Diagramming and Algebraic Word Problem Solving for Secondary Students With Learning Disabilities

Abstract

Algebra is considered an important high school course because it is recognized as the gateway to higher mathematics, college opportunities, and well-paying jobs. In the United States, most secondary schools require students to be proficient in algebra to be able to graduate from high school. One major component of algebra is word problem solving, which is used in algebra courses to teach students mathematical modeling and applied problem-solving skills. However, word problem solving is often a significantly challenging area for students with learning disabilities because it involves computing mathematical equations and implementing a myriad of cognitive processes that require conceptual knowledge. Diagrams are considered an effective and powerful visualization strategy because they help students see the hidden mathematical structure of the problem. The use of diagrams is recommended as students work toward more complex math concepts in middle school and high school.

Keywords

algebra word problem solving diagrams students with learning disabilities

All students in the United States are required to be taught using rigorous academic standards, such as the Common Core State Standards (CCSS; Every Student Succeeds Act, 2015), to meet high school graduation requirements, succeed in postsecondary education, and effectively compete globally. These standards have been designed to increase student academic performance in content areas necessary for college and career success (Common Core State Standards Initiative [CCSSI], 2010). In mathematics, the CCSS emphasize the importance of students not only acquiring a conceptual understanding of problem solving, but also developing the ability to apply recent knowledge to current world situations (Freeman-Green, O’Brien, Wood, & Hitt, 2015). Even more, the CCSS are meant to provide a rigorous mathematical framework for all students with enough flexibility to accommodate students’ future college interests (Wasserman. 2011). The CCSS set grade-specific goals, but do not specify which curriculum, teaching method, or materials should be used to support student learning (Khaliqi, 2016), especially in relation to students with disabilities. Within the CCSS, algebra and algebraic thinking are embedded throughout the K–12 continuum (CCSSI, 2010) as this type of thinking typifies the cognitive complexity required to solve critical problems and engage in college- and career-ready curricula (Morgan et al., 2014).

Algebra is considered an important high school course because it is recognized as the gateway to higher mathematics, college opportunities, and well-paying jobs (Stein, Kaufman, Sherman, & Hillen, 2011; Watt, Watkins, & Abbitt, 2016). In the United States, most secondary schools require students to be proficient in algebra to graduate from high school (Maccini & Rulh, 2000; Watt et al., 2016). Students in algebra courses need opportunities to reason abstractly, make sense of problems, and use graphs, tables, and diagrams to model their thinking (Gavin & Sheffield, 2015) and garner both a deeper understanding of concepts and procedures and the ability to generalize high-level mathematical content using algebraic reasoning (Gavin & Sheffield, 2015; Maccini & Hughes, 2000).

Problem Solving Skills

In the United States, 4.5% of the student population is identified as having a learning disability (LD; U.S. Department of Education, 2016). Generally, students with LD face the challenge of not only correctly identifying strategies and solving problems in algebra classrooms, but also selecting information, monitoring accuracy, and generalizing to other tasks and content areas (Cuenca-Carlino, Freeman-Green, Stephenson, & Hauth, 2016; Maccini & Ruhl, 2000; Xin, Jitendra, & Deatline-Buchman, 2005). Moreover, they often have long-term memory deficits that may affect their ability to store and retrieve information (Cuenca-Carlino et al., 2016). In sum, students with LD struggle not only with basic computation skills, but also with higher-level mathematic skills (Xin et al., 2005).

In most cases, students with LD start having difficulties during elementary school and continue to show mathematical deficits throughout middle and high school (Freeman-Green et al., 2015; Maccini & Ruhl, 2000). As an example, the math academic performance of a 17-year-old student with LD is similar to the expected performance of a 10-year-old student without a disability (Kortering, DeBettencourt, & Braziel, 2005). A national longitudinal transition survey on the academic performance of secondary students in reading and mathematics indicated that 74% of students with LD performed below average on math calculation and 85% of students with LD performed below average on math applied problems (National Center for Learning Disabilities, 2014).

Algebra for Students With Learning Disabilities

Solving word problems is significantly challenging for students with LD (Freeman-Green et al., 2015), but word problem solving is an integral skill covered in algebra courses to target mathematical modeling and applied problem solving (Jitendra et al., 2009). Even though research has demonstrated that all students benefit from taking high school algebra regardless of their math achievement level (Gamoran & Hannigan, 2000), special education teachers often lack enough preparation and knowledge to fulfill the demands of cognitively complex algebra curricula (Foegen, 2008). The Individuals With Disabilities Education Act (IDEA) emphasizes the use of evidence-based practices (EBPs) to support the learning and achievement of students with LD (IDEA, 2006). Effective EBPs for instruction, assessment, progress monitoring, and individualized instruction are vital for the success of students with LD in algebra. Therefore, the education and training of teachers on how to integrate EBPs into their instruction related to college- and career-ready standards is critical to the delivery of more rigorous and efficient mathematical instruction, especially to students with LD (Watt et al., 2016).

