A Comparison of Manipulative Use on Mathematics Efficiency in Elementary Students With Autism Spectrum Disorder

Abstract

Manipulatives are a commonly used intervention that provide visual instruction known to promote mathematical learning; however, the impact on students with autism spectrum disorder (ASD) is less understood. Improving mathematical procedural understanding is important for students with ASD given these skills can help increase access to more advanced mathematics and future opportunities (e.g., postsecondary education). This study expanded upon previous research and compared the ability of students with ASD to solve mathematical problems when using concrete and app-based manipulatives. A single-case alternating treatment design was used to explore differences in steps completed independently per minute (i.e., efficiency) and accuracy when using both types of manipulatives. Two participants were more efficient when using the app-based manipulative while one was more efficient with the concrete manipulative. Similar to previous research, all participants indicated they preferred the app-based condition. Limitations and future research are included.

Keywords

math content/curriculum area autism exceptionality APPS technology perspectives

It is well established that elementary-level mathematics education is important for all students (Ellis & Berry, 2005) including elementary students with autism spectrum disorder (ASD; Yakubova, Hughes, & Shinaberry, 2016). While critical, the research base regarding effective mathematical interventions for students of all ages with ASD is limited (Barnett & Cleary, 2015). It is also recognized that students with ASD may struggle with mathematics due to difficulties with reading, language, and working memory (Root, Browder, Saunders, & Lo, 2017) as well as deficits in executive functioning (Kim & Cameron, 2016). Additionally, understanding early mathematics concepts is critical to assist in conceptualization of more advanced areas (Spooner, Saunders, Root, & Brosh, 2017). Furthermore, elementary-level mathematics is also important for students with ASD as building confidence in basic math skills has an impact on independence beyond school and skills involving managing finances, making purchases, and pursuing employment (Stroizer, Hinton, Flores, & Terry, 2015).

One tool that can improve mathematical learning is manipulatives, defined as “objects designed to represent explicitly and concretely mathematical ideas that are abstract” (Moyer, 2001, p. 176); examples of manipulatives include Base 10 blocks, algebra tiles, and fraction pieces. Two recent reviews explored mathematical education for elementary through postsecondary-aged students with ASD (Barnett & Cleary, 2015; Gevarter et al., 2016), and both identified visual instructional strategies, including manipulatives, as an instructional tool that can be used to support teaching mathematics to students with ASD. A third review similarly identified manipulatives as an effective strategy for students with ASD and emphasized the utility of treatment packages in mathematics instruction that include other components such as prompting and positive consequences (King, Lemons, Davidson, 2016). Previous benefits of incorporating manipulatives for students with and without disabilities included improved understanding of abstract mathematical concepts (Goodman, Seymour, & Anderson, 2016), increased student engagement and motivation (Moyer, 2001), and enhanced performance (Carbonneau, Marley, & Selig, 2013). While the use of manipulatives was previously found to be effective, less is known about the specific impact the different types of manipulatives may have on mathematic instruction for elementary students with ASD.

Types of Manipulatives

Concrete manipulatives are one type of manipulative that provides students with physical representations of abstract concepts in order to improve understanding of the underlying processes of solving problems (Marley & Carbonneau, 2014) and are well-supported by the research literature (Bouck & Park, 2018). In regard to effectiveness for students with ASD, previous research with concrete manipulatives focused on teaching students the concrete–representational–abstract (CRA) framework (Flores, Hinton, Strozier, & Terry, 2014; Stroizer et al., 2015; Yakubova et al., 2016). This framework provides a systematic progression where students solve math problems first with concrete manipulatives (e.g., Base 10 blocks), then drawings (e.g., lines), and finally, abstractly (e.g., numerical strategies; Agrawal & Morin, 2016). Previous researchers found the CRA framework to be beneficial for students with ASD, in that a functional relation was found between interventions involving concrete manipulatives and students’ accuracy in solving mathematical problems (Donaldson & Zager, 2010; Flores et al., 2014; Stroizer et al., 2015; Yakubova et al., 2016). However, there are two challenges associated with concrete manipulatives and students with ASD. First, the handling of multiple physical pieces may distract students’ thought process, and second, the multiple pieces may add to cognitive load resulting in a lack of mathematical concepts (Suh & Moyer-Packenham, 2008). Since students with ASD struggle with cognitive load and executive functioning impairments (Ozonoff, Pennington, & Rogers, 1991), it is important to consider how concrete manipulatives impact mathematical learning in these students and to consider alternatives.

