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Essential Components of Math Instruction

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Title: Essential Components of Math Instruction
Language: English
Authors: Powell, Sarah R. (ORCID 0000-0002-6424-6160), Bouck, Emily C. (ORCID 0000-0002-0515-7627), Sutherland, Marah (ORCID 0000-0002-6108-8515), Clarke, Ben, Arsenault, Tessa L., Freeman-Green, Shaqwana
Source: TEACHING Exceptional Children. 2023 56(1):14-24.
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: 2023
Document Type: Journal Articles
Reports - Descriptive
Descriptors: Mathematics Instruction, Instructional Design, Teacher Competencies, Mathematics Achievement, Evidence Based Practice
DOI: 10.1177/00400599221125892
ISSN: 0040-0599
2163-5684
Abstract: In this article, the authors focus on five instructional approaches with a strong evidence base for the teaching and learning of math: (a) plan, model, and practice: systematic and explicit instruction; (b) focus on the language of math; (c) use multiple representations, including number lines; (d) build fluency; and (e) provide word problem instruction. These evidence-based practices have been identified in practice guides about math intervention from the "What Works Clearinghouse" (Fuchs et al., 2021; Gersten et al., 2009), through the "National Center on Intensive Intervention" (2018, 2019), and through a number of research reviews, see Powell, Doabler, et al., (2020); Powell, Mason, et al., 2021; Jitendra, Nelson, et al., 2016; Lein et al., 2020; Peltier et al., 2020). For each instructional approach, the authors provide a section about the approach and details about how to use the approach when providing math instruction.
Abstractor: ERIC
Entry Date: 2023
Accession Number: EJ1402373
Database: ERIC
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  Value: <anid>AN0173887406;tec01sep.23;2023Dec01.04:45;v2.2.500</anid> <title id="AN0173887406-1">Essential Components of Math Instruction </title> <p>Graph</p> <p>When it comes to how to teach math to students who experience difficulty with math, reliance on evidence-based practices is essential ([<reflink idref="bib13" id="ref1">13</reflink>]). Over the past few decades, hundreds of studies have been conducted with students with or at risk for disability and their educators to learn about which instructional strategies lead to improved outcomes in math, such as those related to early numeracy, whole numbers, solving word problems, fractions, and algebra. Many research reviews about these efforts help provide the foundation for essential components of math instruction ([<reflink idref="bib6" id="ref2">6</reflink>]; [<reflink idref="bib12" id="ref3">12</reflink>]; [<reflink idref="bib32" id="ref4">32</reflink>]; [<reflink idref="bib33" id="ref5">33</reflink>]; [<reflink idref="bib36" id="ref6">36</reflink>]; [<reflink idref="bib40" id="ref7">40</reflink>]; [<reflink idref="bib83" id="ref8">83</reflink>]; [<reflink idref="bib87" id="ref9">87</reflink>]).</p> <p>In this article, we focus on five instructional approaches with a strong evidence base for the teaching and learning of math: (a) plan, model, and practice: systematic and explicit instruction; (b) focus on the language of math; (c) use multiple representations, including number lines; (d) build fluency; and (e) provide word-problem instruction. These evidence-based practices have been identified in practice guides about math intervention from the What Works Clearinghouse ([<reflink idref="bib20" id="ref10">20</reflink>]; [<reflink idref="bib26" id="ref11">26</reflink>]), through the [<reflink idref="bib49" id="ref12">49</reflink>], [<reflink idref="bib50" id="ref13">50</reflink>]), and through a number of research reviews (beyond those listed in the preceding paragraph, see [<reflink idref="bib62" id="ref14">62</reflink>]; [<reflink idref="bib68" id="ref15">68</reflink>]; [<reflink idref="bib37" id="ref16">37</reflink>]; [<reflink idref="bib43" id="ref17">43</reflink>]; [<reflink idref="bib55" id="ref18">55</reflink>]). For each instructional approach, we provide a section about the approach. We then provide detail about how to use the approach when providing math instruction.</p> <hd id="AN0173887406-2">Plan, Model, and Practice: Systematic and Explicit Instruction</hd> <p></p> <hd id="AN0173887406-3">What?</hd> <p>As defined by [<reflink idref="bib20" id="ref19">20</reflink>], systematic instruction is the use of instructional elements that "intentionally build students' knowledge over time toward an identified learning outcome(s)" (p. 5). Systematic instruction includes planning for instruction as well as delivering instruction. Another term often used in this space is "explicit instruction," which [<reflink idref="bib30" id="ref20">30</reflink>] described as "an instructional design and delivery approach characterized as unambiguous, structured, systematic, and scaffolded" (p. 140). In this article, we consider the term "systematic and explicit" to mean done according to a fixed plan (i.e., systematic) and taught clearly and in detail (i.e., explicit).</p> <p>Planning involves developing an appropriate scope for math instruction and a sequence to the scope in which easier skills build to more difficult skills. Appropriate time is built into the schedule for students to develop a strong understanding of essential math concepts and procedures (e.g., the math content necessary for success with algebra). Systematic and explicit delivery of instruction revolves around modeling of different math problems, engaging students to multiple practice opportunities, and supporting students during both modeling and practice. Systematic and explicit instruction is often described as foundational to math instruction for students experiencing math difficulty ([<reflink idref="bib68" id="ref21">68</reflink>]; [<reflink idref="bib36" id="ref22">36</reflink>]; [<reflink idref="bib48" id="ref23">48</reflink>]; [<reflink idref="bib82" id="ref24">82</reflink>]), and it has been shown to benefit students in both in-person and virtual environments ([<reflink idref="bib5" id="ref25">5</reflink>].