Vygotsky’s theory of cognitive development

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The systematic literature search, abstract screening process and full-text screening process resulted in five included articles. Data was extracted for the three research questions (1) What are the descriptions of MLD used in these studies? (2) What interventions have been studied for students with MLD in algebra? and (3) What were the outcomes on the algebraic learning of students with MLD?

Search results

After the abstract screening, 12 articles were included for review at the full-text level. During the ab-stract peer-reviewing process one additional article was found that was included for full-text review. After the full-text review process, the references of the included articles were hand searched for articles relevant to this study and four additional articles were found that were included for full-text review. This resulted in a total of 17 articles that were reviewed at the full-text level. During the full-text screening process arti-cles were excluded because (1) the article did not include classroom-based interventions, (2) the article did not include algebra and (3) it was a review.
Following the systematic review process, five studies were analyzed. Of the five studies two imple-mented single subject designs (Scheuermann, Deshler, & Jean, 2009; Strickland & Maccini, 2013), two used a random assignment of clusters design (Witzel, Mercer, & Miller, 2003; Witzel, 2005) and one used a two-group comparison experimental design (Ives, 2007). See Appendix D for study design and number of participants. The number of participants included in the studies varied significantly. The lowest number of students with disabilities in a study was three (Strickland & Maccini, 2013) and the highest number of stu-dents with disabilities was 34 (Witzel et al., 2003). The studies were also conducted in different settings and students were taught by either the researcher or the general education mathematics teacher (see Table 5).
Two studies were conducted in special education only classroom settings (Ives, 2007; Scheuermann et al., 2009), two studies were conducted in an inclusive classroom setting (Witzel et al., 2003; Witzel, 2005) and one was conducted in a separate classroom setting (i.e., a separate room the instructor used within the school) (Strickland & Maccini, 2013). In three of the studies the researcher was the instructor (Ives, 2007; Scheuermann et al., 2009; Strickland & Maccini, 2013) and in two of the studies the students were taught by the general education mathematics teacher (Witzel et al., 2003; Witzel, 2005).

Description of MLD

In each of the five included articles the researcher used different criteria to determine the eligibility of participants based on their description of MLD (see Table 6).
Most authors (Scheuermann et al., 2009; Strickland & Maccini, 2013; Witzel, Mercer, & Miller, 2003; Witzel, 2005) used the more general term learning disability instead of MLD. In addition to having an identified learning disability, three authors (Strickland & Maccini, 2013; Witzel et al., 2003; Witzel, 2005) required that the participants had mathematics goals indicated on their Individual Education Programs (IEPs). Scheuermann et al. (2009) did not explicitly require mathematics goals, but did require that stu-dents had a diagnosed learning disability, score below the 25th percentile in mathematics on a standardized test and earned below 50% on a pretest measure before being included in the study. The description used by Ives (2007) was the most unique in that language-related disabilities (reading, writing and/or general language) were used as the descriptor; there was no direct mention of mathematics.


After analyzing all of the included studies it was found that they all focused on the same algebra con-cept, solving equations. See Table 7 for an overview of the algebra content and interventions used. In the five included articles two interventions emerged: the concrete-representational-abstract (CRA) model and graphic organizers.

