Using the Virtual-Abstract Instructional Sequence to Support Acquisition of Algebra

Abstract

Although often considered a more advanced area of mathematics, principles of algebra are taught throughout different mathematical concepts, and algebra is often considered gateway mathematical knowledge for more advanced mathematical concepts. For this reason, attention is needed toward making algebraic instruction more accessible to all types of learners, including students with disabilities who often struggle learning mathematics. Using a multiple probe across behaviors replicated across participants single-case design, this study examined whether an intervention sequence consisting of a virtual manipulative and then abstract (i.e., numerical strategies) instruction could support the acquisition of three algebraic behaviors (i.e., one-step division, two-step addition, two-step subtraction, and/or three-step addition) for four middle school students with disabilities. All four students acquired each of the linear algebra behaviors but struggled to maintain their learning once instruction was not provided prior to performance. These findings and their implications are discussed further.

Keywords

math content/curriculum area middle school age/grade level methodologies instructional technology technology perspectives tablets/iPad

Algebra is more than just a mathematics class in secondary school. Algebra is a fundamental mathematical concept, applicable to almost every individual as well as used throughout one’s daily life (Root, Browder, & Jimenez, 2016). Learning algebra develops problem-solving skills, which are applicable to both advanced mathematics and daily living skills (e.g., grocery shopping; Browder et al., 2018; Root et al., 2016). Although often considered a more advanced mathematical concept, principles of algebra are commonly taught early in children’s mathematical careers and embedded within elementary-level mathematics standards (The Common Core State Standards; Council of Chief State School Officers & National Governors Association Center for Best Practices, 2010). For this reason, attention must be directed toward making algebraic instruction more accessible to all types of learners, including those who often struggle learning mathematics such as students with high-incidence disabilities.

The literature on teaching algebra to students with disabilities is typically focused on two populations—teaching algebra to students with learning disabilities and teaching algebra to students with moderate and severe intellectual disability. In a review of algebra instruction to students with learning disabilities, Watt, Watkins, and Abbitt (2016) found 15 articles published between 1980 and 2014. Watt et al. identified five primary interventions for providing algebra instruction across the 15 studies: the concrete–representational–abstract (CRA) instructional sequence, enhanced anchored instruction (EAI), graphic organizers, cognitive strategy instruction, and tutoring. Of these five interventions, the most frequently examined method of providing algebra instruction was the CRA sequence. In analyzing effect sizes, Watt et al. determined all five interventions resulted in moderate or large (i.e., CRA [g = 0.53], EAI [g = 0.80], tutoring [g = 0.40], graphic organizers [g = 0.57], and cognitive strategy instruction [g = 0.83]) effects. The overall effect size for the 15 studies reviewed was g = 0.48. While an exclusive systematic review of algebraic interventions for students with moderate and severe disabilities does not exist, Hudson, Rivera, and Grady (2018) found 10 studies published between 2006 and 2016 focused on teaching algebra for this population and identified the following approaches were used to teach algebra: task analysis, prompting systems (i.e., simultaneous prompting, system of least prompts, constant time delay), and graphic organizers.

The CRA Instructional Sequence

The CRA instructional sequence is an evidence-based practice for teaching mathematics to students with learning disabilities (Bouck, Satsangi, & Park, 2018) and focuses on teaching students mathematical concepts in a three-step sequential order, beginning with concrete models (i.e., manipulatives), followed by representational drawings, and concluding with an abstract notation of the idea and mathematical strategies. Although the literature on the CRA sequence has focused mostly on students with learning disabilities, researchers also examined the use of this strategy to support students with other disabilities in mathematics. For example, Stroizer, Hinton, Flores, and Terry (2015) and Flores, Hinton, Stroizer, and Terry (2014) effectively taught elementary students with autism to solve computation problems involving basic operations using the CRA approach. At the secondary level, Bouck, Park, and Nickell (2017) taught middle school students with mild intellectual disability and learning disabilities to solve mathematics problems focused on making change with coins via the CRA instructional sequence.

In algebra, multiple researchers have examined the CRA instructional sequence for students with disabilities. Maccini and Hughes (2000) and Maccini and Ruhl (2000) both found positive effects in single-case studies using the CRA instructional sequence with algebra tiles in unison with a problem-solving strategy to teach secondary students with learning disabilities to represent and solve problems with integers. Strickland and Maccini (2012) explored a modification to the CRA instructional sequence in which researchers concurrently, as opposed to sequentially, exposed students to the concrete, representational, and abstract phases when teaching them to solve multiplication of linear expressions and garnered positive results. Witzel (2005) and Witzel, Mercer, and Miller (2003) examined the CRA instructional sequence through group design studies with middle school students with disabilities. Using sticks as their concrete manipulatives, students in the CRA group improved to a greater extent than students in the non-CRA traditional instructional group in both studies. Both studies by Witzel examined effect size through a point biserial, finding instruction type (e.g., CRA vs. non-CRA) 24.5% and 56.3% of posttest variance. The effect size for the group who received the CRA instruction in the Witzel (2005) study was g = 0.43.

