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Tackling Complex Math: Five Instructional Practices to Support Middle School Students

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Title:	Tackling Complex Math: Five Instructional Practices to Support Middle School Students
Language:	English
Authors:	Danielle O. Lariviere (ORCID 0000-0002-6170-4701), Katie B. MacLean (ORCID 0009-0005-9607-0437), Jessica Mao (ORCID 0009-0007-5582-7143), Sarah R. Powell (ORCID 0000-0002-6424-6160)
Source:	TEACHING Exceptional Children. 2026 58(4):244-257.
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:	14
Publication Date:	2026
Document Type:	Journal Articles Reports - Descriptive
Education Level:	Junior High Schools Middle Schools Secondary Education Elementary Education Grade 6 Intermediate Grades Grade 7 Grade 8
Descriptors:	Middle School Students, Mathematics Instruction, Teaching Methods, Middle School Mathematics, Common Core State Standards, Students with Disabilities, Individualized Education Programs, At Risk Students, Learning Problems, Grade 6, Grade 7, Grade 8, Mathematics Achievement, Difficulty Level
DOI:	10.1177/00400599251346713
ISSN:	0040-0599 2163-5684
Abstract:	According to national assessment data, only 28% of eighth-grade students in the United States perform at grade level in math (National Center for Education Statistics [NCES], 2024). Among students with disabilities, only 7% of eighth-grade students demonstrate proficiency in math. Middle school students with a range of disability types are at increased risk of experiencing "mathematics difficulty," or MD, a term used to describe students with disabilities who have individualized education program goals in math and students with persistent difficulty in math who might be at risk for disabilities. For all middle school students with MD, challenges with foundational math concepts taught in elementary school can exacerbate difficulties learning increasingly abstract, complex math content in middle school (Cirino et al., 2016; Moser Opitz et al., 2017). In this article, the authors highlight five instructional practices that have emerged as research-validated practices for use with middle school students with MD. These suggested practices are based on research studies addressing math instruction and intervention in the middle school grades (Gersten et al., 2009; Jitendra, Nelson, et al., 2016; Lariviere et al., 2024; Marita & Hord, 2017; Powell et al., 2021a; Star et al., 2015; Woodward et al., 2018).
Abstractor:	ERIC
Entry Date:	2026
Accession Number:	EJ1502538
Database:	ERIC
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FullText	Links: – Type: pdflink Url: https://content.ebscohost.com/cds/retrieve?content=AQICAHj0k_4E0hTGH8RJwT4gCJyBsGNe_WN95AvKlDbXJGqwxwFbV2QTEAAOGBQdtU5VxyvNAAAA4jCB3wYJKoZIhvcNAQcGoIHRMIHOAgEAMIHIBgkqhkiG9w0BBwEwHgYJYIZIAWUDBAEuMBEEDIpguANzPer2K-DM6gIBEICBmmypuLtBXjnKVlnCaEHFvTKVoUVwG7PwUpZgAOqNuZ5nRKMhCi9Ca9pm4rzl74veOlk2uCpwRmC1foYUl9X1Nf74H_x47qawWcnsl0_tPtOp0Ck9ISF6JrRwPTv-ijCz7rPUtaKQ-sCYzwtBy36LzIJKcPa1772EWy6oGA6Q7PBaYv3YcYfGhKTO-dk1mTWg03ErpdtqUYqTktQ= Text: Availability: 1 Value: <anid>AN0192768411;tec01mar.26;2026Apr07.05:50;v2.2.500</anid> <title id="AN0192768411-1">Tackling Complex Math: Five Instructional Practices to Support Middle School Students </title> <p>Graph</p> <p> <emph>Ms. Chord, a special education teacher, was reassigned from the elementary level to the middle school level. She found herself facing new challenges as she taught math to her Grade 7 students with learning disabilities. She looked at the first problem on students' worksheets, −6 = −2 (3×−9). "To solve this equation, we have to isolate the variable. We will have to use some math properties and operations to do this," she said. Her students appeared overwhelmed. "There are so many steps!" one of them exclaimed. "Are those negative numbers?" asked another. Ms. Chord paused, recognizing she was unsure which instructional strategies would best support her middle school students.</emph> </p> <p>In middle school (i.e., Grades 6–8), the Common Core State Standards for math are grouped into six domains: the number system, expressions and equations, geometry, statistics and probability, ratios and proportional relationships, and functions (National Governors Association Center for Best Practices &amp; Council of Chief State School Officers [[<reflink idref="bib36" id="ref1">36</reflink>]). With emphasis placed on ratios, proportions, and algebra in middle school, math content in these six domains becomes increasingly complex and abstract for students. Throughout this article, these domains are referenced through the lens of suggestions for teaching students with or at risk for disabilities. Table 1 displays the standards for each domain.</p> <p>Table 1. Middle School Common Core State Standards for Mathematics</p> <p>Graph</p> <p> <ephtml> &lt;table&gt;&lt;colgroup&gt;&lt;col align="left" /&gt;&lt;col align="char" char="." /&gt;&lt;/colgroup&gt;&lt;tbody&gt;&lt;tr&gt;&lt;td rowspan="6"&gt;The number system&lt;/td&gt;&lt;td&gt;Grade 6&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;&amp;#8226; Compute with multidigit numbers, find common factors and multiples, solve problems including the division of fractions, and extend the notion of number to the system of rational numbers.&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Grade 7&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;&amp;#8226; Develop understanding of operations with rational numbers.&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Grade 8&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;&amp;#8226; Understand nonrational numbers and use rational approximations to approximate irrational numbers.&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td rowspan="6"&gt;Expressions and equations&lt;/td&gt;&lt;td&gt;Grade 6&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;&amp;#8226; Extend previous understandings of arithmetic to write and interpret expressions and equations.&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Grade 7&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;&amp;#8226; Generate equivalent expressions using properties of operations and apply knowledge of expressions and equations to solve problems.&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Grade 8&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;&amp;#8226; Reason with expressions and equations and apply knowledge to work with radicals, integer exponents, and proportional relationships.&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td rowspan="6"&gt;Geometry&lt;/td&gt;&lt;td&gt;Grade 6&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;&amp;#8226; Extend knowledge of area, surface area, and volume to solve real-world and mathematical problems.&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Grade 7&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;&amp;#8226; Construct geometrical figures, analyze relationships between them, and solve problems involving angles, area, surface area, and volume.&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Grade 8&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;&amp;#8226; Use the Pythagorean theorem to analyze two- and three-dimensional space and figures using congruence, similarity, angles, and distance.&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td rowspan="6"&gt;Statistics and probability&lt;/td&gt;&lt;td&gt;Grade 6&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;&amp;#8226; Develop understanding of statistical thinking.&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Grade 7&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;&amp;#8226; Draw informal comparative inferences about two populations using random sampling.