Intensive Word Problem Solving for Students With Learning Disabilities in Mathematics

Challenges Faced by Students With Learning Disabilities in Math

Elementary school students’ ability to solve word problems is critical to their success in advanced mathematics (Gersten et al., 2009) and workforce preparation (Uhalde et al., 2006). Despite the importance, solving word problems remains one of the most difficult tasks to master for students, especially those with LD in mathematics.

There are several reasons why students with LD in mathematics face significant challenges in word problem solving. First, word problems require these students to use cognitive skills to extract essential information and connect this information to an appropriate solution plan to generate an answer (Verschaffel et al., 2020). However, students with LD in mathematics often display cognitive deficits, such as poor working memory and delayed processing speed (Witzel & Mize, 2018). Some word problems require students to use multiple steps and critical thinking to eliminate extraneous information (Verschaffel et al., 2020). However, many of these students lack operational fluency in basic operations and metacognitive skills needed to select and apply appropriate strategies (Van Luit & Toll, 2018). In addition, to solve word problems successfully, students cannot merely rely on their computation facility. Before they can solve the problem, they must decode the complex vocabulary and comprehend expository or narrative text structures (Pongsakdi et al., 2020). The reading difficulties that many students with LD in mathematics experience limit their ability to accurately set up the mathematical equation needed for solving the word problem. Compounding this issue, word problems are often presented several grade levels beyond students’ reading levels (Witzel, 2009).

Students with LD in mathematics require effective core and supplemental instruction using evidence-based practices to improve their word problem performance (Alghamdi et al., 2020; Powell et al., 2020). Based on the findings from meta-analyses (Lein et al., 2020; Peltier & Vannest, 2017), there are key instructional strategies and methods that teachers and schools should use to improve the word-problem-solving proficiency of elementary school students with LD in mathematics.

Empirically Validated Word-Problem-Solving Approaches

To help elementary students with LD in mathematics solve word problems, educators can apply key strategies and methods such as direct instruction with guided practice, strategy instruction, schema-based instruction (SBI), model-based problem solving (MBPS), and computer-assisted instruction (CAI). Notably, these instructional approaches include multicomponents, comprising various high-leverage practices. This article highlights MBPS.

Schema-based instruction engages students in semantic analysis of word problems to develop schematic diagrams (Marshall, 1995) that are specific to different problem types (e.g., change, group, and compare), which aids language processing often associated with students with LD. An extension of SBI is an intervention approach categorized as MBPS. With MBPS, students engage in problem solving that is driven by mathematical models. Model-based problem solving differs from SBI in that it uses a unified algebraic model equation to work across problem types, whereas SBI uses a unique schematic diagram for each different problem type.

One common feature across both models is the use of explicit instruction components, such that each includes modeling, scaffolding, guided and independent practice, and corrective feedback. Every effective word problem intervention reviewed in the meta-analytic work was delivered through an explicit instruction model rather than an inquiry-based approach. At this point, “the extant research on inquiry-based mathematics instruction for students with math learning disabilities is not sufficient to support a recommendation for its use in classrooms” (Krawec & Steinberg, 2019, p. 32).

Another instructional component common across these strategies is the use of visual representations. Visual representations help students with LD see and interact with relevant information within mathematics word problems (Fuchs et al., 2021). Both interventions involve visual representations of problem structure, which helps build the visuospatial working memory of students with LD. Connecting problem structure to visual representations improves the word problem performance of students with LD (Xin, 2012).

Model-Based Problem Solving

Conceptual MBPS is a highly effective word-problem-solving approach for students with LD in mathematics (Lein et al., 2020). Model-based problem solving extends research in SBI. In SBI, students are taught to find the problem type, organize the information in a semantic diagram, apply rules to create a mathematical sentence to solve it, and then check the practicality of the solution obtained (Jitendra et al., 2013). In contrast, MBPS promotes generalized problem solving by using a unified algebraic mathematical model equation. For instance, the part–part–whole model equation (P + P = W) represents a range of addition and subtraction word problems that connect different problem types (e.g., change, combine, and compare problems). Equal-group model equations (i.e., unit value × number of units = product) are used to represent a range of multiplication and division word problems that connect different problem types (e.g., rate × quantity, fair share, measurement division, and compare problems). For sample MBPS lessons and examples, see the free-access book by Xin (2012). By facilitating flexible problem solving, this strategy potentially helps with transitions to abstract reasoning and mathematics equation development.

Empirical evidence demonstrates that when elementary students with LD in mathematics are explicitly taught to use SBI or MBPS, their word-problem performance increases. In their empirical synthesis of the literature, Lein et al. (2020) found that studies implementing SBI or MBPS had the most considerable effect on these students’ word-problem performance. Furthermore, meta-analytic findings (Peltier & Vannest, 2017) support the efficacy of SBI and MBPS for increasing the word-problem accuracy of elementary students with LD in mathematics. Multiple types of visual problem-solving approaches are possible using MBPS or SBI strategies.

For the following problem, a teacher helps students with LD in mathematics use MBPS to make sense of the problem and solve it using the DOTS mnemonic strategy (i.e., Detect, Organize, Translate, Solve) validated for upper elementary grades (Xin, 2012).

At The Donut Shop, glazed donuts cost forty-five cents each and muffins cost one dollar twenty-five cents each. Eduard wants to buy five donuts and three muffins for his family. How much money does he need?

