Virtual Manipulative-Based Intervention Package to Teach Multiplication and Division to Secondary Students With Developmental Disabilities

Abstract

Mathematics instruction for students with intellectual disabilities and autism is important. However, it is imperative for researchers and practitioners to focus on the maintenance of mathematical concepts and not just acquisition for these students. Through a single-case multiple probe across participants study, researchers explored an intervention package consisting of a manipulative-based instructional sequence involving virtual manipulatives and then representations (i.e., drawings; referred to as the virtual-representational instructional sequence), explicit instruction, the system of least prompts, overlearning, and support fading to support students with intellectual disability and autism to acquire and maintain multiplication or division skills. The three middle school students who completed the entire intervention acquired and maintained their targeted mathematics skill—in multiplication or division. The results have implications for use of intervention packages to teach foundational mathematics skills to students with developmental disabilities.

Keywords

middle school age instruction prompts special education

In research on mathematics—as in other domains—a current focus exists on determining evidence-based practices to teach students with disabilities, including those with intellectual disability and autism (e.g., King et al., 2016; Spooner et al., 2019). In a recent review of mathematics research for students with moderate and severe developmental disabilities, Spooner et al. (2019) determined manipulatives, technology-aided instruction, systematic instruction, explicit instruction, and graphic organizers or heuristics to be evidence-based practices. However, many interventions delivered to students in mathematics are actually treatment packages, consisting of multiple interventions (Spooner et al., 2019), as opposed to stand-alone interventions. One such treatment package identified by Spooner et al. (2019) consisted of a manipulative-based instructional sequence taught via explicit instruction (Agrawal & Morin, 2016; Bouck, Satsangi, & Park, 2018).

Manipulative-Based Instructional Sequences

Historically, the use of manipulatives in instructional sequences involved concrete manipulatives, such as base-ten blocks, typically presented in the concrete-representational-abstract (CRA) instructional sequence (Agrawal & Morin, 2016). With the CRA model, concrete manipulatives are used first, followed by pictorial representations or drawings, and finally numerical strategies, all taught via explicit instruction (Bouck, Satsangi, & Park, 2018). While the CRA is an evidence-based practice for students with learning disabilities (Bouck, Satsangi, & Park, 2018), significantly less of a research base exists for examining the CRA instructional framework for students with intellectual disability and/or autism. Of the relevant research in this area (see Bouck, Park, & Nickell, 2017; Flores et al., 2014; Stroizer et al., 2015), the results suggest the elementary and middle school students learned the mathematical skill. Bouck, Park, and Nickell (2017) found a functional relation between the CRA and making change with coins for middle school students. Flores et al. (2014) determined elementary students made progress in solving addition and subtraction problems after being taught via the CRA plus a strategic instruction model. Finally, Stroizer et al. (2015) determined a functional relation existed between the CRA and the basic operations mathematics skills of addition, subtraction, and multiplication for elementary students with autism.

Recently, researchers replaced concrete manipulatives with virtual manipulatives in the initial phase of the CRA instructional sequence. The shift toward virtual manipulatives—particularly for secondary students with disabilities—is supported by researchers discussing that virtual manipulatives are more age appropriate and less stigmatizing; past research also suggested student preference toward virtual manipulatives (Satsangi & Bouck, 2015; Satsangi et al., 2016; Satsangi & Miller, 2017). The framework in which virtual manipulatives replace concrete manipulatives is referred to as the virtual-representational-abstract (VRA) or virtual-abstract (VA) instructional sequences. Bouck, Bassette, et al. (2017) explored the efficacy of the VRA instructional sequence in teaching equivalent fractions to three secondary students with disabilities through a single-case design study. Bouck, Park, et al. (2018) also examined the VRA instructional sequence to teach place value and basic operations to two secondary students with mild intellectual disability via a single-case design study. In both studies, researchers found a functional relation between the VRA instructional sequence and student accuracy. However, in the Bouck, Park, et al. (2018) study, the students struggled to maintain their accuracy when instruction did not precede performance.

Despite success in acquisition with regards to the VRA instructional sequence, researchers found students with disabilities struggle when drawing (i.e., pictorially representing) particular mathematical concepts, such as fractions (Bouck, Bassette, et al., 2017). Previously, researchers suggested the representational phase is not necessary for every mathematical concepts, such as area and perimeter (Cass et al., 2003). As such, researchers examined a virtual manipulative-based intervention without a representational phase for particular mathematical areas (i.e., the VA instructional sequence). Bouck, Park, et al. (2017) examined the VA instructional sequence though a single-case design study to teach adding fractions with unlike denominators to three middle school students with disabilities. Although, they found a functional relation between the intervention and student’s accuracy, students did not always maintain the skill. Bouck, Park, et al. (2019) taught three different linear algebra skills to four middle school students with disabilities using the VA instructional sequence and found a functional relation in the single case study. Yet, similarly, students did not always maintain their performance when instruction did not precede performance.

