Abstract
Basic mathematic skills at the early age are foundational for later learning. Many students with autism spectrum disorder (ASD) struggle in academic learning without sufficient support. Research in the area of concrete manipulatives—tangible representations of abstract concepts—has been found effective. In addition, promising research has emerged in the area of virtual manipulatives—virtual representations of abstract concepts—as tools to support mathematics skill acquisition. Using a multiple baseline across participants with an embedded alternating treatment design, this study presents a replication of previous research comparing the effects of concrete and virtual manipulatives in the acquisition of double-digit addition and word problem-solving abilities of three elementary students with ASD. Findings indicate that while both interventions produce better outcomes than baseline, the virtual manipulative condition appear to be more supportive than concrete manipulatives.
Basic skills in mathematics, learned at an early age, are critical for further development of mathematics skills and performance in other academic content areas, such as literacy and science (Claesens & Engel, 2013; Watts et al., 2014). While individual mathematics skill levels can be highly variable (Keen et al., 2016), students with autism spectrum disorder (ASD) often struggle with sensory and cognitive processing as well as executive functioning, which can create barriers to learning academic content (e.g., Fleury et al., 2014). Nearly half of all students with ASD perform at or under one standard deviation below the mean in mathematical achievement (Wei et al., 2015). In addition to academic learning, deficits in academic skills, such as mathematics, are negatively correlated with such adult outcome achievements as independent living, post-secondary education, and employment (Levy & Perry, 2011; Test et al., 2009).
In a review by Fleury et al. (2014), six evidence-based instructional practices were identified to support academic instruction for students with ASD: priming, peer support, video modeling, explicit strategy instruction, self-management, and graphic organizers. For those with more significant support needs, Fleury et al. (2014) suggested the use of task analysis, time delay, system of least prompts, the model-lead-teach framework, and graphic organizers. Reviews of mathematic instruction, in particular, for students with ASD found an evidence base for the use of visual supports, cognitive strategies, and explicit instruction (Barnett & Cleary, 2015; King et al., 2016). Furthermore, mathematics research in general education as well as that focused on students with significant disabilities, including those with autism, found concrete manipulatives (CM)—tangible representation of abstract numeracy concepts—effective in instruction of various mathematical ideas and skills (Bouck et al., 2014; Carbonneau et al., 2013; Sowell, 1989; Spooner et al., 2019). With the support of CM, students with ASD were able to experience success in performing basic mathematics skills such as addition and subtraction (e.g., Bouck et al., 2014; Jimenez & Kemmery, 2013).
Recently, there is growing evidence for the use of a technology-based version of manipulatives, referred to as virtual manipulatives (VM). VM are a computer-generated version of tangible objects which can be used to support student understanding of mathematical operations (Bouck & Sprick, 2019). Similar to CM such as base-10 blocks or tangrams, VM can be physically moved and controlled, to represent mathematical processes, albeit on a computer screen (Bouck & Flanagan, 2010). While not yet considered to meet the stringent criteria set forth by researchers to warrant the classification of VM as an evidence-based practice (see Cook et al., 2015), emerging research demonstrated positive effects of VM in mathematics instruction for a wide range of students such as those with learning disability (e.g., Satsangi et al., 2018; Shin et al., 2017), intellectual disability (e.g., Bouck, Park, et al., 2018), and ASD (e.g., Bassette, Bouck, Shurr, Park, & Creamans, 2019; Bassette, Bouck, Shurr, Park, Creamans, Rork, et al., 2019; Bouck et al., 2014; Root et al., 2017). Mathematic skills targeted with this instructional strategy include area and perimeter (Satsangi & Bouck, 2015), adding fractions (Bouck, Park, et al., 2017), and subtraction (Bassette, Bouck, Shurr, Park, Cremeans, Rork, et al., 2019; Bouck et al., 2014).
Beyond evaluating the efficacy of VM, researchers also examined differences in mathematics skill performance as related to the type of manipulatives used (i.e., VM vs. CM; Bassette, Bouck, Shurr, Park, & Creamans, 2019; Bassette, Bouck, Shurr, Park, Cremeans, Rork, et al., 2019; Bouck, Chamberlain, & Park, 2017; Bouck et al., 2014; Bouck, Shurr, et al., 2018; Satsangi & Bouck, 2015). While researchers found both strategies effective in increasing independence and accuracy in completing mathematical skills, little substantial differences exist. However, with these positive results in mind and the unique characteristics of VM—ability to capitalize on student interest in technology, availability at a low cost, and effectiveness as a support tool (Bouck & Sprick, 2019)—it is clear that the intervention holds promise.
