Exploring the Virtual-Representational-Abstract Instructional Sequence Across the Learning Stages for Struggling Students

Abstract

In recent years, virtual manipulatives have been explored and used as an alternative to concrete manipulatives in mathematics for students on their own and as part of manipulative-based instructional sequences. Researchers examining virtual manipulative-based instructional sequences tend to focus on students documented with disabilities, as opposed to students at-risk or struggling with mathematics, as well as students’ acquisition of the target skill, despite students experiencing learning in four stages: acquisition, fluency, maintenance, and generalization. This study explored the virtual-representational-abstract (VRA) instructional sequence across four stages of learning for three elementary students struggling in mathematics. In the single-case design study, researchers found a functional relationship between the VRA instructional sequence delivered online via explicit instruction and students’ computational accuracy in their targeted area of mathematics need. Researchers also found limited influence on fluency rate or generalization to word problem accuracy but that students did maintain.

Keywords

elementary manipulatives mathematics online instruction single-case struggling students

Virtual manipulatives are increasingly being explored and used as an alternative to concrete manipulatives for K-12 students, including students with learning disabilities or those at-risk (Park et al., 2022). Historically, Moyer-Packenham and Westenskow (2013) found virtual manipulatives beneficial to students without disabilities. More recently, Park et al. (2022) determined the existing research base on virtual manipulatives for students with learning disabilities was positive. Despite the increase in attention to virtual manipulatives, the majority of the literature focuses on two populations: (a) secondary students with disabilities, which includes examination of the efficacy of virtual manipulatives as an intervention or intervention package and virtual manipulative based instructional sequences and (b) general education students aggregated (Moyer-Packenham & Westenskow, 2013; Park et al., 2022; Shin et al., 2021). Across the studies examining virtual manipulatives, students have improved from baseline to intervention or from pretest to posttest (Moyer-Packenham & Westenskow, 2013; Park et al., 2022; Peltier et al., 2020; Shin et al., 2021).

Virtual Manipulative Based Instructional Sequences

One means of using virtual manipulatives to support students’ mathematical understanding is through a virtual manipulative-based instructional sequence (Bouck & Sprick, 2019). The virtual manipulative-based instructional sequence includes four variations to date. The first is the virtual-representational-abstract (VRA) instructional sequence—an adaptation of the widely examined concrete-representational-abstract (CRA) instructional sequence that involves concrete manipulatives, then drawings, and finally numerical strategies (e.g., Bouck et al., 2018)—that transitions students from virtual manipulatives to pictorial representations and finally numerical strategies (e.g., repeated addition, partial products, and counting on). The second is the virtual-abstract (VA) instructional sequence, which differs from the VRA through the removal of the representational phase (i.e., drawings phase) when representing the mathematics with pictures is challenging or might not be needed to build conceptual understanding (e.g., fractions, algebra; Bone et al., 2021; Bouck et al., 2019). A third approach is the virtual-representational (VR) instructional sequence, which ends students after the representational phase rather than moving them into solving problems with numerical strategies (e.g., Bouck et al., 2020). The VR instructional sequence questions the need for students, such as students with extensive support needs (ESN), to solve computational problems in the abstract and recognizes that representations, such as drawings, are an evidence-based practice (Jitendra et al., 2016). The last variation is the VRA-integrated (VRA-I), in which the future phases (e.g., representational and abstract) are previewed within the current phase (Root et al., 2021). In other words, when learning to solve the problems with virtual manipulatives, students are also exposed to solving with drawings, as well as abstractly via numerical strategies.

VRA Instructional Sequence

The majority of studies within the VRA instructional sequence literature focused on middle school students with intellectual and developmental disabilities (Long et al., 2022). This stands in contrast to the CRA instructional sequence literature, which predominantly involved students with learning disabilities or those at risk (e.g., Bouck et al., 2018; Bouck & Park, 2018; Flores et al., 2018; Hinton et al., 2019). Researchers determined the CRA instructional sequence to be an evidence-based practice for students with learning disabilities (Bouck et al., 2018). Of studies examining the CRA, the majority focused on elementary students and basic operations (e.g., subtraction and multiplication; Bouck et al., 2018).

However, a limited and emerging literature exists examining virtual manipulative based instructional sequences with younger students as well as students with a learning disability or at-risk for a disability. Bouck et al. (2021) examined the online delivery of the VRA instructional sequence, in combination with the system of least prompts (SLP), to three elementary students with mathematics difficulties (i.e., not receiving special education services but struggling in mathematics and/or receiving targeted interventions in mathematics). The researchers found a functional relation between the intervention package of the VRA instructional sequence and the SLP and students’ accuracy in solving problems involving double-digit subtraction with regrouping. The students were also generally independent (i.e., did not need prompts to complete). Although not the VRA, Bouck and colleagues did examine the VA instructional sequence with students with mathematical difficulties in two studies. Bouck and Long (2021) examined the VA instructional sequence plus the SLP to teach solving equivalent fractions to three upper elementary students with a disability or at-risk; all three were successfully acquired and maintained the skill. Similarly, Bouck et al. (2022) successfully taught three upper-elementary students with mathematical difficulties to solve teen-digit by teen-digit multiplication problems via partial products with the VA instructional sequence and the SLP.