&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Grade 8&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;&amp;#8226; Investigate patterns of association in bivariate data.&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td rowspan="4"&gt;Ratios and proportional relationships&lt;/td&gt;&lt;td&gt;Grade 6&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;&amp;#8226; Apply understanding of ratio concepts with multiplication and division to solve problems.&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Grade 7&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;&amp;#8226; Understand and apply proportional relationships.&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td rowspan="2"&gt;Functions&lt;/td&gt;&lt;td&gt;Grade 8&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;&amp;#8226; Understand functions and model quantitative relationships.&lt;/td&gt;&lt;/tr&gt;&lt;/tbody&gt;&lt;/table&gt; </ephtml> </p> <p>1 <emph>Note.</emph> Adapted from [<reflink idref="bib36" id="ref2">36</reflink>].</p> <hd id="AN0192768411-2">Mathematics Difficulty in Middle School</hd> <p>According to national assessment data, only 28% of eighth-grade students in the United States perform at grade level in math ([<reflink idref="bib35" id="ref3">35</reflink>]). Among students with disabilities, only 7% of eighth-grade students demonstrate proficiency in math. We use the term "mathematics difficulty," or MD, to describe students with disabilities who have individualized education program goals in math and students with persistent difficulty in math who might be at risk for disabilities.</p> <p>Middle school students with a range of disability types are at increased risk of experiencing MD. Research indicates that by the time students reach middle school, those without disabilities consistently outperform those with identified disabilities ([<reflink idref="bib35" id="ref4">35</reflink>]; [<reflink idref="bib64" id="ref5">64</reflink>]). Students with certain disability profiles are especially likely to experience MD in middle school. For instance, students with autism show slower growth rates in developing proficiency with middle school math than students with other disability types and those without disabilities ([<reflink idref="bib64" id="ref6">64</reflink>]). Middle school students with multiple disabilities perform lower on measures of calculation and applied problem solving than students with one or zero disability identifications. Middle school students with other disability profiles, such as learning disabilities and intellectual disabilities, also frequently require targeted support in learning math ([<reflink idref="bib28" id="ref7">28</reflink>]; [<reflink idref="bib57" id="ref8">57</reflink>]).</p> <p>"<bold>Incoming challenges with concepts and procedures addressed in elementary math can therefore negatively impact students' readiness to learn new, complex content in middle school.</bold></p> <p>In addition to students with identified disabilities, middle school students at risk for disabilities also frequently experience MD ([<reflink idref="bib45" id="ref9">45</reflink>]; [<reflink idref="bib35" id="ref10">35</reflink>]). Special education researchers thus commonly design and test instructional programs to support middle school students with MD, including students with and at risk for disabilities who benefit from targeted math support ([<reflink idref="bib45" id="ref11">45</reflink>]; [<reflink idref="bib21" id="ref12">21</reflink>]; [<reflink idref="bib28" id="ref13">28</reflink>]; [<reflink idref="bib58" id="ref14">58</reflink>]; [<reflink idref="bib65" id="ref15">65</reflink>]). For all middle school students with MD, challenges with foundational math concepts taught in elementary school can exacerbate difficulties learning increasingly abstract, complex math content in middle school ([<reflink idref="bib6" id="ref16">6</reflink>]; [<reflink idref="bib32" id="ref17">32</reflink>]).</p> <hd id="AN0192768411-3">Impact of Incoming Math Proficiency</hd> <p>Middle school students with or at risk for disabilities are likely to have a history of MD beginning in elementary school ([<reflink idref="bib32" id="ref18">32</reflink>]). Because earlier math knowledge is essential for later math knowledge, students who come to middle school with MD are at risk of not developing proficiency with middle school math content ([<reflink idref="bib6" id="ref19">6</reflink>]). Consider a student whose understanding of the number system is not fully developed. This student struggles to compare different number types, such as whole numbers, fractions, and decimals. They may have difficulty responding to questions like, "Can a fraction be greater than 1?" or "How can you represent ¾ as a decimal?" When this student works with negative integers in sixth grade and irrational numbers in eighth grade, they are likely to have trouble understanding and operating with these unfamiliar number types.</p> <p>Students who begin middle school with MD are also less likely to be fluent in math facts and procedures ([<reflink idref="bib6" id="ref20">6</reflink>]; [<reflink idref="bib32" id="ref21">32</reflink>]). They might struggle to recall 18 ÷ 3 or have trouble regrouping when solving 314 + 289. These students may get bogged down with computation demands when learning new content in middle school, such as solving proportions. Incoming challenges with concepts and procedures addressed in elementary math can therefore negatively impact students' readiness to learn new, complex content in middle school.</p> <hd id="AN0192768411-4">Understanding Middle School Math</hd> <p>Besides the level of math knowledge students bring to middle school, there are several other reasons students with MD can struggle with middle school math. Math concepts in middle school become increasingly abstract, such as solving for a variable, <emph>x</emph>, in −6 = −2 (3×−9). Word problems also become more complex, with many multistep problems and new problem types (e.g., problems about scale models that use proportions). The vocabulary in word problems can also be unfamiliar to students, such as "Use a <emph>linear equation</emph> to model this scenario." For these reasons, it is important to provide research-based supports in math to middle school students with MD.</p> <hd id="AN0192768411-5">Instructional Practices to Use in Middle School Math</hd> <p> <emph>Motivated to support her students with the abstract, multistep processes she was teaching, Ms. Chord investigated research-based instructional practices in middle school math. She accessed multiple math-focused practice guides from the What Works Clearinghouse and journal articles from</emph> TEACHING Exceptional Children. <emph>She noticed several instructional practices were mentioned repeatedly across these resources, such as explicit instruction and multiple representations. Ms. Chord wondered how to teach concepts such as algebraic equations more explicitly. She considered how to make her demonstrations clearer and add more opportunities for practice. She also thought about incorporating manipulatives and drawings to help make concepts more concrete.</emph></p> <p>In this article, we highlight five instructional practices that have emerged as research-validated practices for use with middle school students with MD. These suggested practices are based on research studies addressing math instruction and intervention in the middle school grades ([<reflink idref="bib16" id="ref22">16</reflink>]; [<reflink idref="bib21" id="ref23">21</reflink>]; [<reflink idref="bib26" id="ref24">26</reflink>]; [<reflink idref="bib28" id="ref25">28</reflink>]; [<reflink idref="bib45" id="ref26">45</reflink>]; [<reflink idref="bib58" id="ref27">58</reflink>]; [<reflink idref="bib65" id="ref28">65</reflink>]).