While there is emerging evidence for the use of VM in supporting the acquisition of mathematics skills, the question of differences in effectiveness between the two methods specifically for young students with ASD remains. The present study represents an extension of the research using VM to support the mathematical learning of elementary students with ASD. Specifically, the authors sought to compare the effect of CM versus VM in supporting double-digit addition computation alone and embedded in word problem for three elementary students with ASD. The specific research question was: Are CM or VM more effective in supporting the acquisition of mathematics skills by students with ASD (i.e., double-digit addition computation and word problems)?
Method
Participants
The participants in this study were three elementary boys with autism. Two students were educated in the same school, the third student in a different school. Participants for this study were selected on the basis of (a) having an Individualized Education Program (IEP) with the eligibility for ASD; (b) experiencing difficulty in addition operations, as reported by the teacher and indicated by the KeyMath-3 assessment; and (c) receiving parent or guardian permission to participate. All names of individuals used hereafter are pseudonyms.
Both boys at the same elementary school, Henry and Max, received their primary mathematic instruction from a special education teacher within a resource classroom while accessing other subjects in general education classrooms with support from a paraprofessional. Both students had most recently been working on single-digit addition operations as well as number-object correspondence using manipulatives such as connecting blocks and base-10 blocks. Dane received his mathematics instruction in the general education setting with the assistance of a one-on-one paraprofessional; he was also supported by a resource room teacher and an autism consultant. Dane had been most recently working on both grade level mathematics content as well as increasing the accuracy of basic skills (i.e., addition and subtraction).
Max
Max was a 10-year-old African American boy with ASD. As indicated by teacher report and researcher observation, Max experienced difficulty in grade level mathematics skills, as well as basic operations such as adding and subtracting single digit values. According to his IEP, Max was able to independently solve addition and subtraction problems with sums of 20 or less. Upon teacher report, it was understood that Max was rote counting in operations, without counting on from the higher number, and would often skip a number resulting in a close but incorrect answer. Max qualified for the state alternate assessment, as the state assessment was deemed not developmentally appropriate considering his current needs and skills. Supports listed in Max’s IEP included: visual strategies, AAC, adult supervision, and sensory strategies. His special educational programming needs included 3 to 4 sessions lasting from 1 to 3 hours daily. Results of the KeyMath-3 assessment conducted prior to the study identified double-digit addition as an area in need of support, and therefore became the focus skill for Max’s trials.
Henry
Henry was a 9-year-old Caucasian boy with ASD. According to his most recent IEP, he was able to identify and verbally name numbers to 100 and use manipulatives to add single-digit problems with some support. Participation in the mandated state assessment was deemed not developmentally appropriate for Henry. Instead, he qualified to take the state’s modified alternate assessment, designed for students with significant cognitive impairment. In terms of supports, Henry’s IEP listed the need for visual strategies, augmentative and alternative communication (AAC), adult supervision, and sensory strategies. His special educational programming needs included 3 to 4 sessions lasting from 1 to 3 hours daily. Results of the KeyMath-3 assessment conducted prior to the study identified double-digit addition as an area in need of instruction and support, and therefore became the focus skill for Henry’s trials.
Dane
Dane was a 9-year-old third-grade Caucasian boy with ASD. His teacher indicated that Dane experienced difficulty with consistently and independently demonstrating grade level mathematics skills, particularly in addition and subtraction. Researcher observation concluded Dane had some success adding and subtracting with regrouping (60% accuracy) but could not solve word problems involving addition or subtraction. Dane was educated primarily in the general education setting, but received support from a one-on-one paraprofessional. During mathematics instruction, Dane worked with his paraprofessional on both the presented grade level content as well as his math-related IEP goals involving independence and fluency relative to basic operations. Researchers administered the KeyMath-3, but determined the results to be unreliable. The research team believed that this was due to the horizontal presentation of addition and subtraction problems in the KeyMath-3; Dane had only been exposed to these problems presented vertically. He was unable to solve any of the problems during the assessment. However, when given similar problems presented vertically he experienced greater success. These results, combined with observation and discussion with his educational support team determined that applied addition via word problems including double digit addition would be an appropriate and timely instructional target and was therefore selected.
Setting
The study was conducted in two public elementary schools in rural towns within a Midwestern state. One school served students in Grades 4 to 6. The population comprised of 388 students, including 76% Caucasian, 6% American Indian, 5% African American, 4% Asian and 4% Hispanic. Of the students enrolled, 18% received special education services and 37% were eligible for free and reduced-cost lunch. The other school included Grades K-5 school and was comprised of 334 students, including 77% Caucasian, 17% Hispanic and less than 1% American Indian and African American. Of the students enrolled, 25% received special education services and 68% were eligible for free and reduced lunch.