Learning Stages

Researchers examining the CRA, VRA, and other virtual manipulative based instructional sequences tend to focus on student learning, such as if students learned (i.e., acquired) the targeted mathematics (Bouck et al., 2018; Long et al., 2022; Peltier et al., 2020). Yet, students experience learning in stages. Different but similar stages of learning frameworks exist. Some scholars situate learning in four stages: acquisition (i.e., performing a new behavior), fluency (i.e., performing a behavior with appropriate speed and accuracy), maintenance (i.e., performing a behavior without prior instruction over time), and generalization (i.e., performing the behavior in different contexts; Alberto & Troutman, 2009; Collins, 2012; Shurr et al., 2019). Others suggest five stages of learning: acquisition (i.e., initial stage of new learning), proficiency (i.e., becoming more fast [fluent] and accurate), (c) maintenance (i.e., engaging in behavior mastered previously), (d) generalization (i.e., demonstrating behavior in different context), and (e) application (i.e., applying learning behavior in new situations or new ways; Bos & Vaughn 2002; Bryant et al., 2003; Burns, 2004). Regardless of one’s orientation, it is clear that learning is complex and educators need to plan for learning beyond initial stages (e.g., acquisition) of demonstrating an understanding or performing the skill once.

Current Study

Virtual manipulative instructional sequences, including the VRA, have a positive emerging research base. Virtual manipulatives in and of themselves are considered an evidence-based practice for educating students with learning disabilities as well as students with intellectual and developmental disabilities (Long et al., 2022; Park et al., 2022). Yet, the majority of the research with the VRA instructional sequence occurs in face-to-face learning environments. While face-to-face represents the typical educational setting, it fails to account for one of the main benefits of virtual manipulatives over concrete manipulatives—they can easily be used within online teaching and learning. Further, while there is emerging attention to consideration of the VRA instructional sequence to support students across the learning stages, too much attention exists to just acquisition (e.g., Park et al., 2020). Finally, the research base on virtual manipulatives is primarily limited to students with identified disabilities, rather than students with mathematical difficulties, who may be served exclusively in general education instructional settings, without or without intervention support.

This study sought to explore the online teaching of three students identified with mathematical difficulties and to examine the impact of the VRA instructional sequence across four stages of learning: acquisition, fluency, maintenance, and generalization. As such, researchers attempted to answer the following research questions: (a) Does a functional relationship exist between students’ computational accuracy and learning to solve targeted mathematical computational problems via the VRA instructional sequence, (b) Do increases in students’ computational accuracy with solving targeted mathematical computational problems following the VRA instructional sequence generalize to word problem accuracy of the same targeted mathematical area?, (c) Do increases in student computational accuracy with solving targeted mathematical computational problems following the VRA instructional sequence generalize to greater fluency rate in solving the same targeted computational mathematical problems in a timed probe, (d) Are students able to maintain their accuracy after instruction with the VRA instructional sequence ends, and (d) What are student perceptions of the VRA instructional sequence and learning mathematics online?

Method

Participants

The study included three elementary students whose parents identified them as having mathematical difficulties. The students were recruited via social media ads and parents contacted researchers to express interest in their child participating in a math study. Parents self-reported the disability or at-risk eligibility; researchers confirmed mathematical struggles via the researcher-administered KeyMath-3 assessment (Connolly, 2007). To be included in the study, students needed to be in grades 2 to 6; either receive special education services, be considered at-risk (i.e., receiving Tier response to intervention [RtI] services) in the area of mathematics, or experience difficulties in mathematics; have researcher-administered KeyMath-3 scores in an area (e.g., basic operations, fractions) that show below-grade level functioning; answer zero questions correctly across all baseline sessions; have both parental consent and student assent to participate; and possess the ability to participate in the research virtually (i.e., access and navigate zoom, the virtual manipulatives, and sharing screens relatively independently).

Erica

Erica was a 9-year-old, white, female student enrolled in the fourth grade. Her parents consented to her participation as they were concerned about her mathematically and wanted additional support. Per parental reports, Erica was identified with autism spectrum disorder, ADHD, and a math learning disorder. However, she did not quality for an IEP, as her parents reported that the school found her disability did not adversely affect her educational attainment. Erica also did not receive any additional supports for mathematics in her school. She was participating in mostly in-person schooling during the year of the study, however her school participated in remote learning for 2 weeks in the middle of the study. Erica participated weekly in the late afternoons with the researchers. On the researcher administered KeyMath-3 assessment (Connolly, 2007), Erica had a raw score of 17 on numeration (grade equivalency 3.2). Her raw scores were 11 (grade equivalency 2.8) on the mental computation and estimation subtest, 12 (grade equivalency 2.7) on the addition and subtraction subtest, and 2 (grade equivalency 2.6) on the multiplication and division subtest. Erica scored 26 for total operations (grade equivalency 2.6). On the KeyMath-3, Erica consistently struggled with multiplication telling researchers she could only recall or determine multiplication facts involving two or five. The researchers selected single-digit multiplication as Erica’s targeted area.

