Abstract
Increasing the performance of secondary students with learning disabilities (LD) in mathematics on word problem-solving tasks involving ratios and proportions is challenging for secondary math teachers. Teachers must use evidence-based practices to enhance secondary students’ problem-solving proficiency and math achievement. Schema-based instruction (SBI) is a powerful evidence-based practice to improve students’ understanding and approach to solving word problems involving ratios and proportions. This article describes the protocol for implementing SBI with examples specific to ratios and proportions. Also discussed are the ways teachers can provide additional assistance to students with more intensive instructional needs when implementing SBI.
Ms Rosales teaches in an inclusive eighth-grade mathematics classroom. Her students with learning disabilities (LD) in mathematics persistently struggle to accurately solve problems with ratios and proportions, particularly with word problems. She noticed that her students with LD in mathematics experienced challenges in the following two main areas when solving ratios and proportions word problems: (a) creating diagrams to represent a given problem and (b) following the steps to solve the problems. Ms. Rosales is unsure how to help her students overcome these barriers, but she is committed to using evidence-based practices to increase their word-problem performance. 1
The challenges Ms Rosales’ students displayed in solving ratios and proportions word problems are not atypical for secondary students with LD in mathematics. These difficulties are well-documented. Recent data from the National Assessment of Educational Progress (NAEP) mathematics assessment, which requires students to use proportional reasoning to solve a problem in context, show that 38% of eighth-grade students without disabilities scored at or above the proficient level, while 25% scored below basic (National Center for Education Statistics [NCES], 2020). However, the NAEP performance for students with disabilities, most of whom were students with LD, was comparably worse. Only about 9% scored at or above proficient, whereas 68% scored below basic (NCES, 2020). Recent research on word problems involving proportional reasoning shows that secondary students with LD in mathematics consistently showed smaller gains than their peers without disabilities in mathematics (e.g., Jitendra et al., 2017). Improving the performance of secondary students with LD in mathematics on ratios and proportions word problems is critical. Knowledge of solving ratios and proportions is essential for developing an understanding of algebra and crucial for (a) learning more advanced mathematics concepts, (b) trigonometry, (c) calculus, and (d) success in college and career outcomes (Moore & Shulock, 2010).
Challenges in Solving Ratios and Proportions Word Problems
There are several reasons for the performance challenges faced by secondary students with LD in mathematics in solving ratios and proportions word problems. Students with LD in mathematics have deficiencies in essential cognitive processes, such as working memory, attentional function, and information processing (Fuchs et al., 2020a). These deficiencies interfere with a student’s ability to perform essential problem-solving tasks, such as retrieving basic facts, selecting efficient mathematical strategies, and applying cognitive techniques to monitor and check their work (Fuchs et al., 2020a). They may also inhibit proficiency in using proportional reasoning, selecting suitable solution strategies, and checking the appropriateness of their answers (Jitendra et al., 2017). Some students with LD in mathematics have limited proficiency in essential mathematical foundational skills for understanding ratios and proportions, such as computations with rational numbers (National Mathematics Advisory Panel, 2008).
Many secondary students with LD in mathematics also experience reading challenges that interfere with their ratios and proportions word-problem performance. These students often face challenges involving reading comprehension and vocabulary, which inhibit their ability to decode and understand words in a text to extract essential information (Fuchs et al., 2020b; Verschaffel et al., 2020). As a result, in solving word problems, they face difficulties formulating visual representations, generating mathematical equations, and selecting the most appropriate algorithm(s) to obtain the correct answer (Peterson et al., 2017). To help secondary students with LD in mathematics improve their knowledge of and proficiency in solving ratios and proportions word problems, teachers must provide targeted instruction using evidence-based practices (Alghamdi et al., 2020).
Research of Evidence-Based Practices for Ratios and Proportions Word Problems
Researchers have conducted studies (Lein et al., 2020) to identify evidence-based practices for increasing the word-problem performance of students with LD in mathematics. Findings showed that schema-based instruction (SBI) is one of the most effective strategies for improving students’ understanding and performance in solving mathematical word problems (Lein et al., 2020; Myers et al., 2021). Importantly, research supports the efficacy of SBI for increasing the knowledge and performance of secondary students with LD in mathematics on ratios and proportions word problems (Jitendra et al., 2017).