As noted, the research involving the VRA and VA instructional sequences are positive with regards to mathematics acquisition. However, a question worth considering is why the abstract phase—or only numerical strategies—is needed or desirable to achieve for students with intellectual disability and autism. For example, an instructional sequence that helps students develop conceptual understanding via manipulatives (Miller & Hudson, 2007) and transitions to the representation or drawing phase but ends there provides students with a strategy they can use to independently solve mathematical problems (Jitendra & Woodward, 2019; Reuter et al., 2015). Related, Jitendra et al. (2016) found use of representations in mathematics to be an evidence-based practice for students with disabilities.

Learning Stages

While the previous research regarding the VRA and VA instructional sequences for students with autism and intellectual disability demonstrated gains, researchers focused on the acquisition stage of learning. While this is an important focus, further advancement of student understanding and performance is necessary for regular skill use and application. Stages of learning as a concept presents a framework to describe the processes of student learning. The four stages, in order from early to advanced learning, include: acquisition—performing a new skill with some accuracy, fluency—performing a skill correctly with a reasonable speed, maintenance—fluent performance over time, and generalization—application across contexts and variables (Alberto & Troutman, 2009; Collins, 2012; Shurr et al., 2019). To support mathematics learning from initial (i.e., acquisition) to advanced (i.e., maintenance and generalization), researchers can explore intervention packages consisting of manipulative-based instructional sequences and additional research-based practices that support learning beyond acquisition for students with developmental disabilities. Examples of such practices for students with intellectual disability include the system of least prompts, overlearning, and support fading (Shepley et al., 2019; Shurr et al., 2019; Spooner et al., 2019).

The system of least prompts involves prompting a student who fails to respond or responds incorrectly with a series of increasingly intrusive prompts, such as moving from support through providing gestures to verbal support, and followed by physical assistance as needed (Shurr et al., 2019). The system of least prompts is considered an evidence-based practice for students with developmental disabilities (Shepley et al., 2019; Spooner et al., 2012) and particularly for students with developmental disabilities in the area of mathematics (Spooner et al., 2019). Overlearning involves providing students with additional structured learning opportunities with the skill after they become fluent (Alberto & Troutman, 2009; Collins, 2012; Shurr et al., 2019). Support fading, which is naturally embedded in both the CRA and VRA sequences, involves gradually shifting from more support to less support as students increase their performance accuracy (Shurr et al., 2019).

Current Study

For this study, researchers explored the use of an intervention package involving a virtual manipulative-based instructional sequence (i.e., the virtual-representational [VR])—in conjunction with the system of least prompts (SLP) as well as overlearning and support fading—to support the acquisition and maintenance of solving multiplication or division problems by middle school students with intellectual disability and autism educated in a special education mathematics class. This study sought to answer the following research questions: (a) Does a functional relation exist between the use of an intervention package consisting of a virtual manipulative-based instructional sequence (the VR) and the SLP and students’ accuracy in solving multiplication or division problems?; and (b) What are the perceptions of the secondary students toward virtual manipulatives and a virtual manipulative based instructional sequence?

Method

Participants

Four students with developmental disabilities participated in this research study. All four were educated in the same special education middle school mathematics class, taught by the same special education teacher. Students were selected for participation based on teacher recommendation of struggling in mathematics, as well as parental consent and student assent to participate. For this study, the presence of student performance problems in multiplication or division, as measured by the KeyMath-3 and baseline probes, was used to confirm participation. Brandon, Kelly, and Donna has been exposed to virtual manipulatives before but not with multiplication or division or the particular app used; Dylan had no previous exposure.

Brandon

Brandon was a seventh-grade Caucasian male student who was 14-years-of age at the time of data collection. Brandon’s Individualized Education Program (IEP) indicated his eligibility to receive special education services was under the category of intellectual disability. According to the Multidisciplinary Evaluation Team (MET) in his file, Brandon’s intelligence quotient (IQ) was 60 as determined on the Wechsler Preschool and Primary Scale of Intelligence III (WPPSI-III; Wechsler, 2002). Brandon received his mathematics education, language arts instruction, and life skills instruction in the middle school mild intellectual disability program. The researcher-administered KeyMath-3 indicated a grade equivalency of 3.1 for numeration, a 2.8 for basic operations, and a 2.8 for multiplication and division specifically.