</p> <hd id="AN0192768411-6">Use Explicit Instruction</hd> <p>When using explicit instruction, a teacher designs instruction systematically using direct modeling and student practice opportunities ([<reflink idref="bib18" id="ref29">18</reflink>]). Explicit instruction can be used to break down complex skills (e.g., solving an algebraic equation) into manageable segments. The teacher begins by modeling a skill through thinking aloud, activating and building background knowledge, and posing questions to students. After modeling, the teacher provides opportunities for student practice. Practice opportunities are heavily guided at first, and teachers gradually reduce this guidance as students become more independent with the target skill. During both modeling and practice, teachers engage in supporting practices, including asking high-level and low-level questions, providing students frequent opportunities to respond, offering immediate and specific feedback, and maintaining a brisk pace ([<reflink idref="bib23" id="ref30">23</reflink>]).</p> <p>"<bold>Questions should be strategically chosen to build and activate students' background knowledge.</bold></p> <hd id="AN0192768411-7">Why is this important in middle school math?</hd> <p>Explicit instruction is a key feature of many math interventions that have improved outcomes for middle school students with MD, including those at risk for disabilities ([<reflink idref="bib45" id="ref31">45</reflink>]), those with learning disabilities ([<reflink idref="bib28" id="ref32">28</reflink>]), and those with extensive support needs ([<reflink idref="bib57" id="ref33">57</reflink>]). Research shows middle school students with MD can benefit from explicit instruction on fractions, integers, ratios and proportions, geometry, and algebraic equations ([<reflink idref="bib26" id="ref34">26</reflink>]; [<reflink idref="bib28" id="ref35">28</reflink>]; [<reflink idref="bib45" id="ref36">45</reflink>]; [<reflink idref="bib57" id="ref37">57</reflink>]). For instance, [<reflink idref="bib34" id="ref38">34</reflink>] used explicit instruction to support Grade 6 students with MD in breaking down the steps needed to solve for a variable in an algebraic equation. [<reflink idref="bib20" id="ref39">20</reflink>] showed improved outcomes for Grade 7 students with MD after they received explicit instruction on solving word problems involving proportions. Because students with MD frequently struggle with complex processes in middle school math, explicit instruction can help students break down the myriad of multistep skills addressed ([<reflink idref="bib36" id="ref40">36</reflink>]).</p> <hd id="AN0192768411-8">Recommended strategies</hd> <p>When using explicit instruction to introduce middle school math content to students with MD, it is recommended to first engage in modeling. Modeling should center on teacher think-alouds and demonstrations that build students' conceptual understanding and procedural skill. Think-alouds include verbalizations of statements, self-questions, and questions posed to the class. Questions should be strategically chosen to build and activate students' background knowledge. For instance, a think-aloud about solving two-step equations may sound like,</p> <p>I see the problem –14 = 5<emph>x</emph> + 4. I can read that as –14 is the same as a variable multiplied by 5 to which 4 is added. Remind me, what does the term variable mean? That is right! Variables are symbols that show unknown values. In this problem, my goal is to figure out the value of the variable. How can I figure out this value? I know I need to isolate the variable. That means I need to get the variable by itself on one side of the equation.</p> <p>Demonstrations should include step-by-step procedures and multiple representations (e.g., algebra tiles).</p> <p>The second recommendation when implementing explicit instruction is to engage students in ample practice opportunities ([<reflink idref="bib18" id="ref41">18</reflink>]). Students can engage in practice using a range of modalities, including paper-based activities, discussion, and technology (e.g., apps). When a concept or skill area is relatively new to students, teachers should emphasize guided practice (i.e., working alongside students). For instance, to solve –14 = 5<emph>x</emph> + 4 in a guided-practice setting, both the teacher and students should show their step-by-step work to identify the value of <emph>x</emph>. Determining the series of steps to follow should include a back-and-forth exchange between the teacher and students (e.g., "What is my first step to isolate the variable? Why do I subtract 4 from both sides of the equation?"). Guided practice can occur in a whole-class setting or in small groups. Guided practice with small groups can be especially useful to address misconceptions and procedural errors. When reviewing concepts with which students are mostly or fully proficient, prioritize independent practice. Even when students practice independently, it is important to provide them with affirmative and corrective feedback to validate understanding and address misconceptions ([<reflink idref="bib18" id="ref42">18</reflink>]).</p> <p>The third recommended strategy when using explicit instruction is to incorporate supporting practices into both modeling and practice ([<reflink idref="bib23" id="ref43">23</reflink>]). Supporting practices include (a) asking a balance of high-level (i.e., conceptual) and low-level (i.e., procedural or factual) questions, (b) providing immediate and specific feedback, (c) maintaining a brisk pace, and (d) incorporating examples and nonexamples ([<reflink idref="bib18" id="ref44">18</reflink>]; [<reflink idref="bib23" id="ref45">23</reflink>]). Asking both high- and low-level questions not only provides students with frequent opportunities to respond but also strengthens connections between conceptual and procedural understanding. Table 2 provides examples of high- and low-level questions in each Common Core domain in middle school math. During question posing and student practice, it is important that teachers provide immediate and specific feedback to students. Maintaining a brisk pace throughout instruction supports student attention and engagement. Last, incorporating examples and nonexamples into explicit instruction helps students learn when to apply newly acquired skills.</p> <p>Table 2 Questioning Examples by Domain</p> <p>Graph</p> <p> <ephtml> &lt;table&gt;&lt;colgroup&gt;&lt;col align="left" /&gt;&lt;col align="char" char="." /&gt;&lt;col align="char" char="." /&gt;&lt;/colgroup&gt;&lt;thead&gt;&lt;tr&gt;&lt;th align="center"&gt;Domain&lt;/th&gt;&lt;th align="center"&gt;High-level questions&lt;/th&gt;&lt;th align="center"&gt;Low-level questions&lt;/th&gt;&lt;/tr&gt;&lt;/thead&gt;&lt;tbody&gt;&lt;tr&gt;&lt;td&gt;The number system&lt;/td&gt;&lt;td&gt;Why are &lt;p&gt;&lt;math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"&gt;&lt;mrow xmlns=""&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;&amp;#8722;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/p&gt;, &lt;p&gt;&lt;math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"&gt;&lt;mrow xmlns=""&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo&gt;&amp;#8722;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/p&gt;, and &lt;p&gt;&lt;math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"&gt;&lt;mrow xmlns=""&gt;&lt;mo&gt;&amp;#8722;&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/p&gt; all equivalent? (Grade 7)&lt;/td&gt;&lt;td&gt;Is the equation true: 18.2 + (&amp;#8211;72.9) = 