All sessions were conducted at a small table within the special education resource room. During the sessions, there were typically between one-and-six other students, one teacher, and one-to-three paraprofessionals present. The other students were engaged in either small group instruction or independent academic or break activities during the mathematic sessions.
Materials
Materials in this study included one-to-two sets of base-10 blocks (when two, each were a different color), a place value mat, a set of unique double-digit addition worksheet probes with five problems each, and an iPad loaded with the “Base Ten Blocks” app. For Henry and Max, the double-digit addition probes were used in all conditions and contained five unique double-digit addition computation problems with sums from 20 to 99 presented vertically on a worksheet. No two worksheets were repeated over the course of the study. For Dane, the double-digit addition probes were embedded in five unique word problems (e.g., Barb had 40 notebooks. Her father gave her 13 more notebooks. How many notebooks does Barb have now?), with sums from 20 to 99; no problem or number combinations repeated across the study. Three problems were presented on one side of the paper and two on the other. Researchers based the story problems on released items from the state standardized assessment for third grade. All three students were presented with the paper probes and a pencil to record their answers.
In the CM condition, students had access to concrete base-10 blocks and the place value mat. For Henry and Max, the blocks included both ones and tens in a single color. For Dane, each part of the addition problem (each addend) was represented by different colored base-10 blocks (i.e., one green and one blue). The place value mat had a printed grid replicating the visual format of the VM app. This included a table with two columns, one labeled “tens” and one labeled “ones” and three rows—two for the addition problem and the remaining for the solution. The place value mat was printed on an 8.5 x 11 inch sheet of white paper and covered in a clear plastic page sleeve. For Henry and Max, the researcher used a dry erase marker to write the problem on the sleeve in the corresponding place value boxes along with an addition symbol and an equal sign to mirror the problem printed on the worksheet. The CM were then placed on top of their corresponding numbers on the place value mat. For Dane, researchers set out more than sufficient ones and tens blocks—at the start of each session to the side of the place value sheet. Dane was responsible for reading each problem, writing the computation problem, and setting up the blocks to represent the problems.
In the VM condition, the app “Base Ten Blocks,” from the company Brainingcamp (2018), was presented on an iPad for all three students. Henry and Max used a more recent version of the app; Dane used an older version. For Henry and Max, the app had a table, identical to the place-value mat with columns for ones and tens, but also included a column for the hundreds value. The app allowed students to move manipulatives by touching and dragging them along the screen with a finger. Ones squares can be linked together by placing one next to another. Once a group of ones contained 10 units, it automatically transformed into one tens-block and changed from yellow to blue (the ones blocks were yellow and the tens blocks blue on the app). Once changed, this tens block can be dragged to the tens column. Dragging any non-matching manipulatives (e.g., 9 ones blocks to the tens column), was not possible in the app; when attempted, the blocks immediately returned to the original spot. For each problem, the researcher used a virtual marker embedded in the program to create two horizontal lines of the table—one between the first and second number in the operation and one to divide the problem from the answer, to match the place value mat. In addition, for Henry and Max, the researcher wrote the double-digit addition problem with each number in the appropriate ones or tens cell. The corresponding VM were then set up in the correct column and row by the researcher to ready the problem by using the drag and drop feature of the app, similar to the CM condition.
For Dane, the app also had a place value chart, but the researcher could restrict to showing just ones and tens. The app presented two rows—one for each addend. Dane set up the virtual base-10 blocks to represent each problem. As with the newer version of the app that Henry and Max used, when one combined 10 ones blocks together, they formed a 10 block and could be moved to the tens column. Each addend was a separate color—the first one was green and the second was red.
Independent and Dependent Variables
In this study, the independent variable consisted of the manipulative-based support sequence using either the concrete base-10 blocks or virtual base-10 blocks in solving the double-digit addition problems. The dependent variables included percentage of problems answered correctly and independently of the five double-digit addition problems on the probes. Correct answers that required prompting of any sort were recorded as incorrect. Scores ranged from 0%—no independently produced correct answers, to 100%—5 independently produced correct answers, in increments of 20 percentage points per correct answer. Answers spoken aloud, written, or both were accepted as the dependent variable for Max and Henry. When answers were spoken, the researcher would use verbal or gestural prompts to assist the student in writing the spoken number. Dane independently wrote his answers on the worksheet.