The Virtual–Representational–Abstract (VRA) and Virtual–Abstract (VA) Instructional Sequences

While the CRA is an effective intervention, researchers recently explored adaptations to the CRA for secondary students with disabilities. Research sought to determine whether an adaptation involving virtual manipulatives in place of concrete manipulatives would be effective based on concern of the potential stigmatization of concrete manipulatives for secondary student as well as the potential for virtual manipulatives to provide additional feedback and scaffold for students (Bouck, Working, & Bone, 2018). Based on research suggesting students with disabilities were equally effective in using concrete and virtual manipulatives to solve mathematical problems as well as generally report a preference for virtual manipulatives over concrete manipulative (e.g., subtraction, fraction equivalence, algebra; Bouck, Chamberlain, & Park, 2017; Bouck, Shurr et al., 2018; Satsangi, Bouck, Doughty, Bofferding, & Roberts, 2016), researchers examined the VRA instructional sequence. The VRA instructional sequence, similar to the CRA sequence, is a gradual sequence of instruction in which students transition from using virtual (in place of concrete) manipulatives, to representational drawings, and finally to abstractly solving mathematical problems. All phases are taught using explicit instruction, incorporating modeling, guiding or cueing, and independent practice (Agrawal & Morin, 2016; Bouck & Sprick, 2019).

The research based regarding the VRA is emerging; to date, researchers have explored the VRA instructional sequence to support student acquisition of such mathematical concepts as factions and basic operations (Bouck, Bassette, et al., 2017; Bouck, Park, Shurr, Bassette, & Whorley, 2018). Although researchers found success with the VRA instructional sequence, it was also noted that some mathematical concepts are challenging to represent pictorially when students are responsible for drawing all components. Specifically, Bouck, Bassette, et al. (2017) found middle school students with disabilities struggle with the representational phase when solving equivalent fractions. Specifically, the students struggled to draw the fractions and repeated multiple sessions within the representational phase. In a subsequent study, Bouck, Park, et al. adapted the VRA instructional sequence to a VA instructional sequence, removing the representational phase. The researchers found a functional relation between the VA instructional sequence and students’ acquisition of the mathematical behavior of adding fractions with unlike denominators. The students were successful with both virtual and abstract phases, suggesting perhaps the representational phase was not needed for all mathematical concepts. Cass, Cates, Smith, and Jackson (2003) previously successfully removed the representational phase within the CRA instructional sequence when teaching students to solve area and perimeter problems, concluding the representational phase may not be essential.

Algebra Instruction

Researchers found multiple interventions effective in supporting algebra instruction for students with disabilities (Watt, Watkins, & Abbitt, 2016). What makes the CRA, VRA, and VA instructional sequences effective and desirable is that they allow for the application of effective teaching practices relative to algebra, such as representing the problem (Allsopp, van Ingen, Simsek, & Haley, 2016), while being paired with evidence-based strategies for students with disabilities (e.g., explicit instruction; Agrawal & Morin, 2016). While researchers suggest the CRA instructional sequence is effective in supporting student acquisition of algebraic concepts, limited research to date examines the VRA or VA instructional sequences for teaching algebra. Yet, the VRA or VA instructional sequences may offer advantages over the CRA approach through their very use of virtual manipulatives. Researchers argued virtual manipulatives might be advantageous over concrete manipulatives, especially for secondary students (Satsangi et al., 2016). Virtual manipulatives are socially desirable (i.e., not stigmatizing for older students, such as those in middle school) and often come with built-in constraints or supports that are not present in concrete manipulatives such as preventing errors (i.e., inability to ungroup from a subtrahend; Bouck, Working, et al., 2018). Additionally, virtual manipulatives can reduce students’ cognitive load to a greater extent than concrete manipulatives (Suh & Moyer, 2007). Virtual manipulatives, in contrast to concrete manipulatives, can provide two representations on the same space, such as providing the virtual manipulative (e.g., base 10 blocks, algebra tiles) as well as the problem (e.g., 39 + 12 or 2x + 2 = 10). Concrete manipulatives are not naturally able to provide these parallel perspectives (Suh & Moyer, 2007).

When comparing the VRA and VA instructional sequences in the context of advanced mathematics such as algebra, educators must question if it is essential for students to draw representations of the mathematical concepts (e.g., variables and constants in equations). The authors in this study, based on previous research suggesting the representational phase may not be necessary (Bouck, Park, et al., 2017; Cass, Cates, Smith, & Jackson, 2003), determined it more appropriate to teach students linear algebra via the VA instructional sequence. The research questions for this study included the following: (a) to what extent do middle school students with disabilities acquire linear algebra skills using the VA sequence? (b) to what extent do middle school students with disabilities maintain their performance on solving linear algebraic problems? and (c) to what extent do middle school students with disabilities find the VA sequence beneficial?

Method

Participants

Four middle school students with various disabilities participated in the study. The four were selected because they all came from the same special education program, were educated in the same mathematics and language arts classes by the same special education teacher, and all were not proficient in linear algebra. The state in which the study occurred licensed teachers by disability categories and special education programs is referred to by such categories of licenses (e.g., mild cognitive impairment) rather than a cross-categorical. In the program, which consists of students with a range of disabilities, the special education teacher delivered core content instruction; she did not support general education instruction (i.e., not a resource room).

The inclusion criteria for the study included (a) identification of a student with a disability, (b) identified by their teacher as a student who is not proficient in grade-level mathematics, (c) adequate fine motor skills to operate a touch screen, and (d) inability to solve linear algebra problems as demonstrated on a KeyMath-3 assessment as well as baseline assessments. Each student was administered four subtests of the KeyMath-3: numeration, mental computation and estimation, addition and subtraction, and multiplication and division.