Jeff

Jeff was a white, male student in the fifth grade who was 10 years of age. Per his mother, Jeff was identified with ADHD and Dyslexia but he did not have an IEP, as the school determined he did not qualify. Jeff’s mom was also concerned about processing challenges and felt mathematics was more challenging due to his processing challenges. Jack did not receive any additional supports for math in school but was given additional time to complete his math assignments. Jack was participating in in-person schooling during the year of the study and participated weekly in the late afternoon with the researchers. On the researcher administered KeyMath-3 (Connolly, 2007), Jeff’s raw numeration score was 30 (grade equivalency 6.8). On subtests within the operations portion (total raw of 49; grade equivalency 4.5), Jeff scored a 20 (grade equivalency 5.0; mental computation and estimation), 24 (grade equivalency 5.8; addition and subtraction), and 4 (grade equivalency 3.2; multiplication and division). Jeff struggled to solve multiplication and division problems on the KeyMath-3. In further probing, Jeff was able to solve basic multiplication and division problems but struggled when the quotient on division problems was not a whole number (i.e., division with remainders). Researchers targeted division with remainders as Jeff’s targeted area.

Eva

Eva was a while, female student in the third grade. During the study, she turned 9 years old. Per her parental report, Eva had not been identified with a disability and had no IEP. She also received no additional supports for math in school. Eva was participating in in-person schooling during the year of the study and participated weekly in the late afternoons with the researchers. On the researcher administered KeyMath-3 (Connolly, 2007), Eva’s raw numeration score was 18 (grade equivalency 3.5) and mental computation and estimation subtest was 14 (grade equivalency 3.5). For the addition and subtraction subtest, Eva’s raw score was 10 (grade equivalency 2.2) and on the multiplication and division subtest she had a raw score of 4 (grade equivalency 3.0). Eva scored 28 for total operations (grade equivalency 2.8). On the KeyMath-3, Eva consistently struggled with adding and subtracting that involved regrouping. Researchers targeted double-digit subtraction with regrouping as Eva’s targeted area.

Setting

All activities for the study occurred on Zoom. Researchers met with students one-on-one—or two-on-one during sessions in which interobserver agreement data (IOA) were collected—from their respective homes. Students were located in two different midwestern states and participated by joining a secure and passcode-protected Zoom link during a mutually agreeable set time each week. Each weekly meeting was limited to 30 minutes, although sessions generally took less than 30 minutes. Specifically, Erica’s average baseline session took 2:46, Jeff’s 2:43, and Eva’s 3:33. For intervention, the students ranged from around 5 minutes to less than 11 minutes per session. Erica’s sessions were an average of 10:02 during the virtual phase, 6:45 during the representational phase, and 5:13 during the abstract phase. Jeff’s average duration was 9:00 during the virtual phase, 10:24 during the representational phase, and 6:12 during the abstract phase. For Eva, her average durations included 6:54 during the virtual phase, 6:14 during the representational phase, and 3:59 during the abstract phase. During maintenance, Erica’s average duration was 4:03, Jeff’s was 9:08, and for Eva it was 7:13. During the Zoom sessions, students individually logged onto the computer; parents were present in the home but did not appear on screen unless the student needed assistance with technology such as audio, video, or screensharing difficulties.

Materials

For the study, researchers used Zoom, Google Jamboard, virtual manipulative apps from Brainingcamp, a virtual whiteboard from the Math Learning Center, and probes. Given students had different targeted mathematical areas, based on identified need from the KeyMath-3 and further probing, different virtual manipulative apps were used. During the virtual portion of the VRA instructional sequence, students used the Brainingcamp Color Tiles app (see https://app.brainingcamp.com/) to solve multiplication and division problems and the Brainingcamp Base 10 Blocks app to solve subtraction with regrouping problems (see Figure 1). Each app was a digital representation of its related concrete manipulative—color tiles and base 10 blocks. The background of each app was a virtual whiteboard and included pens for writing on the workspace. During the representational and abstract portions of the VRA instructional sequence, all students used the Math Learning Center virtual whiteboard (see https://apps.mathlearningcenter.org/whiteboard/) to draw pictures or solve problems abstractly. The virtual whiteboard app allowed students to write on a digital whiteboard with virtual pens without any accompanying manipulatives (see Figure 1).

Figure 1.

Screenshots of virtual manipulatives and apps used across VRA instructional sequence. For the virtual phase, the researchers used both Color Tiles and Base Ten Blocks by Brainingcamp (https://app.brainingcamp.com/). For the representational and abstract phases, the researchers use Whiteboard by The Math Learning Center. To learn more visit www.mathlearningcenter.org

During baseline, generalization to word problems, and fluency, researchers used Google Jamboard to present the probe. For baseline and maintenance probes (i.e., computational accuracy), three computational math problem were presented on the Google Jamboard and students solved. For generalization to word problems (i.e., word problem accuracy), students solved three word problems presented on a Google Jamboard page, and for fluency (i.e., fluency rate) students were asked to solve as many of the 30 computational math problems presented on a Google Jamboard page in one minute. For intervention probes (i.e., computational accuracy), students solved three computational problems on either the aforementioned apps or a virtual whiteboard app; researchers orally presented the three computational problem probe during the independent portion of explicit instruction. During the modeling and guided practice portions—one computational problem each—of explicit instruction, researchers used the same app or virtual whiteboard students used during the independent portion (i.e., intervention probe). When modeling, researchers shared their screen and when students engaged in the guided or independent practice, students shared their screens. Students also shared their screens during the baseline, maintenance, generalization to word problems, and fluency probes.

