400599241231228-img5.tif" content-type="Graph" /><bold><italic>Dr. Brave</italic>:</bold><italic>What do you notice about this image?</italic><bold><italic>Maria</italic>:</bold><italic>There is a number line on top of the strips.</italic><bold><italic>Mason</italic>:</bold><italic>There are three orange 1/4 strips under the number line.</italic><bold><italic>Dr. Brave</italic>:</bold><italic>You are right! Think back to our paper strips. The number line represents the number of whole races, and the orange strips show how far the snail ran. So, how much of the race do you think that the snail completed?</italic>Dr. Brave values and acknowledges all ideas. Then, she shows the following image:<graphic href="10.1177_00400599241231228-img6.tif" content-type="Graph" /><bold><italic>Dr. Brave</italic>:</bold><italic>If the 1 on the number line represents the length of the whole race, did the snail finish the whole race?</italic><bold><italic>Maria</italic>:</bold><italic>No, she did not make it to the end! She only made it part of the way!</italic>Dr. Brave then encourages the students to count the unit fractions aloud (1/4, 2/4, 3/4...) because their understanding of spatial relations is critical to maintaining one-to-one correspondence and connecting counting to cardinality. She then compares the length that the snail ran to the length of the race using mathematical language (e.g., 3/4 is less than 1, and 1 is greater than 3/4).Over the next few days, students will use the Didax site to pair fraction tiles to number lines and compare length models. The following example shows how Maria created a model to compare 1/3 of a licorice string to 3/8 of a licorice string.<graphic href="10.1177_00400599241231228-img7.tif" content-type="Graph" />As the students continue working, they clear up common misconceptions. They also notice that fractions "with the smallest denominators" are greater when comparing fractions with the same numerators. They discuss why this is the case, reinforcing the denominator as the number of equal parts in the whole. When comparing these parts to the whole, they also identify how many more fractional pieces are needed to make an equivalent whole.</td></tr></tbody></table> </ephtml> </p> <hd id="AN0185908517-6">Planning for the Abstract Phase</hd> <p>The abstract stage of instruction is the final phase in the manipulative-based instructional sequence. During this phase of instruction, concrete models and semi-concrete representations are paired to numerals and mathematical symbols to describe mathematical concepts and/or procedures.</p> <p>In the case of fractions, it is important to recognize the role of meaning when applying abstract notation. Without the intuitive meaning of fractions, students may revert to whole number bias, viewing symbols and numerals as nothing more than two distinct digits separated by a line or bar ([<reflink idref="bib11" id="ref55">11</reflink>]). For this reason, teachers may choose to introduce this phase simultaneous to or after the concrete or semi-concrete representational phase. [<reflink idref="bib35" id="ref56">35</reflink>] emphasized the fact that a manipulative-based instructional model is not linear but rather, an integrated model where there is parallel modeling throughout to describe the relationships manipulatives, semi-concrete representations, and equations. This means that although some students may connect models and semi-concrete representations to mathematical equations at the same time, others may rely solely on concrete models to communicate their ideas. Consider the UDL framework in planning for representation, action, and expression ([<reflink idref="bib6" id="ref57">6</reflink>]). This allows students to access and exhibit learning in ways that are unique to their needs.</p> <p>Prior to implementing the abstract phase, consider the likelihood of various misconceptions. Abstraction involves Arabic numerals (0, 1, 2, 3,...) and mathematical symbols (+, =, –, <emph>n</emph>,...) rather than real-world objects. This may pose challenges for students with MLDs. Students with MLDs may struggle to identify numerals and symbols, connect numerals and symbols to semi-concrete representations, and apply numerals and symbols in a procedural manner ([<reflink idref="bib18" id="ref58">18</reflink>]). These challenges coupled with the misconceptions related to whole number bias suggest that teachers may choose to provide students with unique scaffolds when implementing the abstract phase. These scaffolds include the revisitation of the concrete or semi-concrete representational stage, the use of authentic context, and the modification of grade-level content.