18.2 &amp;#8211; 72.9? (Grade 7)&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Expressions and equations&lt;/td&gt;&lt;td&gt;Why does a point of intersection of two linear equations show a solution to their system of equations? (Grade 8)&lt;/td&gt;&lt;td&gt;What is the solution to the following system of equations? (Grade 8)&amp;#8211;2x+y = 134x&amp;#8211;y = &amp;#8211;15&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Geometry&lt;/td&gt;&lt;td&gt;How can you decompose the polygon below into other shapes to help you find its area? (Grade 6) &lt;graphic href="10.1177&amp;#95;00400599251346713-img3.tif" content-type="Graph" /&gt;&lt;/td&gt;&lt;td&gt;What is the area of the shape below? (Grade 6)&lt;graphic href="10.1177&amp;#95;00400599251346713-img4.tif" content-type="Graph" /&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Statistics and probability&lt;/td&gt;&lt;td&gt;Why can random sampling help you make inferences about a general population? (Grade 7)&lt;/td&gt;&lt;td&gt;Based on the sample below of teenagers' sleep amounts, what do you expect is teenagers' average amount of sleep? (Grade 7) 7 hours, 6.5 hours, 10 hours, 9.5 hours, 8.5 hours, 7 hours, 7.5 hours&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Ratios and proportional relationships&lt;/td&gt;&lt;td&gt;How can calculating unit rates help you save money while grocery shopping? (Grade 6)&lt;/td&gt;&lt;td&gt;What is the cost per apple if a bag of 11 apples costs $5.25? (Grade 6)&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Functions&lt;/td&gt;&lt;td&gt;Why is the equation x = &amp;#8722;3 not a function? (Grade 8)&lt;/td&gt;&lt;td&gt;What is the rate of change in the equation 2x&amp;#8722;5y = 10? (Grade 8)&lt;/td&gt;&lt;/tr&gt;&lt;/tbody&gt;&lt;/table&gt; </ephtml> </p> <p>2 <emph>Note.</emph> These questions align with the Common Core State Standards of Mathematics from the [<reflink idref="bib36" id="ref46">36</reflink>].</p> <hd id="AN0192768411-9">Incorporate Multiple Representations</hd> <p>While using explicit instruction, teachers may want to use multiple representations in lessons. Multiple representations involve the use of concrete manipulatives, virtual manipulatives, pictorial drawings, the number line, and gestures to demonstrate math concepts. Another term often used to describe multiple representations is the "concrete-representational-abstract" (CRA) framework. The CRA framework is an evidence-based practice that supports students' development of mathematical reasoning ([<reflink idref="bib15" id="ref47">15</reflink>]). We refer to CRA as a "framework" rather than "sequence" because "framework" provides more liberty in how the phases (i.e., concrete, representational or pictorial, and abstract) are ordered in instruction; another term often used to describe this is the CRA-integrated (CRA-I) approach ([<reflink idref="bib60" id="ref48">60</reflink>]). In this section, we provide strategies for each type of representation.</p> <hd id="AN0192768411-10">Why is this important in middle school math?</hd> <p>As an evidence-based practice with robust effects, multiple representations can support the mathematical reasoning of middle school students with MD ([<reflink idref="bib4" id="ref49">4</reflink>]; [<reflink idref="bib21" id="ref50">21</reflink>]; [<reflink idref="bib45" id="ref51">45</reflink>]; [<reflink idref="bib57" id="ref52">57</reflink>]). Multiple representations can be especially useful for introducing and practicing math content in middle school as concepts become increasingly abstract (e.g., interpreting irrational numbers; [<reflink idref="bib36" id="ref53">36</reflink>]). Representations (e.g., two-color counters, fraction circles, algebra tiles, the number line, pictures) allow students to perceive and interact mathematically and spatially with applicable principles, properties, and magnitudes ([<reflink idref="bib1" id="ref54">1</reflink>]). Additionally, representations allow students to actively observe patterns and generalizations. As students engage with multiple representations, they can better retain the learned concepts and procedures necessary to understand and show proficiency with math content.</p> <p>Multiple representations are also helpful for students of culturally and linguistically diverse backgrounds ([<reflink idref="bib19" id="ref55">19</reflink>]). Because multiple representations do not always require the English language in their use, they can benefit students whose primary language is not English. Using multiple representations also opens more avenues for students to communicate their thought process. Finally, representations such as manipulatives are linked to increased student engagement and motivation ([<reflink idref="bib33" id="ref56">33</reflink>]).</p> <hd id="AN0192768411-11">Recommended strategies</hd> <p>This section covers four types of representations that teachers can use in their math instruction for middle school students with MD. The recommended strategies are not ordered in a manner that suggests importance; rather, teachers should use and explore representations most applicable to their classrooms.</p> <p>In middle school math, some of the first concepts taught are fractions and using the number line to identify rational numbers ([<reflink idref="bib36" id="ref57">36</reflink>]). Use of the number line supports students' understanding of the similarities and differences among whole numbers, integers, fractions, and other rational numbers ([<reflink idref="bib56" id="ref58">56</reflink>]; [<reflink idref="bib61" id="ref59">61</reflink>]). When using explicit instruction to teach a concept, teachers should use the number line to demonstrate numerical magnitudes and how numbers relate to other numbers. For example, students can use the number line to solve a fractional subtraction problem (e.g., – <ephtml> &lt;math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/math&gt; </ephtml> – <ephtml> &lt;math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/math&gt; </ephtml> ), where the number starts at – <ephtml> &lt;math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/math&gt; </ephtml> and moves to the left another <ephtml> &lt;math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/math&gt; </ephtml> units. Figure 1 shows content that can be taught using the number line and the different ways the number line can be presented ([<reflink idref="bib52" id="ref60">52</reflink>]).</p> <p>"<bold>Aside from fluency with whole numbers, middle school students need to build fluency with fractions, decimals, negative integers, and procedures like solving equations for unknown variables.</bold></p> <p>Graph: Figure 1 Representations with the number line</p> <p>Although middle school students may initially show an aversion to it, concrete manipulatives are still favorable and effective in building students' conceptual and procedural knowledge in math ([<reflink idref="bib7" id="ref61">7</reflink>]). And to date, studies have shown there is no significant difference in math learning outcomes when using concrete manipulatives versus virtual manipulatives (e.g., [<reflink idref="bib3" id="ref62">3</reflink>]). To teach using either concrete or virtual manipulatives, the teacher can start by modeling how to use the manipulatives while explaining the concept, introduce additional planned examples to reinforce the concept, and provide opportunities for students to use the manipulatives through guided and independent practice.</p> <p>In addition to manipulatives, teachers can use pictorial drawings to represent math concepts, which is also part of the CRA framework. Teachers often use drawings on whiteboards or printouts to demonstrate concepts. It is important to note, however, that research indicates teachers should pair concrete and pictorial representations with abstract symbols and numbers to support the learning outcomes of students with MD ([<reflink idref="bib5" id="ref63">5</reflink>]).