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

Abstract

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.

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 & Council of Chief State School Officers [NGA & CCSSO], 2010). 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.

Table 1.

Middle School Common Core State Standards for Mathematics

The number system	Grade 6
	• 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.
	Grade 7
	• Develop understanding of operations with rational numbers.
	Grade 8
	• Understand nonrational numbers and use rational approximations to approximate irrational numbers.
Expressions and equations	Grade 6
	• Extend previous understandings of arithmetic to write and interpret expressions and equations.
	Grade 7
	• Generate equivalent expressions using properties of operations and apply knowledge of expressions and equations to solve problems.
	Grade 8
	• Reason with expressions and equations and apply knowledge to work with radicals, integer exponents, and proportional relationships.
Geometry	Grade 6
	• Extend knowledge of area, surface area, and volume to solve real-world and mathematical problems.
	Grade 7
	• Construct geometrical figures, analyze relationships between them, and solve problems involving angles, area, surface area, and volume.
	Grade 8
	• Use the Pythagorean theorem to analyze two- and three-dimensional space and figures using congruence, similarity, angles, and distance.
Statistics and probability	Grade 6
	• Develop understanding of statistical thinking.
	Grade 7
	• Draw informal comparative inferences about two populations using random sampling.
	Grade 8
	• Investigate patterns of association in bivariate data.
Ratios and proportional relationships	Grade 6
	• Apply understanding of ratio concepts with multiplication and division to solve problems.
	Grade 7
	• Understand and apply proportional relationships.
Functions	Grade 8
Functions	• Understand functions and model quantitative relationships.

Note. Adapted from National Governors Association Center for Best Practices & Council of Chief State School Officers (2010).

Mathematics Difficulty in Middle School

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. 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.

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 (NCES, 2024; Wei et al., 2013). 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 (Wei et al., 2013). 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 (Marita & Hord, 2017; Spooner et al., 2019).

“Incoming challenges with concepts and procedures addressed in elementary math can therefore negatively impact students’ readiness to learn new, complex content in middle school.

In addition to students with identified disabilities, middle school students at risk for disabilities also frequently experience MD (Powell et al., 2021a; NCES, 2024). 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 (Powell et al., 2021a; Jitendra, Nelson, et al., 2016; Marita & Hord, 2017; Star et al., 2015; Woodward et al., 2018). 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).

Impact of Incoming Math Proficiency

Middle school students with or at risk for disabilities are likely to have a history of MD beginning in elementary school (Moser Opitz et al., 2017). 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 (Cirino et al., 2016). 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.

Students who begin middle school with MD are also less likely to be fluent in math facts and procedures (Cirino et al., 2016; Moser Opitz et al., 2017). 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.

Understanding Middle School Math

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, x, 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 linear equation to model this scenario.” For these reasons, it is important to provide research-based supports in math to middle school students with MD.

Instructional Practices to Use in Middle School Math

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 TEACHING Exceptional Children. 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.

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 (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).

Use Explicit Instruction

When using explicit instruction, a teacher designs instruction systematically using direct modeling and student practice opportunities (Hughes et al., 2017). 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 (Ketterlin-Geller et al., 2019).

“Questions should be strategically chosen to build and activate students’ background knowledge.

Why is this important in middle school math?

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 (Powell et al., 2021a), those with learning disabilities (Marita & Hord, 2017), and those with extensive support needs (Spooner et al., 2019). Research shows middle school students with MD can benefit from explicit instruction on fractions, integers, ratios and proportions, geometry, and algebraic equations (Lariviere et al., 2024; Marita & Hord, 2017; Powell et al., 2021a; Spooner et al., 2019). For instance, Namkung and Bricko (2021) 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. Jitendra, Dupuis, and colleagues (2016) 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 (NGA & CCSSO, 2010).

Recommended strategies

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,

I see the problem –14 = 5x + 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.

Demonstrations should include step-by-step procedures and multiple representations (e.g., algebra tiles).

The second recommendation when implementing explicit instruction is to engage students in ample practice opportunities (Hughes et al., 2017). 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 = 5x + 4 in a guided-practice setting, both the teacher and students should show their step-by-step work to identify the value of x. 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 (Hughes et al., 2017).