Experimental Design
Researchers used a multiple baseline with embedded alternating treatment design study across three participants. Researchers deemed this particular design most fitting as the goal was comparison of independent accurate performance of mathematic skills between the CM and VM as well as baseline. Immediately following comparison of the two conditions, researchers isolated the most effective treatment for three final sessions to ensure consistency in the data.
Procedures
Baseline
During each baseline session, students were given a unique 5-problem double-digit addition probe—with pure computation problems for Max and Henry and word problems for Dane. Per his teacher’s preferences and targeted goals, Dane was required to answer the problems by writing the answers on the worksheet independently. Max and Henry could answer verbally or written; if they answered verbally, they were subsequently prompted to write their response under the problem. Prompts to solve the problem, such as pointing to the space under the addition line, were given in the form of least-to-most (i.e., gesture, indirect verbal, direct verbal, model, partial physical, and full physical), as needed following a 10-s pause of inactivity. Adherent to multiple baseline design, all students started the baseline condition simultaneous, however entered the intervention condition one at a time as the previous student achieved a stable baseline. Students were order from first to last as: Max, Henry, and Dane.
Training
Following baseline, students were individually trained by the authors to use both the VM and CM to solve the addition problems. Each training session was randomly ordered and centered around one unique five-problem, double-digit addition probe using either the VM or CM. Each student completed multiple concrete and virtual manipulative training sessions. A five-step task analysis was used to teach Max and Henry the proper sequence of steps for both conditions. Steps included (a) add the ones, (b) remove set of 10 from one’s column and replace with a tens block, (c) move tens block to the tens column, (d) count and write the number of ones, and (e) count and write the number of tens. Dane’s task analysis for the word problems was 13 steps: (a) read the word problem, (b) write down the first tens of the addition problem, (c) write down the first ones of the addition problem, (d) write down the second tens of the addition problem, (e) write down the second ones of the addition problem, (f) write down the sign, (g) set up the tens blocks for the first number of the problem, (h) set up the ones blocks for the first number of the problem, (i) set up the tens blocks for the second number of the problem, (j) set up the ones blocks for the second number of the problem, (k) regroup ones, (l) count the number of tens and ones, and (m) write the answer.
The instructor modeled solving the problem using the task analysis and the appropriate manipulatives and then allowed the student to complete the same problem using the system of least-to-most prompts for support. In the VM training sessions, the students were taught to drag the appropriate number of virtual ones or 10 blocks to the appropriate cell until the number in the addition problem was accurately represented. Then, students were taught to, starting with the ones column, add the numbers up and regroup as necessary prior to adding the tens column. Similarly, in the CM training session, students were taught to set up the problem with the appropriate ones and tens blocks on the place value mat and then add starting with the ones, followed by any necessary regrouping and adding the tens. Training was conducted until the student was able to achieve 80% accuracy independently on the task analysis across one probe.
Intervention
When training was deemed successful, intervention commenced. Before starting intervention, the treatments (i.e., VM and CM) were randomly assigned for order using an online random integer generator (Haahr, 2020). Max completed a total of 16 sessions with 7 in the CM condition and 9 in VM. Henry completed a total of 18 sessions with 10 in the CM condition and 8 in VM. Dane completed a total of 13 sessions with 5 in the CM condition, and 8 in VM. For Max and Dane, the final three sessions were conducted with the best fit intervention—that is the intervention that resulted in the better performance. Due to scheduling conflicts, Henry’s sessions ended before best fit sessions could begin. While the conditions were randomly assigned, the total number of sessions completed, and therefore the number within each condition, was a product of student availability for the study. Similar to baseline, students were given a unique five-problem, double-digit addition probe and prompted to solve each problem. Students were provided with the appropriate materials for their condition. In the case that a student waited longer than 10 s to initiate a step or that the student completed an incorrect or out of sequence step, the researcher provided support on the appropriate next step using the system of least prompts.
Virtual manipulatives
In the VM condition, students were presented with a mathematics probe and the iPad opened to the app. For Max and Henry, the researcher and student worked together to first set up the problem in the app by replicating the numbers and their corresponding virtual ones and tens blocks in the appropriate cells. Once the problem was set up, students were asked to solve the problem. Dane set up his own problems. Any assistance required to complete the task analysis correctly were prompted and recorded as such. Once the student solved the problem, regardless of whether or not it was correct, he transferred this answer to the appropriate addition problem on the probe, with prompting as necessary.
