Using Virtual Manipulative Instruction to Teach the Concepts of Area and Perimeter to Secondary Students With Learning Disabilities

Abstract

Secondary students with a learning disability in mathematics often struggle with the academic demands presented in advanced mathematics courses, such as algebra and geometry. With greater emphasis placed on problem solving and higher level thinking skills in these subject areas, students with a learning disability in mathematics often fail to keep pace with their general education peers. This study sought to address the lack of existing empirical research targeting viable interventions for learning the concepts of area and perimeter for secondary students with a learning disability in mathematics. Through the use of a multiple baseline design across three participants, virtual manipulatives were found to be an effective tool to acquire, maintain, and generalize the concepts of area and perimeter. Results from this study provide new evidence showing virtual manipulatives to be a viable and accessible technology to teach students with learning disabilities advanced mathematical concepts.

Keywords

virtual manipulatives learning disabilities mathematics area perimeter

Mathematics is a fundamental core content area in secondary education. Given the relationship between mathematics performance and in-school and post-school outcomes, an emphasis remains on high academic standards and expectations for secondary students in education today (Common Core State Standards Initiative [CCSSI], 2010; National Council of Teachers of Mathematics [NCTM], 2000). However, mathematics poses a number of challenges to students. Although many students may struggle with mathematics, approximately 5% to 8% of students are identified with a learning disability in mathematics (Geary, 2004; Lerner, 2003; Maccini, Mulcahy, & Wilson, 2007). For students with a learning disability in mathematics, academic challenges can range from basic skills acquisition (Algozzine, O’Shea, Crews, & Stoddard, 1987; Geary, 2004) to higher level thinking, such as developing and applying problem-solving skills (Geary, 2004; Huntington, 1994; Maccini & Hughes, 2000; Maccini & Ruhl, 2000; Witzel, 2001). Students with a learning disability in mathematics often commit procedural errors in multi-step problems, and possess poor organizational skills and deficits with their working memory and long-term memory (Geary, 2004). As students grow older, the lack of such skills can serve as a significant impediment to their future success in mathematics (Maccini et al., 2007).

The limitations students with a learning disability in mathematics possess with skills such as computation, organization, and working memory magnify in significance when learning advanced mathematical concepts due to their emphasis on completing tasks that require such skills as prerequisites (CCSSI, 2010; Maccini et al., 2007; NCTM, 2000). For example, learning how to calculate the geometric area and perimeter of a given shape requires students to complete a series of steps, many of which assume basic knowledge of skills (e.g., arithmetic operations, proportionality, and equivalence). In the case of area and perimeter, each requires an internalization of the procedures to calculate each concept, as well as conceptual understanding of the properties of various shapes—as emphasized by the CCSSI (2010) and the NCTM (2000). As greater attention is offered to subject areas such as algebra and geometry, and the emphasis within those areas stresses conceptual learning and problem solving skills, the significance of identifying strategies and instructional practices to learn such skills increases. One example of an instructional practice with proven success in the classroom for students with disabilities is the use of manipulative instruction (Bouck & Flanagan, 2009; Maccini & Gagnon, 2002).

Manipulative Instruction in Mathematics

Manipulatives are objects used to learn abstract mathematical properties, concepts, or processes (Bouck & Flanagan, 2010; Moyer, Bolyard, & Spikell, 2002). As instructional tools, mathematics manipulatives are found in two forms: concrete and virtual. Concrete manipulatives are generally low in cost, and do not require a power source to be used. Such examples may include pattern blocks, algebra tiles, fraction strips, and geoboards. Alternatively, virtual manipulatives are typically associated with computer technology—relying heavily on software programs and/or Internet accessibility (Bouck & Flanagan, 2010; Moyer et al., 2002).

For students identified with a learning disability in mathematics, concrete manipulative instruction is considered an effective instructional practice—shown to successfully help students understand a variety of mathematical concepts, including computation (Funkhouser, 1995; Mercer & Miller, 1992; Miller, Harris, Strawser, Jones, & Mercer, 1998), place value (Peterson, Mercer, & O’Shea, 1988), fractions (Butler, Miller, Crehan, Babbitt, & Pierce, 2003; Jordan, Miller, & Mercer, 1998), and word problem solving (Huntington, 1994; Maccini & Hughes, 2000, Maccini & Ruhl, 2000; Marsh & Cooke, 1996; Witzel, 2001). A significant portion of the literature available on concrete manipulative instruction focuses on mathematical standards targeting numbers and operations. Although these standards are important, their emphasis on mathematics curriculum is associated with younger grade levels. As students progress through their schooling, there is a notable shift in curriculum focus (CCSSI, 2010; NCTM, 2000). For secondary curriculum, a larger focus is placed on algebraic and geometric principles (CCSSI, 2010; NCTM, 2000).