</p> <hd id="AN0173887406-4">How?</hd> <p>For the planning of systematic and explicit instruction, consider several aspects. First, a thorough understanding of student strengths and weaknesses is essential. Where do students excel? Where do students need support? Knowing the students will allow for focused math instruction for the group of students. Second, evaluate the math expectations in your state and determine the essential math content to include in math instruction. Students with or at risk for disability often require support across a lot of math content and several grade levels. Therefore, you cannot decide to teach all the math content; instructional time will have to focus on the essential math. Third, once math content is chosen for instruction, appropriately sequence the instruction. Sequencing involves planning for when content will be modeled and practiced and for how long.</p> <p>When delivering systematic and explicit instruction, use a combination of modeling and practice. When introducing math content, use modeling to provide a step-by-step explanation for how to solve a math problem. For example, model how to subtract 4 from 10, how to compare fractions with unlike denominators, or how to calculate the lateral surface area of a figure. Carefully plan the examples used for modeling. Open-ended problems are often used for demonstration, but worked examples (both those correctly solved and incorrectly solved) can also be used. During some lessons, four to five problems might be modeled. For more complex math, there may only be time to model one or two problems. The number of modeled problems may also depend on student need ([<reflink idref="bib16" id="ref26">16</reflink>]).</p> <p>The educator-led modeling lays the groundwork for how to solve a problem, but practice allows students to engage with the math and learn the math. Involve students in practice in a variety of ways. During guided practice, practice doing problems together with students. Perhaps all are sitting around a table and working the same problem on whiteboards. With virtual instruction, use screen sharing to share screen and show your work while students work on the same problem from a worksheet or collaborative whiteboard. Guided practice provides scaffolding for students as they start to do math problems but may need prompts for next steps and more examples of how to do the problems. Guided practice may also occur in small groups of students or in partner situations in which students are working together to solve problems. Also engage students in independent practice. Here, students work on math problems on their own, but you are nearby to answer questions and provide feedback.</p> <p>In Figure 1 ([<reflink idref="bib50" id="ref27">50</reflink>]), the bottom of the figure shows four supports to use during both modeling and practice. First, as you model and engage students in guided practice, involve students in math learning by asking a mix of high- and low-level questions. High-level questions may ask about "why" or "how" (e.g., "How did you calculate the unit rate?"). Such questions help you learn how well students understand the conceptual underpinnings of math problems, whereas low-level questions provide you with quick checks for understanding small parts of a problem. Low-level questions may ask "what," "which," or "when" (e.g., "Which fraction is greater?"). Second, ask these questions often (e.g., every 30 to 60 seconds) to elicit frequent responses from students. Third, as students respond to questions and prompts, it is necessary to provide feedback ([<reflink idref="bib16" id="ref28">16</reflink>]). Feedback may be affirmative to encourage students (e.g., "See how using your checklist of steps helped with solving that word problem."). Feedback may also redirect students when they have made an error or have a misconception (e.g., "Let's look at that again. Let's talk through writing the multiples of 3."). Finally, be prepared and have all materials ready to go so the lesson can move along at an appropriate pace.</p> <p>Graph: Figure 1 Delivery of systematic instruction</p> <hd id="AN0173887406-5">Focus on the Language of Math</hd> <p></p> <hd id="AN0173887406-6">What?</hd> <p>The language of math is embedded in almost all fields (e.g., science, business, technology, medicine), underscoring its importance in our everyday lives. At the secondary level, learning the language of math is imperative to accessing higher-level math courses (e.g., Algebra I) and developing the declarative, conceptual, and procedural knowledge ([<reflink idref="bib44" id="ref29">44</reflink>]) needed to meaningfully engage in academic discourse. Although there are stark differences between social and academic language, both play an important role in the math development of students with or at risk for disabilities.</p> <p>One method for focusing on the language of math is to help students learn essential math vocabulary. Across the elementary and middle school grades, students may see, hear, and use hundreds of math vocabulary terms at each grade level ([<reflink idref="bib61" id="ref30">61</reflink>]. There are several types of math vocabulary students should learn ([<reflink idref="bib45" id="ref31">45</reflink>]). Students have to learn technical vocabulary, such as "trapezoid" or "coefficient," that is only used in math. Students also use many subtechnical terms. These have multiple meanings in math (e.g., "square" as a shape or "square" as in 4<sups>2</sups>) or in and outside of math (e.g., "divide" with the operations or "divide" as in Continental Divide; [<reflink idref="bib75" id="ref32">75</reflink>]). To fully participate in the math classroom, students also need to know general terms, such as "measure" or "solve." Furthermore, math vocabulary features symbolic vocabulary, such as "four" or "½." What is important is to use math terms with precision and provide multiple practice opportunities for learning math terms ([<reflink idref="bib31" id="ref33">31</reflink>]; [<reflink idref="bib58" id="ref34">58</reflink>]).