Ms Rosales studied research-based strategies for word-problem solving and concluded that SBI strategies were the most appropriate for addressing her students’ specific problem-solving difficulties. Now, she needs to understand the components of SBI, details for implementation, and potential implementation challenges.
Schema-Based Instruction
Schema-based instruction is an evidence-based practice that assists students in solving word problems by identifying the underlying structure of the problem (i.e., schema) and using this information to select an appropriate visual representation, known as a schematic (Marshall, 1995). Strategies that help students with LD in mathematics generate schemas of ratios and proportions word problems are critical. Identifying these underlying structures helps students overcome cognitive deficits (e.g., working memory), understand structural relations among quantities, and select appropriate solution strategies (Fuchs et al., 2021; Xin et al., 2011). Schema-based instruction involves a sequential, multi-component instructional strategy that guides students through a systematic process to monitor and reflect on the problem-solving process (Jitendra et al., 2009). Furthermore, it emphasizes creating a visual representation of a word problem that highlights its mathematical quantities and relationships (Jitendra et al., 2009). Using SBI, teachers help students: (a) first read the problem, (b) apply metacognitive strategies to extract critical information from it, (c) select a schematic diagram for the problem, (d) use their previous knowledge of relevant content and experiences to determine the appropriateness of the selected schema, and (e) provide an estimated solution to the problem (Jitendra et al., 2009). In the final stage, students also translate their visual representations into a mathematical sentence, choose a proper solution strategy, and check the appropriateness of their answers (Jitendra et al., 2017).
Schema-based instruction provides explicit instruction in using self-monitoring plans, such as mnemonic device checklists, to assist students in applying each step of the problem-solving process sequentially, which reduces demands on their working memory (Jitendra et al., 2009). The FOPS checklist shown in Figure 1 is a commonly used mnemonic device for SBI, and has been used with students who display mathematical challenges, including secondary students with LD in mathematics (Jitendra et al., 2017). The FOPS strategy includes a systematic and sequential four-step process:
FOPS instructional checklist.
Importantly, to increase students’ efficiency in implementing the problem-solving process and increase solution accuracy, teachers are encouraged to explicitly model instruction using the checklist and demonstrate the application of cognitive strategies (e.g., think-aloud) to guide the problem-solving process.
Implementing Schema–Based Instruction With Ratios and Proportions
In the following example, the teacher explicitly models the implementation of SBI using the FOPS checklist to guide students through solving a word–problem involving ratios and proportions. Mr Browne sold a total of 30,000 eggs produced at his farm this year. If he sold 25,000 eggs the previous year, what is the percentage change in the number of eggs sold from last year to this year?
Step 1: Find the Problem Type
After pre-teaching the different schemas, the teacher thoroughly and carefully reads the word problem to understand its requirements. In the process, the teacher identifies key mathematical terms (e.g., percentage and change) that will provide context clues in solving the problem. However, the teacher should not overemphasize keywords (e.g., sum, total) in identifying the problem type. Focusing primarily on keywords can cause students to overgeneralize solution strategies and commit methodological errors, resulting in incorrect answers (Powell & Fuchs, 2018). For example, in the sample question, the students may focus on the word “total,” leading them to add the two values to arrive at a solution. Therefore, the teacher must underscore the need for students to read the problem to understand its meaning and structure comprehensively and use keywords to recognize the context of the problem. The teacher models how to summarize or restate the problem to comprehend it at this stage. For example, the teacher can translate the problem as “Someone sold 30,000 eggs this year and sold 25,000 last year. Was there an increase or decrease in the number of eggs sold? What was the percentage increase or decrease in the number of eggs?”
After demonstrating how to understand the problem, the teacher concludes that the problem includes a two-step sequence, which involves the combination of two schemas, including a multiplicative (see Figure 2, Panel A) and an additive (see Figure 2, Panel B). The teacher confirms that the problem uses an additive schema to find the difference (i.e., change) in the number of eggs between the 2 years and then a multiplicative schema to divide this value by the previous years’ total to determine the percentage change in the number of eggs. The teacher also explains that the change can represent an increase or a decrease. At this point, the teacher would have extracted all the relevant information to derive an estimated solution for the problem.