Dylan

Dylan was a seventh-grade Caucasian male who turned 14 during the study. He was identified with an intellectual disability in elementary school. On the WPPSI-III (Wechsler, 2002), Dylan’s full-scale IQ was 63. Dylan’s IEP also indicated he had Attention Deficit/Hyperactivity Disorder (ADHD) and Oppositional Defiance Disorder (ODD). He was educated in the secondary program for students with mild intellectual disability at his school for 10 to 15 hr a week, inclusive of mathematics, language arts, and life skills instruction. Dylan’s researcher-administered KeyMath-3 assessment indicated a 2.2 grade equivalency for numeration, a 2.1 for basic operations, and a 1.8 for multiplication and division.

Donna

Donna was a 13-year-old, seventh-grade Caucasian female. Her IEP indicated Individuals with Disabilities Education Act (IDEA) eligibility as Other Health Impairment (OHI). She was evaluated for autism but deemed not to qualify. Donna’s MET file indicated an IQ of 70. Donna spent three class periods a day in the middle school mild intellectual disability program: mathematics, language arts, and life skills. Donna’s KeyMath-3 scores, administered by researchers indicated, a grade equivalence of 2.8 for numeration, 2.9 for basic operations, and 3.2 for multiplication and division. During the course of the study, specifically at the December break, Donna unexpectedly moved to another state; data collection was discontinued but what were collected is reported in the results.

Kelly

Kelly was a 14-year-old, seventh-grade Caucasian female. Per her IEP, Kelly was eligible to receive special education services for autism. Kelly’s IEP file contained no information regarding an IQ test or additional diagnostics, aside from her eligibility determination of autism. Kelly was educated in the mild intellectual disability middle school class for three class periods a day: mathematics, language arts, and life skills. On the researcher-administered KeyMath-3 assessment, Kelly’s scored indicated a 2.5 grade equivalence for numeration, a 2.6 grade equivalence for operations, and a 2.8 grade equivalence for multiplication and division.

Setting

The study occurred at a middle school located in a rural district in a midwestern state. Just over 450 students in grades sixth through eighth attended the school and its special education rate was about 10%. All research sessions occurred outside of the special education mild intellectual disability program classroom, generally at a work table in the hallways shared by two classrooms within that small wing of the building. Researchers worked with students during their mathematics class period (first semester) and life skills period (second semester), due to scheduling changes and teacher preference.

Materials

The materials in the study included the Color Tiles app by Brainingcamp (2018), learning sheets, probes, and a prompting data collection sheet. The virtual manipulative—Color Tiles—was used to solve both multiplication and division problems. The app consists of a whiteboard screen with four different colored color tiles on the left (yellow, red, blue, and green) and then a row of whiteboard markers and an eraser at the bottom. Fifteen learning sheets were used throughout the study; each learning sheet consisted of two problems for modeling, two for guided instruction, and five for independent practice of the targeted mathematics (multiplication or division). Probes were designed to be similar to the independent practice portion of the learning sheet: five targeted mathematic problems. All multiplication problems involved single-digit multiplication (e.g., 3 × 2 or 7 × 4), resulting in a single-digit or double-digit product. All division problems involved a single-digit or double-digit dividend with a single-digit divisor (e.g., 24 ÷ 4 or 6 ÷ 3 Learning sheets were used during intervention and probes during baseline and maintenance—including extended maintenance; each learning sheet was unique, although individual problems were repeated.

Independent and Dependent Variables

The intervention package was the VR instructional sequence and the SLP, in conjunction with overlearning and fading support, resulting in VR+ as the independent variable. The dependent variable was student accuracy in solving multiplication or division problems, as measured out of five possible problems. Depending on each students’ mathematical needs, he or she solved either multiplication or division problems. Accuracy was determined by the correctness of the answer, regardless of any prompts delivered on earlier steps of the task analysis; no prompting as provided on the last step for determining the answer.

Experimental Design

Researchers used a multiple probe across participants design for this study (Gast & Ledford, 2014). Each student started baseline for their respective targeted mathematical skill simultaneously. The first student completed five baseline sessions before starting intervention; to enter intervention, each student’s baseline data needed to be both stable and zero-celerating, at a minimum of the last three baseline sessions. Once the first student achieved 80% accuracy for three intervention sessions, which involved using explicit instruction and virtual manipulatives to solve problems and supported by the SLP, the second student entered intervention. This same pattern continued for the other students, until the fourth student entered intervention. Each student completed a minimum of five sessions of each intervention phase; this reflected overlearning as typically manipulative-based graduated sequences of instruction involved three sessions before transitioning (Agrawal & Morin, 2016). Hence, after a minimum of five sessions with 80% accuracy or higher, each student transitioned from solving the problems with explicit instruction, virtual manipulatives, and the SLP to solving problems with explicit instruction, representations (i.e., drawings), and the SLP, and finally to explicit instruction and representations. This last phase provided the fading of support. After each student completed five sessions in the representational phase without prompting, they were done with intervention and maintenance sessions occurred a minimum of 2 weeks later. Throughout intervention, and continuing into maintenance, each student was probed for maintenance—solving the multiplication or division problems without any explicit strategies or instruction prior to solving—every other session.