</p> <p>Finally, teachers can use hand gestures to represent spatial or abstract concepts ([<reflink idref="bib62" id="ref64">62</reflink>]). For example, when teaching fraction magnitudes on the number line, teachers can use chunking gestures to help students visualize the magnitude of unit fractions or find equivalent fractions ([<reflink idref="bib52" id="ref65">52</reflink>]). When teaching fractions using chunking gestures, teachers should emphasize keeping the magnitude (i.e., the space between the thumb and the index finger) the same as students move their hand along the number line.</p> <p> <emph>Ms. Chord began to include explicit modeling and practice opportunities in her lessons. She showed students how to use double number lines and algebra tiles to help them solve algebraic equations. Now, her students knew how to get started and had strategies to help them visualize problems. However, Ms. Chord noticed that several of her students had not mastered their math facts and got bogged down with the computation required to solve equations. "What is 2 × 3 again?" one student wondered aloud while attempting to multiply –2 by 3. She wondered how to support students with multiplying negative integers when they continued to struggle with fact recall.</emph> </p> <hd id="AN0192768411-12">Build Fluency</hd> <p>Fluency means doing math accurately and efficiently. Building fluency is important because it decreases the load on students' working memory and increases their ability to solve problems that are more complex. In many cases, fluency relates to the operations (i.e., addition, subtraction, multiplication, and division) and students' retrieval of math facts. Building fluency with the operations helps boost students' confidence in math and is critical for their success in math ([<reflink idref="bib37" id="ref66">37</reflink>]), particularly as content increases in complexity in middle school. Aside from fluency with whole numbers, middle school students need to build fluency with fractions, decimals, negative integers, and procedures like solving equations for unknown variables.</p> <hd id="AN0192768411-13">Why is this important in middle school math?</hd> <p>Fluency in math frees up students' cognitive resources to move to higher-order thinking and complex mathematical content presented in middle school ([<reflink idref="bib37" id="ref67">37</reflink>]). Unfortunately, many middle school students with MD are not fluent with math facts and procedures and have a limited understanding of foundational math concepts, thus highlighting the importance of fluency-building instruction ([<reflink idref="bib6" id="ref68">6</reflink>]; [<reflink idref="bib32" id="ref69">32</reflink>]; [<reflink idref="bib38" id="ref70">38</reflink>]). According to standards, students should be able to fluently add and subtract by the end of the second grade and fluently multiply and divide by the end of the third grade ([<reflink idref="bib36" id="ref71">36</reflink>]). Students who lack foundational fluency might struggle to meet these grade-level standards, consequently making middle school math content challenging and frustrating. Without automaticity with math facts, middle school students spend more time with computation in problem solving. This hinders their ability to develop conceptual understanding of new content, such as proportional reasoning and probability. By targeting fluency in instruction, teachers can increase students' access to higher-level math content addressed in middle school.</p> <hd id="AN0192768411-14">Recommended strategies</hd> <p>There are multiple recommended strategies for teachers to consider when working on fluency with middle school students with MD. First, daily math instruction should include several minutes focused solely on fluency building ([<reflink idref="bib15" id="ref72">15</reflink>]). For students to build fluency with math facts and procedures, it is important that they receive regular opportunities to practice. In this daily practice, it is important that teachers emphasize concepts alongside procedures ([<reflink idref="bib47" id="ref73">47</reflink>]).</p> <p>The second recommended strategy is that this regular fluency practice incorporates opportunities for students to develop automaticity with sums, differences, products, and quotients. Practicing automaticity can be done using flash cards, games, technology-based apps, and worksheets ([<reflink idref="bib15" id="ref74">15</reflink>]). Teachers should consider the following structure during fluency practice: (a) present a fact, (b) provide appropriate wait time, (c) allow for student response and focus on automatic recall, and (d) provide corrective, immediate feedback. Automaticity practice should be done daily for 1 to 2 minutes, and it should include both known and unknown facts. A ratio of 9 known facts to 1 unknown fact is effective for helping students with MD learn and retain math facts ([<reflink idref="bib51" id="ref75">51</reflink>]). As students' progress with fact fluency improves, gradually decreasing this ratio from 9:1 to 3:1 is appropriate. When practicing automaticity, sets of facts should be carefully selected and individualized to build students' accuracy and efficiency ([<reflink idref="bib8" id="ref76">8</reflink>]).</p> <p>Third, fluency practice should include reasoning strategies or strategy instruction, helping students learn efficient methods for deriving facts and highlighting patterns to help organize and generalize their fact knowledge ([<reflink idref="bib2" id="ref77">2</reflink>]). Often, strategy instruction includes graphic organizers and multiple representations (e.g., 10 frames, number lines) and teaches students fundamental properties and patterns (e.g., times 10, doubles). When building fluency and learning strategies, students' understanding of math concepts should move beyond solely memorization and fact recall. Fluency should target both knowledge of strategies and conceptual understanding. This helps students understand quantities and relations among numbers alongside their retrieval of math facts. As a result, strategy instruction can help students transfer fact knowledge to more complex math procedures and tasks, such as multidigit computation and word-problem solving. Figure 2 shows how fact fluency connects with both conceptual and procedural fluency. In this case, a student's fluency with facts (i.e., 3 × 5 = 15) and reasoning strategies (i.e., half of 30 is 15) helps them compute the given expression in two different ways, thus demonstrating conceptual and procedural fluency.</p> <p>Graph: Figure 2 Fluency integration.</p> <p> <emph>Ms. Chord began to prioritize building fluency, dedicating 10 minutes each day to practicing math facts and procedures with her students. After making this instructional change, she noticed her students making fewer computation errors. Because they were getting less bogged down with recalling math facts and procedures, they began to grasp new concepts, such as solving algebraic equations, more easily.</emph> </p> <p> <emph>To extend their learning, Ms. Chord started to incorporate more real-world examples and word problems, such as determining the unit rates for different items at a supermarket. She noticed students struggled to understand some of the language used in these problems, such as "rate" and "proportional." Her students were not applying their knowledge as readily as she hoped.