FOPS (Step 1): Find the problem type: Combination of a multiplicative schema (Panel A) and an additive schema (see Panel B).
Step 2: Organize the Information Using Appropriate Schema Diagram
At this stage, the teacher demonstrates that the problem consists of two sequential steps involving a combination of a multiplicative and an additive schema. Next, the teacher models the selection of the appropriate schematic diagrams and uses them to organize the information before creating a plan to solve the problem. In solving the problem, the teacher first recognizes that the number of eggs is an unknown quantity (i.e., represented by a letter). Next, the teacher shows students to apply their knowledge of number operations and number sense with the additive schema to calculate the “change in the number of eggs” (see Figure 3, Panel B).
FOPS (Step 2): Organize information for the multiplicative (Panel A) and additive (Panel B) schemas.
Once the teacher has determined the value representing the change, the teacher shows students how to apply their prior knowledge of fractions to organize the multiplicative schematic diagram (see Figure 3, Panel A). Here, the teacher demonstrates how to analyze math terminology and language to understand the problem’s underlying structure and organize the given details. The teacher acknowledges that the “change in the number of eggs” represents the numerator, and the word “previous” means original, suggesting that the denominator should be the last year’s total. Finally, the teacher emphasizes that percentage change is the unknown quantity (i.e., represented by a letter) expressed out of 100. This information is used to create an open mathematical sentence (i.e., includes at least one unknown) that represents the problem. Finally, the teacher shows students that they can further use the additive schema (see Figure 3, Panel B) later to check the appropriateness of their solutions.
Step 3: Plan to Solve the Problem
At Step 3, the teacher shows that the change in the given problem represents an increase rather than a decrease in the number of eggs since the total for the previous year was smaller than the current year’s total. This information shows how to use these data to create a series of mathematical sentences to solve the problem. First, the teacher makes a mathematical sentence to calculate the change by finding the difference between the two totals (see Figure 4, Equation 1). The second mathematical sentence (see Figure 4, Equation 2) is used to illustrate that the problem can be written as equivalent fractions where the change divided by the original number (x/25,000) equals the percentage change expressed as a fraction (y/100). Finally, the teacher shows how the third mathematical sentence can be used for checking the answer in the solution at the end (see Figure 4, Equation 3).
FOPS (Step 3): Plan to solve the problem.
Step 4: Solve the Problem and Check Solution
In Step 4, the teacher demonstrates how to attain the solution and evaluate its appropriateness. The teacher shows that the second mathematical sentence (see Figure 4, Equation 2), which will be used to solve the problem, has two unknown values, reaffirming that the situation requires the following two sequential steps to obtain a solution: (a) solve the first equation in Figure 4 to obtain x, the change, and (b) substitute the x value into the second mathematical sentence in Figure 4 to complete the equation to solve for the second unknown quantity, the percentage change (y). The teacher solves Equation 1 in Figure 4 to calculate x by subtracting the previous year’s total eggs from the current year’s total (i.e., 30,000−25,000 = 5,000). The teacher then substitutes 5,000 for x in Equation 2 to obtain equivalent fractions (i.e., 5,000/25,000 = y/100). The teacher tells students the importance of simplifying fractions (i.e., reducing computational complexity) and proceeds to show that the fraction
To obtain the solution, the teacher models the application of a suitable algorithm. The teacher recognizes that the solution requires solving equivalent fractions and demonstrates two different methods for getting the answer. First, the teacher uses a cross-multiplication strategy, where the fractions are cross-multiplied to create an algebraic expression (see Figure 5, Panel A). The teacher solves the resulting equation (100 = 5y) by dividing both sides by 5 to obtain y = 20. Using the Equivalent fraction approach (see Figure 5, Panel B), the teacher shows how to solve the problem by identifying a number to multiply the denominator of the left side by (i.e., 20) that will make it equal to the denominator on the right side (i.e., 100) and then multiply the numerator on the left side by the same number. In the example, the teacher multiples the fraction on the left by 20 to obtain 20/100 = y/100, which produces a solution y = 20. Although it is helpful for teachers to encourage students to use various means to solve the problem, they must ensure that approaches used include logical and mathematically appropriate steps and not suboptimal strategies, such as using tricks, checking, and guessing.