To date, seven studies exist in peer-reviewed publications investigating concrete manipulatives with secondary students with a learning disability (Butler et al., 2003; Cass, Cates, Smith, & Jackson, 2003; Maccini & Hughes, 2000; Maccini & Ruhl, 2000; Strickland & Maccini, 2013; Witzel, 2005; Witzel, Mercer, & Miller, 2003). However, only four of these studies measured the effectiveness of manipulatives for secondary students to learn advanced mathematical concepts, such as those found in algebra and geometry (Cass et al., 2003; Maccini & Ruhl, 2000; Witzel, 2005; Witzel et al., 2003). Witzel et al. (2003) and Witzel (2005) used manipulative instruction to teach algebra and pre-algebra standards to middle school students. The results of both studies found student performance increased when concrete manipulatives were used to learn basic algebraic concepts in comparison with control groups. Cass et al. (2003) also used concrete manipulative materials to successfully teach area and perimeter problem solving skills to high school students with a learning disability in mathematics.

The extensive research based on concrete manipulatives for students with disabilities—particularly those with a learning disability—is in stark contrast to the limited amounts of literature examining the effectiveness of virtual manipulatives for elementary and secondary students. Although no studies were found measuring the impact of virtual manipulatives for secondary students, three published studies were found targeting elementary students identified with a disability. Moyer-Packenham, Ulmer, and Anderson (2012) and Reimer and Moyer (2005) both incorporated students with learning disabilities in group designs studying the use of virtual manipulatives to solve fractions for elementary students; both studies reported positive findings in favor of the use of virtual manipulatives. Bouck, Satsangi, Doughty, and Courtney (2013) examined the use of virtual manipulatives to learn single- and double-digit subtraction skills with three elementary-aged students with autism spectrum disorder. When compared alongside concrete manipulatives, the authors found increased skill attainment for each student using virtual manipulatives to solve subtraction problems.

Researchers offer multiple rationales explaining why virtual manipulatives may be better suited for use by students with learning disabilities as compared with concrete manipulatives. One prevalent theory supporting this argument highlights the proposed impact of the cognitive load placed on learners when they use concrete manipulatives to solve multi-step problems (Kaput, 1989; Suh & Moyer, 2008). As designed, concrete manipulatives offer students limited structure in the manner in which they are used. As a result, a significant cognitive load is placed on students when asked to manipulate numerous physical objects in a continuous sequence. From the cognitive load theory, researchers assert that the process of keeping track of the many procedural actions students complete using physical objects often results in their failure to relate the physical manipulations to the mathematical ideas taught (Suh & Moyer, 2008). In contrast, virtual manipulatives offer users greater organization on screen in the manner in which each manipulative is presented and the procedural steps incorporated within each (Bouck et al., 2013). Virtual manipulatives provide built in constraints that lessen much of the cognitive load experienced by students when solving mathematical problems (Moyer, Niezgoda, & Stanley, 2005; Suh & Moyer, 2008). However, such benefits for students with learning disabilities can only be speculated, as limited research to date examines the use of virtual manipulatives with students with disabilities.

Current Study

Given the suggestion of concrete manipulatives as best practice for secondary students with disabilities in conjunction with the increased use of computer-based technology in schools, it is important to examine the effectiveness of virtual manipulatives (Bouck et al., 2013). This study aimed to further the research base in the area of virtual manipulative instruction to assist secondary students identified with a learning disability in mathematics to learn the concepts of area and perimeter. The research questions addressed in this study included the following:

Research Question 1: What percentage of area problems do students with a learning disability in mathematics solve accurately when using virtual manipulatives?

Research Question 2: What percentage of perimeter problems do students with a learning disability in mathematics solve accurately when using virtual manipulatives?

Research Question 3: What are the attitudes and perceptions of students with a learning disability in mathematics about using virtual manipulatives to solve mathematics problems?

Method

Participants

Three male high school students—Xavier, Mark, and Jake—participated in this study, and were chosen from one charter school located in the downtown area of a Midwestern city. Each participant was identified with a learning disability in mathematics and participated in general education mathematics courses. Participants were chosen based on the following criteria: (a) identified learning disability in mathematics, (b) inability to solve area and perimeter problems at a 50% success rate as determined by a researcher-created pre-assessment, and (c) adequate fine/gross motor ability to operate a laptop/desktop computer.

Xavier

Xavier was a 14-year-old Caucasian male enrolled in the ninth grade. He was diagnosed at the age of 10 with a learning disability in mathematics. Based on the Wechsler Intelligence Scale for Children–IV (WISC-IV), Xavier had a full-scale IQ score of 95, placing his general thinking and reasoning skills within the average range of the assessment. Xavier’s school psychological assessment stated that his learning disability diagnosis was determined on the basis of a discrepancy between ability and achievement according to the state’s definition of learning disability. As defined by the Indiana State Board of Education Special Education Rules (511 IAC 7-41-1), a learning disability in mathematics could be evidenced through a pattern of strengths and weaknesses in performance or achievement, or both, relative to (a) age, (b) state-approved grade level standards, or (c) intellectual development (Indiana Department of Education [IDOE], 2010). Xavier received all of his educational instruction in general education classes and was earning the second of his two Algebra 1 credits at the time of this study. As a ninth grader, Xavier had not yet taken the End of Course Assessment (ECA) in Algebra 1. The ECAs were statewide criterion-referenced assessments administered to students completing their instruction in Algebra 1, Biology 1, and English 10; passage of these exams was required for graduation. Xavier was receiving tutoring within his resource periods in preparation for the Algebra 1 ECA.

