Abstract
Mrs. Liddik, a general education teacher, is struggling to support students with disabilities in her Algebra I class. At the end of the first 9-week marking period, it is clear that although students sometimes compute correct solutions, they often struggle with reasoning and explaining their work. The students’ difficulty with reasoning is even more apparent when Mrs. Liddik’s class completes a few questions that require explanations, similar to the constructed response items that students will encounter on the end-of-course assessment. Mrs. Liddik suspects the struggling students are able to solve the problems accurately, but their explanations are not logical or reasoned and do not demonstrate an understanding of the process, even though she provided multiple model demonstrations with think-alouds during initial instruction and before assigning practice. Mrs. Liddik is surprised to find that several of her higher-performing students also experience trouble with reasoning and explaining. After careful analysis of all of the students’ responses, it is clear that many of her students are probably memorizing procedural steps and have an incomplete understanding of why and how the problem-solving procedures work. Mrs. Liddik requests a meeting with the mathematics coach and her special education co-teacher to identify strategies to help students develop a stronger understanding of the processes used to solve algebra problems (i.e., algebraic reasoning). During the meeting, the special education teacher mentions that she recently saw a tweet from @IESResearch about a strategy that uses solved problems to help students improve their algebraic reasoning. She explains that IES, which stands for the Institute of Education Science, is the nation’s engine for independent educational research, evaluation, and statistics and publishes practice guides for educators on research-supported practices. She locates the document on the IES web site ( https://ies.ed.gov/ ). The recommendation focuses on teaching students to reason and explain problems that are already solved. Mrs. Liddik is intrigued and eager to learn more about how to use solved problems in her algebra class.
Students with learning disabilities (LD) struggle in many aspects of mathematics, but one especially problematic area is algebra (Hughes, Witzel, Riccomini, Fries, & Kanyongo, 2014). Recent developments in mathematics standards have amplified challenges in algebra for students with disabilities and their teachers by emphasizing the need for students to demonstrate deeper understanding of algebraic principles. Rather than stress learning and application of procedures and algorithms to accurately solve problems, current standards and goals focus on understanding, explaining, and making connections within and across the procedures (Linewand et al., 2014). Tasks that require understanding, explaining, and making connections fall under the umbrella of algebraic reasoning.
The challenges students with LD encounter during mathematics learning are substantial and well-documented (Cirino, Fletcher, Ewing-Cobbs, Barnes, & Fuchs, 2007). In addition, challenges begin early and persist throughout students’ school years (Morgan, Farkas, & Wu, 2011). In effect, deficits in mathematics content learning compound as students progress toward secondary-level mathematics courses, such as algebra and geometry. Mathematics learning gaps for students with LD span many areas, including basic facts, subtraction, fractions, word problem solving, and algebra (Clarke et al., 2016; Hughes et al., 2014). Each of these deficit areas covers important prerequisite skills for algebra, so it is no surprise students with LD struggle with developing algebraic reasoning. In addition, many students with LD experience low motivation and confidence in algebra as a result of the content challenges and the lack of mathematics success they have experienced previously (Kortering, de Bettencourt, & Braziel, 2005). The combination of multiple content deficits, an ongoing lack of success, and low motivation complicates the challenges faced by students with LD in developing algebraic reasoning and necessitates more intensive instructional supports for this group of students.
Thus, many students with LD enrolled in algebra courses will require significantly different instructionally designed supports that are more intensive and engaging to access the abstract content associated with algebra (National Center on Intensive Intervention, 2013). Although there are multiple ways for teachers to redesign instruction, scaffolding is one high-leverage practice (HLP; McLeskey et al., 2017) that can be used effectively to intensify instruction to provide access to more advanced mathematics courses. Using worked solutions represents a significant change in typical instructional design and is a promising instructional scaffold that is responsive to the needs of students and designed to support the development of algebraic reasoning.
Scaffolding Through Worked Solutions
A growing body of research is demonstrating that using worked solutions (also referred to as worked examples) as an instructional scaffold can engage students and improve outcomes (e.g., Booth et al., 2015; Clark, Nguyen, & Sweller, 2011; Sweller & Cooper, 1985). Worked solutions are a prime example of a scaffold because they can be designed to provide decreasing levels of support as students become more competent in solving complex, multistep algebraic problems. By definition, scaffolded supports “provide temporary assistance to students so they can complete tasks that they cannot yet do independently and with a high rate of success” (McLesky et al., 2017, p. 23). When used as a scaffolded support, worked solutions are matched carefully to students’ current understanding of learning objectives and are systematically faded to include fewer supports as students learn to solve problems independently with procedural accuracy and understanding (i.e., demonstrate understanding of how and why problem-solving processes work).