</p> <p>The use of authentic context is critical in transferring mathematical concepts to abstract symbols. Researchers ([<reflink idref="bib13" id="ref59">13</reflink>]; [<reflink idref="bib30" id="ref60">30</reflink>]) recommend teachers begin by introducing students to a story that is "cognitively accessible" ([<reflink idref="bib30" id="ref61">30</reflink>]). The teacher and students then summarize the story using concrete manipulatives, semi-concrete representations, and abstract numerals and symbols. For example, in the snail race (see Tables 3 and 4), students are asked to consider the information that is presented, relate this information to what is unknown, and develop a strategy for solving that makes sense within the given context ([<reflink idref="bib20" id="ref62">20</reflink>]). This not only allowed for an introduction to the length model but also set the stage for future problem solving.</p> <p>Instruction can also be scaffolded by deliberately choosing fractions to increase accessibility. For example, when using abstract symbols and numerals to add fractions, the teacher may choose to modify an assignment by substituting fractions with like denominators in the place of those with unlike denominators. To reinforce the connection between conceptual understanding and abstract reasoning ([<reflink idref="bib25" id="ref63">25</reflink>]), students may also use visual fraction calculators to solve equations. Sites such as GeoGebra (https://<ulink href="http://www.geogebra.org/m/DV6Ehjnx#material/aEsvBzN2">www.geogebra.org/m/DV6Ehjnx#material/aEsvBzN2</ulink>) and DadsWorksheets (https://<ulink href="http://www.dadsworksheets.com/fraction-calculator.html">www.dadsworksheets.com/fraction-calculator.html</ulink>) include visual components and are free to educators.</p> <p> <emph>Dr. Brave begins by asking the students if they have ever cut a ribbon or string. They tell her that they have done so in art class to create self-portraits. She then asks them to "visualize" as she reads the problem:</emph> </p> <p> <emph>Maria has 3/4 of a yard of ribbon. She cuts 2/8 of the ribbon to make her self-portrait. How much ribbon does she have left?</emph> </p> <p> <emph>Maria immediately tells the group that she would have less ribbon because she is "cutting away ribbon and using it" in this scenario. Together, they pair the Didax fraction tiles with the number line to create a semi-concrete representation of the problem. Then, they use gestures to mimic the act of cutting and discuss how the act of "separating" could be represented with the subtraction symbol. After this, they record the expression under the visual (3/4 – 2/8).</emph> </p> <p> <emph>After brainstorming, the students discover that 3/4 is equivalent to 6/8. So, they begin lining up unit fractions with eighths on the second row of the canvas to create an equivalent fraction (see Figure 4).</emph> </p> <p>Graph: Figure 4 The students modeled an equivalent fraction using unit fractions with eighths on the second row</p> <p> <emph>After creating the equivalent fraction, they separate the minuend from the second row. This results in the difference (see Figure 5).</emph> </p> <p>Graph: Figure 5 The students separated the minuend (2/8) from the unit fractions in the second row to calculate the difference (4/8) and then drew on previously learned information to tell us that 4/8 was equivalent to 1/2.</p> <hd id="AN0185908517-7">Conclusion</hd> <p>Studies have shown that students with MLDs are more likely to exhibit misunderstandings when conceptualizing and calculating fractions ([<reflink idref="bib16" id="ref64">16</reflink>]; [<reflink idref="bib22" id="ref65">22</reflink>]; [<reflink idref="bib33" id="ref66">33</reflink>]). This article described common misconceptions and provided educators with examples of how manipulative-based instructional sequences can be used to support the fractional understanding of students with MLDs. Guidelines for assessing fraction knowledge and planning for each stage of the sequence are also included. Although recommendations and examples apply to the learning of fractions, they also provide a structure for promoting conceptual understanding and procedural fluency in mathematics.