Mark

Mark was a 16-year-old Caucasian male in the 11th grade and was diagnosed with a learning disability in mathematics. His IQ scores were not available to researchers at the time of data collection. Mark’s Individualized Education Program (IEP) team determined his learning disability based on the school’s response-to-intervention process, which incorporated a three-tier model of referral for struggling students (Fuchs & Fuchs, 2006; IDOE, 2010). Based on the Wechsler Individual Achievement Test–II (WIAT-II), Mark’s mathematics composite standard score was 92, with a below average standard score of 82 on the Math Reasoning subtest, which—among other measures—assessed geometric measurement concepts. Mark took all of his classes in general education settings, and was not enrolled in a mathematics course during the semester in which this study took place. He had successfully earned both of his Algebra 1 credits and had just recently passed the ECA in Algebra 1 the previous semester.

Jake

Jake was an 18-year-old African American male in the 11th grade, diagnosed with a learning disability in mathematics. Based on the Reynolds Intellectual Assessment Scales (RIAS), Jake’s composite intelligence index (CIX) score was 83—placing his cognitive ability within the low average range of the assessment. Jake was expelled from two high schools prior to his attendance at the charter school for behavioral reasons. Jake’s IEP team determined his diagnosis based on the school’s response-to-intervention process, which incorporated a three-tier model for referral of struggling students (Fuchs & Fuchs, 2006; IDOE, 2010). Similar to the other participants, he also received all instruction in general education classes. At the time of data collection, Jake was enrolled in a geometry course. Although Jake had earned his two Algebra 1 credits the year prior, he had not yet passed the ECA in Algebra 1.

Setting

This study occurred in a public charter high school located in a large metropolitan, Midwestern city. The high school enrolled approximately 300 students in Grades 9 through 12. Approximately 71% of the student body identified themselves as African American, 17% Caucasian, 9% Multiracial, and 2% Hispanic. Of the total population, roughly 27% received special education services. This charter school offered entirely inclusion-based instruction for all academic classes for students, with additional resource periods available as needed. Data collection for all phases of this study—baseline, intervention, maintenance, and generalization—occurred in an empty conference room within the school.

Materials

Materials for this study included written assessments via paper and pencil, and virtual manipulatives. The intervention of virtual manipulatives to solve area and perimeter problems was offered to students using a 13.3-inch laptop computer and wireless mouse. For each session in the study, students were presented with a writing utensil and a sheet of paper containing five shapes—with corresponding side lengths for each—presented vertically down the left side of the page. All shapes presented were regular and irregular in design, and were adapted based on ninth grade Algebra 1 standards. Lengths of sides for each shape were exclusively single-digit (i.e., less than nine), whereas solutions to the problems were both single-digit and double-digit. All numbers representing the lengths of sides were presented without measurement units (e.g., cm, mm). All shapes of problems given to students possessed exclusively 90-degree angles; this was done because the virtual manipulatives used in this study were not suitable for solving problems with angles greater or smaller than 90 degrees (see Figure 1 for an example assessment). In total, each session contained five area and five perimeter problems. For all sessions of the study, students were permitted to use a four-function calculator for computation purposes if desired.

Figure 1.

Sample assessment for area and perimeter.

The virtual manipulatives were accessed through an online program—the National Library of Virtual Manipulatives (NLVM; http://nlvm.usu.edu). This free website offered users the ability to create a series of three-dimensional visual representations of shapes and objects to be manipulated (e.g., rotate, combine, enlarge) at the user’s discretion (Moyer et al., 2002; Reimer & Moyer, 2005). For this study, students were directed to use the polynomials manipulative found in the “Algebra: Grades 9–12” section of the website (see Figure 2 for a screenshot of the virtual manipulatives). For this manipulative, users were able to create regular and irregular shapes by combining square blocks on their screen, while also altering the colors of individual blocks and/or completed shapes, as well as altering sizes of all blocks in unison (i.e., all blocks on screen could be magnified or minimized but were of the same size at all times). Once finished, completed shapes visually retained the four sides of the individual blocks comprising the shape—allowing users to visually count the exact number of blocks needed to create the completed shape (i.e., the area), as well as the exact number of sides of the individual blocks needed to create each side of the completed shape (i.e., the perimeter).

Figure 2.

Screenshot of virtual manipulatives accessed from the NLVM website.

Independent and Dependent Variables

The independent variable for this study was the virtual manipulative (i.e., the polynomial tiles) used to solve area and perimeter mathematics problems. Using virtual manipulatives was defined as using a computer and the NLVM website to visually manipulate virtual blocks to solve area and perimeter problems. The dependent variables for this study included (a) the percentage of correctly solved area problems and (b) the percentage of correctly solved perimeter problems.