Worked solutions show a solved problem with visual prompts (e.g., text, arrows, color coding or highlighting) that provide information about the problem-solving process. Figure 1 contains a side-by-side comparison of a typical practice structure with one that has embedded worked solutions in the practice problems. In this example, worked solutions are provided in an alternating format as to provide students an opportunity to review the worked example and then solve a similar problem. The worked solution provides a scaffold, or a supported practice opportunity, which is an essential element of explicit instruction (see Hughes, Riccomini, & Morris, 2019).
Comparison of typical and worked-solutions-embedded practice structures
In Figure 1, the worked solutions are provided by the teacher and align with the instruction provided in class. The level of detail in each solution is based on learner characteristics. Note that the set of problems provided is a subset of a practice activity that would generally include eight to 12 problems depending on grade level and learners.
The provided solutions illustrate and describe, step-by-step, what the student should do to solve the problem and, in some cases, may include notes or annotations explaining the rationale for using a particular strategy or step. Prompts are positioned next to the problem on the paper or screen. Figure 2 illustrates a worked solution scaffold applied to problems involving parallel lines cut by transversals. In this example, students are learning to apply their knowledge of angle relationships (i.e., adjacent angles and corresponding angles) to set up algebraic equations used to solve for unknown variables. The teacher provides the solution and the corresponding explanation in a side-by-side structure to support the connection between the algebraic manipulations and the angle relationships (how and why). After the students have reviewed the solution steps and explanations, they should have an opportunity to solve a similar problem with no solution provided. The process of fading the number of steps or amount of information provided to students within worked problems decreases students’ dependence on the prompts. Fading of supports continues until students are able to solve problems accurately with minimal guidance.
A worked solution accompanied by an explanation of each solution step
In Figure 2, both the solution steps and the explanation are provided to the students. The student reviews the solution steps and the corresponding explanation. This problem is then followed by a similar problem with no solution steps or explanations. Note that the explanations should align with the classroom instruction provided.
Using worked examples during instructional activities or independent practice supports students in developing algebraic reasoning and deeper understanding of procedures (Pashler et al., 2007; Star et al., 2015). Classroom discussions focused on worked examples promote understanding by focusing students’ attention on the problem-solving processes rather than on application of a set of memorized procedures. In effect, analyzing worked examples scaffolds the development of algebraic reasoning more so than typical practice that involves solving multiple problems independently with no support or feedback.
After reading some articles, Mrs. Liddik is beginning to recognize the importance of scaffolding students’ learning through the use of worked solutions but does not fully understand why the process works. She is a bit confused about why providing students with a worked problem to study followed by a similar problem to solve helps students develop algebraic reasoning and deepen their understanding of how and why problem-solving processes work. She remembers her own mathematics teachers assigning her large numbers of problems to practice and believes that she developed algebraic reasoning through that repetition.
Why Does Studying Solutions Improve Algebraic Reasoning?
Theorists posit that the use of worked solutions during instructional activities and practice opportunities can improve student outcomes by accommodating working memory limitations and supporting critical thinking through self-explanation (Sweller & Cooper, 1985; Sweller, Merrienboer, & Paas, 1998; Zhu & Simon, 1987). First, working memory has a limited capacity for the number of units of information that an individual can hold in mind and use simultaneously. When that capacity is reached, learning stops or drastically slows down. Providing worked solutions supports learning by reducing the working memory load students experience during practice (Sweller et al., 1998). Algebraic reasoning can lead to working memory overload because it requires processing many pieces of information simultaneously. Worked examples can help lighten the working memory burden associated with solving algebra problems by removing responsibility for computing and by providing a clear visual model of the problem-solving steps and solution. In other words, providing worked examples to students with LD may reduce the cognitive load they experience during initial problem-solving practice.