Virtual manipulatives (e.g., app based) may serve as an appropriate replacement for concrete manipulatives given: the increase of available technology (e.g., iPad), their similarity to concrete manipulatives but availability in a digital format, and their potential to reduce cognitive load for students with ASD. App-based manipulatives can be flipped and rotated like concrete manipulative, but images are viewed on tablets (Satsangi & Miller, 2017). Previous researchers identified virtual manipulatives as analogous to concrete manipulatives, meaning researchers found students to be just as effective in solving mathematical problems with virtual manipulatives as concrete manipulatives, and the majority of students with disabilities report a preference for virtual manipulatives in comparison to concrete manipulatives. Specifically, the efficacy and preference for virtual manipulatives in comparison to concrete manipulatives has been found when teaching secondary students with disabilities such mathematics as adding fractions with unlike denominators (Bouck, Shurr, Bassette, Park, & Whorley, 2018), subtraction with regrouping (Bouck, Chamberlain, & Park, 2017), and linear algebra (Satsangi, Bouck, Taber-Doughty, Bofferding, & Roberts, 2016).

Virtual Manipulatives and Elementary Students With ASD

Emerging research exists which compares the use concrete and virtual manipulatives for teaching mathematics to students with ASD (Bassette et al., 2019; Bouck, Satsangi, Taber-Doughty, & Courtney, 2014; Root et al., 2017). Bouck, Satsangi, Taber-Doughty, and Courtney (2014) examined the ability of elementary students with ASD to solve subtraction with regrouping problems when using concrete manipulatives as compared to virtual manipulatives and found independence was higher and students preferred the virtual manipulatives. Root, Browder, Saunders, and Lo (2017) found three elementary students with ASD and moderate intellectual disability experienced an increase in independence in solving problems with app-based manipulatives as opposed to concrete manipulatives. Bassette et al., (2019) compared the efficiency of students with ASD to solve subtraction problems using concrete manipulatives as compared to app-based manipulatives and found students completed a higher number of task analysis steps independently per minute when using the app-based manipulative. These studies, similar to the studies with students with other types of disabilities, found the majority of participants reported a preference toward the app-based manipulatives.

In addition to the promising previous research, there are logistical reasons (e.g., greater access to multiple types of manipulatives, easy differentiation) to further explore the use of app-based manipulatives in mathematic instruction for students with ASD. First, app-based manipulatives are able to be utilized on tablets (e.g., iPad) which are frequently available to students with disabilities including ASD (Satsangi & Miller, 2017). The convenience of tablets also yields practical benefits, such as the potential for greater student engagement, increased social acceptability/less stigmatization, and mobility/transportability (Bouck et al., 2012, Satsangi & Miller, 2017). App-based manipulatives may also eliminate concerns such as physical manipulatives getting lost, not having enough concrete manipulatives available for all students, or students playing with or being distracted by the concrete manipulatives instead of using them for their math (Bassette et al., 2019; Bouck, Park, et al., 2017). The purpose of the current study is to expand the literature base of manipulatives to support mathematics education for elementary students with ASD by comparing the efficiency (i.e., task analysis steps completed independently per minute) of concrete manipulatives versus app-based manipulatives when teaching mathematics (i.e., addition with regrouping, fraction equivalency, and fraction addition with unlike denominators). Specifically, this study sought to explore students’ efficiency when using the manipulatives to solve fraction and addition problems as well as accuracy.

Method

Participants

Convenience sampling was used to recruit three elementary students with ASD who received special education services. Inclusion criteria for participation in the study included (a) eligible and currently receiving special education services due to ASD diagnosis, (b) ability to manipulate concrete blocks and app on an iPad, (c) recommendation by teacher or ABA therapist, and (d) parental/guardian consent and student assent to participate.

Alex

Alex was a 9-year-old, third-grade, Caucasian student. Alex’s individualized education program (IEP) confirmed he received special education services due to his ASD diagnosis and also had a secondary language impairment diagnosis. No ASD assessment data or IQ information was available. Alex participated in general education for 80% or more of the day. Mathematics was reported to be his strength; however, his teacher reported he was inconsistent in his mathematical performance. Beginning of the year benchmark testing indicated Alex performed at an “at-risk” level in math. Alex’s math goals included improving “his ability to solve math story problems, problems involving estimation, and timed math tests with 70% accuracy.” Alex was on track to participate in state standardized testing but had not completed these at the time of the study. Alex’s accommodations included extended time, tested individually, tests read aloud, and a scribe for standardized testing. Alex typically was compliant, but at times, he was perseverated on preferred topics (e.g., Minecraft) and needed to be redirected. Talking about Minecraft and watching Minecraft YouTube videos on the iPad were used as reinforcement after sessions. On the researcher-administrated KeyMath assessment, Alex’s grade equivalency (GE) was 4.0 and age equivalency (AE) was 9:5; results indicated he struggled with fractions; therefore, during the study, his intervention focused on finding equivalent fractions.