</p> <hd id="AN0173887406-7">How?</hd> <p>Use visual aids such as graphic organizers ([<reflink idref="bib15" id="ref35">15</reflink>]; [<reflink idref="bib34" id="ref36">34</reflink>]) to support students in acquiring the conceptual knowledge necessary to engage with the math language being taught across multiple grade bands. Graphic organizers may help alleviate executive function deficits, such as processing information and organizing materials and time ([<reflink idref="bib74" id="ref37">74</reflink>]; [<reflink idref="bib86" id="ref38">86</reflink>]). Vocabulary graphic organizer strategies such as the Frayer model ([<reflink idref="bib3" id="ref39">3</reflink>]) have been used to teach students more discipline-specific math terms (e.g., "polynomial," "quadrilateral") because they provide a visual summary of the definition, characteristics, examples, and nonexamples for more difficult vocabulary (i.e., academic language) being used in the lesson. Figure 2 is a completed Frayer model for the vocabulary word polynomial that can assist students in engaging in academic discourse in the classroom. The Frayer model serves as an example of a well-known research-based strategy that can cover the scope of several grade levels and core content areas (e.g., math, science). The Frayer model can help students understand the language of math using multiple heuristic strategies that build conceptual knowledge in earlier grades that will serve as declarative knowledge in later grades.</p> <p>Graph: Figure 2 Example of graphic organizer following the Frayer model</p> <p>In addition to graphic organizers, provide instruction on math vocabulary, especially when such instruction is embedded in math teaching ([<reflink idref="bib71" id="ref40">71</reflink>]). When providing instruction, introduce a new term, provide the meaning for the term, and engage students in multiple practice opportunities with the term ([<reflink idref="bib71" id="ref41">71</reflink>]). Additional strategies for building math vocabulary knowledge included use of mnemonics, building fluency with flash cards or games, and practicing math vocabulary through technology-based opportunities ([<reflink idref="bib71" id="ref42">71</reflink>]).</p> <hd id="AN0173887406-8">Use Multiple Representations Including Number Lines</hd> <p></p> <hd id="AN0173887406-9">What?</hd> <p>The use of multiple representations in math means to employ appropriate and effective representations, which are observable ways to depict math concepts, including numbers and relationships ([<reflink idref="bib27" id="ref43">27</reflink>]). Often representations are classified as concrete (e.g., base-10 blocks and other three-dimensional objects), semiconcrete (e.g., drawings of base-10 blocks, number lines, 10-frame, and other two-dimensional pictorials; sometimes they are referred to representational), and abstract (e.g., %, ≥, 5 or 2x + 3 = 9, and other examples of written numbers or mathematical ideas, expressions, and equations; [<reflink idref="bib20" id="ref44">20</reflink>]; [<reflink idref="bib27" id="ref45">27</reflink>]; see Figure 3). Representations, including the multiple ways in which they can exist, support the math understanding of students with and without disabilities ([<reflink idref="bib6" id="ref46">6</reflink>]; [<reflink idref="bib51" id="ref47">51</reflink>], [<reflink idref="bib52" id="ref48">52</reflink>]; [<reflink idref="bib55" id="ref49">55</reflink>]; [<reflink idref="bib68" id="ref50">68</reflink>]).</p> <p>Graph: Figure 3 Examples of multiple representations across mathematical concepts</p> <p>One important representation is a number line. The development of a mental number line is hypothesized as a central construct in how students learn and integrate new math information ([<reflink idref="bib24" id="ref51">24</reflink>]). The mental number line enables students to develop an understanding of number magnitude, first with whole numbers and then with rational numbers ([<reflink idref="bib81" id="ref52">81</reflink>]). Several studies have explored how mental number lines are initially developed and leveraged to solve math problems ([<reflink idref="bib42" id="ref53">42</reflink>]), and performance on number line tasks is highly related to math performance across grades ([<reflink idref="bib77" id="ref54">77</reflink>]). Support and build on the development of mental number lines by using number lines in your instruction.</p> <hd id="AN0173887406-10">How With Multiple Representations?</hd> <p>One common instructional approach to teaching students how to use multiple representations in math is through the lens of systematic and explicit instruction ([<reflink idref="bib4" id="ref55">4</reflink>]. Model solving the math problem with the representation(s) ([<reflink idref="bib20" id="ref56">20</reflink>]; [<reflink idref="bib30" id="ref57">30</reflink>]). The modeling includes both physical demonstrations as well as a think-aloud or verbal narration ([<reflink idref="bib16" id="ref58">16</reflink>]). Next, the student works to solve math problems with the representation while you guide the student, providing cues or prompts as needed. Finally, the student engages independently in solving similar problems without any support ([<reflink idref="bib16" id="ref59">16</reflink>]). Another instructional approach for multiple representations is through task analytic instruction. This is the use of a series of ordered steps that one does to complete a task ([<reflink idref="bib79" id="ref60">79</reflink>]). With task analytic instruction, develop a series of steps (i.e., task analysis) to solve problem types with a representation. Students are then taught to solve the math with the task analysis.