Kya

Kya was a 14-year-old, eighth-grade, White female, who was identified with an intellectual disability on her Individual Education Program (IEP). From the assessment data available, Kya’s full-scale IQ was 53 from the Wechsler Intelligence Scale for Children-Fourth Edition (WISC-IV; Wechsler, 2004). Her composite achievement scores on the Kaufman Test of Educational Achievement, Second Edition (KTEA-II; Kaufman & Kaufman, 2004) were 73 for mathematics and 72 for reading, both below average. On the Vineland-II adaptive behavior assessment (Sparrow, Cicchetti, & Balla, 2005), both Kya’s parent and teacher, at the time, rated her in the low range. Kya was served in a secondary mild cognitive impaired (i.e., the term for intellectual disability in the state) program for 10 hr per week; she took mathematics and language arts in a special education setting taught by a special education teacher. On the KeyMath-3 (Connolly, 2007) administered by researchers, Kya’s numeration score was 20 (4.1 grade equivalency), and her total operations score was 36 (3.4 grade equivalency).

Sadie

Sadie was a 13-year old, eighth-grade, White female. Her IEP indicated she was currently eligible for special education services under the category of other health impairment, specifically for Attention-Deficit/Hyperactivity Disorder (ADHD). From her most recent WISC-IV assessment (Wechsler, 2004), Sadie’s full-scale IQ was 85. On the Woodcock-Johnson-III Normative Update, Sadie scored in the low average for math calculations and low standard in mathematics reasoning. During the academic year of the study, Sadie was served in a secondary mild cognitive impaired program for 10 hr per week; she too had her mathematics and language arts instruction delivered in the special education setting by a special education teacher. On the researcher-administered KeyMath-3, Sadie’s numeration subtest score was 17 (3.2 grade equivalency), and her total operations score was 33 (3.1 grade equivalency).

Eli

Eli was a 15-year-old, eighth-grade, White male. He received special education services through the eligibility of intellectual disability. According to the most recent WISC-IV administered (Wechsler, 2004), Eli’s full-scale IQ was 70. The KTEA-II (Kaufman & Kaufman, 2004) indicated Eli’s mathematics achievement was in the lower extreme (a composite score of 66). On the Vineland-II (Sparrow, Cicchetti, & Balla, 2005), Eli was rated in the low or moderately low levels by his teachers. Like the others, Eli received language arts and mathematics in a secondary mild cognitive impaired program; he received instruction in these areas by a certified special education teacher. Eli’s KeyMath-3 numeration score was 20 (4.1 grade equivalency), and his total operations score was 36 (3.4 grade equivalency).

Cole

Cole was a 13-year-old, seventh-grade, White male. He was eligible to receive special education services through the identification of a learning disability, specifically in reading, reading comprehension, and mathematics calculation. Cole’s IEP also indicated he struggled with ADHD and Obsessive-compulsive disorder (OCD). On his most recent administered WISC-IV (Wechsler, 2004), Cole’s full-scale IQ was 71. At the same time, the TEMA-3 indicated he was in the fifth percentile for mathematics. Consistent with the other students in this study, Cole received his language arts and mathematics instruction in a secondary mild cognitive impaired program. On the KeyMath-3 administered by the researchers, Cole’s numeration score was 20 (4.1 grade equivalency) and his total operations score was 46 (4.2 grade equivalency).

Setting

The setting for this study was a public middle school located in a small town in a Midwestern state. As assessed by the state’s fall count, 452 sixth-grade through eighth-grade students were enrolled in the school. According to the most recent data available for the district (2015–2017 academic year), 95% of the students identified as White, 3% Latino/Latina, and 2% two or more races. At the district level, 10.0% as students with a disability and 27.6% of students were identified as economically disadvantaged. The district’s graduation average was 85.2%.

Materials

The following materials were used in this study: learning sheets, pencil, and an app-based algebra tiles manipulative. Each learning sheet consisted of three pages to represent the modeled, guided, and independent portions of each lesson, consistent with explicit instruction and CRA, VRA, and VA administration. Each learning sheet was unique to the algebraic behavior examined. For each lesson, there were two problems for the researcher to model, two problems for the researcher to cue a student as she or he completed them, and five independent problems. The learning sheets were each printed on standards 8 in. × 11.5 in. paper. Students used a pencil to write their answer to each problem on the learning sheet.

To create the learning sheets, the researchers determined all possible problems for each algebra behaviors examined, given the following parameters: (a) the coefficient was 6 or less, (b) no more than 40 ones tiles were needed for the one-step division algebra problems (e.g., 4x = 8), and (c) no more than 16 ones blocks were needed for the two-step addition or subtraction (e.g., 2x + 2 = 10 and 3x − 2 = 7) and three-step addition algebra problems (e.g., 5x + 3 = 4x + 4). These constraints were added to reduce the complexity of setting up the virtual manipulatives (i.e., control for the number of tiles on screen), while still allowing students to explore a range of problems. Each individual problem was randomly assigned to a maximum of three learning sheets, although researchers ensured the independent portion of each learning sheet was unique.

For all algebra behaviors examined, students used the Algebra Tiles app (Brainingcamp, 2018; see Figure 1). On the left side of the app contains tiles to solve problems: 1 cube, −1 cube, x rectangle, −x rectangle, and larger x ² and −x ² cubes; for the purposes of this study, only the 1, −1, x, and −x tiles were used. The app allows the screen to be broken into two sides, divided by an equal sign to create the equation. To use, a user pulls out the correct tiles onto the screen and places them on their respective side of the equal sign to represent the equation. For example, for the equation 2x + 2 = 10, on the left side of the equal sign, a user would pull out two x rectangles and two 1 cubes, and on the right side of the equal sign, one would pull out 10 individual 1 cubes. A user would then solve by adding two −1 cubes to both sides, resulting in the equation 2x = 10. The app then allows a user to create rows—in this case, two rows—and evenly distribute the x rectangles on the left side into the rows and the 1 cubes into the right, resulting in an answer of x = 5. Note, the virtual algebra tiles—from the app used in this study—are similar to existing concrete algebra tiles, in terms of function and form (i.e., color, shapes). Some differences between the concrete and virtual algebra tiles include that a teacher would need to secure or make an eqution map, as depicted with the divided sides and equal sign in the app (see Figure 1). Also, there are unlimited pieces with the app, while there is a finite number of each pieces with the concrete option. Finally, the virtual manipulative algebra tile pieces are labeled (e.g., x, −1), while the concrete ones are not naturally. A teacher could write on each pieces, but that is not how they are purchased. Hence, a student needs to remember what is the value of each piece.