The third recommended strategy when using explicit instruction is to incorporate supporting practices into both modeling and practice (Ketterlin-Geller et al., 2019). 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 (Hughes et al., 2017; Ketterlin-Geller et al., 2019). 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.

Table 2

Questioning Examples by Domain

Domain	High-level questions	Low-level questions
The number system	Why are $\frac{2}{- 3}$ , $\frac{- 2}{3}$ , and $- \frac{2}{3}$ all equivalent? (Grade 7)	Is the equation true: 18.2 + (–72.9) = 18.2 – 72.9? (Grade 7)
Expressions and equations	Why does a point of intersection of two linear equations show a solution to their system of equations? (Grade 8)	What is the solution to the following system of equations? (Grade 8) –2x+y = 13 4x–y = –15
Geometry	How can you decompose the polygon below into other shapes to help you find its area? (Grade 6)	What is the area of the shape below? (Grade 6)
Statistics and probability	Why can random sampling help you make inferences about a general population? (Grade 7)	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
Ratios and proportional relationships	How can calculating unit rates help you save money while grocery shopping? (Grade 6)	What is the cost per apple if a bag of 11 apples costs $5.25? (Grade 6)
Functions	Why is the equation x = −3 not a function? (Grade 8)	What is the rate of change in the equation 2x−5y = 10? (Grade 8)

Note. These questions align with the Common Core State Standards of Mathematics from the National Governors Association Center for Best Practices & Council of Chief State School Officers (2010).

Incorporate Multiple Representations

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 (Fuchs et al., 2021). 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 (Strickland & Maccini, 2013). In this section, we provide strategies for each type of representation.

Why is this important in middle school math?

As an evidence-based practice with robust effects, multiple representations can support the mathematical reasoning of middle school students with MD (Bouck et al., 2020; Jitendra, Nelson, et al., 2016; Powell et al., 2021a; Spooner et al., 2019). 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; NGA & CCSSO, 2010). 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 (American Institutes for Research, 2022). 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.

Multiple representations are also helpful for students of culturally and linguistically diverse backgrounds (Jao, 2012). 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 (Moyer-Packenham & Westenskow, 2013).

Recommended strategies

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.

In middle school math, some of the first concepts taught are fractions and using the number line to identify rational numbers (NGA & CCSSO, 2010). Use of the number line supports students’ understanding of the similarities and differences among whole numbers, integers, fractions, and other rational numbers (Siegler et al., 2010; Sutherland et al., 2024). 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., – $\frac{1}{2}$ – $\frac{3}{4}$ ), where the number starts at – $\frac{1}{2}$ and moves to the left another $\frac{3}{4}$ units. Figure 1 shows content that can be taught using the number line and the different ways the number line can be presented (Rodrigues et al., 2024).

“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.

Figure 1

Representations with the number line

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 (Clarke et al., 2022). And to date, studies have shown there is no significant difference in math learning outcomes when using concrete manipulatives versus virtual manipulatives (e.g., Bouck et al., 2018). 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.

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 (Butler et al., 2003).

Finally, teachers can use hand gestures to represent spatial or abstract concepts (Wakefield et al., 2019). 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 (Rodrigues et al., 2024). 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.

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.

Build Fluency

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 (National Mathematics Advisory Panel [NMAP], 2008), 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.

Why is this important in middle school math?

Fluency in math frees up students’ cognitive resources to move to higher-order thinking and complex mathematical content presented in middle school (NMAP, 2008). 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 (Cirino et al., 2016; Moser Opitz et al., 2017; Nelson et al., 2016). 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 (NGA & CCSSO, 2010). 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.

Recommended strategies

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 (Fuchs et al., 2021). 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 (Powell et al., 2015).

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 (Fuchs et al., 2021). 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 (Riccomini et al., 2017). 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 (Clarke et al., 2016).

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 (Baroody et al., 2009). 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.

Figure 2

Fluency integration.

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.

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.

Teach Math Vocabulary

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”; Monroe & Panchyshyn, 1995; Rubenstein & Thompson, 2002).

Why is this important in middle school math?

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 (Lin et al., 2021). Researchers have demonstrated this association across several math domains, including algebra, fractions, and word-problem solving (Lin et al., 2021; Peng & Lin, 2019). Math-vocabulary knowledge gives students access to teachers’ instruction in math, and it provides students with the language to articulate their own mathematical reasoning.