Procedures

Researchers conducted all sessions working individually with each participant; during sessions involving interobserver agreement (IOA), two researchers were present. Each student completed a minimum of five baseline sessions, 15 intervention sessions, eight maintenance sessions, and four extended maintenance sessions—two with the virtual manipulative available and two without. Each intervention session lasted less than 15 min; baseline and maintenance sessions went faster. Two researchers delivered the sessions—a special education faculty member whose research and teaching involves mathematics for students with disabilities and an advanced doctoral student, trained by the first author, whose research was also focused on math.

Baseline

During baseline, each student completed a minimum of five probes. For each probe, students answered five multiplication or division questions, depending on their targeted mathematical skill. During the baseline, the app-based manipulative Color Tiles by Brainingcamp (2018) was available, but no instruction was provided. Students received no prompting or assistance during any baseline session.

Intervention

The VR+ instructional sequence is an adaption of the VRA instructional sequence, in which students gradually transition from solving mathematical problems using virtual manipulatives, to drawing (i.e., representations), and finally to solving them abstractly—or just with numerical strategies. The researchers adapted the VRA instructional sequence to better support students with intellectual and other developmental disabilities and be responsive that use of representations (i.e., drawings) is itself an evidence-based practice and beneficial strategy for solving mathematical problems (Jitendra et al., 2016). In other words, the researchers questioned the value of abstract (i.e., no support but numerical strategies) problem solving for students with developmental disabilities. The researchers then added prompting (i.e., the SLP), given research suggesting prompting is an evidence-based practice in teaching academic skills for students with disabilities (Shepley et al., 2019; Spooner et al., 2019).

The VR+ intervention consisted of three phases: virtual manipulative plus SLP, representational plus SLP, and representational. Students completed a minimum of five sessions per phase, for a minimum of 15 sessions of intervention. To transition from one phase to the next, each student had to achieve a minimum of 80% for each session for the five lessons. If a student did not achieve at least 80% accuracy, she or he repeated the lesson the next session. Each session—regardless of phase —involved use of explicit instruction. The researcher started each session modeling two problems, then providing prompts and cues as needed for two problems, and finally allowing the student to solve five problems (see Figure 1). For the virtual manipulative sessions, the researcher modeled how to solve the multiplication or division problems with the Brainingcamp (2018) Color Tiles apps and the student used the app to solve the problems. For the representational sessions, the researcher modeled how to solve the multiplication or division problems with drawings (i.e., circles and lines or Xs; see Figure 1).

Figure 1.

Visual depiction of explicit instruction with the VR instructional sequence.

For the virtual manipulative phase plus the SLP and the representational phase plus SLP, the researcher administered the SLP during the independent portion of the explicit instruction; the use of prompting was faded for the final intervention session of just representation. The SLP involved five levels: independent (i.e., no prompts), gesture (i.e., pointing or referencing without words), indirect verbal (i.e., providing a non-descriptive verbal cue, such as “what do you so now”), direct verbal (i.e., providing a direct verbal of what to do, such as “first you need to draw your groups”), and modeling (i.e., demonstrating how to complete the next step). For each mathematical skill—multiplication or division—a task analysis was created for both the virtual and representational phases of intervention (see Table 1 for the task analyses). Note that the task analyses were used by the researchers to teach the steps of each skill to students as well as to record data. For division, the virtual manipulative phase involved seven steps and for the representational six; the difference in steps involved a clear the screen step for the virtual manipulatives that was not needed with drawings with paper and pencil. The multiplication skill involved six steps for the virtual manipulative phase and five for the representational phase, the difference again being the clear step for the app. The researcher applied the SLP for each step of the task analysis during the independent portion of explicit instruction for the first two phases of intervention after a 10-s time delay or when a student made an error. The only error not corrected was a counting error, meaning the student was counting all representations or tiles she or he had drawn or pulled out on the app and she or he had the correct number, but made a counting error. Hence, a student’s accuracy could be less than 100%.

Table 1.

Task Analysis for Multiplication and Division.