Figure 1.

Screenshot of Algebra tiles app (Brainingcamp, 2018).

Independent and Dependent Variables

The VA instructional sequence served as the independent variable for this study. During the virtual (V) phase, students used the Algebra Tiles app from Brainingcamp (2018) to solve one-step division, two-step addition, two-step subtraction, and/or three-step addition linear algebra problems. In the abstract (A) phase, the students solved the same types of problems as presented in the (V) phase but without the app (i.e., numerically). The dependent variable represented problem-solving accuracy, more specifically the raw number of linear algebra problems solved correctly during the independent portion of the learning sheet out of five.

Experimental Design

This study used a multiple probe across behaviors replicated across participants single-case design to examine the functional relation between the VA instructional sequence and student performance in solving linear algebra equations (Gast & Ledford, 2014). A multiple probe was selected to reduce the number of baseline sessions, especially for the later behaviors for each student (Gast & Ledford, 2014). The study was designed following the Council for Exceptional Children’s (2014) quality indicators and standards for single-case design studies (see also Cook et al., 2015).

Three of the students’ first behavior was solving one-step division algebra equations (e.g., 5x = 10), followed by two-step with addition algebra equations (e.g., 2x + 2 = 10) and then two-step with subtraction algebra equations (e.g., 2x − 2 = 10). Kya was proficient at one-step division equations and therefore started with the two-step with addition equations and ended with three-step with addition equations (e.g., 4x + 2 = 3x + 4). Each of the algebraic behaviors explored was determined based on students being unable to solve the problems; the students were exposed to, and proficient at, elementary concepts of algebra (e.g., 3 + X = 7) but not the more advanced concepts explored in the study.

All four students started baseline for each of their three behaviors consecutively. When each had a zero-celeration, stable baseline for his or her first behavior across a minimum of three sessions, each one started intervention for his or her first behavior. This consisted of starting with the virtual phase for the first lesson. Consistent with previous administrations of the instructional sequences (e.g., Agrawal & Morin, 2016; Bouck, Bassette, et al., 2017; Bouck, Park, et al., 2017), once each student’s accuracy was 80% or higher for three sessions, each student moved into the abstract phase for the first behavior as well as the virtual phase for the second behavior. At least one baseline session was completed for subsequent behaviors for the student after she or he completed his or her first virtual lesson for their first behavior but before their first virtual lesson for their second behavior. The same procedures were used when students moved from virtual to abstract for their second behavior and started the virtual phase for their third behavior. For each behavior, maintenance occurred 2 weeks after the last abstract lesson.

Procedures

Data collection for the study occurred at a table outside of the students’ special education classroom. All data occurred during the students’ 61-min instructional period. All students worked one-on-one with a researcher 1–2 days per week for 13 weeks, excluding school holidays. Students completed no more than two sessions any day. Two researchers delivered the intervention—one a university faculty member in special education with a decade of experience researching mathematics for students with disabilities and the other a doctoral candidate in special education. Both previously delivered CRA, VRA, and VA interventions in prior research. The first author ensured the doctoral student was well-trained in delivering the intervention, including watching the first author deliver it within the setting with a student and then demonstrating that she or he could deliver it prior to working with the actual participants.

Baseline

During baseline for each mathematical behavior, students were given a sheet a paper with five algebra problems and a pencil to solve the problems. Students were not provided any instruction, manipulatives, or prompting. To move from baseline to intervention, students needed at least three baseline sessions in which a zero-celeration or deceleration trend occurred. For longer baselines, researchers examined the last three baseline sessions for each behavior for the zero-celeration or deceleration trend and stability (i.e., 80% of the data points falling within 25% of the median; Gast & Spriggs, 2014).

Intervention

The VA instructional sequence consisted of a minimum of six sessions for each student—a minimum of three for the virtual manipulative phase and three for the abstract phase. Each intervention session—regardless of behavior or phase—began with explicit instruction. The researcher modeled two problems of the mathematical behavior in question using either the app or notation, depending on the phase (first virtual, then abstract). During the modeling portion, the researcher engaged in think alouds, meaning she or he narrated every step of the problem-solving process with the app and/or the notation (see Figure 2 for a sample of explicit instruction). During the virtual phase, the researcher modeled with the virtual manipulative and during the abstract phase, the researcher modeled with numerical strategies.

Figure 2.

Example modeling portion of explicit instruction for the virtual phase of the virtual–abstract instructional sequence.