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 (Forsyth & Powell, 2017). By Grade 8, students have been exposed to more than 1,200 math-vocabulary terms (Powell et al., 2021b). 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; Rubenstein & Thompson, 2002). For those with MD who are emergent bilingual students, navigating false cognates and unfamiliar formal mathematics language may be especially challenging (Lariviere et al., 2025). 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.

Table 3

Challenging Features of Middle School Mathematics Vocabulary

Challenging features	Examples
Multiple academic meanings	• “Inequality” as greater or less than (math) vs. unequal rights (social studies) • “System” of equations (math) vs. the digestive system (science) • “Translation” as movement of a figure (math) vs. translation into English (foreign language)
Meanings in mathematics and general language	• “Face” of a three-dimensional figure vs. a person’s face • “Negative” as a value below zero vs. unfavorable • “Similar” figures as proportional vs. generally alike
Homophones and near homophones	• “Axis” vs. “access” • “Chord” vs. “cord” • “Rational” vs. “rationale”

Recommended strategies

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 (Powell et al., 2019). 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.

Additionally, preteaching is a recommended strategy for teaching math vocabulary to students with MD, including emergent bilingual students with MD (Namkung & Bricko, 2021; Orosco, 2014; Sanford et al., 2020). 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 (Wright & Cervetti, 2017). When teachers preteach important math vocabulary, students are primed to recognize pretaught terms in math content lessons.

A third strategy for teaching math vocabulary is using visuals (Prediger et al., 2022; Sanford et al., 2020). 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 (Stevens et al., 2023). See Figure 3 for an adapted version of a commonly used graphic organizer called the “Frayer model.”

Figure 3

Frayer model

Finally, teachers should provide students with repeated exposures to math-vocabulary terms (McKeown et al., 2018). 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.

Provide Word-Problem Strategy Instruction

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 (NCES, 2024; Powell et al., 2022).

Why is this important in middle school math?

In the United States, most high-stakes math test items are word problems (Powell et al., 2022). 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 (NCES, 2024; Wei et al., 2013). 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 (Fuchs et al., 2015; Kingsdorf & Krawec, 2014; Peake et al., 2015; Wang et al., 2016). Underlying difficulties related to working memory and language may exacerbate these challenges (Peng et al., 2020).

Recommended strategies

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 (Powell et al., 2022). 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?”

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 (Freeman-Green et al., 2015; Jitendra & Star, 2012). 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 (Polya, 2015). 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 (Montague et al., 2011).

Figure 4

Word-problem attack strategies

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” (Cox & Root, 2020; Griffin et al., 2019). 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 (Fuchs et al., 2021; Peltier et al., 2020). Schema instruction has a robust evidence base, including for culturally and linguistically diverse students with MD (Driver & Powell, 2017; Freeman-Green et al., 2021). 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 (Griffin et al., 2019; Jitendra & Star, 2012).

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.

Figure 5

Word problems by schema

Importantly, in middle school, word problems regularly involve multiple steps or multiple schemas, for example,

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?

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.

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.

Conclusion

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 (Fuchs et al., 2021). Thus, these instructional practices can—and should be—adapted to the complex math concepts taught in the middle school grades.

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; Powell et al., 2021a; Root et al., 2023). 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 (Hughes et al., 2017; Root et al., 2023).

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Danielle O. Lariviere

Katie B. MacLean

Jessica Mao

Sarah R. Powell

Danielle O. Lariviere, Katie B. MacLean, Jessica Mao, and Sarah R. Powell, Department of Special Education, The University of Texas at Austin, Austin, TX, USA

References

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.

Baroody

A. J.

Bajwa

N. P.

Eiland

(2009). Why can’t Johnny remember the basic facts? Developmental Disabilities Research Reviews, 15(1), 69–79. https://doi.org/10.1002/ddrr.45

Bouck

E. C.

Shurr

Bassette

Park

Whorley

(2018). Adding it up: Comparing concrete and app-based manipulatives to support students with disabilities with adding fractions. Journal of Special Education Technology, 33(3), 194–206. https://doi.org/10.1177/0162643418759341

Bouck

E. C.

Shurr

Park

(2020). Virtual manipulative-based intervention package to teach multiplication and division to secondary students with developmental disabilities. Focus on Autism and Other Developmental Disabilities, 35(4), 195–207. https://doi.org/10.1177/1088357620943499