CRA model

Different variations of the CRA model were used in four (Scheuermann et al., 2009; Strickland & Maccini, 2013; Witzel et al., 2003; Witzel, 2005) of the five included articles. The CRA model is used to help students make a connection between conceptual and procedural mathematics knowledge (Agrawal & Morin, 2016). It provides a framework for teachers that allows them to teach abstract concepts while con-currently providing concrete and visual representations (Agrawal & Morin, 2016; Steele & Steele, 2015). Mathematical concepts are taught through three consecutive phases: first, the concrete phase; second, the representational phase and third, the abstract phase.
During the concrete phase of the CRA model students used physical objects to represent the terms of the equation and physically manipulated them to solve the equation (Scheuermann et al., 2009; Strickland & Maccini, 2013; Witzel et al., 2003; Witzel, 2005). Strickland and Maccini (2013) used algebra blocks and Scheuermann et al. (2009) used buttons and unifix cubes as the manipulative objects. In the studies by Witzel et al. (2003) and Witzel (2005) the representation of the variables was unique in that the coefficient and the variable were represented separately. Instead of using one colored block to represent 1x the same value was represented using one coefficient marker and one x marker (Witzel et al., 2003; Witzel, 2005) (see Figure 1).
In the representational phase, visual representations, in the form of drawings, other pictorial repre-sentations, graphic organizers or even virtual manipulatives, are used to help students move from the con-crete use of manipulatives to more abstract thinking (Agrawal & Morin, 2016). In the studies by Witzel et al. (2003) and Witzel (2005) the pictorial representations closely resembled the manipulatives used during the concrete phase. Students drew the coefficient and the variable separately, and then drawings were made of how to correctly solve the equation. Students used drawings with tallies or dots, either alone or in combination with a graphic organizer, to represent the equations and the solution process (Scheuermann et al., 2009). Strickland and Maccini (2013) introduced the concrete and representational phases simulta-neously and students drew the algebra blocks that were used during the concrete phase.
In the abstract phase, Arabic, or mathematical, symbols were used in all four of the studies (Scheuermann et al., 2009; Strickland & Maccini, 2013; Witzel et al., 2003; Witzel, 2005). Students wrote each step when solving the problem using only Arabic symbols (Witzel, 2005).

Graphic organizers

The fifth included article (Ives, 2007) used graphic organizers as an intervention when teaching solv-ing systems of equations. Graphic organizers are visual representations, usually in the form of diagrams or charts, which show the relationships between ideas (Strickland & Maccini, 2010). Graphic organizers use symbols, or other visual representations, in place of language and can help to organize information or steps in multistep problems (Strickland & Maccini, 2010).
Ives (2007) presented two related studies involving the use of the same graphic organizer when solv-ing systems of equations. In the first study, the graphic organizer was used to solve two linear equations with two variables, and in the second study, it was used to solve three linear equations with three variables. The graphic organizer consisted of three columns and two rows with Roman numeral column headings (see Figure 2).
Students were to begin in the upper left hand corner and work from cell to cell in a clockwise direction (Ives, 2007). The top row was used to combine equations until a one variable equation was produced and the bottom row was used to solve that one variable equation.