</p> <p>Beyond the common pedagogical approach to teaching students to understand and use different representations to solve math problems with systematic and explicit instruction, use of multiple representations is often taught as a graduated sequence of instruction ([<reflink idref="bib1" id="ref61">1</reflink>]; [<reflink idref="bib6" id="ref62">6</reflink>]). This instruction—most commonly referred to as the concrete-representational-abstract (CRA) or concrete-semiconcrete-abstract (CSA) instructional framework—systematically transitions students from learning to make sense and solve math problems with concrete manipulative representations and semiconcrete (e.g., pictorial) representations as they learn about abstract representations that rely on numerical strategies ([<reflink idref="bib1" id="ref63">1</reflink>]). The CRA instructional framework is considered an evidence-based practice for students with learning disabilities ([<reflink idref="bib6" id="ref64">6</reflink>]).</p> <p>In addition to the traditional CRA or CSA, in which students individually transition between using different types of representation to engage with math problems on the basis of mastery (e.g., 100% accuracy for three sessions), some researchers and practitioners use CRA-integration (CRA-I), in which each form of representation is presented simultaneously (i.e., concrete manipulatives with pictorial representations and abstract notation and numerical strategies; [<reflink idref="bib85" id="ref65">85</reflink>]). The multiple forms of representation are then systematically removed to end with the abstract phase ([<reflink idref="bib85" id="ref66">85</reflink>]). Regardless of the use of the CRA-I or the more traditional CSA, make sure students understand the relationship to each other between concrete, semiconcrete, and abstract representations (see Figure 3 for examples). In other words, help students connect that the 10s-rod represents the number 10 and that tally marks can represent the same thing as the 10s-rod and the math notation of 10.</p> <p>When selecting multiple representations, it is important to carefully consider which representations to use and which can be challenging because often multiple options exist to represent different math concepts. First, the representation should support the targeted math and be mathematically accurate. Second, consider the use of bland versus perceptually rich concrete manipulative representations, meaning those that look more boring and basic (e.g., dull colored tiles or cubes) compared to those that are more realistic in appearance, bright, and aesthetically appealing (e.g., counting bears, plastic money; [<reflink idref="bib57" id="ref67">57</reflink>]). Researchers have suggested bland manipulatives may support student learning to a greater extent ([<reflink idref="bib10" id="ref68">10</reflink>]; [<reflink idref="bib9" id="ref69">9</reflink>]; [<reflink idref="bib55" id="ref70">55</reflink>]). In addition, the consistency of the representation choices (across both math concepts in a grade and as a transition to more advanced math topics) represents a potential consideration. For example, base-10 blocks can be used for place value and addition and subtraction of whole numbers and in later grades for addition and subtraction of decimals.</p> <hd id="AN0173887406-11">How With Number Lines?</hd> <p>To illustrate how you can strategically use number lines, we provide examples in both whole and rational number contexts. The number line is an important early number representation that can help students learn the number list, understand relations among whole numbers, and engage in operations such as addition and subtraction ([<reflink idref="bib25" id="ref71">25</reflink>]; [<reflink idref="bib41" id="ref72">41</reflink>]). As students begin to map quantities onto numerals (e.g., "••••" can be represented by the numeral "4"), the number line helps to reinforce the idea that the number list is fixed (i.e., the numbers always appear in the same order) and that each successive number on the number line represents a quantity that is one more than the previous number. Once students understand this concept, they can readily use the number line to compare number magnitudes (e.g., comparing 13 and 16), knowing that numbers further away from 0 represent larger quantities.</p> <p>Students may also use a physical or mental number line to represent and solve addition and subtraction problems. For instance, students may use a number line to count up or count down to solve a problem such as 9 – 5 (e.g., counting up from 5 to 9 or counting down from 9 to 5 to find the difference of 4). When solving more complex problems, number lines can support students in using more advanced strategies, such as breaking apart or decomposing an addend (i.e., a number being added) or subtrahend (i.e., a number being subtracted), to make a 10. For example, to solve a problem that involves crossing over 10, such as 8 + 6, students may decompose the "6" into 2 and 4, add 2 to 8 to make a 10, then add the remaining 4 to equal 14 (see Figure 4). The number line can nicely illustrate this type of thinking for students and can be extended to composing and decomposing numbers using similar strategies in more advanced math content, such as 38 + 6 or 44 – 6.