Butler

F. M.

Miller

S. P.

Crehan

Babbitt

Pierce

(2003). Fraction instruction for students with mathematics disabilities: Comparing two teaching sequences. Learning Disabilities Research and Practice, 18(2), 99–111. https://doi.org/10.1111/1540-5826.00066

Cirino

P. T.

Tolar

T. D.

Fuchs

L. S.

Huston-Warren

(2016). Cognitive and numerosity predictors of mathematical skills in middle school. Journal of Experimental Child Psychology, 145, 95–119. https://doi.org/10.1016/j.jecp.2015.12.010

Clarke

Nelson

N. J.

Ketterlin Geller

Kosty

Smolkowski

Lesner

Furjanic

Fien

(2022). Investigating the promise of a tier 2 sixth-grade fractions intervention. Learning Disability Quarterly, 45(5), 252–266. https://doi.org/10.1177/0731948720972411

Clarke

Nelson

Shanley

(2016). Mathematics fluency—More than the weekly timed test. In Cummings

K. C.

Petscher

(Eds.), The fluency construct: Curriculum–based measurement concepts and applications (pp. 67–89). Springer. https://doi.org/10.1007/978-1-4939-2803-3_3

Cox

S. K.

Root

J. R.

(2020). Modified schema-based instruction to develop flexible mathematics problem-solving strategies for students with autism spectrum disorder. Remedial and Special Education, 41(3), 139–151. https://doi.org/10.1177/0741932518792660

10.

Driver

M. K.

Powell

S. R.

(2017). Culturally and linguistically responsive schema intervention: Improving word problem solving for English language learners with mathematics difficulty. Learning Disability Quarterly, 40(1), 41–53. https://doi.org/10.1177/0731948716646730

11.

Forsyth

S. R.

Powell

S. R.

(2017). Differences in the mathematics-vocabulary knowledge of fifth-grade students with and without learning difficulties. Learning Disabilities Research and Practice, 32(4), 231–245. https://doi.org/10.1111/ldrp.12144

12.

Freeman-Green

Driver

M. K.

Wang

Kamuru

Jackson

(2021). Culturally sustaining practices in content area instruction for CLD students with learning disabilities. Learning Disabilities Research and Practice, 36(1), 12–25. https://doi.org/10.1111/ldrp.12240

13.

Freeman-Green

S. M.

O’Brien

Wood

C. L.

Hitt

S. B.

(2015). Effects of the SOLVE strategy on the mathematical problem solving skills of secondary students with learning disabilities. Learning Disabilities Research and Practice, 30(2), 76–90. https://doi.org/10.1111/ldrp.12054

14.

Fuchs

L. S.

Fuchs

Compton

D. L.

Hamlett

C. L.

Wang

A. Y.

(2015). Is word-problem solving a form of text comprehension? Scientific Studies of Reading, 19(3), 204–223. https://doi.org/10.1080/10888438.2015.1005745

15.

Fuchs

L. S.

Newman-Gonchar

Schumacher

Dougherty

Bucka

Karp

K. S.

Woodward

Clarke

Jordan

N. C.

Gersten

Jayanthi

Keating

Morgan

(2021). Assisting students struggling with mathematics: Intervention in the elementary grades. National Center for Education Evaluation and Regional Assistance (NCEE), Institute of Education Sciences, U.S. Department of Education. https://ies.ed.gov/ncee/wwc/PracticeGuide/26

16.

Gersten

Beckmann

Clarke

Foegen

Marsh

Star

J. R.

Witzel

(2009). Assisting students struggling with mathematics: Response to Intervention (RtI) for elementary and middle schools (NCEE 2009-4060). National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. http://ies.ed.gov/ncee/wwc/publications/practiceguides/

17.

Griffin

C. C.

Gagnon

J. C.

Jossi

M. H.

Ulrich

T. G.

Myers

J. A.

(2019). Priming mathematics word problem structures in a rural elementary classroom. Rural Special Education Quarterly, 37(3), 150–163. https://doi.org/10.1177/8756870418772164

18.