Concrete manipulatives
In the CM condition, students were again presented with the addition worksheet and the tools—base-10 blocks, both in ones and tens, and the place value mat. For Max and Henry, the researcher and student worked cooperatively to build the problem using the blocks on the place value mat. When the problem was accurately set up, the student was asked to solve the problem. Dane set up his own problems based on reading the word problem on the probe. Similar to the app condition, assistance with the task analysis steps were given as needed and recorded as prompted responses. Finally, the student transferred the answer from the place value mat to the addition worksheet, independently or with prompts as necessary.
Inter-observer agreement and treatment fidelity
Inter-observer agreement (IOA) data were collected on all of the baseline sessions and six intervention sessions for Henry and Max, with half comprising the CM condition and the other half VM condition. The six intervention sessions of IOA constituted 33% of sessions for Henry and 38% for Max. For Dane, IOA was conducted on 31% of intervention session conditions—with two sessions per intervention condition—and on 43%, or three of baseline sessions. IOA data collected independently by a trained observer (i.e., classroom paraprofessional) and included both the prompt types used in each of the task analysis steps, as well as, whether the answer to the addition problem was correct or incorrect. IOA training included verbal discussion and modeling with the research team followed by an informal assessment (i.e., observation of performance in practice) of the team to determine rated readiness. IOA was determined by the number of agreements divided by the number of opportunities, which included the sum of agreements and disagreements. The result was then converted to percentage for the sake of reporting. IOA for Henry was found to be 100% in baseline, 99% in the CM condition, and 97% in the VM condition. Max’s average IOA was 100% in baseline, 96% in the CM condition, and 95% in the VM condition. For Dane, IOA was 96% at baseline, 97% in the CM condition, and 92% in VM condition.
Treatment fidelity measures were used to describe the accuracy of the intervention as delivered by the researcher. For six sessions across each condition (33% of sessions for Henry, 38% for Max, and 24% for Dane), the researcher was observed by a trained observer (i.e., classroom paraprofessional) to record the accuracy of the researcher’s adherence to the intervention procedures. Treatment fidelity for all sessions for each student was 100%.
Social validity
On completion of the study, the researcher interviewed the special education teacher of Henry and Max and the special education teacher and paraprofessional for Dane to assess the social validity of the intervention. Questions included the individual’s perception of the interventions including feasibility and propensity for use in the classroom, student engagement and progress, and any noted comparisons between the two methods.
Data Analysis
In line with guidelines for singe subject research, visual analysis of the graphed data included analysis of the level, trend, variability, immediacy of effect, overlap, and data consistency across the subjects (Kratochwill et al., 2010). Level was determined by drawing a mean line for each of the conditions for comparison. To find the trend, researchers used the split-middle technique which splits the data for each condition in half and uses the mean of each half to draw a trend line which can be compared across conditions. Variability was described as high, medium, or low, based on visual analysis of the data proximity to the best fit line or trend line (Kennedy, 2005). Immediacy of effect described any change in level between the last and first three data points in subsequent conditions; this only applied to baseline-intervention condition changes. For overlap, conditions and interventions were compared by counting the number of data points not in common and dividing that number by the total points per condition. This number was then multiplied by 100 to obtain a percentage of non-overlapping data (PND) points. A result of 0% indicated a significant amount of overlap in the data and 100% signified no overlap. Researchers assessed data consistency by examining the similar conditions across participants to assess the presence of patterns (Kratochwill et al., 2010). Researchers chose the Improvement Rate Difference (IRD) measure of effect size to further describe the data due to its ability to distinguish the scale of effect between two conditions (Chen et al., 2016). IRD was calculated using the IRD calculator from Single Case Research™ (Vannest et al., 2016).
Results
The results appear to mirror previous research on the effectiveness of using CM or VM to support mathematical performance (see Figure 1). In addition, for one participant—Max—a functional relation was noted for the VM condition in regards to double-digit addition. In the other two cases, the students performed with greater accuracy and independence in the VM over the CM condition, however, due to variability and overlap in the data, a clear functional relation was not established for either of them among the group of students.
Students’ independent and accurate performance with virtual vs. concrete manipulatives.
Max
In baseline, Max was unable to produce any correct, independent answers to the double-digit addition problems. During the intervention phase with both interventions combined, he achieved an average of 45% independent accuracy. With support through the system of least prompts, Max achieved an average of 99% accuracy with a range of 80% to 100%. When given support as needed, Max achieved 97% in the CM condition and 100% in VM condition.