A second explanation for the effectiveness of worked solutions comes from the research on self-explanation (Aleven & Koedinger, 2002; Trafton & Reiser, 1993). Self-explaining refers to the ability to explain a problem and solution to oneself, and extensive research shows that self-explaining supports learning (Siegler, 2002). Giving high-quality explanations to operate on (i.e., supported self-explaining; Fuchs et al., 2015) is helpful for students with LD because they often struggle to independently generate accurate self-explanations. Worked examples, and especially those with annotations, provide prompts and opportunities for supported self-explaining that students may not get otherwise (Atkinson, Renkl, & Merrill, 2003; Carroll, 1994; Mevarech & Kramarski, 2003). The opportunity to self-explain without responsibility for solving is also important for students with LD who often struggle to solve accurately. The synergistic effects of reduced cognitive load and increased capacity for self-explanation make the use of worked examples an important consideration for all students but especially for students with LD.
Mrs. Liddik now connects her past study routines to the worked solution strategy. For example, when studying for math tests in college, she would often study a worked example from her textbook, then cover it up and try to solve it. She now sees that she was using self-explanation to better understand the problem-solving process. Mrs. Liddik is very excited to use worked solutions and starts brainstorming about how to incorporate the strategy into instruction.
How to Use Worked Solutions in Instruction and Practice
Using solved problems promotes algebraic reasoning by shifting students’ focus from finding an answer to reasoning about and explaining the problem-solving process. Focusing on the process draws attention to important relationships and structures within and across problems and can help students develop deeper and more flexible understanding. Effectively introducing worked solutions into instruction is completed in three main phases. First, teach students how to use worked solutions strategically. Next, introduce worked solutions into instruction. Use worked solutions frequently and set the expectation that students will use worked solutions as a problem-solving tool. Finally, embed worked solutions in independent practice materials. Table 1 provides checklists for implementing each phase of this process. In addition, an extended example is provided to illustrate how to turn a traditional worksheet into a worked solution practice sheet (see Figures 1, 2, and 3).
Implementation Checklists for Phases 1, 2 and 3
Sample teacher model with think-aloud and guided questions for whole-class discussion
Phase 1: Teach students to use worked solutions strategically
It is essential to devote instructional time to teaching students how to use worked solutions strategically. When students are not taught why and how to use worked solutions, they are more likely to skip solutions altogether or only identify the how (i.e., procedural steps) without considering the why. Teachers can explain that they are introducing the worked solution strategy because it will help students learn to solve various types of complex, multistep problems. Teachers can also share with students that the strategy is effective because it reduces cognitive load and promotes self-explaining. In addition to explaining why the strategy works, teachers can increase student interest by giving the strategy a name individualized to their classroom or school.
After explaining why worked solutions are useful, teachers should begin to describe how to use the strategy through modeling with think-aloud. See Figure 3 for an example script a teacher might use when modeling how to study a solved problem. Provide several models while thinking aloud to describe and explain the reasoning behind each step in a solution. When modeling, it is also helpful to use self-questioning to ask whether a given solution is efficient, logical, and mathematically correct. Provide several models before involving students in discussion. This will help the students see the importance of identifying the how and why when using worked examples.
When evaluating steps in a solution process, the how will generally be a procedural step or a computation. For example, in solving the equation 4x + 5 = 17; the first step in the solution is subtracting 5 from both sides. Subtraction is one step in how the problem is solved. The why is the reasoning behind subtracting 5 from both sides of the equation. The why, or the reasoning teachers and students use in solving any given problem, should align with the instruction that was provided in class.
When first introducing how to use the worked solution strategy in class, it is important to use problems that are familiar to students. Using familiar content will allow students to focus on using the strategy, whereas using unfamiliar content will cause students to focus on the problem and distract from learning the strategy. In addition, it is important to select problems that require multistep solutions rather than one-step problems. For example, basic integer problems (e.g., –5 × 6) do not require multistep solutions and are not a good fit for this strategy. In contrast, order-of-operations problems involving all operations and grouping symbols are excellent problems to use for teaching students to use solutions.
After modeling proper use of worked solutions, teachers should provide guided practice opportunities. Guided practice is a vital step in building student independence with the strategy. In Phase 1, teachers should continue to use familiar content during guided practice so students can focus on learning the strategy (versus learning new content). Guided practice can occur in multiple formats, including during whole-group instruction, small-group instruction, or partner or group activities. See Figure 3 for examples of guided questions that a teacher might pose during practice to support student reasoning while studying a solved problem. During guided practice, remind students to identify the how and why as they study the solution, and monitor students carefully to ensure that they do not skip these important steps. Posting prompts on the board, such as “Tell me how the problem was solved? Tell me why the problem was solved?” is an excellent support to help students learn the process of using worked solutions. Before moving on to Phase 2, teachers should verify that all students are using solutions properly and are accurately identifying the how and why of problem-solving processes. Any student who is not correctly using the strategy should receive additional modeling and guided practice.