Steps	Multiplication	Division
1	V and R: State the problem, putting into mathematical terms (e.g., 4 × 3 is “I have 4 groups of 3”) [OPTIONAL]	V and R: State the problem, putting into mathematical terms (e.g., 16 ÷ 4 is that I have 16 objects and want to evenly divide them among 4 groups) [OPTIONAL]
2	V and R: Draw the number of groups	V: Pull out the number of tiles needed for the dividend (e.g., 16) R: Draw the number of objects
3	V and R: Drag out or draw the number of objects in each group	V: Draw the number of groups (e.g., circle) to divided into (e.g., divisor; 4) R: Draw the number of groups (i.e., circles) to be divided into
4	V and R: Check that there is the correct # of group and each group has the same # of objects/tiles	V: Equally distribute the tiles into each group (i.e., circle)—putting one in each group before putting multiple in any one group R: Equally distribute the objects into each group (i.e., circle)—putting one in each group before putting multiple in any one group
5	V and R: Add all the tiles together (can use count all)	V: Check that there are no leftover tiles and there are tiles in each group/circle R: Check that there are no leftover objects and there are tiles in each group/circle
6	V and R: Write down answer	V and R: Count the number of tiles/objects in one group
7	V: Clear	V and R: Write down answer
8		V: Clear

Note. V = virtual; R = representational.

Virtual manipulative plus prompting

The first intervention phase was virtual manipulative plus SLP; students completed a minimum of five sessions. Students used the Brainingcamp (2018) Color Tile app to solve multiplication or division problems and researchers applied the SLP to the independent portion of explicit instruction following the modeling and guiding of two problems each with the app (see Figure 1). To model and guide how to use the app to solve multiplication and division problems, the researcher began by discussing what it meant to do multiplication or division, respectively (e.g., that 16 ÷ 4 means I have 16 objects and want to evenly divide them among 4 groups and that 4 × 3 means I have 4 groups and 3 objects in each group). With multiplication, the researcher physically demonstrated, while providing a verbal narration, 4 × 3, for example, by drawing four circles on the white screen of the Color Tile app with a marker within the app and then by dragging out three tiles into each of the circles. The researcher modeled adding all of the tiles within the circles and/or skip counting all the tiles. With division, the researcher also physically demonstrated while providing a verbal narration, 16 ÷ 4, for example, by first dragging out 16 tiles of any color to the top of the app whiteboard space. Next, the researcher drew four circles and then dragged tiles one-by-one into each circle, distributing a tile into each circle before adding additional tiles.

Representational plus prompting

The second intervention phase was representational plus SLP; students also completed a minimum of five sessions. During the representational plus prompting phase, students were taught to draw pictures to represent multiplication or division, using circles and Xs or lines, depending on preference. Researchers implemented the SLP during the independent portion of explicit instruction, following the modeling and guiding of two problems each. As with the virtual phase, each session began with a discussion of what it meant mathematically to multiply or divide. With multiplication, the researcher physically demonstrated, while providing a verbal narration, 5 × 6, for example, by drawing five circles on the paper with a pencil and then by drawing six Xs (or lines) into each of the circles. The researcher then demonstrated adding all of the Xs (or lines) in all the circles and/or skip counting. With division, the researcher also physically demonstrated while providing a verbal narration, 20 ÷ 5, for example, by first drawing 20 Xs (or lines) on the paper with a pencil. Next, the researcher drew five circles underneath and then distributed the Xs (or lines) one-by-one into each circle, distributing one into each circle before adding additional ones, and crossing out each X (or line) at the top after distribution.

Representational

The third, and final intervention phase was representational; students also completed a minimum of five sessions. During the representational phase, instruction was similar to the representational plus SLP phase. The only difference was that the SLP was not implemented during the independent portion of explicit instruction.

Maintenance

Maintenance probes were conducted during intervention. During maintenance, students solved five problems—multiplication or division—but instruction was not provided before and students were not provided with the app. Maintenance sessions began before the second intervention session (i.e., after one intervention session was completed) and occurred every other session; maintenance probes were always conducted prior to intervention sessions and students each completed eight.

Extended Maintenance

All students completed four maintenance sessions, which occurred at a minimum of 2 weeks after the last intervention session. The researchers felt 2 weeks provided an opportunity to determine if students could maintain their learning but not drag out the study. For two of the extended maintenance probes, students were given access to the app-based manipulative but were allowed to solve the five problems with whatever method he or she desired (i.e., app, representational, numerically). For two other extended maintenance probes, the app-based manipulative was not made available. No prompts or instruction were provided during maintenance sessions.