</p> <p>Graph: Figure 4 Using number lines to support whole and rational number understanding</p> <p>As students transition to learning about rational numbers beginning in Grade 3, the number line serves as a consistent representation to help students understand rational number magnitudes and where rational numbers are situated in comparison to whole numbers ([<reflink idref="bib17" id="ref73">17</reflink>]; [<reflink idref="bib22" id="ref74">22</reflink>]; [<reflink idref="bib80" id="ref75">80</reflink>]). Researchers increasingly support the use of number line or "measurement" representations of fractions to help struggling learners understand the meaning of rational numbers ([<reflink idref="bib22" id="ref76">22</reflink>]; [<reflink idref="bib29" id="ref77">29</reflink>]; [<reflink idref="bib47" id="ref78">47</reflink>]). When instruction only focuses on teaching fractions as a relation between parts and a whole (e.g., representing <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></math> </ephtml> by splitting a pie into three equal parts and shading one part), students may have a difficult time understanding that fractions have numerical value and that they can exceed the value of 1 ([<reflink idref="bib80" id="ref79">80</reflink>]). Number lines can be used to correct these misconceptions and to show equivalency and differences in the magnitude of various fractions and decimals. One proven strategy to build students' magnitude understanding is to help students identify "benchmark" numbers on a number line, such as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mrow><mn>0</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></math> </ephtml> or 0, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mrow><mn>2</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mrow><mn>4</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></math> </ephtml> (or 1), and then compare other fractions to determine whether they are greater or less than the benchmark fractions (see Figure 4 for an example). Equivalent fractions, such as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mrow><mn>2</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math> </ephtml> , should be displayed vertically rather than side by side to show their precise location on the number line (see Figure 4; [<reflink idref="bib20" id="ref80">20</reflink>]). Decimals can also be represented alongside fractions to build understanding of relations among different types of rational numbers. In addition to comparing fraction magnitudes, number lines are good representational tools for modeling fraction addition and subtraction.</p> <hd id="AN0173887406-12">Build Fluency</hd> <p></p> <hd id="AN0173887406-13">What?</hd> <p>Fluency in math means doing math easily and accurately. Most conversations related to fluency relate to the operations of addition, subtraction, multiplication, and division, but students need to establish fluency in many other areas of math, including counting, identifying benchmark fractions, or applying formulas in geometry. In this article, we focus on fluency related to the operations. When students have fluency with the operations, it makes higher-level math easier ([<reflink idref="bib69" id="ref81">69</reflink>]), provides less stress on working memory ([<reflink idref="bib39" id="ref82">39</reflink>]), and helps students build confidence in math. Instruction related to fluency must emphasize concepts alongside procedures ([<reflink idref="bib63" id="ref83">63</reflink>].</p> <hd id="AN0173887406-14">How?</hd> <p>In this article, we focus on building fluency with number combinations. This includes 100 addition (with single-digit addends; 5 + 8), 100 subtraction (with single-digit subtrahends and differences; 13 – 9), 100 multiplication (with single-digit factors; 6 × 5), and 90 division (with single-digit divisors and quotients; 42 ÷ 7) facts. Building fluency with the number combinations involves teaching students the conceptual interpretations of each of the operations. For example, addition means to put together or add on. Subtraction means to take away or compare. Conceptually, multiplication involves equal groups or multiplying a set. Division can be partitive or quotative. To help students with these concepts, model the different concepts with multiple representations and number lines. Use precise math language and present the concepts of the operations in story problems.</p> <p>As described by [<reflink idref="bib72" id="ref84">72</reflink>], effective instruction related to fluency involves modeling, multiple opportunities to practice, immediate feedback, and an appropriate ratio of known to unknown number combinations. With the number combinations, students should learn both the concepts and procedures that accompany strategies. For example, with addition, students should learn the concept that addition may mean putting amounts together or the concept of joining to a set ([<reflink idref="bib11" id="ref85">11</reflink>]) alongside procedures for using representations or numbers to add. Of note, when students have developed fluency with number combinations, focused instruction may not be necessary ([<reflink idref="bib8" id="ref86">8</reflink>]); however, students need to continually practice number combinations to retain their fluency.