</p> <ref id="AN0185908517-8"> <title> References </title> <blist> <bibl id="bib1" idref="ref4" type="bt">1</bibl> <bibtext> Alkhateeb M.A. (2019). Common errors in fractions and thinking strategies that accompany them. International Journal of Instruction, 12(2). 399–416. <ulink href="http://doi.org/10.29333/iji.2019.12226a">http://doi.org/10.29333/iji.2019.12226a</ulink></bibtext> </blist> <blist> <bibl id="bib2" idref="ref7" type="bt">2</bibl> <bibtext> Bouck E. C., Anderson R. D., Long H., Sprick J. (2022). Manipulative-based instructional sequences in mathematics for students with disabilities. TEACHING Exceptional Children, 54(3), 178–190. doi:10.1177/0040059921994599</bibtext> </blist> <blist> <bibl id="bib3" idref="ref40" type="bt">3</bibl> <bibtext> Bouck E. C., Mathews L. A., Peltier C. (2019). Virtual manipulatives: A tool to support access and achievement with middle school students with disabilities. Journal of Special Education Technology, 35(1), 51–59. https://doi.org/10.1177/0162643419882422</bibtext> </blist> <blist> <bibl id="bib4" idref="ref51" type="bt">4</bibl> <bibtext> Bove S. P. (1995). Place value: A vertical perspective. Teaching Children Mathematics, 1(9), 542–546. https://doi.org/10.5951/TCM.1.9.0542</bibtext> </blist> <blist> <bibl id="bib5" idref="ref5" type="bt">5</bibl> <bibtext> Braithwaite D. W., Siegler R. S. (2018). Developmental changes in whole number bias. Developmental Science, 21(2), 1–13. https://doi.org/10.1111/desc.12541</bibtext> </blist> <blist> <bibl id="bib6" idref="ref37" type="bt">6</bibl> <bibtext> Brown N., Bostic J. D., Folger T., Folger L., Hicks T., Nafziger S. (2022). Revising word problems to address UDL and standards. Mathematics Teacher: Learning & Teaching PK–12, 115(4), 252–264. https://doi.org/10.5951/MTLT.2020.0365</bibtext> </blist> <blist> <bibl id="bib7" type="bt">7</bibl> <bibtext> Clark D. M., Roche A., Mitchell A. (2008). 10 Practical tips for making fractions come alive and make sense. Mathematics Teaching in the Middle School, 13(7), 373–380. https://doi.org/10.5951/MTMS.13.7.0372</bibtext> </blist> <blist> <bibl id="bib8" idref="ref9" type="bt">8</bibl> <bibtext> Common Core State Standards Initiative. 2021a. Grade 1 >> Geometry. <ulink href="http://www.thecorestandards.org/Math/Content/1/G/">http://www.thecorestandards.org/Math/Content/1/G/</ulink></bibtext> </blist> <blist> <bibl id="bib9" idref="ref34" type="bt">9</bibl> <bibtext> Common Core State Standards Initiative, 2021b. Grade 3 >> Numbers and operations fractions. <ulink href="http://www.thecorestandards.org/Math/Content/3/NF/">http://www.thecorestandards.org/Math/Content/3/NF/</ulink></bibtext> </blist> <blist> <bibtext> Cramer K. A., Wyberg T. (2009). Efficacy of different concrete models for teaching the part-whole construct for fractions. Mathematical Thinking and Learning, 11(4), 226–257. https://doi.org/10.1080/10986060903246479</bibtext> </blist> <blist> <bibtext> Dehaene S. (2011). The number sense: How the Mind creates mathematics (Rev. ed.). Oxford University Press.</bibtext> </blist> <blist> <bibtext> Fosnot C.T., Dolk M. (2002). Young mathematicians at work: Constructing fractions, decimals, and percents. Heinemann.</bibtext> </blist> <blist> <bibtext> Fuchs L. S., Newman-Gonchar R., Schumacher R., Dougherty B., Bucka N., Karp K. S., Woodward J., Clarke B., Jordan N. C., Gersten R., Jayanthi M., Keating B., Morgan S. (2021). Assisting students struggling with mathematics: Intervention in the elementary grades (WWC 2021006). National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. https://ies.ed.gov/ncee/wwc/Docs/PracticeGuide/WWC2021006-Math-PG.pdf</bibtext> </blist> <blist> <bibtext> Geller E., Son J. Y., Stigler J. (2017). Conceptual explanations and understanding fraction comparisons. Learning and Instruction, 52, 122–129. https://doi.org/10.1016/j.learninstruc.2017.05.006</bibtext> </blist> <blist> <bibtext> Gersten R. M., Schumacher R. F., Jordan N. C. (2017). Life on the number line: Routes to understanding fraction magnitude for students with difficulties learning mathematics. Journal of Learning Disabilities, 50(6), 655–657. https://doi.org/10.1177/0022219416662625</bibtext> </blist> <blist> <bibtext> Im S., Jitendra A. K. (2020). Analysis of proportional reasoning and misconceptions among students with mathematical learning disabilities. The Journal of Mathematical Behavior, 57, Article 100753. https://doi.org/10.1016/j.jmathb.2019.100753</bibtext> </blist> <blist> <bibtext> Izsák A., Tillema E. S., Tunç-Pekkan Z. (2008). Teaching and learning fraction addition on number lines. Journal for Research in Mathematics Education, 39(1), 33–62.