IOA and Procedural Fidelity

Researchers calculated both IOA and procedural fidelity during the study. IOA for accuracy was conducted for 20% of baseline, intervention (each phase), and maintenance sessions and 50% for extended maintenance sessions. To determine accuracy IOA, a second member of the researcher team scored the probes; the original researcher scored each one during the session. IOA for prompts was also conducted for 20% of virtual and SLP and representational and SLP sessions. To determine prompting IOA, two researchers were present for those sessions and each recorded the SLP on a data collection sheet (available upon request). Researchers calculated IOA by dividing the number of agreements by the sum of the number of agreements and disagreements for both accuracy and independence. Accuracy IOA was 100% for each student for each phase. Independence IOA was 100% for each student for each for both intervention phase with the exception of Donna, which was 97.1% for the virtual and SLP phase. Researchers determined procedural fidelity via a checklist. For a minimum of 20% of intervention sessions per phase, researcher checked if the intervention was implemented as written. Specifically, for Brandon, Donna, and Kelly, procedural fidelity was 20% (three intervention sessions), and 30% for Dylan (six intervention sessions). Procedural fidelity was 100% for all students for both phases.

Social Validity

The researchers conducted social validity interviews regarding the intervention. Following the conclusion of the intervention phase of the study, each student was interviewed by one of the researchers. The researcher asked the students about the preference of phases as well as their perception of the app-based manipulative and the importance of learning multiplication or division. The researcher also asked the teacher regarding the mathematical performance of each student and the generalization of the mathematics skills and strategies (e.g., representational) into the mathematics classroom.

Data Analysis

To analyze the data, researchers graphed data and then conduct visual analysis as well as calculated metrics. With the analysis, researchers determined level, trend, immediacy of effect, and effect size. Level was found for baseline, intervention, and maintenance by calculating if 80% of the data for each phase fell within 25% of the median (Gast & Spriggs, 2014). Researchers used the split-middle method to calculate trend, meaning researchers found the middle point, mid-rate, and mid-date for the baseline, intervention, and maintenance phases. Researchers then determined if the trend was accelerating, decelerating, or zero-celerating by drawing a line between the mid-rate and mid-date (White & Haring, 1980). Immediacy of effect was determined by comparing the last baseline phase to the first intervention phase. Researchers selected Tau-U as the measure of effect size and used an online calculator to compute (Vannest et al., 2016). Using the metrics provided by Vannest and Ninci (2015), researchers interpretation the Tau-U as a very large effect if greater than 0.80, large effect between 0.60 and 0.0, and a moderate effect between 0.20 and 0.60.

Results

For the three students who were able to complete the entire study, a functional relation was determined between the intervention package—the VR instructional sequence plus (VR+)—and students’ accuracy in solving multiplication or division problems (see Figure 2 and Table 2). Students acquired and maintained their targeted mathematical skill.

Figure 2.

Student accuracy in multiplication and division across sessions.

Table 2.

Data Analysis Summary of the VR + P Intervention Package Across Participants.

Overall			Intervention			Maintenance
Measure	Baseline	Intervention	V + SLP	R + SLP	R	Maintenance
Brandon
Range	0%	80–100%	80–100%	100%	100%	0–60%
M	0%	98.7%	96%	100%	100%	22.5%
Stability	S	S	S	S	S	V
Trend	Z	Z	Z	Z	Z	A
Tau-U^a	—	1.0	—	—	—	0.5
Sessions	5	15	5	5	5	8
Repeated	—	0	0	0	0	—
Dylan
Range	0%	60–100%	60–100%	60–100%	69–100%	0–100%
M	0%	80%	90%	86.7%	77.5%	47.5%
Stability	S	S	S	S	S	V
Trend	Z	D	Z	Z	Z	A
Tau-U^a	—	1.0	—	—	—	0.625
Sessions	7	20	6	6	8	8
Repeated	—	5	1	1	3	—
Donna
Range	0–20%	100%^a	100% ^a	100% ^a	—	0–40% ^a
M	15%	100% ^a	100% ^a	100% ^a	—	20% ^a
Stability	V	S^a	S^a	S^a	—	V^a
Trend	D	Z^a	Z^a	Z^a	—	A^a
Tau-U^b	—	—	—	—	—	—
Sessions	8	9^a	5	4^a	—	4^a
Repeated	—	0	0	0	—	—
Kelly
Range	0–20%	80–100%	100%	100%	80–100%	0–100%
M	6.7%	98.7%	100%	100%	96%	87.5%
Stability	V	S	S	S	S	S
Trend	A	Z	Z	Z	Z	Z
Tau-U^a	—	1.0	—	—	—	0.833
Sessions	9	15	5	5	5	8
Repeated	—	0	0	0	0	—

Note. VR = virtual-representational; V = variable; SLP = system of least prompts; S = stable; Z = zero-celerating; A = accelerating; D = decelerating; P = prompting.

Only completed nine intervention sessions before moving. ^bDenotes Tau-U between baseline and intervention (overall).