Experimental Design

A multiple baseline design, across three students and two dependent variables—area and perimeter problem solving—was used to measure student performance using virtual manipulatives. This design allowed researchers to assess the effectiveness of the independent variable (i.e., virtual manipulatives) to solve area and perimeter problems for one participant as the other participants’ baseline scores served as controls for comparison (Kennedy, 2005). After baseline data were collected, the first student started training and then intervention, while the second and third students remained in the baseline phase, with their area and perimeter performance regularly probed. Immediately after the first student exhibited three sessions of stable performance with the intervention, the second student began training and then intervention, while the third student remained in baseline. This process was repeated between the second and third participants, until all three students were engaged in the intervention (Kennedy, 2005). Students continued with intervention until their data within this phase exceeded at least five sessions, and deemed stable. Following the intervention phase and a 2-week break, each student participated in three consecutive sessions of maintenance and generalization. From the start of baseline through the completion of generalization, the current study lasted approximately 8 weeks in length.

Data Collection

Event recording was used to measure the effectiveness of virtual manipulatives to accurately solve area and perimeter mathematics problems. Percentage accuracy was recorded for each participant by measuring the total number of correctly completed area and perimeter problems in each session. The percentage of prompts needed (i.e., verbal, gestural, and physical) for each student during intervention, maintenance, and generalization phases was also recorded. For each phase, a member of the research team was responsible for administering assessments and recording data.

Procedures

Baseline

Baseline was conducted with each participant in an empty conference room. Students were presented with a writing utensil and a single sheet of paper with five shapes displayed vertically down the left side of the page. For each shape, students were asked to solve for area and perimeter, in whichever order they preferred. In total, each session contained five area and five perimeter problems. No training was provided to students in advance on how to solve area or perimeter problems, and no prompting was offered to students if they were unable to solve a problem.

Intervention

Following baseline, a researcher taught each student individually the concepts of area and perimeter in a 40-min instructional lesson completed within 1 day. To begin, the researcher used 8.5″ × 11″ blue and white sheets of paper to cover the entire surface area of the top of a conference table. Using a scaled drawing of the table on a white board, students learned why the blue sheets placed along the edges of the table represented the perimeter, while the white sheets covering the remainder of the table—in addition to the blue sheets—represented the area. Through verbal instruction, the researcher briefly reviewed the terms length and width for the students. Finally, through visual illustrations on the board, students were shown how to identify length and width dimensions of rectangular and square objects.

Next, students were trained on how to use the virtual manipulatives on a laptop computer to calculate the area and perimeter of a given shape. To begin, students were introduced to the NLVM program and shown how to create a shape on screen, how to increase and decrease the size and color of the blocks, and how to delete unwanted blocks and/or entire shapes. Once given a shape on a sheet of paper, students were instructed on how to select the correct number of blocks on their screen to coincide with the values depicting the length and width of the given shape. Next, students were shown how to construct the shape on their screen by arranging the blocks in an identical pattern as the shape shown on the sheet of paper. Students were then shown which blocks to count on their screen to determine the area and perimeter of the object they had created. Finally students were instructed to write their answers on their sheet of paper next to the corresponding shape. Throughout the training period, students were re-taught any concepts they struggled with while using the virtual manipulatives to solve problems. Re-teaching continued until students were capable of completing four out of five perimeter and area problems independently using the virtual manipulatives. Once this criterion was met during training, any additional re-teaching requested by the students was recorded as a prompt.

Once trained on how to use the virtual manipulatives, students continued solving problems as done in baseline—with each session requiring students to solve for area and perimeter of five shapes on a single sheet of paper while using the virtual manipulatives. To begin, students looked at the problem presented on the sheet of paper and identified the number of sides to the shape, as well as the numerical length stated for each side. Students were instructed to identify one side of the shape and select the corresponding number of blocks on their computer screen. Students arranged the blocks on their screen to mirror the selected side of the shape on their sheet of paper. For example, if the shape possessed a right side measuring four units vertically, then students selected four blocks on their screen and stacked them to create a vertical line. Students continued this process of re-creating each side, remembering to connect each of the individual sides to one another, until an outline of the shape took form. Next, students filled the remaining empty area within the outline with blocks, until one uniform solid figure was created on their computer screen to mirror the shape on their paper. Finally, with the chosen shape created on their screen, students counted the total number of blocks comprising the shape to calculate the area. To calculate the perimeter, students counted the sides of each individual block comprising the outer sides of the shape. Students were permitted to use the calculation process of their choosing (i.e., addition and/or multiplication) when computing each solution. Students solved for area and perimeter concurrently for each shape, before moving on to the next problem.

Maintenance

Following intervention, students were given a 2-week hiatus from the study. Afterward, researchers returned to the school and conducted three sessions of maintenance data collection. Identical to intervention, students were asked to solve the area and perimeter of five shapes per session while using the virtual manipulatives to assist them.