Hughes

C. A.

Morris

J. R.

Therrien

W. J.

Benson

S. K.

(2017). Explicit instruction: Historical and contemporary contexts. Learning Disabilities Research and Practice, 32(3), 140–148. https://doi.org/10.1111/ldrp.12142

19.

Jao

(2012). The multicultural mathematics classroom: Culturally aware teaching through cooperative learning & multiple representations. Multicultural Education, 19(3), 2–10.

20.

Jitendra

A. K.

Dupuis

D. N.

Star

J. R.

Rodriguez

M. C.

(2016). The effects of schema-based instruction on the proportional thinking of students with mathematics difficulties with and without reading difficulties. Journal of Learning Disabilities, 49(4), 354–367. https://doi.org/10.1177/0022219414554228

21.

Jitendra

A. K.

Nelson

Pulles

S. M.

Kiss

A. J.

Houseworth

(2016). Is mathematical representation of problems an evidence-based strategy for students with mathematics difficulties? Exceptional Children, 83(1), 8–25. https://doi.org/10.1177/0014402915625062

22.

Jitendra

A. K.

Star

J. R.

(2012). An exploratory study contrasting high- and low-achieving students’ percent word problem solving. Learning and Individual Differences, 22, 151–158. https://doi.org/10.1016/j.lindif.2011.11.003

23.

Ketterlin-Geller

L. R.

Powell

S. R.

Chard

D. J.

Perry

(2019). Teaching mathematics in middle school: Using MTSS to meet all students’ needs. Paul H. Brookes Publishing Company.

24.

Kingsdorf

Krawec

(2014). Error analysis of mathematical word problem solving across students with and without learning disabilities. Learning Disabilities Research and Practice, 29(2), 66–74. https://doi.org/10.1111/ldrp.12029

25.

Lariviere

D. O.

Arsenault

T. L.

Payne

S. B.

(2025). A literature review: Mathematics-vocabulary interventions for students with mathematics difficulty. School Science and Mathematics, 125(1), 6–17. https://doi.org/10.1111/ssm.12684

26.

Lariviere

D. O.

Powell

S. R.

Akther

S. S.

(2024). A synthesis of prealgebraic reasoning interventions for students with mathematics difficulty in grades 6 through 8. Learning Disabilities Research and Practice, 39(1), 4–17. https://doi.org/10.1177/09388982231222179

27.

Lin

Peng

Zeng

(2021). Understanding the relation between mathematics vocabulary and mathematics performance: A meta-analysis. The Elementary School Journal, 121(3), 504–540. https://doi.org/0013-5984/2021/12103-0007

28.

Marita

Hord

(2017). Review of mathematics interventions for secondary students with learning disabilities. Learning Disability Quarterly, 40(1), 29–40. https://doi.org/10.1177/0731948716657495

29.

McKeown

M. G.

Crosson

A. C.

Moore

D. W.

Beck

I. L.

(2018). Word knowledge and comprehension effects of an academic vocabulary intervention for middle school students. American Educational Research Journal, 55(3), 572–616. https://doi.org/10.3102/0002831217744181

30.

Monroe

E. E.

Panchyshyn

(1995). Vocabulary considerations for teaching mathematics. Childhood Education, 72, 80–83. https://doi.org/10.1080/00094056.1996.10521849

31.

Montague

Enders

Dietz

(2011). Effects of cognitive strategy instruction on math problem solving of middle school students with learning disabilities. Learning Disability Quarterly, 34(4), 252–272. https://doi.org/10.1177/071948711421762

32.

Moser Opitz

Freesemann

Prediger

Grob

Matull

Hußmann

. (2017). Remediation for students with mathematics difficulties: An intervention study in middle schools. Journal of Learning Disabilities, 50(6), 724–736. https://doi.org/10.1177/0022219416668323

33.

Moyer-Packenham

P. S.

Westenskow

(2013). Effects of virtual manipulatives on student achievement and mathematics learning. International Journal of Virtual and Personal Learning Environments, 4, 35–50. http://doi.org/10.4018/jvple.2013070103

34.

Namkung

J. M.

Bricko

(2021). The effects of algebraic equation solving intervention for students with mathematics learning difficulties. Journal of Learning Disabilities, 54(2), 111–123. https://doi.org/10.1177/0022219420930814

35.