Brandon

Brandon answered zero division questions correctly during baseline; his baseline data were stable and zero-celerating. Brandon experienced an immediate effect in terms of accuracy from his last baseline session (0%) to his first intervention session (80%). Brandon’s average accuracy during the virtual and SLP intervention phase was 96%; he repeated zero sessions. Brandon also repeated zero sessions during the representational and SLP phase; he achieved 100% accuracy for all five sessions. Brandon maintained 100% accuracy for all the representational phase sessions and also repeated zero sessions. Brandon’s intervention data across the phases were zero-celerating and stable. The Tau-U between baseline and intervention was 1.0. In terms of maintenance, Brandon struggled to a greater extent. Although his first three maintenance sessions were 0% accuracy, his maintenance data had an accelerating trend. Brandon’s extended maintenance average was 90% across the four sessions. Although provided for two sessions, Brandon chose not to use the virtual manipulatives during maintenance, rather he drew representations of the problems (e.g., circles and lines).

In terms of independence, Brandon’s average was 96%. His first intervention session—virtual manipulative and SLP—was his lowest (80.6%). For this session, Brandon was prompted across four different steps, including use of gestures, indirect verbal, and direct verbal. All subsequent intervention sessions were 90% or greater for independence, including five sessions with 100% independence—three during the representational and SLP intervention phase.

Dylan

Dylan answered zero multiplication questions correctly across his six sis baseline sessions; his baseline data were stable and zero-celerating. Dylan experienced an immediate effect in terms of accuracy from his last baseline session (0%) to his first intervention session (80%). Dylan’s average accuracy during the virtual and SLP intervention phase was 90%; he repeated one session. Dylan also repeated one session during the representational and SLP phase; his average was 86.7% during this phase. Dylan’s average accuracy decreased during the representational phase (77.5%) and he completed the same session four time before achieving 80%. When Brandon answered questions incorrectly, it was always a counting error, such as putting out the correct number of tiles or drawing the correct number of lines to represent the multiplication problem but then counting the tiles or lines incorrectly. The Tau-U between baseline and intervention was 1.0 and his overall intervention data were variable with a decelerating trend. In terms of maintenance, Dylan struggled at the beginning during the virtual + prompting intervention phase as well as during the representational phase. His average accuracy for maintenance was 47.5%, although he did have three sessions with 100% accuracy during intervention. Dylan’s extended maintenance average was 65% across the four sessions. When the app was provided, he did worse. Both times he chose to use the virtual manipulatives and attempted to do division rather than multiplication. Without using the app, he scored 100%.

Dylan’s average independence was 99%. He was 100% independent for eight sessions, including for all sessions in the virtual and SLP phase. When he needed prompting—all sessions occurring in the representational and SLP phase, Dylan’s prompts were generally with drawing the number of objects needed for the dividend and generally just a gesture was sufficient.

Donna

Donna baseline data were decelerating; she answered zero division questions correctly her first and last baseline session (session 8), but one question correctly for the other baseline sessions. Donna experienced an immediate effect in terms of accuracy from her last baseline session (0%) to her first intervention session (100%). For the nine intervention sessions Donna completed prior to moving away during the winter break, she answered all problems correctly. Hence, her averaged 100% for the five virtual and SLP intervention sessions she completed and 100% for the four representational and SLP intervention sessions she completed. In terms of maintenance, Donna’s average for the four maintenance sessions she completed was 20%.

Donna’s average independence across the nine intervention sessions she completed was 96.1%. When she needed prompting, Donna’s prompting was generally related to the first step of the division task analysis—pulling out or drawing the number of tiles (or objects) for the dividend. Most of the time, a gesture was a sufficient prompt.

Kelly

Kelly’s average accuracy across her nine baseline sessions was 6.7%; her accuracy on division problems was 0% for her first five sessions and then we answered one correct for three of her last four baseline probes. Kelly experienced an immediate effect in terms of accuracy from her last baseline session (20%) to her first intervention session (100%). Kelly’s average accuracy during the virtual and SLP intervention phase as well as the representational and SLP phase was 100%; she repeated zero sessions in either phase. She also repeated zero sessions in the representational phase; her average accuracy was 96%, as she incorrectly answered one division problem on one probe. The Tau-U between baseline and intervention was 1.0 and her overall intervention data were stable with a zero-celerating trend. After the first maintenance session, Kelly’s accuracy was 100%, resulting in an average accuracy for all eight maintenance sessions of division. Kelly’s extended maintenance was 100% with or without the app.

Kelly’s average independence was 99.5%. After the first intervention session in which Kelly’s independence was 95.2%, she was 100% independent. For the one session in which Kelly needed prompting (i.e., first virtual), the prompting was for the first step.