Generalization

The final phase of this study assessed whether students could apply the same skills they gained using virtual manipulatives to solve for area and perimeter of static shapes to abstract word problems. Because each of the three participants had recently participated, or were preparing to participate, in the ECA for Algebra 1 (which contains problems of varying degrees of difficulty in multiple formats), area and perimeter questions were presented as strictly word problems. For three sessions, students were given five word problems printed on a single sheet of paper. Each word problem described specific shapes accompanied by length and width dimensions of either the entire shape in whole or individual parts that required combining to form the final shape. Each word problem was approximately four sentences in length, with no accompanying visual illustration of the shape described. With the use of the virtual manipulatives, students were asked to create each shape on their computer screen based entirely on their understanding of the figure described in each word problem and then calculate its area and perimeter. All word problems were read aloud to students multiple times by a researcher for each of the three sessions.

Inter-Observer Agreement (IOA) and Treatment Integrity

The researcher and a second trained observer concurrently but separately collected IOA and treatment integrity data across the four phases of this study. For IOA data, percentage accuracy was collected for 33.3% or more of all baseline and intervention sessions, and exactly 33.3% of all maintenance and generalization sessions for each participant. This calculation was determined by dividing the number of agreements by the total number of agreements plus disagreements and multiplying by 100. IOA for this study was 100% for each of the three participants.

Treatment integrity data were collected for 33.3% or more of all baseline and intervention sessions, and exactly 33.3% of all maintenance and generalization sessions for each participant through the use of a checklist. The checklist determined whether students were given a writing utensil and access to the virtual manipulatives via computer for each session. Treatment integrity was 100% for all three students.

Social Validity

Interviews were conducted with each participant prior to the start of the study as well as at its conclusion. The four questions asked of students focused on attitudes and personal preferences toward past learning experiences, mathematics instruction, the use of technology, and virtual manipulatives. Each interview lasted approximately 10 min in length.

Data Analysis

Visual analysis and calculations were used to assess within- and between-phase patterns across the following measures: level, trend, variability, overlapping data, and uniformity of patterns across data in similar phases with similar conditions (Kennedy, 2005). Stability of student data were determined based on the following criteria: (a) once 80% of the data points collected for each measure fell within a 20% range of the overall mean for that phase (Gast & Spriggs, 2010); (b) if, however, a student’s data collection exceeded eight sessions and stability was not achieved, intervention would then conclude once a student earned identical scores for three consecutive sessions for both area and perimeter measures. Trends evident in student data within and between conditions were identified using the split-middle method technique (Gast & Spriggs, 2010), which is calculated by dividing data for each phase in halves, determining the intersections of the mid-rate and mid-date for each half, drawing the quarter-intersect line, and then adjusting this line to create the split-middle line of progress.

The effect-size index of percentage of non-overlapping data (PND; Scruggs, Mastropieri, & Casto, 1987) was used to assess the between-phase performance differential of baseline and intervention data for each student. PND was calculated by determining the highest value within the baseline phase(s), and then calculating the percentage of intervention data points that surpass this value (Wolery, Busick, Reichow, & Barton, 2010). PND was used to illustrate the intervention’s effectiveness and was reported as a percentage from 0 to 100; scores above 90 suggested a very effective treatment, 70% to 89% were deemed an effective treatment, 50% to 69% were considered questionable, and scores less than 49% showed an ineffective treatment (Scruggs & Mastropieri, 1998). Tau-U, a non-parametric statistical measure of effect size that combines non-overlap data between phases with trend from within the intervention phase, was also calculated (Parker, Vannest, Davis, & Sauber, 2011); Tau-U was calculated using the online calculator from www.singlecaseresearch.org (Vannest, Parker, & Gonen, 2011). Tau-U scores are reported between 0 and 1, with 0.93 to 1 considered a large effect, 0.66 to 0.92 a medium effect, and 0 to 0.65 a small effect (Parker & Vannest, 2009). For the purposes of the multiple baseline design used in this study, the combined weighted average Tau-U score and individual Tau-U scores were calculated and reported.

Results

The use of virtual manipulatives increased the performance of all three participants in solving area and perimeter mathematics problems (see Figure 3 for graphical representation of the data). Students demonstrated improved performance from their respective baseline scores during intervention, maintenance, and generalization phases. Assessing the effect size of the intervention, the combined weighted average Tau-U score (Parker et al., 2011) was 1.0 (95% confidence interval [CI] [0.6069, 1.3931]) for both area and perimeter problem solving, showing the virtual manipulatives to be highly effective. Furthermore, for two of the three participants, their performances on area problems were substantially higher in comparison with their scores on perimeter-based problems.

Figure 3.

Percentage of correctly solved area and perimeter problems.

Xavier

Xavier did not answer any area or perimeter problems correctly across five consecutive baseline sessions, demonstrating a stable baseline of 0%. Immediately after receiving training on the virtual manipulatives, his intervention scores reached 100% accuracy for area and perimeter problem solving, which represented an abrupt change in level between conditions (Gast & Spriggs, 2010). Xavier then scored 100% on both measures across five consecutive intervention sessions, illustrating no variability in his performance. With greater than 80% of his scores for area and perimeter falling within 20% of the overall mean for this phase, Xavier’s intervention data were identified as stable (Gast & Spriggs, 2010). He further maintained 100% accuracy performance for area and perimeter across three maintenance sessions and three generalization sessions.