</p> <p>At some point, it is helpful if students develop automaticity with sums, differences, products, and quotients. Note that accuracy with number combinations develops before speed (i.e., automaticity), so different students may be ready to work on automaticity at different times ([<reflink idref="bib70" id="ref87">70</reflink>]). Automaticity practice should be brief (e.g., 1–2 minutes), but it should occur regularly. [<reflink idref="bib78" id="ref88">78</reflink>] noted distributed practice—practicing the number combinations several times in the same day—was effective for developing automaticity with addition number combinations. Ways to practice automaticity include games (e.g., Smath, Mobi), activities (e.g., wrap-ups, dice, cover-copy-compare, incremental rehearsal; [<reflink idref="bib7" id="ref89">7</reflink>]; [<reflink idref="bib84" id="ref90">84</reflink>]), and technology-based apps. As an example, [<reflink idref="bib60" id="ref91">60</reflink>] used flashcards to build automaticity with addition and subtraction number combinations. Students answered with sums or differences during two, 1-minute timings and then graphed their higher score (of correctly answered flashcards) on a graph. Each lesson, before starting the flashcard activity, students would review their previous lesson's score and aim to beat that score. Automaticity can be practiced in untimed and timed situations, but the timing should be secondary to the focus on accuracy. Beyond number combinations, practice fluency with whole-number computation (e.g., 25 + 189 or 1056 / 14) and rational-number computation (e.g., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></math> </ephtml> − <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></math> </ephtml> or 4.5 × 7.9).</p> <hd id="AN0173887406-15">Provide Word-Problem Instruction</hd> <p></p> <hd id="AN0173887406-16">What?</hd> <p>To successfully solve word problems, a math problem in a story, students must interpret the language and then solve the math problem ([<reflink idref="bib14" id="ref92">14</reflink>]). The language in word problems includes complex general and math language in the story or embedded in graphs and charts that may be either relevant or irrelevant information ([<reflink idref="bib18" id="ref93">18</reflink>]; [<reflink idref="bib76" id="ref94">76</reflink>]). After identifying the relevant information, students then must set up the necessary equations and solve for the unknown ([<reflink idref="bib64" id="ref95">64</reflink>]). This step in solving a word problem can be challenging due to calculation complexity and the position of the unknown in an equation and position of the unknown ([<reflink idref="bib66" id="ref96">66</reflink>]; [<reflink idref="bib23" id="ref97">23</reflink>]; [<reflink idref="bib54" id="ref98">54</reflink>]).</p> <hd id="AN0173887406-17">How?</hd> <p>To support students in solving word problems, provide schema instruction paired with an attack strategy ([<reflink idref="bib43" id="ref99">43</reflink>]). An attack strategy is a step-by-step process for working through a word problem, and this attack strategy can be used to solve all different types of word problems ([<reflink idref="bib19" id="ref100">19</reflink>]; [<reflink idref="bib38" id="ref101">38</reflink>]). Through schema instruction, students learn to classify problems into problem types ([<reflink idref="bib73" id="ref102">73</reflink>]). Pairing schema instruction with an attack strategy supports students to make gains in solving word problems because it provides students with the tools to identify the problem type and a metacognitive strategy for working through the problem-solving process ([<reflink idref="bib46" id="ref103">46</reflink>]).</p> <p>Using an attack strategy builds automatic self-regulation strategies for students during the problem-solving process ([<reflink idref="bib46" id="ref104">46</reflink>]). When choosing an attack strategy, (<reflink idref="bib1" id="ref105">1</reflink>) choose one that is easy to remember, (<reflink idref="bib2" id="ref106">2</reflink>) use an attack strategy that starts with reading the word problem, and (<reflink idref="bib3" id="ref107">3</reflink>) always use the same attack strategy. Some examples of attack strategies include RUN, UPS-Check, and FOPS ([<reflink idref="bib65" id="ref108">65</reflink>]; [<reflink idref="bib38" id="ref109">38</reflink>]). For RUN, students Read the problem, Underline the label and cross out the irrelevant information, and Name the problem type ([<reflink idref="bib19" id="ref110">19</reflink>]). In UPS-Check, students attack a word problem by Understanding by reading, Plan how to solve the problem, Solve, and Check their answer ([<reflink idref="bib59" id="ref111">59</reflink>]). To attack a problem using FOPS, students Find the problem, Organize the information using a diagram, Plan to solve the problem, and Solve the problem ([<reflink idref="bib38" id="ref112">38</reflink>]).</p> <p>The use of schema instruction leads to growth in solving word problems ([<reflink idref="bib43" id="ref113">43</reflink>]; [<reflink idref="bib56" id="ref114">56</reflink>]). Word problems can be categorized as additive or multiplicative schemas ([<reflink idref="bib65" id="ref115">65</reflink>]). In schema instruction, students are taught to identify schemas with the support of the schema definitions and how to set up an equation for each schema ([<reflink idref="bib64" id="ref116">64</reflink>]; [<reflink idref="bib53" id="ref117">53</reflink>]).</p> <p>Additive schemas, typically introduced first, include total, difference, and change schemas ([<reflink idref="bib21" id="ref118">21</reflink>], [<reflink idref="bib20" id="ref119">20</reflink>]). In the total schema, parts are put together for a total. In total problems, students solve for either an unknown part or total. For the difference schema, a greater amount and an amount that is less are compared for a difference. For the change schema, a starting amount increases or decreases for a new amount. Multiplicative schemas include equal groups, compare, and ratio/proportion problems. For the equal groups schema, there are groups with an equal number in each group ([<reflink idref="bib2" id="ref120">2</reflink>]). In the compare schema, two sets are compared ([<reflink idref="bib28" id="ref121">28</reflink>]). Last, ratio and proportion schemas are multiplicative comparisons of quantities. Ratio problems involve one situation, and proportion problems involve two ratios ([<reflink idref="bib35" id="ref122">35</reflink>]). Figure 5 provides an example of how to use an attack strategy with schema instruction to identify the schema type and solve for the unknown.