National Center for Education Statistics. (2024). The nation’s report card. U.S. Department of Education, Institute of Education Sciences. https://www.nationsreportcard.gov/

36.

National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Mathematics standards. Common Core State Standards Initiative. https://www.corestandards.org/math

37.

National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. U.S. Department of Education.

38.

Nelson

P. M.

Parker

D. C.

Zaslofsky

A. F.

(2016). The relative value of growth in math fact skills across late elementary and middle school. Assessment for Effective Intervention, 41(3), 184–192. https://doi.org/10.1177/1534508416634613

39.

Orosco

M. J.

(2014). Word problem strategy for Latino English language learners at risk for math disabilities. Learning Disability Quarterly, 37(1), 45–53. https://doi.org/10.1177/0731948713504206

40.

Peake

Jiménez

J. E.

Rodríguez

Bisschop

Villarroel

(2015). Syntactic awareness and arithmetic word problem solving in children with and without learning disabilities. Journal of Learning Disabilities, 48(6), 593–601. https://doi.org/10.1177/0022219413520183

41.

Peltier

Sinclair

T. E.

Pulos

J. M.

Suk

(2020). Effects of schema-based instruction on immediate, generalized, and combined structured word problems. The Journal of Special Education, 54(2), 101–112. https://doi.org/10.1177/0022466919883397

42.

Peng

Lin

(2019). The relation between mathematics vocabulary and mathematics performance among fourth graders. Learning and Individual Differences, 69, 11–21. https://doi.org/10.1016/j.lindif.2018.11.006

43.

Peng

Lin

Ünal

Z. E.

Lee

Namkung

Chow

Sales

(2020). Examining the mutual relations between language and mathematics: A meta-analysis. Psychological Bulletin, 146(7), 595–634. https://doi.org/10.1037/bul0000231

44.

Polya

(2015). How to solve it: A new aspect of mathematical method. Princeton University Press.

45.

Powell

S. R.

Mason

E. N.

Bos

S. E.

Hirt

Ketterlin-Geller

L. R.

Lembke

E. S.

(2021a). A systematic review of mathematics interventions for middle–school students experiencing mathematics difficulty. Learning Disabilities Research and Practice, 36(4), 295–329. https://doi.org/10.1111/ldrp.12263

46.

Powell

S. R.

Bos

S. E.

Lin

(2021b). The assessment of mathematics vocabulary in the elementary and middle school grades. In Fritz-Stratmann

Gürsoy

Herzog

(Eds.), Diversity dimensions in mathematics and language learning (pp. 313–330). De Grutyer.

47.

Powell

S. R.

Fuchs

L. S.

Cirino

P. T.

Fuchs

Compton

D. L.

Changas

P. C.

(2015). Effects of a multitier support system on calculation, word problem, and prealgebraic performance among at-risk learners. Exceptional Children, 81(4), 443–470. https://doi.org/10.1177/0014402914563702

48.

Powell

S. R.

Namkung

J. M.

Lin

(2022). An investigation of using keywords to solve word problems. The Elementary School Journal, 122(3), 452–473. https://doi.org/10.1086/717888

49.

Powell

S. R.

Stevens

E. A.

Hughes

E. M.

(2019). Math language in middle school: Be more specific. TEACHING Exceptional Children, 51(4), 286–295. https://doi.org/10.1177/0040059918808762

50.

Prediger

Erath

Weinert

Quabeck

(2022). Only for multilingual students at risk? Cluster-randomized trial on language-responsive mathematics instruction. Journal for Research in Mathematics Education, 53(4), 255–276. https://doi.org/10.5951/jresematheduc-2020-0193

51.

Riccomini

P. J.

Stocker

J. D.

Morano

(2017). Implementing an effective mathematics fact fluency practice activity. TEACHING Exceptional Children, 49(5), 318–327. https://doi.org/10.1177/0040059916685053

52.

Rodrigues

Locke

S. L.

Singell

Mirielli

L. G.

(2024). Teaching fraction magnitude using the number line. Intervention in School and Clinic, 59(3), 165–172. https://doi.org/10.1177/10534512231156885

53.

Root

Saunders

Jimenez

Gilley

(2023). Essential components for math instruction: Considerations for students with extensive support needs. TEACHING Exceptional Children, 56(1), 34–43. https://doi.org/10.1177/00400599221120882

54.