For the multiplication probes, the researchers used numbers 3, 4, and 6 to 9 to create the single-digit multiple problems across the computational (i.e., baseline, intervention, and maintenance), word problems (i.e., generalization), and fluency probes (i.e., fluency). Problems could be repeated across baseline, intervention, maintenance, fluency, or generalization but problems or their inverse (e.g., $3 \times 4$ and $4 \times 3$ ) were not repeated in the same session. Each product for the multiplication problems on the fluency probes was a double-digit number, resulting in a total possible digits for the fluency probe of 60. For the division with remainder probes, the whole number quotient for each problem was a single-digit, as was the divisor and the remainder; thus, each dividend was a teen number. Each division with remainder fluency probe had a possible 60 digits. For the subtraction with regrouping problems, each difference was a double-digit number and digits 1 to 9 were used to create the problems. Each double-digit subtraction with regrouping fluency probe also had 60 possible digits.

Experimental Design

Researchers employed a multiple probe across participants single case design methodology (Ledford & Gast, 2018). Researchers selected a multiple probe across participants design as the behavior (learning to solve math problems) was non-reversible, the pace and change in conditions was controlled by the students, and researchers desired not to frustrate students with continuous baseline measurements of problems they struggled to solve (Ledford & Gast, 2018). In the design, all students started baseline simultaneously but entered intervention in a staggered fashion. The first student—Erica—transitioned from baseline to intervention after three baseline sessions in which the accuracy was 0% in each session, resulting in a stable and zero-celerating trend (Cook et al., 2014). Subsequent students (e.g., Jeff) entered intervention after the previous student (e.g., Erica) had two intervention—virtual—sessions with 100% accuracy. Students transitioned between phases of the VRA (i.e., virtual to representational and representational to abstract) after two consecutive sessions with 100% accuracy in each phase. Students exited intervention and moved into maintenance after two consecutive sessions with 100% accuracy in the abstract phase. Researchers probed for both generalization to word problems and fluency every other session, with at least one probe for both in each condition and phase (i.e., baseline; intervention—virtual, representational, and abstract; and maintenance).

Independent and Dependent Variables

The primary dependent variable in this study was accuracy, which researchers reported as a percentage of the problems answered correctly. Accuracy data were collected for each baseline, intervention, maintenance, and generalization session; they were not collected for fluency sessions. During baseline, intervention, and maintenance sessions, researchers collected computational accuracy and during generalization they collected word problem accuracy. Each students’ accuracy for all phases was based on their targeted mathematics (e.g., single-digit multiplication [Erica], double-digit subtraction with regrouping [Eva], and division with remainders [Jeff]). For baseline, intervention, and maintenance, computational accuracy was the percentage out of three computational problems (e.g., $3 \times 4$ , $6 \times 6$ , and $5 \times 7$ for multiplication; 13 ÷ 3, 17 ÷ 5, and 14 ÷ 4 for division with remainders; and 45 – 18, 31 – 12, and 52 – 27 for subtraction with regrouping). For generalization, which was probed every other session, word problem accuracy was the percentage out of three word problems of the same mathematical focus as the computational problems. To graph accuracy for acquisition, maintenance, and generalization, the percentage correct was calculated out of the number possible (n = 3).

The researchers also collected fluency data, which was collected every other session. Researchers calculated a fluency rate of correct digits per minute (cdpm; out of 60 possible digits from 30 computational problems) each time a fluency probe was administered. The researchers chose digits, meaning if an answer was incorrect but one of the digits was correct, students would be given credit. For example, with $3 \times 4$ in the fluency probe, if a student answered 13, they would have one digit correct. To graph fluency, researchers converted the cdpm to a percentage by dividing that number by 60.

For the independent variable, researchers employed the VRA instructional sequence, which involved students first learning to solve their targeted mathematical computational problems with virtual manipulatives, then representational drawings, and finally numerical strategies (i.e., abstract). Researchers delivered the VRA instructional sequence via explicit instruction. With explicit instruction, the researchers demonstrated how to solve (i.e., modeling) one problem with the appropriate materials for that phase (e.g., virtual manipulatives, drawings, numerical symbols) along with verbal narration (i.e., think aloud). Researchers then had students solve one problem, while they provided prompts and cues as needed (i.e., guided). Finally, students solved three problems independently, which served as the intervention probe (i.e., computational accuracy data). Researchers provided no prompting or feedback during the independent probes.

Procedures

Three researchers participated in data collection. One researcher was a faculty member with over a decade of experience delivering mathematical interventions, including virtual manipulatives, to students with disabilities or those at-risk. Another was a doctoral student, trained by the faculty member, who had several years of experience delivering mathematical interventions, including virtual manipulatives, to students with disabilities and those at-risk. The last researcher—an advanced education doctoral student—collected IOA data after being trained by the other two researchers.