Ms. Liddik decides to introduce worked solutions during whole-class instruction using familiar problems. She decides to use order-of-operations problems because they require multistep solutions and because her students mastered solving them several weeks ago. She is concerned with the amount of class time it may take to teach students how to use worked solutions, so she decides to take 10 to 15 minutes at the beginning of class across 10 days to teach the strategy. She will introduce the strategy and model how to use worked solutions across 3 or 4 days, then provide guided practice across the next 5 or 6 days. Although she has a plan for getting started, Mrs. Liddik is still wondering about how to use worked solutions in instruction once students are familiar with the strategy.
Phase 2: Incorporate worked solutions into instruction and set the expectation that students will use the strategy as a problem-solving tool
There are at least two ways to incorporate worked solutions into instruction to scaffold student development of algebraic reasoning. Worked solutions can be used during in-class activities and embedded in independent practice activities, such as homework. Providing frequent opportunities to use worked solutions in class with teacher guidance helps students become more proficient and confident in their ability to used work solutions independently. In order to ensure that students interact with the solutions in a purposeful manner during instruction, we recommend that teachers plan instructional activities with worked solutions that (a) use problems aligned with the instructional aim of the lesson, (b) provide students with frequent opportunities to discuss worked problems and solutions and use guided questioning to help focus students’ attention on important aspects of problems and solutions, and (c) use errors to highlight common problem-solving mistakes and misconceptions (Pashler et al., 2007).
First, teachers should select solved problems that match the lesson’s instructional aim. By carefully matching worked examples to lesson objectives, teachers can facilitate students’ ability to transfer their learning. Worked examples that align closely with objectives can be pulled from current or past student work, borrowed from the text or other curricular materials, or created from scratch. Using several solved problems that have a similar structure and that can be solved using the same strategy can help students recognize patterns and apply effective solution strategies. Teachers can display analogous problems simultaneously and use guided questioning to prompt students to notice similarities.
In addition, instruction using worked examples should provide students with frequent opportunities to discuss solutions and should use guided questioning to focus students’ attention on important features of worked problems and solutions. Teachers can prompt students to examine solution strategies by asking them to describe solution steps, to explain the reasoning behind each step, and to explain why the steps lead to a correct solution. As students become proficient in describing solution steps and the reasoning behind them, teachers can begin to pose questions to prompt deeper examination. For example, ask the following questions: “Why do the steps in this solved problem work in this order?” “Would the steps work in a different order?” Questions will vary with content, but the key is to ask questions that require students to reason through the solution. Questions can also help focus attention on structural features of problems.
Teachers can also help students to examine problem structure by asking guided questions focused on quantities (i.e., numbers and variables), relationships (e.g., equality and inequality, multiplicative), operations, and symbolic notation. Teaching students to identify the structure and components of an initial expression or equation will help them to make connections across problems, and teaching students to analyze the structure of each successive solution step will help them to recognize the problem-solving sequence and anticipate next steps. Here are several questions to help focus students on structural features: “What quantities can you identify in the problem?” “What operations does the problem involve?” and “What are the relationships between the quantities in the problem?” Teachers should tailor questions to draw attention to the fundamental attributes of the problem at hand. Table 2 provides additional questions to pair with worked solutions as well as a list of important verbs for students to think about when using worked solutions.
Potential Questions and Verbs for Students to Think About When Using a Worked Solution
After students practice using correctly solved problems, teachers can consider carefully introducing worked problems with errors to highlight and remediate common mistakes and misconceptions. In order to avoid confusion, incorrect solved problems should always be clearly labeled. Teachers can introduce this strategy by presenting a correctly solved problem side-by-side with the same problem with an error in the solution, and use prompting and guided questioning to help students discover and explain the error. After describing an error, teachers should support students in explaining why the error leads to an incorrect solution. Depending on the problem type and students’ needs, teachers can choose to highlight errors in reasoning, strategy use, or problem-solving procedures.