Discussion

This study explored an intervention package consisting of a manipulative-based instructional sequence involving virtual manipulatives and then drawings, the SLP, explicit instruction, overlearning, and support fading to help students with intellectual disability and autism acquire and maintain multiplication and division problems. The results of the single-case design study suggest the middle students acquired the skills as well as maintained. All three students who completed the intervention package drew pictorial representations to solve the problems when instruction did not immediately precede the probe.

Previous research regarding virtual manipulative-based instructional sequences found students with intellectual disability and/or autism struggle with maintaining the mathematics skills after the intervention ends (Bouck, Bassette, et al., 2017; Bouck, Park, et al., 2018). This is problematic for multiple reasons, including that mathematical concepts build upon each other as well as such skills are needed in daily living (Test et al., 2009). Hence, interventions need to focus on students not just acquiring skills or knowledge but also maintaining. This study suggests a comprehensive intervention package involving a virtual manipulative-based instructional sequence, overlearning, explicit instruction, and the SLP aided in students maintaining the skills of solving multiplication and division problems.

Implications for Practice

One implication for practice for this study is the value in using intervention packages. The participants in this study maintained their accuracy after intervention ended and during intervention when instruction was not provided. Students in previous studies involving virtual manipulatives struggled with maintenance (c.f., Bouck, Bassette, et al., 2017; Bouck, Park, et al., 2018), suggesting simple interventions may not be sufficient for students with intellectual disability and autism. All the students in the study had previously been taught multiplication and division, and yet they still struggled to independently solve these problems during baseline. Interventions to target mathematics learning for secondary students with intellectual disability and autism need to be comprehensive, involving strategies for acquisition and maintenance.

Another implication involves the focus on ending the intervention in the representational phase. Previous manipulative-based instructional sequence research exploring the CRA, VRA, or VA all ended the intervention in the abstract phase. In this study, the researchers challenged the need to have middle school students with intellectual disability and autism rely on numerical strategies to solve multiplication or division problems rather than other strategies, such as drawing pictures. Previous researchers suggested students with learning disabilities apply less advanced strategies when solving problems, such as using representations rather than numerical strategies (Geary et al., 2007; Xin et al., 2016). Rather than being viewed as a negative (c.f., Xin et al., 2016), the use of drawings to solve problems independently should be viewed positive for students with intellectual disability and autism. It may not be an efficient strategy with more complex multiplication and division; however, a calculator can be provided to secondary students if they demonstrated a conceptual understanding of the process, even if not a memorization (Woodward et al., 1999). Jitendra et al. (2016) found representations in mathematics to be an evidence-based practice for students with disabilities. Teaching students with intellectual disability and autism to draw pictures to solve problems provides them with a strategy they can apply whenever they face problems (Krawec, 2014; van Garderen, 2007). The students in this study clearly acquired and maintained the strategy of drawing representations to solve multiplication or division problems.

A final implication involves the use of technology, particularly socially acceptable and desirable technology (i.e., iPad; Bouck et al., 2012). As previous researchers suggested concrete manipulatives may be stigmatizing to secondary students with disabilities, teachers’ use of virtual manipulatives—even through an instructional sequence—can meet students where they are at in terms of interest and motivation (Satsangi & Bouck, 2015). Although concrete manipulatives and concrete manipulative–based instructional sequences are effective (see Bouck, Satsangi, & Park, 2018; Carbonneau et al., 2013), secondary students may experience increased engagement and interest with virtual manipulative (Satsangi & Bouck, 2015).

Limitations and Future Directions

One limitation of this study was the inability to determine what particular elements of the intervention packaged contributed to student success. The intervention package was comprehensive but perhaps given the length of the intervention, some components were superfluous. Researchers should determine if all elements of the intervention package contribute, or if there are superfluous ones, would removing as such make the intervention just as effective but perhaps more efficient. A more efficient intervention would result in the same benefit to learning but perhaps provide greater time to address more mathematics. Another limitation was that the dependent variable of accuracy only truly reflected students’ ability to complete the last step of the task analysis—excluding clearing the screen for the virtual phase—correctly (i.e., write down the answer). In future studies, researchers should set mastery as being able to complete all task analysis steps of each problem independently and accurately.

Another limitation was that Donna did the complete the intervention. While out of the researchers’ control, as she moved with only 1 week notice prior to winter break, she was the one student who had failed to make progress on maintenance sessions (i.e., performing the mathematics without instruction). A complete intervention phase with Donna would have allowed researchers to evaluate if with additional exposure and fading of supports in the representational phase, Donna was able to solve the division problems without instruction. Future researchers should seek to replicate this study as well as systematically replicate with students with other disabilities or via other modes, such as small group settings or delivered by a teacher.

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.

ORCID iDs

Emily C. Bouck

Jordan Shurr

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