During the modeling portion, the researcher would first demonstrate—via narration—how to solve the equation, such as 2x + 2 = 10. With the virtual phase, the modeling involved representing the written equation (2x + 2 = 10) via the virtual algebra tiles. This entailed restating the equation and discussing what the equal sign of the equations means by the analogy of a balance scale. The researcher would state that both sides need to stay equal and what one does to one side of the equation (i.e., equal sign), one must also do to the other side of the equation to keep it balanced. The researcher would drag out two x tiles and two 1 tiles and place them on the left side of the equation, which is represented by the left side of the screen and is separated by an equal sign. The researcher also then dragged out ten 1 tiles and placed them on the right side of the equation (i.e., screen). The researcher again reminded the student that the equation needed to stay balanced or equal; in other words, what one does to one side one needs to do the other. Next, the researcher discussed that to figure out x tiles value, one would want to get the x tiles on one side and the numbers (i.e., one tiles) on the other. Thus, the researcher modeled adding two −1 tiles to both sides of the equation (i.e., app screen), to “get rid” of the numbers on the left side of the equation, as adding the negatives is the inverse of adding the positives (i.e., opposite operation). Next, the researcher moved each of the −1 tiles overtop of the 1 tiles one by one on both sides, which resulted in those canceling each other out and disappearing from the screen. The resulting equation became 2x = 8. The researcher verbalized that two x tiles equals 8, but one needs to figure out what one x tile is equal to numerically. To figure this out, the researcher discussed how division was the inverse operation of multiplication. With the app, the researcher demonstrated separating the space into two rows and placing an x tile in each row. Next, the researcher evenly distributed the 1 tiles on the right side of the equation (or app) across the two x tiles, resulting in each x tile being equal to 4.

The modeling was similar during the abstract phase, minus the use of the virtual manipulatives. During the virtual phase, the researcher discussed the numerical strategies that would be present throughout the abstract phase (e.g., inverse operations). During the abstract phase, the researcher demonstrated using those inverse numerical strategies, such as subtracting two from each side and dividing each side by two to solve for x. Again, the researcher physically demonstrated how to record these procedures on paper with numbers and operations as well as verbalized what the researcher was doing for each step.

Next, the student attempted two problems—using either the app or just notation, while the researcher provided prompts as needed (i.e., the guided portion of explicit instruction). Prompts included statements such as “What do you do next?” and “Remember that you are solving for x so you need to get x by itself.” Finally, the student engaged in five mathematical problems for that behavior independently with either the app or just notation, depending on the phase. No prompts or cues were provided during the independent portion.

To transition from the virtual to the abstract phase, a student had to achieve 80% accuracy (i.e., answer four or more correctly) on the independent portion of a learning sheet for three sessions. If a student did not achieve 80% accuracy, the lesson was repeated the next session. When a lesson was repeated, the same learning sheet was used. For all behaviors examined, students wrote their answers on the independent portion of the learning sheet.

Maintenance

For each behavior, each student completed two maintenance probes 2 weeks after their last abstract intervention session. The maintenance probes were administered in an identical fashion as the baseline probes; no manipulatives or prompts were provided. Two maintenance probes were selected to provide data regarding maintenance, although the focus was on acquisition of the mathematical concepts.

Interobserver Agreement (IOA) and Treatment Fidelity

IOA data were collected for at least 33% of the session for all phases of the study, including baseline, intervention (virtual and abstract), and maintenance. To calculate IOA on accuracy data, an undergraduate mathematics education major checked each selected assessment for agreement on the accuracy of students’ answers. IOA was then calculated by finding the agreements on accuracy between the researcher collecting the data and the undergraduate research assistant and dividing that number by the sum of the agreements and disagreements on accuracy between the authors. IOA for Kya and Eli was 100% for all three behaviors for all phases. IOA for Sadie was 100% for baseline and maintenance for all three behaviors and 93.3% for intervention across the three behaviors. IOA for Cole for 100% for intervention and maintenance and 94.3% for baseline across the three behaviors

Treatment fidelity was assessed through a checklist during intervention. The researcher-created checklist included (a) students receiving the correct tool/phase (i.e., the app in the virtual phase and nothing in the abstract phase); (b) students using the tool provided (if applicable); (c) the implementation of explicit instruction, complete with modeling at least two problems, providing prompts for at least two problems, and allowing students to solve five problems independently; and (d) the researcher collecting data recording accuracy on the data collection sheet. Researchers found 100% treatment fidelity for all students for all behaviors. Each student needed only two modeling and two guided problems per learning sheet to progress to the independent phase of explicit instruction.

Social Validity

To assess social validity, researchers conducted brief interviews with the students at the conclusion of data collection. Each student was asked individually whether they enjoyed learning algebra and why it was important to learn this subject area. Regarding the intervention, students were asked if they found the VA instructional sequence beneficial, and whether they preferred the virtual or abstract phase, along with their rationale.

Data Analysis

To analyze the data, researchers conducted visual analysis on the accuracy-dependent variable. To determine the level for each behavior for each student, researchers used the 80–25 rule; they calculated if 80% of the data fell within 25% of the median for each phase (Gast & Spriggs, 2014). To calculate trend for each phase, researchers used the split-middle method technique; they determined the middle point, mid-rate, and mid-date for each phase and analyzed the line between the mid-rate and mid-date as accelerating, decelerating, or zero-celerating (White & Haring, 1980). Researchers also visually analyzed the graphs for the immediacy of effect, comparing the last baseline session to the first intervention session. Finally, researchers calculated a Tau-U effect size, using an online calculator (Vannest, Parker, Gonen, & Adiguzel, 2016); Tau-U was conducted between baseline and the intervention (virtual and abstract) data. The following metrics were used to interpret the effect size: 93% or above represents a large effect, 66–92% represents a medium effect, and less than 65% a small effect (Parker, Vannest, & Brown, 2009).