Baseline

For each baseline probe session, researchers presented students with three problems of their targeted mathematical computational area on a Google Jamboard page to solve. Prior to baseline sessions, students were exposed to Google Jamboard and solved mathematical problems they knew (e.g., 2 + 2) to become familiar with the technology (e.g., virtual pen). During baseline probe sessions, researchers collected computational accuracy data.

Intervention

The intervention was the VRA instructional sequence, which involved students first solving their targeted mathematical computational problems with virtual manipulatives, then pictorial drawings, and finally with numerical symbols and strategies. At each phase of the instructional sequence, researchers used explicit instruction (i.e., one problem modeled, one problem guided, and then students solved three problems independently) to teach students. If a student did not achieve over 80% independence in the guided phase on the computational problem, they did not advance to the independent phase that session. In such a case, a probe for computational accuracy was not administered. Students remained in intervention until they achieved 100% accuracy for two consecutive sessions for each phase: virtual, representational, and abstract. Students completed an accuracy probe at the end of each intervention session; the probe for accuracy (i.e., computational accuracy) was the independent phase of explicit instruction. The probes were the same regardless of VRA phase.

Virtual

During the virtual phase, researchers taught students to solve their targeted mathematical computational problems with an appropriate virtual manipulative. To teach Erica to solve multiplication problems, the researchers used the color tiles app from Brainingcamp to model while providing verbal narration. The researcher started by explaining that multiplication means that you have X groups of Z objects, so 4 ×5 means you have four groups with five objects in each group. The researcher began by drawing four circles to represent the groups and then placed five tiles in each group. The researcher then counted the number of tiles in all the circles to get the product—20 in this example (see Figure 1).

The researchers also used the Brainingcamp Color Tiles app to teach Jeff to solve division with remainders problems. The researchers similarly began by explaining that division means you have X number of objects that you want to evenly distribute or divide across Z groups but that sometimes you have objects leftover as each group needs to have the same amount. Thus, with the problem 13 ÷ 3, the researcher pulled 13 tiles onto the top of the screen and then drew three circles. The researcher then distributed tiles to each of the circles one at a time, stopping when there were fewer than three tiles left. The researcher counted the number of tiles in each group—emphasizing it should be the same—as well as the number left over that could not be evenly distributed. The researcher noted the same number of objects in each group was the quotient and the leftover was the remainder, writing 4R1 (see Figure 1).

To teach double-digit subtraction with regrouping to Eva, the researchers used Brainingcamp Base 10 Blocks app. The researcher explained the blue rods represent 10s and the yellow blocks represent ones. The researcher drew a tens-and-ones chart (i.e., T-chart), and wrote the problem on the screen (e.g., 36 − 19; see Figure 1). The researcher brought out 3 tens blocks and 6 ones, placing them within the respective tens and ones columns of the chart. Starting th the ones, the researcher noted one cannot take nine pieces of candy from someone if they only have six and stated the need to regroup one tens-block into ten ones. After ungrouping one tens block, the researcher crossed out nine ones blocks and then one tens-block with the pen. Researchers summed the blocks left—one ten and seven ones or 17.

Representational

In the representational phase, researchers explicitly taught students how to draw pictorial representations to aid in solving their targeted mathematical computational area. For multiplication and division with remainders, this included drawing circles and lines or Xs to represent groups and objects. For double-digit subtraction with regrouping, this involved lines for tens-blocks and dots for one-blocks. Researchers employed the same instructional approaches as in the virtual phase for each mathematical area, except focused on drawing pictures to represent the virtual manipulatives and all used the Math Learning Center virtual whiteboard so no manipulatives were present (see Figure 1).

Abstract

In the abstract phase, researchers provided explicit instruction to students on solving their respective targeted computational areas with just numerical strategies. For all targeted areas, the researchers used the Math Learning Center virtual whiteboard but no drawings or manipulatives were used (see Figure 1). In the abstract phase, researchers provided strategies for students to employ, such as, for example, repeated addition, repeated subtraction, and the standard algorithm. With multiplication, the researchers modeled that if one did not know the answers, such that 3 × 7 was 21, one could use either skip counting (i.e., count by 3s seven times—6, 9, 12, 15, 18, and 21) or add 3 seven times (e.g., 3 + 3 = 6 + 3 = 9. . . 18 + 3 = 21). For division with remainders (e.g., 15 ÷ 6, researchers also modeled two strategies: repeated subtraction (e.g., 15 − 6 = 9 − 6 = 3; answer is 2R3) or finding the closest multiple of the divisor and the difference between that closest multiple and the dividend (e.g., 6 × 2 = 12 and 15 − 12 = 3, so answer is 2R3). For the double-digit subtraction with regrouping problems, the researchers emphasized the standard algorithm but with attention to regrouping. For example, with 44 − 28, the researcher noted that one cannot take 8 from 4 and get a positive whole number, so they needed to regroup one ten, leaving 3 tens. The researcher wrote 14 − 8, either knowing it as 6 or counting up from 8 to 14 to arrive at 6. The researchers subtracted 3 from 4 to get an answer of 16.