Teachers can create small-group activities using worked solutions in addition to using the strategy during whole-class instruction. For example, teachers can assign students to study a worked solution in pairs or small groups and assign a set of questions for partners or group members to ask each other or to answer together. After students have answered the questions in small groups or with their partner, the students can share their thinking and reasoning. Then, the teacher can present another problem with no solution for the partners or small groups to solve before answering the same set of questions.
Ms. Liddik decides to incorporate worked solutions into instruction during small-group work time every Tuesday and Thursday. Her current curriculum already requires students to work in small groups to solve problems several times a week, so embedding solved problems into small-group work will require only small adjustments to her plans. She decides to start by giving small groups a solution and a set of questions to answer. She will then have each group present their reasoning to the other students. After groups share, she will ask each group to solve a problem from start to finish with no solution steps provided. In addition to using worked solutions during instruction, Ms. Liddik also wants to find a way to include solved problems in independent practice activities and homework assignments.
Phase 3: Embed solutions in independent practice materials
After students gain confidence and competence with using worked solutions as a learning tool during instructional activities, teachers may consider embedding worked solutions in independent practice materials, such as homework, study guides, and review practice sheets. There are four considerations that should guide the creation of independent practice sheets with embedded worked solutions.
First, researchers have found that worked solutions are effective when presented in an alternating format with independent problems (Sweller & Cooper, 1985). If there are 10 practice problems on a worksheet, problems 1, 3, 5, 7, and 9 should include a solution, whereas problems, 2, 4, 6, 8, and 10 should require students to solve the problem. The first problem on a worked solution sheet should be solved so that students begin the worksheet by studying a solution.
The second design consideration involves the degree of “matching” between solved and unsolved problems. This may be the most important feature of worked solution worksheets for students with disabilities. The solution and subsequent problem must be similar in structure in order to allow students to use the solved problem as a guide for solving an independent problem. Researchers have not yet illuminated the most effective amount of variance between a worked solution and a matched independent problem, but given that students with disabilities have well-documented problems with generalization (Taylor, Smiley, & Richards, 2015), teachers should consider using closely matched problems in order to scaffold students’ ability to notice patterns among problems and solution strategies.
The third design consideration involves deciding how to annotate worked solution steps. Annotations can provide students with an additional scaffold. Annotations not only help students reason through a problem but also allow students to refer back to solution details as they work subsequent problems. Researchers have documented the effectiveness of embedded worked solutions with and without annotations (Ward & Sweller, 1990), so teachers should use their professional judgment and knowledge of their students’ strengths and weaknesses when deciding whether or not to include annotations.
The final design consideration for creating worked solution practice sheets involves whether and how to fade solutions. Research is inconclusive regarding the exact amount of support to provide within worked examples (Schwonke, Renkl, Salden, & Aleven, 2011), but fading the number of steps provided within solutions gradually increases student independence and appears to improve learning (Kalyuga, Chandler, & Sweller, 2001; Kalyuga, Chandler, Touvinen, & Sweller, 2001). Fading steps gradually releases responsibility to students and requires them to take on more responsibility in the problem-solving process as they work. When designing independent practice activities using worked examples, teachers should carefully consider students’ current level of understanding and make sheets that they are confident students can complete with a high level of accuracy.
After a few weeks of using worked solution practice during small-group work, Ms. Liddik feels confident that students have learned to use the strategy effectively and decides to embed solutions in homework assignments. As she continues to expand the use of worked solutions across instructional and practice activities, she notices improvements in her students’ reasoning skills and their ability to explain the why and the how of problem-solving processes.
Conclusion
Tasks that require reasoning and explaining in algebra and other advanced mathematics courses are often especially challenging for students with LD. Solving problems correctly is part of the equation, but it is equally important for students to understand and reason through how and why they must execute each step in a problem-solving process in order to develop algebraic reasoning and the ability to generalize problem-solving processes to novel problems. Given the difficulty posed by even basic algebra tasks for students with LD and the importance attributed to reasoning, understanding, and explaining solutions in practice and across mathematics standards and curricula, embedding worked solutions into instructional activities as well as independent practice materials offers a promising technique to scaffold students’ mathematical learning. Using worked solutions in mathematics instruction can reduce cognitive load (e.g., Carroll, 1994) and encourage self-explaining (e.g., Atkinson et al., 2003; Mevarech & Kramarski, 2003), making this method especially useful for students with LD.
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.