</bibtext> </blist> <blist> <bibtext> Karagiannakis G., Baccaglini-Frank A., Papadatos Y. (2014). Mathematical learning difficulties subtypes classification. Frontiers in Human Neuroscience, 8, Article 57. https://doi.org/10.3389/fnhum.2014.00057</bibtext> </blist> <blist> <bibtext> Kunwar R. (2021). Impacts of dyscalculia in learning mathematics: Some considerations for content delivery and support. In Misciagna S. (Ed.), Learning disabilities: Neurobiology, assessment, clinical features and treatments. IntechOpen. https://doi.org/10.5772/intechopen.99038</bibtext> </blist> <blist> <bibtext> Lee J. (2007). Context in mathematics learning: Problems and possibilities. Teaching Children Mathematics, 14(1), 40–44. https://doi.org/10.5951/TCM.14.1.0040</bibtext> </blist> <blist> <bibtext> Lee J., Lee M.Y. (2023). How elementary prospective teachers use three fraction models: their perceptions and difficulties. Journal of Mathematics Teacher Education, 26, 455–480. https://doi.org/10.1007/s10857-022-09537-4</bibtext> </blist> <blist> <bibtext> Mazzocco M. M., Myers G. F., Lewis K. E., Hanich L. B., Murphy M. M. (2013). Limited knowledge of fraction representations differentiates middle school students with mathematics learning disability (dyscalculia) versus low mathematics achievement. Journal of Experimental Child Psychology, 115(2), 371–387. https://doi.org/10.1016/j.jecp.2013.01.005</bibtext> </blist> <blist> <bibtext> Namkung J. M., Fuchs L. (2019). Remediating difficulty with fractions for students with mathematics learning difficulties. Learning Disabilities: A Multidisciplinary Journal, 24(2), 36–48.</bibtext> </blist> <blist> <bibtext> National Center for Education Statistics. (2022). The nation's report card: NAEP data explorer. https://<ulink href="http://www.nationsreportcard.gov/ndecore/landing">www.nationsreportcard.gov/ndecore/landing</ulink></bibtext> </blist> <blist> <bibtext> National Council of Teachers of Mathematics. (2015). Calculator use in the elementary grades. https://<ulink href="http://www.nctm.org/Standards-and-Positions/Position-Statements/Calculator-Use-in-Elementary-Grades/">www.nctm.org/Standards-and-Positions/Position-Statements/Calculator-Use-in-Elementary-Grades/</ulink></bibtext> </blist> <blist> <bibtext> National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for mathematics. https://corestandards.org/mathematics-standards/</bibtext> </blist> <blist> <bibtext> Newton K. J., Willard C., Teufel C. (2014). An examination of the ways that students with learning disabilities solve fraction computation problems. Elementary School Journal, 115(1), 1–21. https://doi.org/10.1086/676949</bibtext> </blist> <blist> <bibtext> NRICH Primary Team. (2019). Exploring fractions. University of Cambridge. https://nrich.maths.org/10496</bibtext> </blist> <blist> <bibtext> Pieters S., Desoete A., Roeyers H., Vanderswalmen R., Van Waelvelde H. (2012). Behind mathematical learning disabilities: What about visual perception and motor skills? Learning and Individual Differences, 22(4), 498–504. https://doi.org/10.1016/j.lindif.2012.03.014</bibtext> </blist> <blist> <bibtext> Root J., Saunders A., Jimenez B., Gilley D. (2023). Essential components for math instruction: Considerations for students with extensive support needs. TEACHING Exceptional Children. 56(1), 34–43. https://doi.org/10.1177/00400599221120882</bibtext> </blist> <blist> <bibtext> Schumacher R. F., Jayanthi M., Gersten R., Dimino J., Spallone S., Haymond K. S. (2018). Using the number line to promote understanding of fractions for struggling fifth graders: A formative pilot study. Learning Disabilities: Research and Practice, 33(4), 192–206. https://doi.org/10.1111/ldrp.12169</bibtext> </blist> <blist> <bibtext> Shanty N. O. (2016). Investigating students' development of learning integer concept and integer addition. Journal of Mathematics Education, 7(2), 57–72. https://doi.org/10.22342/jme.7.2.3538.57-72</bibtext> </blist> <blist> <bibtext> Tian J., Siegler J. (2017). Fraction learning in children with mathematics difficulties. Journal of Learning Disabilities, 50(6), 614–620. https://doi.org/10.1016/j.jecp.2013.01.005</bibtext> </blist> <blist> <bibtext> Van de Walle J., Thompson C. S. (1984). Let's do it: Fractions with fraction strips. Arithmetic Teacher, 32(4), 4–9.