Fluency

Every other session—at least one session per phase (i.e., baseline, intervention, and maintenance) and intervention condition (i.e., virtual, representational, and abstract)—researchers conducted a fluency probe. For fluency probes, students were presented with 30 computational problems, each involving 60 digits, of their mathematical area and were given one minute to solve as many as they could. All fluency probes were presented on Google Jamboard. Researchers determined correct digits per minute (cdpm). Fluency probes to determine fluency rate occurred at the beginning of the sessions in which they were administered (i.e., prior to computational accuracy probes)

Generalization

Researchers also probed for generalization to word problems (i.e., word problem accuracy) every other session, with at least one session per phase (i.e., baseline, intervention, and maintenance) and intervention condition (i.e., virtual, representational, and abstract). During generalization probes, three word problems representing the targeted math area were presented to students on Google Jamboard. Students could use Google Jamboard to solve (e.g., pen to write to draw pictures) and take as much time as needed. Generalization probes for word problem accuracy occurred at the beginning of the sessions in which they were administered (i.e., after fluency probes but prior to computational accuracy probes).

Maintenance

Once a week for 2 weeks following the end of intervention, researchers probed for maintenance. Maintenance probe sessions were similar to baseline, although students completed the problems with the Math Learning Center virtual whiteboard. Researchers orally presented three targeted mathematical computational problems and students solved. Researchers collected computational accuracy data.

Interobserver Agreement and Procedural Fidelity

Researchers collected IOA data and procedural fidelity data throughout the study. For at least one-third of baseline, intervention sessions—including at least for each phase of the VRA instructional sequence, generalization, fluency, and maintenance sessions, a second researcher participated and collected both IOA and procedural fidelity data. IOA data were collected on computational accuracy (baseline, intervention, and maintenance), word problem accuracy (generalization), and fluency rate (fluency). To determine IOA, researchers determined the number of agreements for accuracy, divided it by the sum of the number of possibilities for agreement for each session (n = 3), and multiplied by 100 to get IOA. IOA for each student was 100% for each phase (i.e., intervention, generalization, fluency, and maintenance).

For the one-third of sessions in which IOA data was captured, the second observer also collected procedural fidelity data. Specifically, the second researchers completed a checklist. The checklist included: (a) student is completing the targeted math behavior; (b) researcher models one problem, which involves showing their screen and providing a verbal narration along with the demonstration; (c) student engages in one guided problem in which they independently try to solve and the researcher provides prompts and cues as needed; (d) student progresses from guided to independent portion of explicit instruction if accurate on the guided problem and needed two or fewer prompts; (e) student completes three problems independently without any assistance or support from the researcher by showing their screen; and (f) the student is provided the correct app—virtual manipulative or virtual whiteboard corresponding to the appropriate phase. Researchers determined procedural fidelity was 100% for each student for all phases.

Social Validity

Researchers assessed social validity at the end of the study from the students. The researchers created symbol-based questionnaire that asked students about the VRA instructional sequence as a whole as well as each phase and learning math virtually. The researchers also orally asked students about which approach they preferred to use to solve the problems—the virtual, representational, or abstract, if they felt they learned from working with the researchers, and which technology or app they liked the best.

Data Analysis

Researchers used visual analysis of the graphed data as well as conducted calculations. Researchers graphed the data in Excel and examined the computational accuracy data for the immediacy of effect when students moved from baseline to intervention as well as overlap between phases (Ledford & Gast, 2018). For trend for computational accuracy, researchers used the split middle technique, in which they drew a line between the mid-date and mid-rate of each phase’s data and concluded if that line was accelerating, decelerating, or zero-celerating (White & Haring, 1980). For stability, researchers found each phase’s median and then computed if 80% of the phase’s data fell within 25% of the median; if yes, then stable and if not, variable (Ledford & Gast, 2018). Last, researchers calculated Tau-U for computational accuracy to determine effect size. Researchers entered the baseline and intervention data into an online calculator (http://singlecaseresearch.org/calculators/tau-u). Effect sizes between 0.20 and 0.60 were considered moderate effect, between 0.60 and 0.80 were considered a large effect, and any effect size over 0.80 was considered a very large effect (Vannest et al., 2016).

Results

Overall, researchers found a functional relation between the VRA instructional sequence and students’ computational accuracy (see Figure 2). For two of the three students, consistent increases in fluency rates were found between baseline and intervention but the overall fluency rate for each remained quite low. Generalization to word problems (i.e., word problem accuracy) was less consistent for all three students. Students maintained high levels of computational accuracy in solving the computational problems.

Figure 2.

Graphed data for student acquisition, fluency, maintenance, and generalization.

Erica

During baseline, Erica answered zero problems correctly on her computational probes (see Figure 2). When she entered intervention, her computational accuracy on the multiplication probes was 66.7% for the first virtual session. However, she was 100% accurate for the next two virtual manipulative sessions. She struggled with the first two representational sessions, earning 66.7% computational accuracy for each. However, she earned 100% on her next two representational sessions and maintained at 100% for two consecutive abstract sessions, even with winter break in between. Erica’s Tau-U for computational accuracy between baseline and intervention was 1.0 indicating a very large effect. She was 100% accurate for the multiplication computational probes during her two maintenance sessions.

During her baseline word problem generalization session, Erica’s word problem accuracy was 0%. Her word problem accuracy when generalizing to word problems was also 0% on her first probe during the virtual intervention phase as well as her last two during abstract intervention phase and maintenance. She did achieve 100% word problem accuracy twice—once on her second virtual intervention phase probe and second representational intervention phase probe. In terms of the fluency probes, Eric’s cdpm in baseline were zero, resulting in a 0% fluency rate. During intervention, her highest number of cdpm was 8 (13.3% fluency rate).