The DOTS checklist guides students through the following steps: Detect the problem type, Organize the information using the model diagram, Translate the diagram into a meaningful math equation, and Solve the unknown quantity in the equation and check the answer. The significance of this approach is that it requires the student to understand the problem and more importantly to represent the problem in a mathematical model equation. It is the model equation that drives the solution plan for an accurate answer.

Detect the problem type

The student needs to read the entire word problem story, not just circle the cue words, and focus on the mathematical relationships within the story. It is suggested that the teacher ask the student to retell the story using his or her own words to ensure comprehension of the story. In the case of solving above sample problem, the student understands that there are two different “equal groups” of items with different unit prices, and Eduard, from the story, needs to combine the subtotals from two groups of items to solve the problem. As demonstrated in Figure 1, the student shows the understanding that Eduard is buying five donuts and three muffins and that donuts and muffins cost different amounts. The visual shows an increasing cost based on the number of items bought, such that the total cost = 5 ($0.45) + 3 ($1.25). Similar bar-model-based visuals are used across research to help students with reading needs by providing a visuospatial model of the word problem (Jerman, 2010).

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Figure 1.

Model-based problem solving: bar-model-based visual for comprehension.

The student concludes that this is an equal-group problem containing two groups, muffins and donuts, and a part–part–whole problem in that combining the two subtotals will solve for the total amount of money he would need for the purchase. Figure 2 shows that the student will first use the equal-group model equation (see Figure 2, upper panel) to solve for the subtotal for the cost of five donuts and the subtotal for the cost of three muffins. The student then will use the part–part–whole model equation (see Figure 2, lower panel) to solve the problem.

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Figure 2.

Model-based problem solving: Detect the problem type—Mathematical model visual.

Organize using the model diagram

As shown in Figure 3, equal groups for multiplication use a visual that shows both the number of groups and the number per group (unit price in this case) needed for determining the subtotal. The part–part–whole model shows that combining the two parts (subtotal P1 and P2) will get the total cost. The student organizes the information from the problem into a diagram that helps the student translate the word problem into the mathematical sentence.

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Figure 3.

Model-based problem solving: Organize the information using the model diagram—Visual.

Translate into a meaningful math equation

As this is a two-step problem, in Step 1 the student will “peel off” the bubbles or boxes and rewrite the mathematical equations involving the appropriate numbers for each group of donuts and muffins, separately, to determine the subtotal cost for each set. In Step 2, the student will input the two subtotals solved from Step 1 in the part–part–whole model equation and solve for the grand total for the final answer (see Figure 4).

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Figure 4.

Model-based problem solving: Translate the diagram into a meaningful math equation and Solve for the unknown quantity and check the answer.

The final step in DOTS also involves checking the answer. For this problem, $2.25 + $3.75 = $6.00. This makes sense considering that the cost will increase more than either the cost of the donuts or muffins individually. Therefore, the total amount of money needed by Eduard is $6.00.

To this point, one might wonder if using the bar model only (see Figure 1) will help students solve the problem. For solving simple problems such as the one presented previously, where the product is the unknown, the bar model might be enough for some students to figure out the answer. However, when the students are presented with problems where the unit price (i.e., or number per group) is unknown or the “number of groups” is unknown (i.e., or in general, one of the factors is unknown in the equal-group model equation), then representing the problem in the model equation will help students accurately solve the problem without struggling to decide what operation to apply. The model equation reveals what operation to use for solution. For instance, if one of the factors (i.e., in the “factor [number of units] × factor [unit] = product” model) is unknown, the student will be guided to “undo” the multiplication, through division, to determine the unknown quantity for the answer. With MBPS, students will be able to solve a range of multiplicative or additive word problems regardless of whether the beginning amount is unknown (e.g., missing factor or missing addend) or the product or ending amount is unknown.

Algebraically, missing factors and addends are revealed through student representation of word problems where a question mark or a letter is used to represent the unknown quantity in the equation. For example, the student is asked to solve the following problem: “At The Donut Shop, Eduard paid $2.25 for five donuts. What was the cost of each donut?” The problem diagram would be different (see Figure 5).

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Figure 5.

Model-based problem solving: Algebraic model equation where a is the unknown quantity (the cost per donut).

Solve for the unknown quantity

Students would represent the information in the equal-group diagram equation and then computationally solve for the unknown quantity in the equation. Finally, students check the reasonableness of their answer. By having the student analyze the problem, diagram the problem, and then develop a mathematical equation before computing an answer, students engage in a systematic approach to solving word problems. Otherwise, students with LD or executive-functioning disorder may impulsively select the numbers in the word problem and guess what operation solves the problems.

Conclusion

Several word problem intervention strategies and methods show research effectiveness: (a) direct instruction with guided practice, (b) strategy instruction, (c) SBI, (d) MBPS, and (e) CAI. However, it is up to the teacher to determine which to infuse in their instruction based on the intensity needs of the students. For more advanced mathematical reasoning and problem solving, both SBI and MBPS are beneficial to promote students’ conceptual understanding of mathematical ideas while advancing student strategy use from visual representations to abstract reasoning.

Each model has strong evidentiary support for helping students with LD in mathematics solve word problems. Schema-based instruction attends to specific word problem schemata to help students understand the word problem story structure. Model-based problem solving not only helps students understand the word problem structure through word-problem-specific grammar (Xin et al., 2008) but also advances students’ understanding from comprehending the word problem at the semantic level in the context of the real world to the mathematical world where students engage in representing the word problem in a mathematical model equation for generalized problem solving.