Rubenstein

R. N.

Thompson

D. R.

(2002). Understanding and supporting children’s mathematical vocabulary development. Teaching Children Mathematics, 9(2), 107–112. https://doi.org/10.5951/TCM.9.2.0107

55.

Sanford

A. K.

Pinkney

C. J.

Brown

J. E.

Elliott

C. G.

Rotert

E. N.

Sennott

S. C.

(2020). Culturally and linguistically responsive mathematics instruction for English learners in multitiered support systems: PLUSS enhancements. Learning Disability Quarterly, 43(2), 101–114. https://doi.org/10.1177/0731948719836173

56.

Siegler

Carpenter

Fennell

Geary

Lewis

Okamoto

Thompson

Wray

(2010). Developing effective fractions instruction for kindergarten through 8th grade: A practice guide (NCEE 2010-4039). National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. https://ies.ed.gov/ncee/wwc/docs/practiceguide/fractions_pg_093010.pdf

57.

Spooner

Root

J. R.

Saunders

A. F.

Browder

D. M.

(2019). An updated evidence-based practice review on teaching math to students with moderate and severe developmental disabilities. Remedial and Special Education, 40(3), 150–165. https://doi.org/10.1177/0741932517751055

58.

Star

J. R.

Caronongan

Foegen

Furgeson

Keating

Larson

M. R.

Lyskawa

McCallum

W. G.

Porath

Zbiek

R. M.

(2015). Teaching strategies for improving algebra knowledge in middle and high school students (NCEE 2014-4333). National Center for Education Evaluation and Regional Assistance (NCEE), Institute of Education Sciences, U.S. Department of Education. https://ies.ed.gov/ncee/wwc/docs/practiceguide/wwc_algebra_040715.pdf

59.

Stevens

E. A.

Leroux

A. J.

Mowbray

M. H.

Lee

G. S.

(2023). Evaluating the effects of adding explicit vocabulary instruction to a word-problem schema intervention. Exceptional Children, 89(3), 275–293. https://doi.org/10.1177/00144029221112290

60.

Strickland

T. K.

Maccini

(2013). The effects of the concrete–representational–abstract integration strategy on the ability of students with learning disabilities to multiply linear expressions within area problems. Remedial and Special Education, 34(3), 142–153. https://doi.org/10.1177/0741932512441712

61.

Sutherland

Furjanic

Hermida

Clarke

(2024). Using the number line to develop understanding of whole number magnitude and operations. Intervention in School and Clinic, 59(3), 158–164. https://doi.org/10.1177/10534512231156869

62.

Wakefield

E. M.

Congdon

E. L.

Novack

M. A.

Goldin-Meadow

James

K. H.

(2019). Learning math by hand: The neural effects of gesture-based instruction in 8-year-old children. Attention, Perception, & Psychophysics, 81, 2343–2353. https://doi.org/10.3758/s13414-019-01755-y

63.

Wang

A. Y.

Fuchs

L. S.

Fuchs

(2016). Cognitive and linguistic predictors of mathematical word problems with and without irrelevant information. Learning and Individual Differences, 52, 79–87. https://doi.org/10.1016/j.lindif.2016.10.015

64.

Wei

Lenz

K. B.

Blackorby

(2013). Math growth trajectories of students with disabilities: Disability category, gender, racial, and socioeconomic status differences from ages 7 to 17. Remedial and Special Education, 34(3), 154–165. https://doi.org/10.1177/0741932512448253

65.

Woodward

Beckmann

Driscoll

Franke

Herzig

Jitendra

Koedinger

K. R.

Ogbuehi

(2018). Improving mathematical problem solving in grades 4 through 8: A practice guide (NCEE 2012-4055). National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. https://ies.ed.gov/ncee/wwc/Docs/PracticeGuide/MPS_PG_043012.pdf

66.

Wright

T. S.

Cervetti

G. N.

(2017). A systematic review of the research on vocabulary instruction that impacts text comprehension. Reading Research Quarterly, 52(2), 203–226. https://doi.org/10.1002/rrq.163

67.

Xin

Y. P.

Zhang

(2009). Exploring a conceptual model-based approach to teaching situated word problems. The Journal of Educational Research, 102(6), 427–442. https://doi.org/10.3200/JOER.102.6.427-442