Jake

Jake was a 10-year-old, fourth-grade, Caucasian student whose IEP confirmed he received special education services due to his ASD diagnosis and he also had a secondary language impairment. No ASD assessment data or IQ information was available. Jake participated in general education for 80% or more of the day. Jake completed state standardized testing and was deemed proficient during the year of the study. On the state standardized assessment, he did not pass number sense, computation, geometry and measurement, and mathematic process but did pass algebraic thinking and data analysis. A review of his IEP indicated he performed at a third-grade level in math based on his CBM assessment and he mastered multiplication facts for 0, 1, 2, 5, and 10. IEP goals related to math included demonstrating mastery of multiplication facts for numbers 0–10 when given a 5-min assessment. Jake was typically very compliant and polite during sessions but at times could be a bit lethargic. On the KeyMath assessment, Jake’s GE was 3.8 and AE was 9:1; results indicated he struggled with accurately solving fraction addition problems. Reinforcement for Jake after sessions included free time on the iPad.

Jess

Jess was a second-grade, 7-year-old, Caucasian male student. His IEP confirmed he received special education services due to his ASD diagnosis. No ASD assessment data or IQ information was available. A review of Jess’s IEP indicated he struggled with math. At school, he spent approximately 90% of his time in general education and received resource room instruction in mathematics once a week. In regard to mathematics, on the Woodcock–Johnson Test of Achievement (fifth edition), he scored 63 (first percentile). It was reported that Jess used TouchMath to work on addition and subtraction problems prior to the study, and his teacher reported he had been making progress on single-digit addition problems but struggled with independently solving both addition and subtraction problems and needed prompting to initiate and complete problems. IEP goals related to math included identifying signs associated with subtraction and addition problems and correctly being able to “count on” or “count back,” being able to set up and solve addition or subtraction problems with three or less verbal prompts, and being able to solve an addition or subtraction problem using TouchMath correctly with one or less verbal prompt (may use visual prompt). On the KeyMath assessment, Jess’s GE was less than kindergarten and his AE was less than or equal to 4:6. During baseline probes, he struggled with solving single-digit addition with regrouping problems. Access to the computer (e.g., Om Nom stories) and iPad (e.g., YouTube Kids) served as reinforcement following sessions.

Setting and Materials

During sessions, participants worked independently with researchers at a desk or table with multiple chairs available. Initial data collection (e.g., KeyMath assessment and baseline sessions) for Alex and Jake were collected in the computer room at their school; however, the school year ended prior to the study. Remaining sessions were conducted in quiet spaces available at the researchers’ university (e.g., the university’s Center for Autism Spectrum Disorders). All sessions for Jess occurred at his family home (e.g., family dining table).

This study’s materials consisted of researcher-constructed assessment probe sheets with five problems per sheet (validated by a former mathematics teacher and current mathematics education coach), pencils, task analysis data collection sheets, concrete manipulatives (i.e., Base 10 blocks and Rainbow Fraction Tower equivalency cubes) and app-based manipulatives (i.e., Base 10 blocks by Brainingcamp, 2017) and the Fraction Tiles app (Fraction Tiles by Brainingcamp, 2017) on an iPad. Participants received a worksheet and pencil during sessions.

The Base 10 blocks concrete manipulatives were used in the study for single-digit addition with regrouping problems. These were plastic blocks representative of units (i.e., ones) and rods (i.e., tens). The Base 10 blocks starter kit contained green (used for first addend) and orange (used for second addend) blocks. The blocks provided a clear representation of units that helped create a visual understanding of the number of units in the rods, and problems were solved on a paper place value mat. As with the concrete manipulatives, in the app, green Base 10 blocks were used for the first addend and orange were used for the second. The Base 10 blocks app was purchased from Brainingcamp (2017) for US$1.99 on iTunes. In the app, students set up addition problems by manipulating the virtual Base 10 blocks on the virtual place value mat to represent the problem. Blocks were manipulated by touch on the app by moving them into groups, deleting them, and clearing all from the screen.