</emph> </p> <hd id="AN0192768411-15">Teach Math Vocabulary</hd> <p>Middle school students regularly encounter and use math vocabulary during instruction, on assignments, and on tests. Math vocabulary includes terms that have at least one meaning specific to the context of math. Some of these terms have a singular meaning (e.g., "integer"), and others have multiple meanings (e.g., a "reflection" of a two-dimensional figure vs. a written "reflection"; [<reflink idref="bib30" id="ref78">30</reflink>]; [<reflink idref="bib54" id="ref79">54</reflink>]).</p> <hd id="AN0192768411-16">Why is this important in middle school math?</hd> <p>There are several reasons teachers should emphasize math vocabulary for middle school students with MD. First, students' ability to use math vocabulary is associated with overall proficiency in math ([<reflink idref="bib27" id="ref80">27</reflink>]). Researchers have demonstrated this association across several math domains, including algebra, fractions, and word-problem solving ([<reflink idref="bib27" id="ref81">27</reflink>]; [<reflink idref="bib42" id="ref82">42</reflink>]). Math-vocabulary knowledge gives students access to teachers' instruction in math, and it provides students with the language to articulate their own mathematical reasoning.</p> <p>Students with MD are especially likely to struggle with learning math vocabulary, so it is important to provide targeted support to this group of students ([<reflink idref="bib11" id="ref83">11</reflink>]). By Grade 8, students have been exposed to more than 1,200 math-vocabulary terms ([<reflink idref="bib46" id="ref84">46</reflink>]). Many of these terms have challenging features, such as complex spellings (e.g., "isosceles") and false cognates (e.g., "mil" in Spanish sounds like "million," but it means thousand; [<reflink idref="bib54" id="ref85">54</reflink>]). For those with MD who are emergent bilingual students, navigating false cognates and unfamiliar formal mathematics language may be especially challenging ([<reflink idref="bib25" id="ref86">25</reflink>]). See Table 3 for more examples of challenging features in middle school math vocabulary. With many important yet challenging math-vocabulary terms, it is critical to support middle school students with MD in developing math-vocabulary knowledge.</p> <p>Table 3 Challenging Features of Middle School Mathematics Vocabulary</p> <p>Graph</p> <p> <ephtml> &lt;table&gt;&lt;colgroup&gt;&lt;col align="left" /&gt;&lt;col align="char" char="." /&gt;&lt;/colgroup&gt;&lt;thead&gt;&lt;tr&gt;&lt;th align="center"&gt;Challenging features&lt;/th&gt;&lt;th align="center"&gt;Examples&lt;/th&gt;&lt;/tr&gt;&lt;/thead&gt;&lt;tbody&gt;&lt;tr&gt;&lt;td&gt;Multiple academic meanings&lt;/td&gt;&lt;td&gt;&amp;#8226; "Inequality" as greater or less than (math) vs. unequal rights (social studies)&amp;#8226; "System" of equations (math) vs. the digestive system (science)&amp;#8226; "Translation" as movement of a figure (math) vs. translation into English (foreign language)&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Meanings in mathematics and general language&lt;/td&gt;&lt;td&gt;&amp;#8226; "Face" of a three-dimensional figure vs. a person's face&amp;#8226; "Negative" as a value below zero vs. unfavorable&amp;#8226; "Similar" figures as proportional vs. generally alike&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Homophones and near homophones&lt;/td&gt;&lt;td&gt;&amp;#8226; "Axis" vs. "access"&amp;#8226; "Chord" vs. "cord"&amp;#8226; "Rational" vs. "rationale"&lt;/td&gt;&lt;/tr&gt;&lt;/tbody&gt;&lt;/table&gt; </ephtml> </p> <hd id="AN0192768411-17">Recommended strategies</hd> <p>There are multiple recommended strategies to teach math vocabulary to middle school students with MD. The first is for teachers to focus on formal vocabulary use ([<reflink idref="bib49" id="ref87">49</reflink>]). Many math concepts have both formal and informal terms associated with them (e.g., "denominator" vs. "bottom number"). Although using informal terms may seem to help students in the moment, it does them a disservice in the long run. Students will be expected to understand and use formal math vocabulary in other contexts, such as high-stakes tests and new classrooms, so it is important to use formal vocabulary consistently.</p> <p>Additionally, preteaching is a recommended strategy for teaching math vocabulary to students with MD, including emergent bilingual students with MD ([<reflink idref="bib34" id="ref88">34</reflink>]; [<reflink idref="bib39" id="ref89">39</reflink>]; [<reflink idref="bib55" id="ref90">55</reflink>]). Preteaching math vocabulary involves introducing vocabulary terms prior to embedding them in math content instruction. This can include sharing student-friendly definitions and leading brief discussions about the vocabulary terms ([<reflink idref="bib66" id="ref91">66</reflink>]). When teachers preteach important math vocabulary, students are primed to recognize pretaught terms in math content lessons.</p> <p>A third strategy for teaching math vocabulary is using visuals ([<reflink idref="bib50" id="ref92">50</reflink>]; [<reflink idref="bib55" id="ref93">55</reflink>]). Sometimes, this can be as simple as showing images of two identical triangles to demonstrate the term "congruent." Often, though, embedding visuals into graphic organizers can give students more insight into a term's meaning. Graphic organizers can include information such as student-friendly definitions, examples, and nonexamples ([<reflink idref="bib59" id="ref94">59</reflink>]). See Figure 3 for an adapted version of a commonly used graphic organizer called the "Frayer model."</p> <p>Graph: Figure 3 Frayer model</p> <p>Finally, teachers should provide students with repeated exposures to math-vocabulary terms ([<reflink idref="bib29" id="ref95">29</reflink>]). Students with MD benefit from many opportunities to hear, see, speak, and write new terms before they commit their meanings to memory. This means teachers should embed target vocabulary into learning activities daily. They should also hold students accountable for using the vocabulary terms in whole-class discussions, small-group work, and written tasks. Word walls and vocabulary cards can be helpful tools to provide students with sustained exposure to important math vocabulary throughout the school year.</p> <hd id="AN0192768411-18">Provide Word-Problem Strategy Instruction</hd> <p>Word problems are scenarios in which students must read and understand text to answer a math question or respond to a prompt. Success with word-problem solving not only relates to using math in the real world, but word problems on high-stakes tests are often how middle school students are expected to demonstrate math proficiency ([<reflink idref="bib35" id="ref96">35</reflink>]; [<reflink idref="bib48" id="ref97">48</reflink>]).</p> <hd id="AN0192768411-19">Why is this important in middle school math?</hd> <p>In the United States, most high-stakes math test items are word problems ([<reflink idref="bib48" id="ref98">48</reflink>]). Some of these are directive word problems, in which students read text that provides a direction to do something (e.g., "Which two expressions are equivalent?"). Other word problems are routine, in which students answer a question based on the text (e.g., "Lando has 504 tomatoes to put into boxes for the farmer's market. Each box holds 24 tomatoes. How many boxes does Lando need for all the tomatoes?"). For middle school students with MD, word-problem solving is often difficult. This is the case for students with MD who have a range of identified disabilities and those who are at risk for disabilities ([<reflink idref="bib35" id="ref99">35</reflink>]; [<reflink idref="bib64" id="ref100">64</reflink>]). Middle school students may experience difficulty with word problems for many reasons, including reading and interpreting the text, identifying important information, ignoring irrelevant information, determining a solution strategy, managing multistep problems, or solving with computation ([<reflink idref="bib14" id="ref101">14</reflink>]; [<reflink idref="bib24" id="ref102">24</reflink>]; [<reflink idref="bib40" id="ref103">40</reflink>]; [<reflink idref="bib63" id="ref104">63</reflink>]). Underlying difficulties related to working memory and language may exacerbate these challenges ([<reflink idref="bib43" id="ref105">43</reflink>]).