</p> <p>Graph: Figure 5 Word-problem example with attack strategy and schema focus</p> <hd id="AN0173887406-18">Conclusion</hd> <p>For students who experience difficulty with math, it is essential to provide timely and evidence-based math instruction. Each of the five instructional approaches, (a) plan, model, and practice: systematic and explicit instruction; (b) focus on the language of math; (c) use multiple representations, including number lines; (d) build fluency; and (e) provide word-problem instruction, has a strong evidence base to support their use. You may want to include some or all of these approaches in your math instruction, but we note the importance of data collection and decision-making conversations to ensure each approach is appropriate for each individual student ([<reflink idref="bib67" id="ref123">67</reflink>].</p> <p>There are many excellent resources available to understand which instructional approaches are helpful to use in math instruction, including resources from the What Works Clearinghouse, National Center for Intensive Intervention, IRIS Center at Vanderbilt University, CEEDAR (Collaboration for Effective Educator Development, Accountability, and Reform) Center at the University of Florida, among others. Another resource may be that associated with the Science of Math—an effort analogous to that of the Science of Reading. The Science of Math group aims to put together resources about objective evidence focused on how students learn math in order to make sound decisions about instruction. 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TEACHING Exceptional Children, 49(2), 115–125. https://doi.org/10.1177/0040059916673353</bibtext> </blist> <blist> <bibtext> van Garderen D. (2007). Teaching students with LD to use diagrams to solve mathematical word problems. Journal of Learning Disabilities, 40(6), 540–553.</bibtext> </blist> <blist> <bibtext> Zheng X., Flynn L. J., Swanson H. L. (2013). Experimental intervention studies on word problem solving and math disabilities: A selective analysis of the literature. Learning Disability Quarterly, 36(2), 97–111. https://doi.org/10.1177/0731948712444277</bibtext> </blist> </ref> <ref id="AN0173887406-20"> <title> Footnotes </title> <blist> <bibtext> The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.</bibtext> </blist> <blist> <bibtext> The author(s) received no financial support for the research, authorship, and/or publication of this article.</bibtext> </blist> <blist> <bibtext> Sarah R. Powell</bibtext> </blist> <blist> <bibtext>Graph</bibtext> </blist> <blist> <bibtext>https://orcid.org/0000-0002-6424-6160 Emily C. Bouck</bibtext> </blist> <blist> <bibtext>Graph</bibtext> </blist> <blist> <bibtext>https://orcid.org/0000-0002-0515-7627 Marah Sutherland</bibtext> </blist> <blist> <bibtext>Graph https://orcid.org/0000-0002-6108-8515</bibtext> </blist> </ref> <aug> <p>By Sarah R. Powell; Emily C. Bouck; Marah Sutherland; Ben Clarke; Tessa L. Arsenault and Shaqwana Freeman-Green</p> <p>Reported by Author; Author; Author; Author; Author; Author</p> </aug> <nolink nlid="nl1" bibid="bib13" firstref="ref1"></nolink> <nolink nlid="nl2" bibid="bib12" firstref="ref3"></nolink> <nolink nlid="nl3" bibid="bib32" firstref="ref4"></nolink> <nolink nlid="nl4" bibid="bib33" firstref="ref5"></nolink> <nolink nlid="nl5" bibid="bib36" firstref="ref6"></nolink> <nolink nlid="nl6" bibid="bib40" firstref="ref7"></nolink> <nolink nlid="nl7" bibid="bib83" firstref="ref8"></nolink> <nolink nlid="nl8" bibid="bib87" firstref="ref9"></nolink> <nolink nlid="nl9" bibid="bib20" firstref="ref10"></nolink> <nolink nlid="nl10" bibid="bib26" firstref="ref11"></nolink> <nolink nlid="nl11" bibid="bib49" firstref="ref12"></nolink> <nolink nlid="nl12" bibid="bib50" firstref="ref13"></nolink> <nolink nlid="nl13" bibid="bib62" firstref="ref14"></nolink> <nolink nlid="nl14" bibid="bib68" firstref="ref15"></nolink> <nolink nlid="nl15" bibid="bib37" firstref="ref16"></nolink> <nolink nlid="nl16" bibid="bib43" firstref="ref17"></nolink> <nolink nlid="nl17" bibid="bib55" firstref="ref18"></nolink> <nolink nlid="nl18" bibid="bib30" firstref="ref20"></nolink> <nolink nlid="nl19" bibid="bib48" firstref="ref23"></nolink> <nolink nlid="nl20" bibid="bib82" firstref="ref24"></nolink> <nolink nlid="nl21" bibid="bib16" firstref="ref26"></nolink> <nolink nlid="nl22" bibid="bib44" firstref="ref29"></nolink> <nolink nlid="nl23" bibid="bib61" firstref="ref30"></nolink> <nolink nlid="nl24" bibid="bib45" firstref="ref31"></nolink> <nolink nlid="nl25" bibid="bib75" firstref="ref32"></nolink> <nolink nlid="nl26" bibid="bib31" firstref="ref33"></nolink> <nolink nlid="nl27" bibid="bib58" firstref="ref34"></nolink> <nolink nlid="nl28" bibid="bib15" firstref="ref35"></nolink> <nolink nlid="nl29" bibid="bib34" firstref="ref36"></nolink> <nolink nlid="nl30" bibid="bib74" firstref="ref37"></nolink> <nolink nlid="nl31" bibid="bib86" firstref="ref38"></nolink> <nolink nlid="nl32" bibid="bib71" firstref="ref40"></nolink> <nolink nlid="nl33" bibid="bib27" firstref="ref43"></nolink> <nolink nlid="nl34" bibid="bib51" firstref="ref47"></nolink> <nolink nlid="nl35" bibid="bib52" firstref="ref48"></nolink> <nolink nlid="nl36" bibid="bib24" firstref="ref51"></nolink> <nolink nlid="nl37" bibid="bib81" firstref="ref52"></nolink> <nolink nlid="nl38" bibid="bib42" firstref="ref53"></nolink> <nolink