Implementation of interventions

The intervention duration in the four studies using the CRA model as the intervention varied from three lessons (Strickland & Maccini, 2013) to twenty-three lessons (Scheuermann et al., 2009).
The CRA model was used without any modifications in the studies by Witzel et al. (2003) and Witzel (2005). Students were taught five concepts that became increasingly harder over 19 lessons. Both Witzel et al. (2003) and Witzel (2005) used algebra content that progressed from reducing simple two-statement expressions to solving more complex equations, including transforming equations with single variables. Each lesson in the 19-lesson sequence had the same four steps: introduce the lesson, model the new pro-cedure, guide students through the procedure, and independent practice (Witzel et al., 2003; Witzel, 2005). The first lesson for each of the five concepts was taught concretely, the second lesson was taught repre-sentationally and the third and fourth lessons were taught abstractly (Witzel et al., 2003; Witzel, 2005).
Strickland and Maccini (2013) simultaneously introduced all three phases: concrete, representational and abstract, when teaching multiplication of linear expressions. They referred to this as the concrete-to-representational-to-abstract integration strategy (CRA-I). In addition, they used the support of graphic organizers in combination with the CRA model. The students were taught the concept of multiplication of linear expressions by embedding the expressions in area problems. This was taught over the course of three lessons. In the third lesson, the students moved from using the algebra blocks to a graphic organizer called the box method (Strickland & Maccini, 2013). Using the box method students were first to place the algebra blocks into the corresponding box on the graphic organizer, then they were to write the sym-bolic notation for each box and finally, they created their own box to multiply the linear expressions. This incorporated all three phases of the CRA model (concrete, representational and abstract) into one lesson.
The CRA model was combined with EIR in the study conducted by Scheuermann et al. (2009) to create an intervention strategy to help students solve one-variable equations. The EIR method used in this study consisted of explicit instruction, guided practice and the opportunity to explore a variety of algo-rithms (Scheuermann et al., 2009). “The EIR was comprised of three instructional components: explicit content sequencing, scaffolded inquiry, and systematic use of various modes of illustration” (Scheuermann et al., 2009, p. 106). In the first phase, the explicit content sequencing, students were explicitly taught one-variable equations in an order in which they became increasingly difficult. An example of the simplest was x + 3 = 10 and the most difficult was 3x + 2x – 4 = 51. This order was chosen specifically so that stu-dents would progress through the equations in a predefined manner and that they would be introduced to prerequisite skills and concepts before moving on to more difficult ones (Scheuermann et al., 2009). The scaffolded inquiry process was divided into three phases. During the first phase the teacher modeled the solution of equations by asking students how they would solve them and demonstrated their thinking. The teacher also pointed out “critical insights to the problem or questioned potential challenges the students could encounter” (Scheuermann et al., 2009, p. 107). The second phase of the scaffolded inquiry process was for students to use peer discussion to solve problems and discuss solution methods. Finally, in the third phase of the scaffolded inquiry process students were supposed to use a self-talk process, internal dialogue, to walk themselves through their thought process when solving equations (Scheuermann et al., 2005). During the final phase of this intervention, the various modes of illustration phase, the CRA model was used.
Graphic organizers were used to instruct students in solving systems of equations in the study by Ives (2007). Ives (2007) presented two related studies; both studies used the same graphic organizer. The first study involved two linear equations with two variables and the second study involved three linear equations with three variables. Although the content of the studies was different the execution was the same in both. Teachers used a combination of strategy and direction instruction approaches: they asked questions; provided feedback; administered probes; and included elaborate explanations, verbal modeling, and reminders (Ives, 2007). The first lesson was used to review prerequisite skills, the second presented 17 relatively simple systems of equations and the next two lessons introduced variations of systems of equa-tions (Ives, 2007).