</p> <hd id="AN0192768411-20">Recommended strategies</hd> <p>There are several recommended strategies to help middle school students with MD with setting up and solving word problems. Before discussing those strategies, it is important to make note of one strategy that is not recommended: identifying keywords in word problems and tying a keyword to a specific operation (e.g., "total" means to add). Although prevalent in schools, this strategy is ineffective for many single-step word problems and almost entirely unhelpful for multistep word problems regularly encountered in middle school ([<reflink idref="bib48" id="ref106">48</reflink>]). For instance, the following problem includes the word "total" but would not be correctly solved using addition: "One sixth-grade class has 25 students in total, 8 of whom like soccer. If the total number of sixth-grade students in the school is 75, how many students would you expect like soccer?"</p> <p>The first effective strategy to help middle school students with MD solve word problems is to use an attack strategy. An attack strategy is a general process for working through a word problem from the beginning to the end. Example attack strategies include SOLVE and FOPS ([<reflink idref="bib13" id="ref107">13</reflink>]; [<reflink idref="bib22" id="ref108">22</reflink>]). Figure 4 describes these and other attack strategies. A solid attack strategy encourages students to read the problem, make a plan, solve the problem, and check the work ([<reflink idref="bib44" id="ref109">44</reflink>]). When students know an attack strategy and how to use it every time they see a word problem, it can make approaching word problems easier for students, particularly students with disabilities ([<reflink idref="bib31" id="ref110">31</reflink>]).</p> <p>Graph: Figure 4 Word-problem attack strategies</p> <p>The second strategy important for word-problem solving is to help students understand the conceptual background of the word problem. Often, this is referred to as a problem's "problem type," "structure," or "schema" ([<reflink idref="bib9" id="ref111">9</reflink>]; [<reflink idref="bib17" id="ref112">17</reflink>]). When students can recognize a word problem as belonging to a specific schema and if they have an approach for solving word problems with that schema, it makes word-problem solving easier for students ([<reflink idref="bib15" id="ref113">15</reflink>]; [<reflink idref="bib41" id="ref114">41</reflink>]). Schema instruction has a robust evidence base, including for culturally and linguistically diverse students with MD ([<reflink idref="bib10" id="ref115">10</reflink>]; [<reflink idref="bib12" id="ref116">12</reflink>]). An attack strategy is used alongside a focus on the schemas to help students have a process for working through the problem and a way to understand the problem ([<reflink idref="bib17" id="ref117">17</reflink>]; [<reflink idref="bib22" id="ref118">22</reflink>]).</p> <p>There are several schemas that are common in word problems in the middle school grades. In "total" problems, a word problem features parts put together for a total. In "difference" problems, students compare two amounts for a difference. For "change" problems, an amount increases or decreases. With the "equal groups" schema, the word problem discusses groups with an equal number in each group. In "comparison" problems, a set is multiplied a number of times. And with the "proportion" schema, students identify relationships among quantities. Figure 5 has an example problem of each type of schema.</p> <p>Graph: Figure 5 Word problems by schema</p> <p>Importantly, in middle school, word problems regularly involve multiple steps or multiple schemas, for example,</p> <p>An elementary school had 90 boxes of glue sticks. Each box had 36 glue sticks. Teachers put all the glue sticks into bags to give to the students. They put 6 glue sticks into each bag. What is the number of bags the teachers can fill with these glue sticks?</p> <p>This problem involves the equal groups schema (i.e., 90 groups with 36 in each group for an unknown product) and then the application of the equal groups schema again (i.e., unknown groups with 6 in each group for a product of 3,240). As with single-step problems, this focus on the schemas should be integrated into the attack strategy that guides students through the process of setting up and solving a word problem.</p> <p> <emph>With new insight into best practices to teach math vocabulary and word problems, Ms. Chord continued to refine her approach to supporting her middle school students with disabilities. She built in instructional time for preteaching formal vocabulary terms, such as "inverse" and "proportional." She made sure to provide repeated exposures to the terms in and across lessons. Through vocabulary instruction and consistent use of formal language in class, Ms. Chord observed her students using math terms to articulate their understanding of the concepts. Ms. Chord also taught an attack strategy (UPS Check) and provided schema instruction to help her students navigate word problems effectively. As Ms. Chord focused her instructional efforts on these practices, she felt a sense of pride and satisfaction as she began noticing improved mathematical understanding, success in applying skills to new contexts, and decreased behavioral challenges in her class.</emph> </p> <hd id="AN0192768411-21">Conclusion</hd> <p>Several instructional practices have been shown to support middle school students with MD. These include (a) explicit instruction, (b) multiple representations, (c) fluency-building instruction, (d) math-vocabulary instruction, and (e) word-problem strategy instruction. Although these recommendations are based on middle school math intervention studies, it is interesting to note that several of these practices mirror those that have shown efficacy in the elementary grades ([<reflink idref="bib15" id="ref119">15</reflink>]). Thus, these instructional practices can—and should be—adapted to the complex math concepts taught in the middle school grades.</p> <p>For middle school students with MD, it can be especially effective to integrate multiple recommended instructional practices. Integration of instructional practices is recommended across the range of middle school students with MD (e.g., those at risk for disabilities and those with identified disabilities who have extensive support needs; [<reflink idref="bib45" id="ref120">45</reflink>]; [<reflink idref="bib53" id="ref121">53</reflink>]). For instance, when teaching about the Pythagorean theorem, a teacher may provide explicit vocabulary instruction on the terms "legs" and "hypotenuse" using geoboards. When implementing any instructional practice or combination of practices, it is critical to check for understanding, address misconceptions, and support connections across concepts and contexts ([<reflink idref="bib18" id="ref122">18</reflink>]; [<reflink idref="bib53" id="ref123">53</reflink>]).