Results

A functional relation was found between the VA instructional sequence and students’ accuracy solving linear algebra problems (see Figures 3 –6 and Table 1). Specifically, there was an increase in each student’s raw number of accurately solved linear algebra problems for all three behaviors examined during the VA instructional sequence as compared to baseline.

Figure 3.

Accuracy of Algebra problems for Kya.

Figure 4.

Accuracy of Algebra problems for Saddie.

Figure 5.

Accuracy of Algebra problems for Eli.

Figure 6.

Accuracy of Algebra problems for Cole.

Table 1.

Data Analysis Summary of the Virtual–Abstract Instructional Sequence Across Participants.

Measure	First Behavior					Second Behavior					Third Behavior
	B	Overall I	I		M	B	Overall I	I		M	B	Overall I	I		M
	B	Overall I	V	A	M	B	Overall I	V	A	M	B	Overall I	V	A	M
Kya
Range	0	80–100%	100%	80–100%	100%	0–20%	100%	100%	100%	40–100%	0–40%	80–100%	80–100%	100%	100%
Mean	0	96.7%	100%	93.3%	100%	10%	100%	100%	100%	70%	7.5%	96.7%	93.3%	100%	100%
Stability	S	S	S	V	S	V	S	S	S	V	V	S	V	S	S
Trend	Z	Z	Z	A	Z	A	Z	Z	Z	A	A	Z	A	Z	Z
Tau-U^a		100%					100%					100%
Sessions	3	6	3	3	2	6	6	3	3	2	8	6	3	3	2
Repeated	—	0	0	0	—	—	0	0	0	—	—	0	0	0	—
Sadie
Range	0	80–100%	100%	80–100%	100%	0	80–100%	80–100%	80–100%	0	0	100%	100%	100%	20–40%
Mean	0	93.3%	100%	86.7%	100%	0	93.3%	93.3%	93.3%	0	0	100%	100%	100%	30%
Stability	S	S	S	V	S	S	S	V	V	S	S	S	S	S	V
Trend	Z	D	Z	A	Z	Z	Z	A	A	Z	Z	Z	Z	Z	A
Tau-U^a		100%					100%					100%
Sessions	3	6	3	3	2	5	6	3	3	2	7	6	3	3	2
Repeated	—	0	0	0	—	—	0	0	0	—	—	0	0	0	—
Eli
Range	0	0–100%	100%	0–100%	100%	0	80–100%	80–100%	80–100%	0	0–40%	80–100%	100%	80–100%	0
Mean	0	85.7%	100%	75%	100%	0	93.3%	93.3%	93.3%	0	6%	96.7%	100%	93.3%	0
Stability	S	S	S	V	S	S	V	V	V	S	V	S	S	V	S
Trend	Z	Z	Z	A	Z	Z	Z	D	A	Z	A	Z	Z	D	Z
Tau–U^a		85.7%					100%					100%
Sessions	3	7	3	4	2	5	6	3	3	2	10	6	3	3	2
Repeated	—	1	0	1	—	—	0	0	0	—	—	0	0	0	—
Cole
Range	0	100%	100%	100%	40%–100%	0	60%–100%	100%	60–100%	20–40%	0–40%	80–100%	100%	80–100%	20–60%
Mean	0	100%	100%	100%	70%	0	94.3%	100%	90%	30%	6%	96.7%	100%	93.3%	40%
Stability	S	S	S	S	V	S	V	S	V	V	S	S	S	V	V
Trend	Z	Z	Z	Z	D	Z	Z	Z	A	A	A	Z	Z	D	D
Tau-U^a		100%					100%					100%
Sessions	3	6	3	3	2	5	7	3	4	2	10	6	3	3	2
Repeated	—	0	0	0	—	—	1	0	1	—	—	0	0	0	—

Note. S = stable; V = variable; A = accelerating; D = decelerating; Z = zero-celerating.

^aTau-U between baseline and intervention (overall).

Kya

Kya’s accuracy for her first behavior—two-step with addition—during baseline was 0 for every session (see Figure 2); the data were stable, with a zero-celeration trend. She experienced an immediate effect with the first intervention session. Her accuracy range during intervention was 4–5 (i.e., 80–100%; µ = 4.8 or 96.7%), and she did not repeat any sessions during the virtual or abstract phases. Her overall intervention data were stable with a zero-celeration trend. The Tau-U was 100%, indicating a large effect. Her accuracy was 5 (i.e., 100%) for both maintenance sessions.

Kya’s completed six baseline sessions (range = 0–1 [0–20%]; µ = 0.5 or 10%) for her second behavior—two-step with subtraction. Her baseline had an accelerating trend and was variable. The intervention had an immediate effect on Kya’s accuracy; her first virtual intervention session was 100% (i.e., five). Her accuracy was never below 5 (i.e., 100%) for this behavior during the virtual or abstract sessions; the Tau-U was 100%, indicating a large effect. Kya’s maintenance data range from 2 to 5 (i.e., 40–100%; µ = 3.5).

Kya’s baseline for her third behavior—three-step with addition—ranged from 0 to 2 (i.e., 0–40%, µ = 0.38) across eight sessions. Her baseline data were variable and accelerating. Kya experienced an immediate effect, achieving five correct responses (i.e., 100%) on her first virtual session. She repeated no sessions during intervention, and her accuracy ranged from 4 to 5 (i.e., 80–100%, µ = 4.83 or 96.7%); her Tau-U was 100%, indicating a large effect. Her accuracy was 5 (i.e., 100%) for both maintenance sessions.