Benefits of Using Diagrams

A meta-analysis of the mathematics intervention literature conducted between 1971 and 2007 (Gersten et al., 2009) for students with disabilities addressed effective instructional components to improve mathematical academic performance (e.g., explicit instruction, visual representations, positive feedback). In addition, studies on word problem solving interventions for students with LD (Jitendra et al., 2015; Montague & Dietz, 2009) indicated that visual representations (e.g., graphs, tables, diagrams) are a successful approach to improve the problem-solving skills of students with LD. Moreover, visual representations can enhance students’ conceptual knowledge in problem solving by emphasizing the semantic structure of problems (Jitendra, 2002). In essence, visual representations are mathematical tools that help students with LD clarify their ideas and support their reasoning and understanding (Gersten et al., 2009).

Diagrams are considered a strong visualization strategy, not only because they are used to sort out the structure of a problem, but also because they have the flexibility to be used and implemented on different problem-solving processes and mathematical areas (van Garderen, 2007). Unlike mental images alone, diagrams (see Figure 1) can be easily monitored, evaluated, and improved upon (van Garderen, 2006). Even more, experts recommend the use of diagrams (e.g., Venn diagrams, matrices, map diagrams) as students work toward more complex mathematical concepts in middle and high school grades (van Garderen & Scheuermann, 2015). According to Novick (2004), matrices are a type of diagram that effectively helps students solve algebraic word problems by illustrating abstract concepts and ideas.

Figure 1.

Example of different types of diagrams.

Even though diagrams have been recommended to effectively support the word problem solving skills of secondary students with LD, more teacher guidance on how to implement diagrams in algebra courses is needed. Research suggests that students with LD can be explicitly taught to use diagrams or schemas to solve word problems (Jitendra et al., 2009) by (a) understanding the meaning of the problem, (b) transforming the problem into a relevant representation that emphasizes its numeric features, and (c) developing understanding of the important relationships among individual statements (van Garderen, 2006). Misunderstanding the word problem usually leads to an error in the solution of the problem (Hinsley, Hayes, & Simon, 1977).

Diagram strategies focus on helping students visualize the hidden mathematical structure of the problem (Jitendra et al., 2009). Teachers should consider three main instructional activities to integrate the diagramming process as a support for students with LD: (a) teach students to create and interpret the diagram, (b) support students in using the diagrams to solve problems and check solutions to answers, and (c) teach students metacognitive strategies focused on identifying critical components of algebraic word problems to ensure mastery of the diagramming process.

Creating and Interpreting Diagrams

To successfully help students create and understand diagrams, teachers should incorporate principles of explicit instruction, such as modeling, scaffolding, guided and independent practice, rehearsal, and feedback (Montague & Dietz, 2009; van Garderen, 2007). When teaching diagrams, teachers should first teach (a) the definition of a diagram, (b) reasons to use a diagram in Algebra, specifically word problems, (c) general rules when generating diagrams, such as time management and relevant information, (d) symbols and codes needed to understand the problem, such as letters or numbers, and (e) different types of diagrams (e.g., hierarchies, matrices, networks) to represent relationships embedded in the word problem (Novick, Hurley, & Francis, 1999; van Garderen, 2006; van Garderen & Scheuermann, 2015).

Importantly, students need to understand the unique and distinguishing properties of each type of diagram to use them correctly (van Garderen, Scheuermann, & Jackson, 2012). A matrix is a diagram composed by two or more labeled rows and two or more labeled columns, where cells hold a specific value (Novick, 2004). In Algebra, matrices are especially useful because students have used this type of diagram before (e.g., multiplication tables) and can transfer their representational knowledge to new situations (Novick et al., 1999).

Table 1 shows a sequence of critical steps teachers should follow to effectively teach their students how to build a matrix to solve algebraic word problems. First, the teacher should carefully read aloud the word problem written on the board. Figure 2 shows an algebraic word problem example involving distance, rate, and time, and its corresponding diagram. Next, the teacher should explicitly teach students how to recognize and underline relevant information stated in the problem (e.g., a numeric value like 28 miles per hour). In addition, the teacher should teach how to recognize the problem question, and identify the missing variable (e.g., distance). Specifically, students need to understand what type of problem they have to solve, and what are the variables involved in the problem (e.g., distance = d, rate = r, and time = t). Later, the teacher should demonstrate how many rows and columns are needed to create the matrix (e.g., two rows because there is information about Mary and Laura, three columns because there are three variables). Then, the teacher should demonstrate to the students how to complete the matrix with the known information stated in the problem.

Table 1.

Steps to Create a Diagram to Effectively Solve Algebraic Word Problems.

Step	Description
1	Carefully read the algebraic word problem
2	Underline the relevant information stated in the word problem
3	Determine what the word problem is asking you to do
4	Choose the formula to find the missing variable
5	Determine how many variables the formula contains
6	Define your variables (e.g., t = time, d = distance)
7	Create a matrix with as many columns as variables (including the missing variable)
8	Write the formula at the top of the matrix, placing each variable in a different column
9	Determine how many equations you can build with the information in the problem (e.g., you have the information of 2 different people)
10	Create the rows in your matrix depending on the different equations you will build
11	Start completing your matrix with the known variables stated in the word problem
12	Solve the formula for the problem
13	Write the solution of your formula in the missing variable column in terms of the other variables
14	Solve the equations according to your problem

Figure 2.