The Rainbow Fraction Tower equivalency tiles (i.e., colored connectable blocks that represented fractions) were used for fraction problems. Each tower represented one denominator, such as 4ths, 3rds, and 10ths, and when all pieces of each tower were stacked together, all towers were equal in height, length, and width (i.e., one whole, measuring 4.875″ × 0.75″ × 0.75″). Each tower was a unique color, and all cubes within that tower were in the same color. Each cube within a tower had its fraction value displayed on one of its sides in decimal, percent, and fraction form. The blocks allowed for manipulation by displaying fraction equivalencies (e.g., student could stack blocks to see equality of $\frac{2}{4}$ and $\frac{3}{6}$ ). The Fraction Tiles app was purchased from Brainingcamp (2017) for US$1.99 on iTunes. In the app, students set up fraction equivalency or addition problems by manipulating the virtual blocks that resembled the concrete manipulatives. Problems were solved by dragging blocks to the virtual workspace where they were manipulated by connecting pieces to view equivalency. They could also be grouped, deleted, and cleared. The pen feature also allowed students to write information when solving problems.

Experimental Design, Independent, and Dependent Variables

In order to determine whether the intervention conditions would yield differential effects on the dependent variables, an adaptive alternating treatment design was used (Wolery, Gast, & Ledford, 2014). The independent variables were the manipulatives (i.e., Base 10 blocks, Fraction blocks, Base 10 app, and Fraction Tiles app). The dependent variables for this experiment included number of problems solved correctly and efficiency of manipulative use which were assessed through task analysis steps completed independently per minute. The number of problems solved correctly was out of five problems on each probe. Efficiency referred to the total number of task analysis steps of the procedure the student was able to complete without prompting per minute. Permanent products of probes were used to determine accuracy (i.e., event recording), and raw independence data were determined by recording any prompts provided to students when using the manipulatives to solve problems. Additionally, a stopwatch was used to record durations of total session times. Steps completed independently per minute was calculated from the total steps completed independently (from all five problems) divided by total session time (in seconds) and multiplied by 60.

Procedures

Pre-assessment and baseline

A KeyMath-3 pre-assessment was conducted on the three participants to determine mathematical ability. Each student was probed for types of problems they struggled with, and additional baseline probes confirmed the targeted problems were challenging for students. Baseline session procedures included giving students researcher-created probes consisting of five problems. Students completed problems with pencil and paper and without being prompted. Once at least five baseline sessions were completed and accuracy data showed zero celeration, intervention began.

Manipulative training

Before beginning intervention, participants were taught how to use the manipulatives (McNeil & Jarvin, 2007). Jake and Alex received concrete manipulative training before app training and, due to logistical reasons, Jess was trained on the app-based manipulative first. During all training sessions, researchers first provided direct instruction and modeled how to solve the problems with manipulatives based on the training protocol. Protocols for each type of problem included prerequisite steps (e.g., participants who solved fraction problems first had to demonstrate they could identify the correct concrete block when given a single fraction such as $\frac{1}{2}$ or $\frac{5}{6}$ ). Following mastery of prerequisite steps, the researchers would use explicit instruction to teach how to use the manipulatives. Specifically, the researchers would model how to complete each step of the task analysis while providing a think aloud (i.e., verbal narration; see Table 1). Then, participants received guided instruction where researchers provided prompts and error correction for steps participants struggled with or failed to do correctly as they worked to independently solve. Lastly, participants’ independence in solving problems was probed. Mastery criteria for independence for Alex required him to solve three problems 100% independently (i.e., all steps of the task analysis) before moving to intervention. Mastery criteria for the other participants required them to solve three problems 80% independently (i.e., 9 of the 11 tasks analysis for Jake and 5 of the 6 tasks analysis steps for Jess). Training was restarted if independence criteria were not met. Because participants were initially learning to use the manipulative during training, efficiency data were not recorded during training sessions.

Table 1.

Steps of Task Analysis for Problem Type.