Xavier’s trend for both his baseline and intervention data for both area and perimeter was zero celerating, representing no change in trend direction between the conditions (Gast & Spriggs, 2010). Measuring performance differential (i.e., effect size) between Xavier’s baseline and intervention scores, the PND (Scruggs et al., 1987) was found to be 100% for both area and perimeter problems, demonstrating the immediate effect of the intervention on Xavier’s ability to solve both types of problems, and categorizing the intervention as very effective (Scruggs & Mastropieri, 1998). Xavier’s Tau-U score (Parker et al., 2011) was 1.0 for area and perimeter (90% CI [0.370, 1.630]). In all, Xavier’s data collected during intervention, maintenance, and generalization illustrated no variability, with no visible change in level or trend for both area and perimeter measures.

Mark

Mark struggled to solve both types of mathematics problems correctly during baseline, earning an average score of 3.3% accuracy across six area and perimeter sessions. Once given the intervention of virtual manipulatives to solve area and perimeter problems, his scores on both measures increased significantly; he scored 100% for both area and perimeter during the first intervention session, representing an abrupt change in level (Gast & Spriggs, 2010). Mark scored 100% accuracy on all but one session of area problems during intervention (µ = 97.5%). Although his overall performance was lower with perimeter, Mark scored above 80% on all but one session of perimeter mathematics problems during intervention (µ = 85.0%), demonstrating an accelerating upward trend across his last five sessions. With more than 80% of his scores for area and perimeter falling within 20% of the overall mean for the intervention phase, Mark’s data were deemed stable (Gast & Spriggs, 2010).

Mark’s trend for area during baseline was zero celerating and for perimeter was slightly accelerating. For intervention, Mark’s trend data were again zero celerating for area, representing no change in trend direction. Yet, for perimeter intervention data, Mark’s performance was slightly decelerating. Comparing baseline and intervention scores, Mark’s PND (Scruggs et al., 1987) was calculated to be 100% for both area and perimeter problems—suggesting the intervention was very effective (Scruggs & Mastropieri, 1998). Mark’s Tau-U score (Parker et al., 2011) was 1.0 for area and perimeter (90% CI [0.469, 1.531]). Mark earned an average score of 86.7% accuracy for area problems and 73.3% for perimeter problems across three sessions of maintenance each. When generalized to word problems, Mark earned 100% accuracy across three sessions for area problems; however, his scores for each of the three perimeter sessions illustrated a downward deteriorating trend, finishing with an average score of 60% for this phase.

Jake

Jake did not answer any area or perimeter problems correctly across five consecutive baseline sessions, exhibiting a stable baseline of 0%. When given the opportunity to use virtual manipulatives during intervention, Jake steadily increased his performance solving both types of problems. Solving area problems, Jake reached 100% accuracy by the third session of intervention, holding steady for three continuous sessions, before declining to 80% where his scores remained stable for his final four sessions of intervention (µ = 82.2%). Jake showed greater variability with his performance solving perimeter problems, averaging 68.9% accuracy. However, by the end of intervention, he demonstrated significant improvement, earning 80% or greater accuracy on his final four sessions using virtual manipulatives to solve for perimeter. Jake’s intervention data were deemed stable by researchers based on three identical consecutive scores of 80% earned on both area and perimeter measures. (Data for one additional session of intervention for area and perimeter were collected and reported.)

Jake’s trend for his area and perimeter baseline data were zero celerating, representing no change in trend direction between the conditions. For intervention, Jake’s area data were again zero celerating, while his perimeter data were slightly accelerating. Comparing baseline and intervention scores, Jake’s PND (Scruggs et al., 1987) was calculated to be 100% for both area and perimeter problems, suggesting the intervention was very effective (Scruggs & Mastropieri, 1998). Jake’s Tau-U score (Parker et al., 2011) was 1.0 for area and perimeter (90% CI [0.452, 1.548]). Jake sustained an increased level of performance solving perimeter problems during the maintenance phase, averaging 86.7% accuracy across three sessions. In contrast to intervention, Jake demonstrated greater variability with his area problem solving during maintenance (µ = 40%), with scores ranging from 20% to 80% over three sessions. The trend of greater stability and consistency solving for perimeter over area continued for Jake during generalization when given word problems to solve using the virtual manipulatives. Solving perimeter problems, Jake showed stable performance, earning 60% accuracy for all three sessions; for area, his average accuracy score was 33.3% over three sessions.

Social Validity

When asked to describe how they generally received mathematics instruction in their general education classes, all three students described a traditional approach used by their teachers, where students are asked to sit at their desks and take notes and/or solve problems as the teacher instructs from the board. Jake explained that he struggled a great deal in class because he was easily distracted with this approach. Mark shared that he did not like how fast his teachers generally talk and preferred more self-paced instruction.