</p> <ref id="AN0192768411-22"> <title> References </title> <blist> <bibl id="bib1" idref="ref54" type="bt">1</bibl> <bibtext> American Institutes for Research. (2022). Embedding multiple representations in middle school mathematics classrooms: Expanding middle school students' understanding of algebraic reasoning for students with disabilities. U.S. Department of Education, Office of Special Education Programs.</bibtext> </blist> <blist> <bibl id="bib2" idref="ref77" type="bt">2</bibl> <bibtext> Baroody A. J., Bajwa N. 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MacLean</bibtext> </blist> <blist> <bibtext>Graph</bibtext> </blist> <blist> <bibtext>https://orcid.org/0009-0005-9607-0437 Jessica Mao</bibtext> </blist> <blist> <bibtext>Graph</bibtext> </blist> <blist> <bibtext>https://orcid.org/0009-0007-5582-7143 Sarah R. Powell</bibtext> </blist> <blist> <bibtext>Graph https://orcid.org/0000-0002-6424-6160</bibtext> </blist> </ref> <aug> <p>By Danielle O. Lariviere; Katie B. MacLean; Jessica Mao and Sarah R. 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Items	– Name: Title Label: Title Group: Ti Data: Tackling Complex Math: Five Instructional Practices to Support Middle School Students – Name: Language Label: Language Group: Lang Data: English – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Danielle+O%2E+Lariviere%22">Danielle O. Lariviere</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-6170-4701">0000-0002-6170-4701</externalLink>)<br /><searchLink fieldCode="AR" term="%22Katie+B%2E+MacLean%22">Katie B. MacLean</searchLink> (ORCID <externalLink term="https://orcid.org/0009-0005-9607-0437">0009-0005-9607-0437</externalLink>)<br /><searchLink fieldCode="AR" term="%22Jessica+Mao%22">Jessica Mao</searchLink> (ORCID <externalLink term="https://orcid.org/0009-0007-5582-7143">0009-0007-5582-7143</externalLink>)<br /><searchLink fieldCode="AR" term="%22Sarah+R%2E+Powell%22">Sarah R. Powell</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-6424-6160">0000-0002-6424-6160</externalLink>) – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="SO" term="%22TEACHING+Exceptional+Children%22"><i>TEACHING Exceptional Children</i></searchLink>. 2026 58(4):244-257. – 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: 14 – Name: DatePubCY Label: Publication Date Group: Date Data: 2026 – Name: TypeDocument Label: Document Type Group: TypDoc Data: Journal Articles<br />Reports - Descriptive – Name: Audience Label: Education Level Group: Audnce Data: <searchLink fieldCode="EL" term="%22Junior+High+Schools%22">Junior High Schools</searchLink><br /><searchLink fieldCode="EL" term="%22Middle+Schools%22">Middle Schools</searchLink><br /><searchLink fieldCode="EL" term="%22Secondary+Education%22">Secondary Education</searchLink><br /><searchLink fieldCode="EL" term="%22Elementary+Education%22">Elementary Education</searchLink><br /><searchLink fieldCode="EL" term="%22Grade+6%22">Grade 6</searchLink><br /><searchLink fieldCode="EL" term="%22Intermediate+Grades%22">Intermediate Grades</searchLink><br /><searchLink fieldCode="EL" term="%22Grade+7%22">Grade 7</searchLink><br /><searchLink fieldCode="EL" term="%22Grade+8%22">Grade 8</searchLink> – Name: Subject Label: Descriptors Group: Su Data: <searchLink fieldCode="DE" term="%22Middle+School+Students%22">Middle School Students</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Instruction%22">Mathematics Instruction</searchLink><br /><searchLink fieldCode="DE" term="%22Teaching+Methods%22">Teaching Methods</searchLink><br /><searchLink fieldCode="DE" term="%22Middle+School+Mathematics%22">Middle School Mathematics</searchLink><br /><searchLink fieldCode="DE" term="%22Common+Core+State+Standards%22">Common Core State Standards</searchLink><br /><searchLink fieldCode="DE" term="%22Students+with+Disabilities%22">Students with Disabilities</searchLink><br /><searchLink fieldCode="DE" term="%22Individualized+Education+Programs%22">Individualized Education Programs</searchLink><br /><searchLink fieldCode="DE" term="%22At+Risk+Students%22">At Risk Students</searchLink><br /><searchLink fieldCode="DE" term="%22Learning+Problems%22">Learning Problems</searchLink><br /><searchLink fieldCode="DE" term="%22Grade+6%22">Grade 6</searchLink><br /><searchLink fieldCode="DE" term="%22Grade+7%22">Grade 7</searchLink><br /><searchLink fieldCode="DE" term="%22Grade+8%22">Grade 8</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Achievement%22">Mathematics Achievement</searchLink><br /><searchLink fieldCode="DE" term="%22Difficulty+Level%22">Difficulty Level</searchLink> – Name: DOI Label: DOI Group: ID Data: 10.1177/00400599251346713 – Name: ISSN Label: ISSN Group: ISSN Data: 0040-0599<br />2163-5684 – Name: Abstract Label: Abstract Group: Ab Data: According to national assessment data, only 28% of eighth-grade students in the United States perform at grade level in math (National Center for Education Statistics [NCES], 2024). Among students with disabilities, only 7% of eighth-grade students demonstrate proficiency in math. Middle school students with a range of disability types are at increased risk of experiencing "mathematics difficulty," or MD, a term used to describe students with disabilities who have individualized education program goals in math and students with persistent difficulty in math who might be at risk for disabilities. For all middle school students with MD, challenges with foundational math concepts taught in elementary school can exacerbate difficulties learning increasingly abstract, complex math content in middle school (Cirino et al., 2016; Moser Opitz et al., 2017). In this article, the authors highlight five instructional practices that have emerged as research-validated practices for use with middle school students with MD. These suggested practices are based on research studies addressing math instruction and intervention in the middle school grades (Gersten et al., 2009; Jitendra, Nelson, et al., 2016; Lariviere et al., 2024; Marita & Hord, 2017; Powell et al., 2021a; Star et al., 2015; Woodward et al., 2018). – Name: AbstractInfo Label: Abstractor Group: Ab Data: ERIC – Name: DateEntry Label: Entry Date Group: Date Data: 2026 – Name: AN Label: Accession Number Group: ID Data: EJ1502538
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RecordInfo	BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1177/00400599251346713 Languages: – Text: English PhysicalDescription: Pagination: PageCount: 14 StartPage: 244 Subjects: – SubjectFull: Middle School Students Type: general – SubjectFull: Mathematics Instruction Type: general – SubjectFull: Teaching Methods Type: general – SubjectFull: Middle School Mathematics Type: general – SubjectFull: Common Core State Standards Type: general – SubjectFull: Students with Disabilities Type: general – SubjectFull: Individualized Education Programs Type: general – SubjectFull: At Risk Students Type: general – SubjectFull: Learning Problems Type: general – SubjectFull: Grade 6 Type: general – SubjectFull: Grade 7 Type: general – SubjectFull: Grade 8 Type: general – SubjectFull: Mathematics Achievement Type: general – SubjectFull: Difficulty Level Type: general Titles: – TitleFull: Tackling Complex Math: Five Instructional Practices to Support Middle School Students Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Danielle O. Lariviere – PersonEntity: Name: NameFull: Katie B. MacLean – PersonEntity: Name: NameFull: Jessica Mao – PersonEntity: Name: NameFull: Sarah R. Powell IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 03 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 0040-0599 – Type: issn-electronic Value: 2163-5684 Numbering: – Type: volume Value: 58 – Type: issue Value: 4 Titles: – TitleFull: TEACHING Exceptional Children Type: main
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