Outcomes of interventions

The outcome of the intervention in each of the included studies was evaluated using different meas-urement tools. However, all studies used some variation of pre-, post and follow-up, or maintenance, measurement (Ives, 2007; Scheuermann et al., 2009; Strickland & Maccini, 2013; Witzel et al., 2003; Witzel, 2005), with the exception of the second study conducted by Ives (2007) where only pre- and post-tests were used (see Table 8). The outcomes in all five (Ives, 2007; Scheuermann et al., 2009; Strickland & Maccini, 2013; Witzel et al., 2003; Witzel, 2005) articles were measured using either the number or the percentage of correct responses. Ives (2007), Scheuermann et al. (2009) and Strickland and Maccini (2013) also calculated the effect sizes of their interventions. The CRA model and the graphic organizer interven-tions both showed positive outcomes for students with MLD.
The CRA intervention group and the comparison group, abstract only, were given a pretest, post-test and follow-up test that consisted of 27 items and was the same for all three-time points (see Table 9).
The CRA group outperformed the abstract only group on the posttest, having 23 and 17 correct respons-es respectively. Although, both groups maintained significant gains on the follow-up test, the gap between groups was smaller. The CRA group had 22 correct responses and the abstract only group had 21 correct responses (Witzel et al., 2003). Both groups showed significant improvement on the post and follow-up tests (Witzel et al., 2003). However, the CRA group outperformed the comparison group, abstract only, at a statistically significant level (Witzel et al., 2003).
The treatment group, the group using the CRA model intervention, and comparison group both showed improvement from the pretest to the posttest and on the follow-up test (Witzel, 2005). Students were assessed using a 27-item test with a medium difficulty level and were given the same test at all three measurement points. See Table 10 for an overview of all results.
On the pretest the comparison group (0.57, SD = 1.12) outperformed the treatment group (0.18, SD = 0.53). However, on the posttest the treatment group (8.26, SD = 7.65) outperformed the comparison group (5.36, SD = 5.75). These gains remained on the follow-up test where the treatment group (7.96, SD 84) continued to outperform the comparison group (5.51, SD = 5.97). Although, both groups showed improvement the treatment group outperformed the comparison group at a statistically significant level on both the posttest and follow-up test (Witzel, 2005).
All three participants, in the study by Strickland and Maccini (2013), showed substantial increases pre- to posttest, the pretest average scores ranged from 0-17% and the posttest average scores ranged from 78-93%. The students were tested using domain probes that were developed by the researcher and used as a pretest, posttest and maintenance test (Strickland & Maccini, 2013). All three were different, but covered the same algebra content at the same difficulty level. However, only two students maintained mastery, or a percentage of at least 80% correct, on the maintenance test that was given three to six weeks after the end of the intervention. One student was given the maintenance test three weeks after the inter-vention ended and scored 93%, one was given the maintenance test six weeks after and scored 98% and the third student was given the maintenance test four weeks after and scored 52%. The effect size for the intervention used by Strickland and Maccini (2013) was determined by the percentage of nonoverlapping data points and was calculated to be 100%, which represents a very effective intervention.
The outcomes for students with MLD, on two of the assessments used in the study by Scheuermann et al. (2009), showed improvement in the percentage of correct responses (see Table 11).
The concrete manipulation test was designed to assess students while they concretely solved a one-variable equation embedded in a word problem (Scheuermann et al., 2009). On the concrete manipulation test, students showed an increase in the percentage of correct responses; going from 38% on the pretest to 89% on the posttest and 80% on the maintenance test. The far-generalization test was designed to as-sess students in solving one-variable word problems that were found in school textbooks. On the far-generalization test students also made gains; however, they were not as significant going from 21% on the pretest to 30% on the posttest. On both the concrete manipulation test and the far-generalization test stu-dents showed a statistically significant improvement in their scores (Scheuermann et al., 2009). Scheuer-mann et al. (2009), used Glass’s D to calculate effect sizes. Effect sizes were found between pretest and posttest means and pretest and maintenance means for three of the assessment measures used by the au-thors, the Concrete Manipulation Test, the Far-Generalization Test and KeyMath-Revised. All three showed moderate effect sizes (Scheuermann et al., 2009). According to the conclusions of the authors, the students in this study were able to solve a variety of one-variable equations and generalize the skills they learned to new problems of the same format and maintain these gains for up to 11 weeks. However, less than 60% of the students showed significant gains so the increases in performance did not reach signifi-cant levels (Scheuermann et al., 2009).
The overall outcomes by Ives (2007) showed that all students, both the control group and the group using the graphic organizer, were able to make positive gains when solving systems of equations (see Table 12). Students were given a pretest, two posttests (one generated by the teacher and one generat-ed by the researcher) and a maintenance test. The test was split into two sections the concept section and the system solving section, or problems section. The concept section was designed to measure whether students understood the concept behind the solution process (Ives, 2007). The system solving section was designed to evaluate whether or not students could correctly solve systems of equations. Ives (2007) pre-sented two related studies involving the use of the same graphic organizer.

Table of Contents
1 Introduction 
1.1 Mathematics learning disabilities
1.2 Inclusive classroom setting
1.3 Interventions
1.4 Vygotsky’s theory of cognitive development
1.5 Rationale for this study
1.6 Aim
2 Method 
2.1 Procedure
2.2 Peer-review process
2.3 Data extraction
2.4 Quality assessment
2.5 Analysis
3 Results 
3.1 Search results
3.2 Description of MLD
3.3 Interventions
3.4 Implementation of interventions
3.5 Outcomes of interventions
4 Discussion 
4.1 Description of MLD used in these studies
4.2 Vygotsky’s theory of cognitive development
4.3 Explicit inquiry routine
4.4 Factors that may influence the outcomes of the interventions
4.5 Discussion of interventions
4.6 Practical implications
4.7 Discussion of quality assessment
4.8 Methodological issues
4.9 Limitations
4.10 Future research
5 Conclusion 
Interventions in Solving Equations for Students with Mathematics Learning Disabilities A Systematic Literature Review

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