FOPS (Step 4): Solve for the unknown quantity and check the solution’s appropriateness.
The final step in FOPS requires teachers to model how to evaluate their solution’s correctness and compose a complete sentence to answer the question. For the given word problem, the percentage change is 20%, which is appropriate since the sum of its corresponding value representing the change (5,000) and the original total (25,000 eggs) equals the new total given in the problem, 30,000 (see Figure 5, Panel C). Finally, the teacher constructs a complete sentence to express their answer: “The percentage change in the number of eggs sold from last year to this year is 20%.”
Potential Roadblocks and Solutions
The SBI strategy is a practical and effective strategy with a plethora of research support. Of course, research does not always translate smoothly into the classroom, especially for students with LD in mathematics. To prepare for such difficulties in research to practice, teachers should consider some potential roadblocks and possible solutions, including reading problems and issues with computation.
Reading Challenges
Some students with LD in mathematics struggle intensely with reading comprehension (Verschaffel et al., 2020). Seeing a long paragraph with multiple sentences embedded with numbers and symbols may intimidate the student. To help with the reading comprehension, consider using a paraphrasing strategy that requires the students to read and retell the story problem. Yet, the teacher may have to provide direct reading support to students with reading difficulties who need more intensive instructional supports to comprehend text. Hence, teachers may paraphrase word problems for students who need assistance, especially when problems contain multiple steps.
Furthermore, in pairs or with the teacher, the student clarifies what is happening in the problem. The teacher can ask clarifying questions to help guide the student (e.g., What is the problem about? Was there an increase or decrease in the number of eggs from last year to this year? What operation would you need to use to find the change in the number of eggs sold?). Using guiding and probing questions will help the student identify vital information and the most effective solution strategy. In addition, if the student is too intimidated by a long story problem, the teacher may consider presenting the problem with a line for each sentence. The student can then paraphrase each sentence separately. Significantly, however, teachers should reduce the support level as students become more proficient, increasing their opportunities for independent practice (Powell & Fuchs, 2018).
Computational Challenges
Many students with LD in mathematics face challenges with computation involving rational numbers. In some cases, students will understand and set up the problem correctly but cannot perform even single–digit calculations, adding to their stress and making multiple–step problems seem unbearable. Teachers may consider a few approaches when supporting students who are still developing proficiency with computation. A proportional reasoning problem does not require decimals or large numbers. Hence, one approach is to reduce the computational complexity of word problems. The same mathematical strategy could be applied with 30 eggs rather than 30,000. Another approach is to provide students with computational support tools, such as multiplication charts or calculators. However, when doing so, it is crucial to encourage students to use these tools only to perform calculations that they have not memorized or mastered. This is easier with a multiplication chart because the known problems could be covered or deleted as students gain proficiency throughout the academic year.
Conclusion
Improving the understanding and capacity to apply word-problem strategies of students with LD in mathematics in ratios and proportions are critical in increasing their overall math proficiency and achievement, and by extension, their access to positive post-secondary outcomes. Encouragingly, SBI has strong empirical support for enhancing students’ proficiency in solving ratios and proportions word problems. Yet, in assisting students in using SBI, teachers are strongly encouraged to use explicit instruction to guide students through problem-solving. They must also modify their instruction to meet the needs of students who require more intensive instructional supports, such as students with reading difficulties and those with low computational fluency.
Ms Rosales used SBI to intensify her math instruction and noticed that her students with LD in mathematics improved their performance and confidence in solving ratios and proportions word problems. Remarkably, she noticed that her students became more proficient in identifying the schema type, creating a suitable visual representation, and creating and implementing an appropriate plan to solve word problems. She will continue to provide opportunities for independent practice, monitor her students’ progress, and adjust her instructional strategy based on their performance.
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.