</bibtext> </blist> <blist> <bibtext> Van de Walle J. A., Karp K. S., Bay-Williams J. M. (2019). Elementary and middle school mathematics: Teaching developmentally. Pearson Education, Inc.</bibtext> </blist> <blist> <bibtext> Van Hoof J., Verschaffel L., De Neys W., Van Dooren W. (2020). Intuitive errors in learners' fraction understanding: A dual-process perspective on the natural number bias. Memory & Cognition, 48, 1171–1180. https://doi.org/10.3758/s13421-020-01045-1</bibtext> </blist> <blist> <bibtext> Zhang S., Yu S., Xiao J., Liu Y., Jiang T. (2021). The effects of concrete representational abstract sequence instruction on fractions for Chinese elementary students with mathematics learning disabilities. International Journal of Science and Math Education, 20, 1481–1498. https://doi.org/10.1007/s10763-021-10215-9</bibtext> </blist> </ref> <ref id="AN0185908517-9"> <title> Footnotes </title> <blist> <bibtext> Kathryn Lavin Brave conceived of the idea and conducted preliminary background research. All individuals worked to collaborate in researching and implementing instruction. The proposed article was written by Kathryn Lavin Brave.</bibtext> </blist> <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> The author(s) received no financial support for the research, authorship, and/or publication of this article.</bibtext> </blist> <blist> <bibtext> Kathryn Lavin Brave</bibtext> </blist> <blist> <bibtext>Graph https://orcid.org/0000-0002-6806-4145</bibtext> </blist> </ref> <aug> <p>By Kathryn Lavin Brave; Izzy Berman; Debita Basu and Alexis Szkotak</p> <p>Reported by Author; Author; Author; Author</p> </aug> <nolink nlid="nl1" bibid="bib24" firstref="ref1"></nolink> <nolink nlid="nl2" bibid="bib23" firstref="ref2"></nolink> <nolink nlid="nl3" bibid="bib37" firstref="ref3"></nolink> <nolink nlid="nl4" bibid="bib14" firstref="ref6"></nolink> <nolink nlid="nl5" bibid="bib26" firstref="ref10"></nolink> <nolink nlid="nl6" bibid="bib16" firstref="ref11"></nolink> <nolink nlid="nl7" bibid="bib22" firstref="ref12"></nolink> <nolink nlid="nl8" bibid="bib33" firstref="ref13"></nolink> <nolink nlid="nl9" bibid="bib28" firstref="ref17"></nolink> <nolink nlid="nl10" bibid="bib35" firstref="ref19"></nolink> <nolink nlid="nl11" bibid="bib36" firstref="ref25"></nolink> <nolink nlid="nl12" bibid="bib30" firstref="ref28"></nolink> <nolink nlid="nl13" bibid="bib34" firstref="ref38"></nolink> <nolink nlid="nl14" bibid="bib27" firstref="ref39"></nolink> <nolink nlid="nl15" bibid="bib19" firstref="ref43"></nolink> <nolink nlid="nl16" bibid="bib12" firstref="ref44"></nolink> <nolink nlid="nl17" bibid="bib29" firstref="ref45"></nolink> <nolink nlid="nl18" bibid="bib31" firstref="ref48"></nolink> <nolink nlid="nl19" bibid="bib15" firstref="ref49"></nolink> <nolink nlid="nl20" bibid="bib32" firstref="ref52"></nolink> <nolink nlid="nl21" bibid="bib17" firstref="ref53"></nolink> <nolink nlid="nl22" bibid="bib10" firstref="ref54"></nolink> <nolink nlid="nl23" bibid="bib11" firstref="ref55"></nolink> <nolink nlid="nl24" bibid="bib18" firstref="ref58"></nolink> <nolink nlid="nl25" bibid="bib13" firstref="ref59"></nolink> <nolink nlid="nl26" bibid="bib20" firstref="ref62"></nolink> <nolink nlid="nl27" bibid="bib25" firstref="ref63"></nolink>
Header DbId: eric
DbLabel: ERIC
An: EJ1475234
AccessLevel: 3
PubType: Academic Journal
PubTypeId: academicJournal
PreciseRelevancyScore: 0
IllustrationInfo
Items – Name: Title
  Label: Title
  Group: Ti
  Data: Using Manipulative-Based Instructional Sequences to Increase the Understanding of Fractional Concepts of Students with Mathematical Learning Disabilities
– Name: Language
  Label: Language
  Group: Lang
  Data: English
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Kathryn+Lavin+Brave%22">Kathryn Lavin Brave</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-6806-4145">0000-0002-6806-4145</externalLink>)<br /><searchLink fieldCode="AR" term="%22Izzy+Berman%22">Izzy Berman</searchLink><br /><searchLink fieldCode="AR" term="%22Debita+Basu%22">Debita Basu</searchLink><br /><searchLink fieldCode="AR" term="%22Alexis+Szkotak%22">Alexis Szkotak</searchLink>
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="SO" term="%22TEACHING+Exceptional+Children%22"><i>TEACHING Exceptional Children</i></searchLink>. 2025 57(5):368-378.