When asked whether they enjoyed using the virtual manipulatives, all three students responded affirmatively. Jake highlighted the visual presentation of the manipulatives on screen as a key advantage, explaining,

It helped mainly with area—perimeter you can get on paper [referring to how he could add the dimensions of each side of the shape given on his sheet of paper without using the shape created with virtual manipulatives on screen]. But on word problems, it helps you with perimeter.

Mark cited the ability to highlight and use colors to separate the blocks as his favorite feature, explaining that this allowed him to stay organized and “not get confused.” Xavier also commented favorably with regard to the visual display of the blocks, saying, “I like the lines [of the connected blocks], you just count them for area.” Finally, when asked whether they would like to see such technology used in their mathematics classes, Xavier and Mark both responded yes, as Mark cited the ability to use the technology at his own pace to solve problems as a key benefit. However, Jake mentioned he saw some of the features of the program as a deterrent and stated the program was too slow and cumbersome at times for him.

Discussion

The objective of this study was to determine whether virtual manipulative instruction could effectively support the teaching of area and perimeter to secondary students with a learning disability in mathematics. To address this question, researchers measured the accuracy of area and perimeter problems solved by students with learning disabilities using virtual manipulatives to assist them. Across the intervention phase, results showed the use of virtual manipulatives increased performance for all three participants on both area and perimeter measures as compared with baseline. This trend remained stable for Xavier throughout maintenance and generalization. However, although still higher when compared with their baseline levels, scores for Mark and Jake fluctuated on particular measures during the last two phases.

Results garnered from this study support the practice of using manipulatives as an effective practice to teach mathematical concepts to students with learning disabilities (Butler et al., 2003; Cass et al., 2003; Maccini & Hughes, 2000; Maccini & Ruhl, 2000; Strickland & Maccini, 2013; Witzel, 2005; Witzel et al., 2003). Similar results in this study were obtained in comparison with the findings of Cass et al. (2003). Cass et al. (2003) highlighted the benefits of concrete manipulative instruction as a means to teach area and perimeter to secondary students with a learning disability in mathematics. The present study adds to the literature base in support of virtual manipulatives as a potential alternative to concrete manipulatives for this population. Furthermore, unlike the limited existing research on virtual manipulatives for students with disabilities—such as the work of Bouck et al. (2013) that focused on the lower level mathematics skill of subtraction with elementary-aged students—the findings of the present study support virtual manipulative use for higher level mathematical concepts (i.e., area and perimeter) for older students. Prior to the present study, no existing literature was found examining the viability of virtual manipulatives to teach such concepts.

Looking closely at student scores, trends emerged illustrating the impact of virtual manipulatives on students’ ability to solve area and perimeter problems. When comparing baseline with intervention scores for all three students, the use of virtual manipulatives resulted in immediate improvement in area and perimeter problem solving. In this case, the ability to create and then transform shapes (e.g., rotate, flip, magnify) offered students a better understanding of the shape’s dimensional properties. For example, if a student were presented a shape—similar to the one depicted in Problem 5 in Figure 1—he or she may not fully understand the meaning of the numeral 4—as denoted along the base of a shape—until he or she has had the opportunity to re-create the base using four adjacent blocks on his or her computer screen. This may explain why each of the participants struggled during baseline when provided with only static drawings of shapes on paper to solve for area and perimeter measures.

A second trend evident shows a discrepancy in the students’ ability to solve area problems in comparison with perimeter problems using virtual manipulatives. With the exception of Xavier, for whom area and perimeter performance was identical across all sessions and phases, the use of virtual manipulatives appeared to improve performance on area problems more so than perimeter for Mark and Jake. Both students rose to 100% mastery faster—or sustained such levels longer—with area problems as compared with perimeter. Through visual analysis, both students experienced greater variability in solving perimeter problems, versus relative stability in their area problem-solving scores during intervention. One explanation for the discrepancy between area and perimeter performance for Mark and Jake can be linked back to the theory of cognitive load (Kaput, 1989; Suh & Moyer, 2008). The higher and more stable performance exhibited by these two students solving area problems during intervention suggests the virtual manipulatives simplified the conceptual process for solving area problems to a greater degree than it did for perimeter-based problems. In other words, the number of steps (i.e., manipulations) needed to create a shape and then calculate its area was less burdensome to the students’ ability to make connections to the underlining mathematical concept than it was when determining perimeter.

Inherent features within the virtual manipulative program may have lent themselves better to organizing and solving area problems more so than perimeter for students with learning disabilities. As expressed in their social validity responses, all three students cited the visual presentation of the blocks on screen as a useful attribute of the manipulatives. Each student specifically mentioned the layout of completed shapes on screen as useful to determining area. The ability to color various blocks was also mentioned by Mark as one of the strengths of the program. However, it should be noted that none of the three students used this feature to assist them when solving for perimeter by distinguishing between the blocks forming the perimeter of a created shape and the remaining interior blocks, as demonstrated by the researcher during the training portion of intervention with a table and sheets of blue and white paper. For students with a learning disability in mathematics, the visual presentation of virtual manipulatives on screen may have offered greater cognitive support in terms of their organization of problems containing multiple steps to solve for area more so than perimeter.