Sadie

During baseline for Sadie’s first behavior—one-step with division—she answered zero problems correctly across three sessions (see Figure 3); the data were stable, with a zero-celeration trend. Sadie experienced an immediate effect; her accuracy was 5 (i.e., 100%) during her first virtual intervention session. She repeated zero intervention sessions, and her range for accuracy was 4–5 (i.e., 80–100%, µ = 4.67 or 93.3%). Her Tau-U was 100%, indicative of a large effect. Sadie’s accuracy was 5 or 100% for both maintenance sessions.

Sadie’s baseline accuracy scores for her second behavior—two-step with addition—were all 0 across five sessions; the data were stable and zero-celerating. Sadie’s accuracy experienced an immediate effect, as her accuracy was 4 (i.e., 80%) for the first virtual intervention session. Her overall intervention accuracy ranged from 4 to 5 (i.e., 80–100%, µ = 4.67 or 93.3%); the data were stable with a zero-celeration trend, and she did not repeat any sessions. The Tau-U for her accuracy was 100%, indicating a large effect. Sadie struggled during the maintenance sessions; she answered zero questions correctly for both sessions.

Sadie’s final behavior—two-step with subtraction—produced similar results during baseline (0% accuracy for all seven baseline sessions). She experienced an immediate effect upon the first virtual intervention session (100% accuracy). She repeated zero sessions during intervention and achieved five correct answers (i.e., 100%) for each of the six sessions; her Tau-U was 100%, indicating a large effect. Sadie’s range during maintenance was 1–2 (i.e., 20–40%, µ = 1.5 or 30%).

Eli

Eli’s accuracy for his first behavior—one-step with division—was 0 for each of three sessions; data were stable and zero-celerating (see Figure 3). Eli experienced an immediate effect during his first virtual intervention session, achieving 5 or 100%. His intervention accuracy ranged from 0 to 5 (i.e., 0–100%, µ = 4.29 or 85.7%), and he repeated one abstract intervention session in which he scored 0. On all other intervention sessions, Eli scored 100%; his Tau-U was 85.7% indicating a medium effect. Eli maintained his accuracy at 5 or 100% for two maintenance sessions.

Eli’s baseline accuracy for his second behavior—two-step with addition—was also 0% for each of five sessions. He experienced an immediate effect upon entering intervention with the first virtual session (100% accuracy). He repeated zero sessions during intervention and averaged 4.67 or 93.3% across three virtual and abstract sessions. Eli’s Tau-U was 100%, indicating a large effect. His maintenance for this behavior was 0 for both sessions.

For Eli’s last behavior—two-step with subtraction—the data for his eight baseline sessions were variable (0–2 or 0–40%). He experienced an immediate effect, answered five correctly (i.e., 100%) on his first virtual intervention session and maintaining 100% accuracy for all intervention sessions, except the last abstract session (i.e., 4 or 80%). Eli’s Tau-U was 100%, indicating a large effect, and he repeated zero intervention sessions. His maintenance for this behavior was also 0 for both sessions.

Cole

Cole’s accuracy was 0 for one-step with division problems for each of the three sessions. Cole experienced an immediate effect during the virtual intervention sessions. He scored five correct answers (i.e., 100%) for all his virtual and abstract sessions and repeated zero sessions. His Tau-U was 100%, indicating a large effect. Cole’s maintenance data were variable (ranging from 2 to 5 or 40 to 100%).

Cole’s accuracy during baseline for his second behavior—two-step with addition—was consistently 0. He experienced an immediate effect for his first virtual intervention session; he was 100% accurate for each virtual session (i.e., answered all five correctly). Cole repeated one session—his first abstract session, in which he answered three correct (i.e., 60%). After, he answered five correct (i.e., 100% accuracy) for all remaining abstract sessions. Cole’s Tau-U was 100%, indicating a large effect. Cole’s maintenance ranged from 1 to 2 (i.e., 20–40%).

For Cole’s final behavior—two-step with subtraction—his baseline data were variable (0–2 or 0–40%), although his last three baseline session scores were all 0. He experienced an immediate effect; he answered five correct (i.e., 100% accuracy) for his first virtual intervention session. He maintained answering all correctly (i.e., 100% accuracy) throughout intervention, except his last abstract session where he answered four correct (i.e., 80%). Cole repeated zero sessions, and his Tau-U was 100%, indicating a large effect. His maintenance accuracy ranged from 1 to 3 (i.e., 20–60%.)

Social Validity

The students all stated they preferred the virtual app over the abstract phase, while also indicating they prefer to use the app in the future to help them learn mathematics. Kya and Cole both commented that it was quicker and easier, while Sadie expressed that she was a visual learner, so the app allowed her “to see the pictures.” Three of the students also expressed they enjoyed learning algebra, offering comments such as, “yes, because you can use it in life, such as cooking and stuff” (Cole) and “yes, so you can get good grades in math” (Sadie). Eli indicated he did not like algebra because it was really complicated.

Discussion

Algebra is a key mathematical concept, yet a struggle for many students, including students with disabilities. This study examined whether an intervention sequence consisting of an app-based manipulative and then abstract instruction could support the acquisition of three algebraic behaviors (e.g., one-step with division, two-step with addition) for middle school students with mild intellectual disability, learning disabilities, and ADHD. The results suggest students acquired each of the linear algebra behaviors but struggled to maintain their learning once instruction was not provided prior to performance.