Jeff

Jeff answered zero problems correctly on his computational probes during baseline (see Figure 2). When he entered intervention, his computational accuracy on the division with remainder probes was 100% for all intervention sessions when he advanced to the independent phase. However, on his first virtual and first abstract sessions he did not advance past guided practice as his guided independence rate was insufficient (i.e., needed too many prompts). Jeff’s computational accuracy Tau-U between baseline and intervention was 0.67, indicating a large effect. Jeff was 66.7% and 100% accurate, respectively, for the division with remainders computational probes during his two maintenance sessions.

For his first three generalization to word problem sessions, Jeff’s word problem accuracy was 0%. However, his word problem accuracy was 100% during the representational intervention phase and 66.7% (and then 0%) during the abstract intervention phase. Jeff returned to 100% word problem accuracy on the generalization probe during maintenance. In terms of his fluency, Jeff’s baseline fluency rates were 13.3% and 10%. He increased slightly during intervention (highest was 20% or 12 cdpm); his fluency rate was 16.7% during maintenance.

Eva

Across her five baseline sessions, Eva answered zero double-digit subtraction with regrouping problems correctly (see Figure 2). She was 100% computationally accurate during her first virtual intervention session, before falling to 66.7% and rebounding for her third and fourth virtual intervention sessions. . She started her first representational intervention session at 66.7% for computational accuracy but completed her last two representational and only two abstract intervention session 100%. Eva’s Tau-U for computational accuracy between baseline and intervention was 1, indicating a very large effect. She was also 100% computationally accurate on her two maintenance computational probes.

In terms of her generalization to word problem probes, Eva’s word problem accuracy was 0% for all sessions, expect one—her first virtual intervention generalization probe session (100%). That was also the only session in which Eva had any correct digits for any of the fluency probes (18.3% fluency rate).

Social Validity

Erica and Jeff both liked numerical strategies best, as they indicated the strategy was easier and faster than using virtual manipulative or drawings. Eva indicated she liked virtual manipulatives best because she thought they made sense how to represent the problem and she liked the grouping and ungrouping features. All three students thought they strategies they learned helped them to solve the problems and indicated a change from the beginning of the study from being “bad at math” or not understanding to now being able to solve the problems. All three students stated they thought they would sometimes use the strategies in the future. Erica and Jeff both indicated they liked online and in-person mathematics instruction. Eva indicated a preference for online because she liked that she could receive help on math while remaining safe in her home. While parents were not explicitly asked to respond to social validity questions, Erica’s and Eva’s mothers reported they saw improvement in their daughters’ motivation and confidence in mathematics as a result of their participation in the study.

Discussion

Determining effective mathematics interventions to support students with mathematical difficulties is needed both when delivering interventions traditionally face-to-face as well as within a virtual environment. A common intervention for students with mathematical difficulties or who have a disability in mathematics is the CRA instructional sequence (Bouck et al., 2018; Flores et al., 2018; Hinton et al., 2019). However, virtual manipulatives are an emerging and research-supported alternative to concrete manipulatives (Moyer-Packenham & Westenskow, 2013; Park et al., 2022; Shin et al., 2021). This study explored the online teaching of the VRA instructional sequence to three elementary students identified with mathematical difficulties. It further sought to examine the effects of the VRA instructional sequence across four stages of learning: acquisition, fluency, maintenance, and generalization (Shurr et al., 2019). Researchers found a functional relation between the VRA instructional sequence and students’ computational accuracy. Students acquired and became fluent in solving the problems without time constraints. The three students also generally maintained their computational accuracy after intervention ended. However, all three struggled with consistently generalizing to solve their targeted mathematical skill presented in a word problem (i.e., word problem accuracy) as well as solving for cdpm on a timed assessment of their targeted skill (i.e., fluency rate). The efficacy of the VRA instructional sequence in supporting elementary students with mathematics difficulties in this study supports the emerging prior research regarding virtual manipulative-based instructional sequences as an intervention for students who struggle but are not identified with disabilities in mathematics (Bouck & Long, 2021; Bouck et al., 2021, 2022).

Although all three students successfully solved their targeted mathematics computational problems (i.e., computational accuracy), generalization to word problems (i.e., word problem accuracy) was sporadic and limited. Erica successfully solved word problems for three sessions in the middle of the study before returning to adding—as opposed to multiplying—after winter break. Jeff too was successfully generalized twice—once in the representational and once in the abstract phase—before answering zero correct, and finally achieving 100% word problem accuracy during maintenance. Eva only successfully generalized during one session. While a large goal of learning is generalization (Hwang & Riccomini, 2016; Shurr et al., 2019), the students were only taught to solve problems in the pure computational form rather than solving word problems. Generalization is challenging for students and generalization should be taught rather than assumed it will occur (Ellis et al., 1987a, 1987b; Mercer & Miller, 1992).