nlid="nl39" bibid="bib77" firstref="ref54"></nolink> <nolink nlid="nl40" bibid="bib79" firstref="ref60"></nolink> <nolink nlid="nl41" bibid="bib85" firstref="ref65"></nolink> <nolink nlid="nl42" bibid="bib57" firstref="ref67"></nolink> <nolink nlid="nl43" bibid="bib10" firstref="ref68"></nolink> <nolink nlid="nl44" bibid="bib25" firstref="ref71"></nolink> <nolink nlid="nl45" bibid="bib41" firstref="ref72"></nolink> <nolink nlid="nl46" bibid="bib17" firstref="ref73"></nolink> <nolink nlid="nl47" bibid="bib22" firstref="ref74"></nolink> <nolink nlid="nl48" bibid="bib80" firstref="ref75"></nolink> <nolink nlid="nl49" bibid="bib29" firstref="ref77"></nolink> <nolink nlid="nl50" bibid="bib47" firstref="ref78"></nolink> <nolink nlid="nl51" bibid="bib69" firstref="ref81"></nolink> <nolink nlid="nl52" bibid="bib39" firstref="ref82"></nolink> <nolink nlid="nl53" bibid="bib63" firstref="ref83"></nolink> <nolink nlid="nl54" bibid="bib72" firstref="ref84"></nolink> <nolink nlid="nl55" bibid="bib11" firstref="ref85"></nolink> <nolink nlid="nl56" bibid="bib70" firstref="ref87"></nolink> <nolink nlid="nl57" bibid="bib78" firstref="ref88"></nolink> <nolink nlid="nl58" bibid="bib84" firstref="ref90"></nolink> <nolink nlid="nl59" bibid="bib60" firstref="ref91"></nolink> <nolink nlid="nl60" bibid="bib14" firstref="ref92"></nolink> <nolink nlid="nl61" bibid="bib18" firstref="ref93"></nolink> <nolink nlid="nl62" bibid="bib76" firstref="ref94"></nolink> <nolink nlid="nl63" bibid="bib64" firstref="ref95"></nolink> <nolink nlid="nl64" bibid="bib66" firstref="ref96"></nolink> <nolink nlid="nl65" bibid="bib23" firstref="ref97"></nolink> <nolink nlid="nl66" bibid="bib54" firstref="ref98"></nolink> <nolink nlid="nl67" bibid="bib19" firstref="ref100"></nolink> <nolink nlid="nl68" bibid="bib38" firstref="ref101"></nolink> <nolink nlid="nl69" bibid="bib73" firstref="ref102"></nolink> <nolink nlid="nl70" bibid="bib46" firstref="ref103"></nolink> <nolink nlid="nl71" bibid="bib65" firstref="ref108"></nolink> <nolink nlid="nl72" bibid="bib59" firstref="ref111"></nolink> <nolink nlid="nl73" bibid="bib56" firstref="ref114"></nolink> <nolink nlid="nl74" bibid="bib53" firstref="ref117"></nolink> <nolink nlid="nl75" bibid="bib21" firstref="ref118"></nolink> <nolink nlid="nl76" bibid="bib28" firstref="ref121"></nolink> <nolink nlid="nl77" bibid="bib35" firstref="ref122"></nolink> <nolink nlid="nl78" bibid="bib67" firstref="ref123"></nolink>
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  Data: Essential Components of Math Instruction
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  Data: <searchLink fieldCode="AR" term="%22Powell%2C+Sarah+R%2E%22">Powell, Sarah R.</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-6424-6160">0000-0002-6424-6160</externalLink>)<br /><searchLink fieldCode="AR" term="%22Bouck%2C+Emily+C%2E%22">Bouck, Emily C.</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-0515-7627">0000-0002-0515-7627</externalLink>)<br /><searchLink fieldCode="AR" term="%22Sutherland%2C+Marah%22">Sutherland, Marah</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-6108-8515">0000-0002-6108-8515</externalLink>)<br /><searchLink fieldCode="AR" term="%22Clarke%2C+Ben%22">Clarke, Ben</searchLink><br /><searchLink fieldCode="AR" term="%22Arsenault%2C+Tessa+L%2E%22">Arsenault, Tessa L.</searchLink><br /><searchLink fieldCode="AR" term="%22Freeman-Green%2C+Shaqwana%22">Freeman-Green, Shaqwana</searchLink>
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  Data: <searchLink fieldCode="SO" term="%22TEACHING+Exceptional+Children%22"><i>TEACHING Exceptional Children</i></searchLink>. 2023 56(1):14-24.
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  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
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  Data: 11
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  Data: Journal Articles<br />Reports - Descriptive
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  Data: <searchLink fieldCode="DE" term="%22Mathematics+Instruction%22">Mathematics Instruction</searchLink><br /><searchLink fieldCode="DE" term="%22Instructional+Design%22">Instructional Design</searchLink><br /><searchLink fieldCode="DE" term="%22Teacher+Competencies%22">Teacher Competencies</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Achievement%22">Mathematics Achievement</searchLink><br /><searchLink fieldCode="DE" term="%22Evidence+Based+Practice%22">Evidence Based Practice</searchLink>
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  Data: 10.1177/00400599221125892
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  Data: In this article, the authors focus on five instructional approaches with a strong evidence base for the teaching and learning of math: (a) plan, model, and practice: systematic and explicit instruction; (b) focus on the language of math; (c) use multiple representations, including number lines; (d) build fluency; and (e) provide word problem instruction. These evidence-based practices have been identified in practice guides about math intervention from the "What Works Clearinghouse" (Fuchs et al., 2021; Gersten et al., 2009), through the "National Center on Intensive Intervention" (2018, 2019), and through a number of research reviews, see Powell, Doabler, et al., (2020); Powell, Mason, et al., 2021; Jitendra, Nelson, et al., 2016; Lein et al., 2020; Peltier et al., 2020). For each instructional approach, the authors provide a section about the approach and details about how to use the approach when providing math instruction.
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      – SubjectFull: Teacher Competencies
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      – SubjectFull: Mathematics Achievement
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