Each of the four students improved in their acquisition of the linear algebra behaviors during the course of the VA instructional sequence. Across six sessions for each behavior, only two students (Eli and Cole) needed one session repeated, both in the abstract phase. Otherwise, students scored 4–5 correct (i.e., 80–100%) on the independent portion of the learning sheets for each session. Across all four students, in 84% of the sessions, the students were 100% accurate, and in 13.3% of sessions, the students were 80% accurate. All four students improved over their baseline scores, and each student’s Tau-U results for each behavior suggested a medium to large effect size.

Consistent with previous studies regarding the CRA, VRA, and VA instructional sequences (e.g., Bouck, Bassett, et al., 2017; Bouck, Park, & Nickell, 2017; Bouck, Park, et al., 2017), the four students in the study experienced an immediate effect from their baseline performance to their intervention (i.e., virtual phase) performance for each algebraic behavior. However, consistent with previously published research (e.g., Bouck, Park, et al., 2018), the students in this study struggled with maintaining skills across their three linear algebra behaviors. In other words, while the students acquired each of the behaviors successfully during the intervention portion when explicit instruction was provided, during the maintenance phase in which no instruction preceded students independently solving the linear algebra problems, the students struggled.

While this study, like many single-case design studies, assessed for maintenance, attention to teaching for maintenance was not part of the intervention design. In other words, the intervention—the VA instructional sequence—focused on student acquiring the mathematical skills related to linear algebra (i.e., the use of explicit instruction; Snell & Brown, 2011). Learning occurs across stages, of which maintenance (i.e., “the ability to perform a response over time without reteaching”) is one of them (Alberto & Troutman, 2009, p. 43). The authors did not employ specific interventions or instructional approaches targeting maintenance (e.g., fading supports, overlearning, or distributed trials; Collins, 2012; Snell & Brown, 2011). Given learning mathematics is not just about acquisition, but also about students being fluent, maintaining said mathematical behavior, and generalizing their learning, in future studies, researchers should attend to and specifically target interventions to improve these others stages, including maintenance.

Implications for Practice and Research

The VA instructional sequence can be an effective and efficient means to teach students to acquire mathematical behaviors. All four students in this study successfully acquired three linear algebra behaviors within six to seven instructional sessions, which were each less than 20 min. This suggests the VA instructional sequence can be a useful tool in teachers’ instructional tool kit. The VA sequence may be used in contrast to other effective interventions. For example, Watt et al. (2016) found the CRA instructional sequence to be an effective intervention for teaching algebra to students with disabilities. While researchers have effectively used the CRA instructional sequence for multiple decades, researchers suggested a preference of secondary students for virtual manipulatives over concrete manipulatives (Bouck, Chamberlain, & Park, 2017; Satsangi et al., 2016). Teachers may want to consider the VA instructional sequence as an approach for teaching secondary students with disabilities. Given this is the only study regarding the VA instructional sequence and algebra to date, as well as the somewhat mixed results, researchers should seek to conduct additional studies regarding the VA instructional sequence for algebra.

Yet to go beyond acquisition, teachers should consider how to plan for maintenance and generalization when implementing the VA instructional sequence. To be useful, students need to solve algebra problems even when a teacher does not provide immediate explicit instruction. To address, researchers should consider exploring an adapted instructional sequence of VA+, in which interventions specifically targeting maintenance are included. For example, VA+ might include overlearning, in which the abstract phase extends beyond three successful sessions at 80% accuracy to six to eight sessions. Perhaps an intervention with more duration and/or intensity would be beneficial relative to maintenance. Another example of VA+ might include the fading of supports, thus after 80% accuracy on three sessions of traditional explicit instruction, students receive three additional sessions with only modeling or only cueing. Related assessing students on acquisition not immediately after learning (i.e., explicit instruction), such as the next day, may result in a more accurate depiction of acquisition and thus better support maintenance.

Limitations and Future Directions

This study is not without limitations. One limitation was the diverse disability profiles of the students selected to participate. However, the authors were not targeting a disability category per se but students for who had similar mathematical struggles and who experienced similar educational programming. Per the Individuals with Disabilities Education Act (2004; Yell, 2012), services are not provided to students on the basis of a categorical identification but instead on need. Related to the participants, one could perceive the sample size as a limitation; however, it was appropriate for the methodology. Another limitation may be perceived that Kya’s behaviors were not identical to the other three students. Yet, the researchers worked to support students on his or her mathematical needs. Also, Kya’s second and third intervention behaviors were started before a stable or zero-celerating or decelerating trend could be established, at a minimum of the last three behaviors proceeding intervention. In fact, both of Kya’s second and third behaviors started with an accelerating trend apparent in baseline across all baseline data points, due to a researcher oversight. Although the baseline sessions immediately preceding intervention for her for both these behaviors were zero, intervention should not have occurred and additional baseline sessions taken. This issue was not present with the other three students.

Another limitation was the dependent variable was delivered immediately after the instruction; researchers should seek to deliver the dependent variable—the independent portion of explicit instruction—the following day to measure learning. This would allow for a measure of acquisition to better ensure greater maintenance. Related, there is a potential for overalignment between the dependent measure and the intervention, as they were similar. Another limitation involved that the researcher-created independent portion of each learning sheet served as dependent measure. While they were reviewed prior to use by a secondary mathematics teacher, they were lacking in technical elements, such a reliability and validity measures. Also, related to maintenance, the authors only probed twice during the maintenance phase; given the inconsistent scores of some of the participants during the two maintenance probes, additional probes might have been warranted. Finally, the researchers delivered the intervention and did so via a one-on-one format; in future studies, researchers should seek to explore the implementation of the VA by classroom teachers and/or in larger group settings.

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The first author received a grant from the Learning Disabilities Foundation of America to support this research.

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