Implications for Practice

Conclusions drawn from the findings of this study point to general strategies teachers should consider when adopting virtual manipulative technology for mathematics instruction with students with learning disabilities. Prior to the use of virtual manipulatives, teachers should give their students extensive modeling and guided instruction as offered in the training portion of intervention. Researchers in this study cannot separate the impact the training had on the students from their success using the virtual manipulatives. The importance of modeling, instruction, and continuous practice is worth noting based on Mark and Jake’s maintenance and generalization scores. Mark’s average mean levels for perimeter problems and downward trending scores, as well as Jake’s low mean levels for area problems, suggest both students struggled retaining these respective skills when given a brief hiatus from the study; in fact, the scores suggest below passing rates per classroom setting. This was contrary to the performance exhibited by each student during intervention; for Mark, his data indicated an increasing trend over the last five sessions of intervention with perimeter problems, while Jake exhibited greater and more stable performance during intervention with area than he did with perimeter. However, in both instances, it can be surmised these two students had not yet made a solid connection between the digital manipulations being made on screen with the underpinning conceptual concepts of area and perimeter. In retrospect, Mark and Jake may have benefited from an extended intervention period—offering each student more time to formulate such cognitive associations. When applied to the classroom, teachers should emphasize continuous practice and re-teaching of core concepts linked to each manipulation performed on screen, with the goal of students understanding the significance of each step through the solution of the problem.

To support their organization of multi-step problems, teachers are advised to emphasize the visual customization features (e.g., altering color and size) of any virtual manipulative program to their students in advance, accompanied by specific steps for them to follow when re-creating shapes and figures. For example, teachers can have students use specific colored blocks to represent different areas/properties of shapes they create, and then provide a checklist to follow as they solve for area and perimeter. Teachers may also consider having students create multiple sizes of the same shape on one computer screen, side-by-side, to demonstrate the commonalities and differences of their geometric properties. Higher level problem-solving questions can also be posed using similar practices. For example, visualization of multiple shapes on screen can assist in explaining how two shapes can have the same perimeters but different areas.

Limitations and Future Directions

This study is not without limitations. One limitation was the exclusive use of 90-degree angles in the problems given to students to solve. Due to the design of the NLVM program, shapes could only be constructed with blocks possessing 90-degree angles. In real life scenarios, students will need to be able to measure and calculate shapes containing a variety of obtuse and acute angles. Similarly, area and perimeter problems commonly found on state standardized tests will often possess shapes containing angle sizes that do not lend themselves to be solved with the use of this specific program. However, for students with limited initial understanding of the concepts of area and perimeter, such as those students chosen for this study, it may be best to begin teaching such concepts using shapes with basic uniform properties (e.g., shapes with identical angle and side measures). In this respect, the NLVM program may lend itself best to teaching the beginning principles of area and perimeter, before a student progresses to more complex applications of such concepts. Finally, the findings of this study are limited when generalizing to a larger population due to the relatively small number of participants used. Although this study serves as an initial step toward highlighting the potential of virtual manipulatives to teach advanced mathematics standards to students with learning disabilities, additional research is needed using larger sample sizes of students to validate its findings.

Moving forward, future research should examine the viability of using virtual manipulatives to teach area and perimeter of shapes containing angle measures of varying degrees. Future research should also consider assessing the practicality of virtual manipulative technology in a classroom setting with multiple learners (e.g., classrooms with 20 or more students). Infusion of computer technology in large classroom settings can often serve as a barrier to technology adoption for teachers (Suh, Moyer, & Heo, 2005). For the purposes of this study, the three participants were taken into an empty conference room to work with a researcher and the NLVM program. Through this process, a one-on-one dynamic was established, which may have benefited the students when learning to use this new technology while learning content. Future research should look into whether such technology can be used for large, whole-class instruction, or whether it is best reserved for one-on-one instruction and remediation. Researchers should also examine the influence of varying instructional strategies, such as those used in the training phase of this study (i.e., guided instruction, modeling), on the students’ understanding of area and perimeter as they are paired with virtual manipulative instruction.

To date, researchers identified a concrete–representational–abstract (CRA) teaching sequence to characterize the process by which students learn new concepts manipulating concrete objects, internalize their representations, and then apply them in an abstract manner (Peterson et al., 1988; Witzel, 2005). However, no research exists explaining where virtual manipulatives fall within the CRA sequence. Virtual manipulatives can be characterized as not fully concrete in nature—as they cannot be physically held and manipulated—nor are they entirely representational—as they are not drawings re-created by students based on prior experiences with concrete objects. Virtual manipulatives can be best classified as falling between the concrete and representational stages. The findings of this study serve as a precursor to studying the role of virtual manipulatives within a CRA approach, suggesting virtual manipulatives may replace concrete manipulatives within such a sequence. Further research is needed on virtual manipulatives and the impact they have on the learning process of students with learning disabilities, as they transition into the abstract learning of new concepts using this form of manipulative.

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.

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