Within the intervention phase, Max achieved higher levels of independent and accurate performance in the VM condition (63%), as compare to the CM condition (17%). He was independent and accurate 73% of the time during best fit (VM). Both conditions exhibited upward trends. In the VM condition, variability was found to be high and medium to high in the CM condition. The data indicated a clear distinction between the baseline and VM condition, while the CM condition showed similarity in data. No overlap in the data was noted between the VM and baseline conditions (PND = 100%) and moderate overlap with the CM condition (PND = 57%). Overlap between the two interventions was relatively low (PND = 78%).
In terms of effect size between the interventions and baseline, the CM condition resulted in an IRD of 0.57 and the VM condition an IRD of 1.0, indicating a strong effect of the VM intervention when compared to baseline. IRD comparing the interventions resulted in an IRD of 0.78, indicating a moderate effect of the VM as compared to the CM condition. In terms of support required to perform the steps within the CM and VM conditions, Max demonstrated more independence in the VM condition, with an average of 93% steps unprompted as compared to 69% steps unprompted in the CM. In addition, Max independently completed all steps for each of the five addition problems in two sessions for the VM condition and none for the CM.
Henry
Consistently, during each of Henry’s baseline sessions, he was unable to answer questions independently. During the intervention phase, Henry achieved a combined average of 43% independent and accurate problem solving across treatments. In addition, he demonstrated an overall average of 96% accuracy with a range of 60% to 100% in answering questions correctly with various levels of prompting. Henry answered a slightly higher average of problems correctly in the VM condition as compared to the CM condition (98% and 94%, respectively).
In the intervention phase, Henry’s data showed higher levels of accurate independent performance for the VM condition over the CM, with averages of 65% and 26%, respectively (see Figure 1). Both conditions resulted in upward trends over the sessions. Variability for the VM condition can be described as medium, while that for the CM was high. In terms of immediacy effect, there was a clear distinction between the baseline and VM condition, while the CM condition showed some similarity in data. There was no overlap in data between the VM and baseline condition (PND = 100%) and moderate overlap in the CM condition (PND = 40%). Overlap between the two interventions was very high (PND = 0%; see Figure 1). IRD between the baseline and CM condition was reported as 0, while that of the VM was 1.0, indicating a significant difference in values between the latter condition and no intervention. When compared against one another, the two independent variables resulted in an IRD of 0.60, indicating a moderate to high degree of overlap in data. In terms of support required to perform the steps within the CM and VM conditions, Henry demonstrated more independence in the VM condition, with an average of 81% steps unprompted in each session in the CM compared to 92% in the VM condition. In addition, Henry independently completed all steps for each of the five addition problems in one session for the CM condition and two for the VM condition.
Dane
During baseline, Dane began without independent accurate performance, experienced an acceleration, and then a deceleration (see Figure 1). In the intervention phase with both interventions combined, he achieved an average of 92% independent accuracy in addition to 100% in the best-fit treatment phase. With support through the system of least prompts, as needed, Dane achieved an average of 99% accuracy with a range of 80% to 100%. In comparison between conditions, Dane achieved 97% accuracy in the CM condition and 100% in VM.
Within the intervention phase, Dane achieved higher levels of independent and accurate performance with 96% in the VM and 89% in the CM condition, as well as 100% in the final three best fit sessions for the VM condition. Baseline produced a general upward trend with some deceleration at the end while the CM condition was stable and the VM condition had a slight upward trend. In baseline, the variability was high, while for both of the intervention conditions it was low. There was an immediate effect from baseline to both conditions. Overlap between the interventions individually and baseline was non-existent (PND = 100%) and high between the two interventions (PND = 0%). In terms of effect size between the interventions and baseline, both resulted in an IRD of 1.0, indicating a strong effect when compared to baseline performance. IRD comparing the interventions to each other resulted in an IRD of 0.4, indicating a small effect of the VM as compared to the CM condition.
Data Consistency
Comparing each condition across the three students found a relative unity in baseline conditions, with little independent and accurate success. Despite Dane’s few upward trending data in sessions 4 and 5, his overall baseline level remained below 25% with data peaking at 60% independent accuracy. As a whole, students performed better in the intervention phases than in baseline. While, slight, it appeared that the VM condition was more conducive to successful performance than the CM condition.
Social Validity
Although the students were not directly asked their perspective on the study and materials they were observed to be both willing and interested in participating in the study and readily gave assent at the start of each session. When asked about her thoughts on the study, Max and Henry’s teacher reported that she had used base-10 blocks for mathematics instruction in the past and found them to be helpful for most students. Although, following the study, she reported a strong interest in the VM as a viable alternative and requested information about the app for future use in her classroom. To her, the virtual tool was more appealing because it used an iPad and it offered less of a chance for distraction as the base-ten blocks, with its multiple little pieces. Dane’s teacher and paraprofessional both expressed pleasure at Dane’s engagement during the study. Dane’s teacher asked for the name of the app and specifications, as she was recommending it to be ordered for him to use after the study ended and into the next school year.