– 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: 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: 2025
– Name: TypeDocument
  Label: Document Type
  Group: TypDoc
  Data: Journal Articles<br />Reports - Descriptive
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Teaching+Methods%22">Teaching Methods</searchLink><br /><searchLink fieldCode="DE" term="%22Manipulative+Materials%22">Manipulative Materials</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="%22Students+with+Disabilities%22">Students with Disabilities</searchLink><br /><searchLink fieldCode="DE" term="%22Learning+Disabilities%22">Learning Disabilities</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+Logic%22">Mathematical Logic</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Instruction%22">Mathematics Instruction</searchLink><br /><searchLink fieldCode="DE" term="%22Computation%22">Computation</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1177/00400599241231228
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 0040-0599<br />2163-5684
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Manipulative-based instructional sequences have proven to be successful with students with disabilities. However, instruction must not only support the acquisition of conceptual and procedural knowledge but also build on students' strengths. This article describes how teachers can use manipulative-based instructional sequences to support the fractional understanding of students who are diagnosed with mathematical learning disabilities. We begin by describing common misconceptions that may affect students' abilities to reason with and calculate fractions. We then explain how teachers can use a concrete-semi-concrete representational-abstract (CRA) model to plan and implement lessons that maximize learning and minimize frustration. Readers are also provided with detailed descriptions of how teachers can assess fraction knowledge, reinforce conceptual understanding, and promote procedural fluency.
– Name: AbstractInfo
  Label: Abstractor
  Group: Ab
  Data: As Provided
– Name: DateEntry
  Label: Entry Date
  Group: Date
  Data: 2025
– Name: AN
  Label: Accession Number
  Group: ID
  Data: EJ1475234
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1475234
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1177/00400599241231228
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 11
        StartPage: 368
    Subjects:
      – SubjectFull: Teaching Methods
        Type: general
      – SubjectFull: Manipulative Materials
        Type: general
      – SubjectFull: Fractions
        Type: general
      – SubjectFull: Mathematical Concepts
        Type: general
      – SubjectFull: Concept Formation
        Type: general
      – SubjectFull: Students with Disabilities
        Type: general
      – SubjectFull: Learning Disabilities
        Type: general
      – SubjectFull: Mathematical Logic
        Type: general
      – SubjectFull: Mathematics Instruction
        Type: general
      – SubjectFull: Computation
        Type: general
    Titles:
      – TitleFull: Using Manipulative-Based Instructional Sequences to Increase the Understanding of Fractional Concepts of Students with Mathematical Learning Disabilities
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Kathryn Lavin Brave
      – PersonEntity:
          Name:
            NameFull: Izzy Berman
      – PersonEntity:
          Name:
            NameFull: Debita Basu
      – PersonEntity:
          Name:
            NameFull: Alexis Szkotak
    IsPartOfRelationships:
      – BibEntity:
          Dates:
            – D: 01
              M: 05
              Type: published
              Y: 2025
          Identifiers:
            – Type: issn-print
              Value: 0040-0599
            – Type: issn-electronic
              Value: 2163-5684
          Numbering:
            – Type: volume
              Value: 57
            – Type: issue
              Value: 5
          Titles:
            – TitleFull: TEACHING Exceptional Children
              Type: main
ResultId 1