The three students also did not make large gains with cdpm or fluency defined by accuracy on timed computational assessments. While Erica and Jeff made gains, the fluency rate was far below estimated rates of 40-to-60 correct digits per minute or one answer per second (Hasselbring et al., 1987; Mercer & Miller, 1992). Although the rates are considerably lower, suggesting the VRA instructional sequence is not designed to support fluency as measured on timed computational assessments, one must also consider students took these assessments online. Previous researchers found lower rates of mathematical fluency in elementary students when they solved multiplication or additional problems on a computer-based word document or a on a tablet with a stylus than with paper-and-pencil (Aspiranti et al., 2020; Hensley et al., 2017).

Another potential factor that may have contributed to the low rates of cdpm (i.e., fluency rate) and inconsistent generalization to word problems (i.e., word problem accuracy) was the intensity of instruction or treatment dosage (DeFouw et al., 2019; Hinton & Flores, 2019). In this study, the dose frequency may have been insufficient to adequately address the fluency rate (i.e., cdpm) or generalization, as students only received intervention one day per week. Although the dose form—one-on-one—may have been more intense and the researchers focused on mastery for the dose or length of the intervention, perhaps the VRA instructional sequence would be more effective if students received multiple sessions per week (e.g., 2, 3, and 4).

Implications for Practice

One implication for practice is the efficacy of using the VRA instructional sequence for students with mathematical difficulties. All three students acquired, became fluent without a time limit, and maintained computational accuracy in solving their respective targeted mathematical computational problems, and do so in seven-to-nine sessions, which generally lasted no more than 20 minutes. While students were taught both online and one-on-one, there are implications about the effectiveness in supporting struggling students who might receive response-to-intervention (RtI) tiered services (e.g., tier 2) with the VRA instructional sequence. Another implication is the need to build in a focus on generalization within the intervention. The VRA instructional sequence should provide explicit instruction for students with mathematical difficulties for both computational and word problems of targeted mathematical areas, providing greater support for students within their areas of struggle and focusing on generalization.

Limitations and Future Directions

One limitation for this study was the online nature of recruitment and participation, which occurred as a result of the COVID-19 pandemic. While parents self-reported their children struggled in mathematics, were at-risk, and would benefit from mathematical support, this was not confirmed with any school records or information. However, the researchers did confirm mathematical struggles with areas below grade level in the KeyMath-3 assessment for each student and targeted mathematical skills and concepts within that area for each student. Given the online nature and the one-on-one instruction, researchers should seek to replicate the study within RtI Tier 2 interventions to focus on face-to-face data collection as well as small group instruction. Further, the students completed all assessments online, which may have negatively impacted data collection, particularly for the correct digits per minute on fluency probes. As noted, previous researchers documented students performed less well on computer-based assessments than paper-pencil assessments, which may have contributed to the lower rates of cdpm (Aspiranti et al., 2020; Hensley et al., 2017). Researchers may seek to replicate this study but focus on paper-pencil assessments. Similarly, researchers probed immediately following instruction, which may have inflated scores. In future studies, researchers should consider probing for computational accuracy at the beginning of the next session.

Researchers also acknowledge the limited range in responses (i.e., three) as a limitation. While the researchers made this decision based in the need to keep each session within a 30-minute time frame (modeling, guided, and independent), to be sensitive to “Zoom fatigue” in students, and based on previous research within online environments documenting effecting mathematics interventions with struggling students using three problems per probe (Bouck & Long, 2021; Bouck et al., 2021, 2022), there are fewer data points on which to base decisions. In future studies, researchers should probe for more problems (e.g., 5 or 10) for both computational as well as word problem accuracy. Related, researchers could have also sought to examine the accuracy of digits for computational problems and word problems, outside of just fluency. The authors also acknowledge students progressed through all the phases of the VRA instructional sequence without researchers controlling for sequencing effects. While the researchers implemented the VRA and conducted the research study in similar and consistent fashion with past research regarding the VRA or CRA, they acknowledge that they cannot distinguish the effects of the individual phases.

A final limitation could be seen as the lack of consistent generalization to solving word problems for the students and the researchers lack of teaching for generalization (Mercer & Miller, 1992). The researchers were interested in seeing if students could generalization from the VRA instructional sequence with a focus on computational problems to solving similar word problems. The researchers acknowledge that a limitation in their study include the lack of assessing or otherwise determining students’ reading fluency. As such, the researchers were unable to take students’ reading ability into consideration with regards to students generalization scores. While researchers do not know given the lack of known information, students’ reading ability may have impacted their success with generalizing the mathematics to solving word problems. In future research, researchers should seek to actively teach for generalization as well as explore the VRA instructional sequence for word problem instruction. Other future directions include continuing to explore the VRA instructional sequence at the elementary grades, as the majority of the current research focuses on secondary students, as well as for students at-risk.

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 authors received an internal grant from the College of Education at Michigan State University to support this work.

ORCID iD

Emily Bouck

Author Biographies

Emily Bouck is a Professor in the Special Education Program in the College of Education at Michigan State University. Her research focuses on mathematics interventions and students with disabilities or students struggling in mathematics.

Holly Long is a doctoral candidate in the special education program in the College of Education at Michigan State University. Her research focuses on mathematics and young children with autism.

Yukyung Bae is a doctoral student in the EAD program in the College of Education at Michigan State University.

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