Discussion
This study contributed to the growing research base on the use of VM as a support for students with ASD in learning mathematics. Using a multiple baseline across participants with an embedded alternating treatment design, virtual and concrete base-ten blocks manipulatives were randomly provided as supports for elementary students with autism in completing double-digit computation and double-digit addition word-problems. Independent and accurate performance within both intervention conditions greatly exceeded baseline condition and while close, it appears that the virtual condition may produce more stable and successful performance.
Previous research is idiosyncratic with regards to the greater independence and/or accuracy in mathematics with VM than CM. In this study, students were more independently accurate when using VM than CM. Differences from previous research (i.e., researchers finding CM to produce more independence and/accuracy) could be due to a number of factors, including differences populations (e.g., learning disabilities or intellectual disability vs. autism and elementary vs. secondary) and in the mathematical skills presented (i.e., double-digit addition versus adding fractions, Bouck, Shurr, et al., 2018, or algebraic equations, Satsangi et al., 2016). Of the previous research comparing VM and CM, only three (Bassette, Bouck, Shurr, Park, & Creamans, 2019; Bassette, Bouck, Shurr, Park, Cremeans, Rork, et al., 2019; Bouck et al., 2014) similarly included elementary students with ASD. These particular studies mirrored the favorable outcomes with VM over CM; these particular studies also focused on similar mathematics skills (i.e., addition and subtraction) in addition to similar populations. Collectively, the results lend support that future research needs to examine unique elements of elementary students with autism that result in greater independence or accuracy with virtual in contrast to CM. Researchers may hypothesize that elementary students with autism are engaged with the technology as well as the inherent features built-in within VM provide scaffolds that benefit students.
Regarding the first research question of student ability to use the VM independently on the mathematics problems, students were not only successful in learning the task analysis steps, but had success performing them independently, despite some variability in accuracy of performance. Independent performance is a good indicator of a supportive intervention as students with ASD often have difficulty managing their own interventions and supports; those interventions that can be self-managed are highly useful for this population of students (Southall & Gast, 2011). As for the performance variability observed, this was likely due to the newness of the skill and the probe lengths (5 problems). Despite the ranges in performance, overall visual analysis indicated stronger performance with the VM over CM and baseline. In terms of usefulness of VM for support in addition, results indicated effectiveness of the intervention. These findings together reiterate the previous research on the effectiveness of VM in teaching a variety of mathematics skills to students a variety of with ASD and other disability diagnoses (e.g., Bassette, Bouck, Shurr, Park, & Creamans, 2019; Bassette, Bouck, Shurr, Park, Cremeans, Rork, et al., 2019; Bouck, Park, et al., 2018; Root et al., 2017; Satsangi & Bouck, 2015).
In terms of the comparison between the two interventions, it appears that while both were significantly more effective at producing independent and accurate performance on the mathematics tasks than baseline conditions, the VM condition fared slightly better in terms of independence and accuracy. While the majority of the research comparing VM and CM failed to highlight one method as superior, the results in this present study echo those of Bassette, Bouck, Shurr, Park, and Creamans (2019), Bassette, Bouck, Shurr, Park, Creamans, Rork, et al. (2019), and Bouck et al. (2014), confirming the potential advantage of VM.
There are several possible explanations for the positive results of VM, including student preference toward digital tools, the supportive nature of the specific VM used, and the systematic nature of VM. In terms of student preference, there is no denying the popularity and increasing prevalence of electronics both inside and outside of school (e.g., Delgado et al., 2015). Such mediums have been successfully used to increase student motivation or engagement in school-related tasks (Harper & Milman, 2016). And, students are often quite comfortable using digital devices for a variety of purposes, including learning and play (e.g., Johnson, 2010). The slightly greater success in the VM condition over CM condition could be rooted in student preference toward digital supports, without regard to the embedded content.