Example of diagram of an algebraic word problem involving distance, time, and rate.

The previous step is vital for the accurate solution of the problem; therefore, teachers need to explicitly teach how to make connections with the information in the problem to support students’ understanding of the problem (e.g., left 1 hour after; drove 20 mph faster than). It is very important to remember that students cannot start solving the problem until the whole matrix is filled out, and all cells have their corresponding values. Once the matrix is complete, students can start solving the algebraic equation. As a final suggestion, teachers should promote cooperative learning and peer tutoring to promote students’ spontaneous use of diagrams (Uesaka & Manalo, 2007). Moreover, research by Kortering et al. (2005) suggested that students with LD in mathematics are more likely to succeed in algebra courses if they work with other people (e.g., peers, individual assistance, group work) because they receive help to access specific and general course content information.

Using Diagramming to Check for Accuracy

Once the diagramming process has been taught, teachers should use the generated diagram as a part of the reasoning process students should engage in when checking the accuracy of their solved problem. During this process, teachers should teach their students to (a) understand that diagrams generate consistent answers when the completed information is accurate, (b) compare their diagrams against the word problem, and (c) use diagrams to track the solution of the problem, avoiding misunderstandings in the problem solution process (van Garderen, 2006). In this case, the missing variable is distance, and all the information provided by the word problem is in terms of the time variable. It is important to remember that before students start solving the algebraic equation, they have to understand what the problem is asking them to do.

Incorporating Metacognitive Strategies

Jitendra et al. (2009) included a mnemonic in a schema-based strategy to support student learning as a think-aloud described as (a) find the problem type, (b) organize information using the diagram, (c) plan to solve the problem, and (d) solve the problem (FOPS). Teachers should encourage the use of think-alouds to monitor and direct students’ problem-solving skills development on problem comprehension, representation, planning, and solution (Jitendra et al., 2009). Figure 3 shows the corresponding think-aloud for the previous algebraic word problem involving distance, rate, and time. Teachers must help their students become reflective thinkers by being aware of their thought processes (Lochhead & Whimbey, 1987). At the end of the lesson, during independent practice, teachers should gradually reduce instructional support regarding the generation of diagrams but continue to support learning on understanding word problem structures and connections between quantitative components in a word problem (van Garderen, 2006).

Figure 3.

Example of a student “think aloud” of an algebraic word problem involving distance, time, and rate.

Implementing Diagrams in Algebra Courses

Research suggests that teachers should not wait until students master computational skills to expose them to word problem solving opportunities (Jitendra, 2002). Frequently, diagrams are included in math lessons as an end tool in the problem-solving process. However, diagrams can play an important role during the entire instructional process (van Garderen, 2007). To improve students’ mathematical performance on word problem solving, teachers need to carefully choose well written word problems that will help their students develop their problem-solving skills and also integrate targeted mathematical concepts and skills (Jitendra, 2002). In addition, teachers should incorporate multiple similar word problems in their algebra lessons to facilitate students’ identification of essential components of the problem (Jitendra, 2002). Even more, research recommends culturally inclusive word problems that reflect students’ everyday situations to promote generalization (Novick, 2004). Importantly, students with LD require explicit and systematic instruction with more practice, time, and scaffolding to understand and develop diagrams and conceptual learning (Jitendra et al., 2009).

To help students with LD effectively and strategically use diagrams to solve word problems with accuracy, teachers should first teach diagram use (e.g., why, what, when, how) using explicit and systematic instruction, and then incorporate the use of metacognitive strategies such as think-alouds in their lessons (van Garderen & Scheuermann, 2015). Students tend not to use diagrams spontaneously (Uesaka & Manalo, 2012). That is why, once students know how to create and use a diagram, teachers should promote students’ spontaneous generation and employment of diagrams during the problem-solving process (Uesaka & Manalo, 2012). For example, Uesaka and Manalo (2007) suggested peer tutoring and collaborative learning as a very effective method to promote the use of diagrams in word problem solving. In other words, teachers should give students the opportunity to explain to their peers how they created and interpreted a diagram to solve the word problem.

Conclusion

As the demands of a highly cognitive mathematical coursework increase in the United States, effective mathematical instructional components are key to support students’ academic development and achievement. Diagrams can help students with LD understand the algebraic relationships of quantitative components in a word problem by providing a structure of the critical components of the word problem. Even though it may take students multiple attempts to accurately generate and use diagrams, the results of utilizing diagrams to solve algebraic word problems are important to help students with LD meet the algebra requirement needed not only to graduate from high school, but also to apply to college.

Footnotes

Declaration of Conflicting Interests

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

Funding

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

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