As noted in the method section, the particular VM used included several embedded features that made it more supportive of independent student performance. For instance, the software would not allow students to place more than 10 ones blocks in a line to create a tens block. In contrast, using the CM, students would require adult intervention if they attempted to exchange 9 or 11 ones blocks for a tens block. This additional support in the VM could provide some explanation for higher independent VM scores (see Bouck, Chamberlain, & Park, 2017, for a similar hypothesis discussion). While efforts were made to provide a visual system for the CM conditions that mirrored that of the VM (i.e., use of the number mat), the VM layout and overall system was more predictable and systematic. For instance, it was impossible to misplace a ones-block in the VM system, unlike for CM in which pieces could be dropped or flicked. While evidence appears to identify VM as superior, findings should be considered in context given the potential explanations listed above. Additional research in comparison and manipulation of CM and VM would help to provide more clarity in these areas.
Implications for Practice
The data in this present study and that of Bouck et al. (2014), Bassette, Bouck, Shurr, Park, and Creamans (2019), and Bassette, Bouck, Shurr, Park, Creamans, Rork, et al. (2019) suggest that while both VM and CM can be a highly beneficial teaching tools for basic mathematic operations, VM may have a slight edge for young students with ASD. Students with ASD often have difficulties in mathematics related to deductive reasoning, problem solving, and abstract thinking (Santos et al., 2015). While CM can be part of a healthy mathematics instructional tool kit, VM can help students build conceptual understanding of basic computation and word-problem solving as they are able to visualize and interact with mathematical concepts in a virtual format.
While visual representation in mathematics instruction, such as CM, are commonplace (van Garderen et al., 2018), researchers found technology an evidence-based practice for students with ASD (Wong et al., 2015). In a survey of special educators, Flanagan et al. (2013) found teachers believed technology to be beneficial in support of academic learning, but not without significant barriers such as cost, ease of use, and a lack of sufficient training. Courduff et al. (2016) similarly found special educators were aware of potential difficulties in using technology, yet determined teachers were able to surpass barriers when they recognized opportunities for technology use and believed technology could benefit student learning.
In addition to benefits for student learning, technology use also has been found to positively impact how a teacher spends their time. This could include a shift from whole group instruction to more time on individualized student support and independent student work (McKnight et al., 2016). Such integration of technology can also have a positive effect on teacher assessment of student performance, as technology-based instructional resources often automatically track student progress (McKnight et al., 2016).
Limitations and Future Directions
Despite the findings within this study, several limitations existed. For one, researchers led the intervention. While the teachers were involved in observing sessions, as well as discussions about student progress, the intervention, and potential carry through attempts, neither teacher facilitated the intervention. Researchers assume instructional use of both manipulatives could be easily done by special and general education teachers. However, it is critical for researchers to consider including natural environments when evaluating the usefulness and effectiveness of interventions. In the future, researchers should assess the teachers delivering this intervention to learn about pragmatic adjustments to increase the effectiveness and usefulness in the classroom. In this study, two students did not independently set up their own problem using the manipulatives, rather researchers prepared the problems. As with Dane, students can learn the skills to prepare a problem using VM and CM. While it may take additional training, it is a worthy goal and should be considered in future research. As students gain more independence in the process of learning, the capabilities for learning specific skills and content widens.
Also, this study was not in strict alignment with the What Works Clearinghouse (WWC) standards on presentation of treatments and baseline conditions and it lacked both a maintenance and generalization phase. According to WWC (2020), alternating treatment design should employ no more than two continuous presentations of a treatment. In one case, three presentations occurred. While this can be seen as a conflict with the standards, it is not considered to negate the findings due to the full context of the data including the five minimum required data points as well as other single case guidelines without the specific two or less rule (e.g., Ledford & Gast, 2018). In addition, WWC indicated a need for five baseline data points before intervention. However, quality indicators from the Council for Exceptional Children (Cook et al., 2015) suggest a minimum of three baseline points are sufficient. After three baseline data sessions, Max moved into the intervention phase. Researchers made the decision due to the clear consistency in his performance, combined with his notable frustration completing the task with limited support and tools. In a study such as this, a maintenance phase—skill performance after a break in intervention—can show the ability to maintain the skill over time. Such information can add to the overall understanding of the intervention and identify necessary modifications to improve intervention training for long-lasting performance effect. In addition, a generalization phase—skill performance in a different context—can illustrate a student’s abilities to expand their performance beyond the prescribed activity (i.e., mathematics worksheets with a researcher in a resource room). This could include use of the intervention in a general education setting, with different problems, or some other relevant variation. Information in a generalization phase can assist in understanding of the bounds of the intervention and student learning and performance, and again to improve the intervention procedures. Future research on VM in mathematics should include moth a maintenance and generalization phase.
Footnotes
Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author(s) received no financial support for the research, authorship, and/or publication of this article.
Ethical Approval
All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards.