Single-Digit Addition With Regrouping	Fraction Equivalence	Fraction Addition
CM: Set up the ones blocks for the first number of the problem.VM: Set up the ones blocks on the app for the first number of the problem. CM: Set up the ones blocks for the second number of the problem.VM: Set up the ones blocks on the app for the second number of the problem. CM: Move down green and orange blocks to get a total of 10.VM: Drag down green and orange blocks to get a total of 10. CM: Regroup the 10 ones blocks (both green and orange) to a green tens block by switching out the units for a rod.VM: Regroup ones blocks (both green and orange) to a tens block by selecting the 10 ones and then select highlight to make them blue. CM: Count up tens blocks and remaining ones blocks (orange blocks).VM: Count up tens block and remaining ones blocks (orange blocks). CM: Write or state answer.VM: Write or state answer and clear the iPad.	CM: Set up the block(s) for the first fraction of the problem.VM: Pull the block(s) out on the app for the first fraction of the problem. CM: Get the block(s) out for the second fraction based on the denominator.VM: Pull out the blocks on the app for the second fraction of the problem based on the denominator. CM: Line the blocks up for both fractions until they are equal.VM: Line the blocks up for both fractions until they are equal. CM: Count the number of blocks used for the second fraction to equal the height of the first.VM: Count the number of blocks used for the second fraction to equal the height of the first. CM/VM: Write or state the answer.	CM: Pull out the one whole block.VM: Pull out the one whole block and write problem on screen. CM: Set up the first fraction blocks.VM: Set up the first fraction blocks. CM: Set up the second fractions blocks.VM: Set up the second fractions blocks. CM: Find the common denominator (can use calculator).VM: Find the common denominator (can use calculator). CM: Change the first fraction into blocks with common denominator (if applicable).VM: Change the first fraction into blocks with common denominator (if applicable). CM: Write down equivalent fraction for first fraction (if applicable).VM: Write down equivalent fraction for first fraction (if applicable). CM: Change the second fraction into blocks with common denominator (if applicable).VM: Change the second fraction into blocks with common denominator (if applicable). CM: Write down equivalent fraction for second fraction (if applicable).VM: Write down equivalent fraction for second fraction (if applicable). CM: Write the denominator answer. VM: Write the denominator answer. CM: Write the numerator answer.VM: Write the numerator answer. CM: Clear blocks.VM: Clear app.

Concrete manipulative training

During concrete manipulative training, participants were taught to understand how the blocks represented numbers (e.g., Jess learned a unit was equivalent to 1 and rod was equivalent to 10; Alex and Jake learned how multiple fractions blocks were equivalent to other fractions, such as $\frac{2}{4} = \frac{1}{2}, \frac{4}{4} = 1$ ). Students were then taught to set up problems (e.g., Jess was taught to use the green blocks to set up the first addend and orange blocks to set up the second; Jake was taught to set up the two different fractions in his addition problem using the necessary fractional blocks). Students then learned to complete the remaining task analysis steps required to solve the problems using the blocks (e.g., Jess learned to regroup 10 units for a rod by counting out the orange blocks and then green until he got to 10 and then replace 10 ones blocks with a1 tens rod; Jake learned to replace fraction blocks with those representing the least common denominator and add them together).

App-based manipulative training

Students were taught the same concepts within the associated apps (e.g., Jess learned the ones block represented the number one in the app; Alex learned how fraction tiles in the app represented fractions—four $\frac{1}{6}$ blocks equaled $\frac{4}{6}$ ). Students were then taught how to use the features of the app to set up the problems (e.g., Jess learned to set up the addends, in the app; Jake learned to set up the first and second fractions by manipulating the fraction tiles on the screen). Students then learned to complete the remaining steps to solve problems (e.g., Jess learned to count all green blocks and then any orange blocks needed until he got 10 blocks—he then highlighted these 10 ones block and regrouped them into a tens rod; Jake learned to use the pen feature of the app to write the new fractions after determining the least common denominator of the fractions in his problem).

Intervention

Three alternating conditions comprised the intervention phase: concrete manipulative (Base 10 blocks or fraction equivalency cubes), virtual manipulative (Base 10 blocks app or Fraction Tiles app), and paper/pencil (no manipulative—extended baseline). Fifteen total intervention sessions were conducted with five sessions of each condition, the order of which was randomly assigned before the intervention phase. No more than two consecutive sessions of the same condition were implemented. During the sessions in which a manipulative was used, the student was given 10 s to initiate the first step, and if the student did not initiate solving the problem independently after 10 s, the researchers used the system of least prompts to aid the student. Prompts used, in order, included gesture (e.g., pointing to the $\frac{1}{4}$ blocks), indirect verbal (e.g., “Now what?”), direct verbal (e.g., “Connect two $\frac{1}{4}$ blocks to make $\frac{2}{4}$ ”), model (e.g., showing how to connect two $\frac{1}{4}$ blocks to make $\frac{2}{4}$ ), partial physical (e.g., gently guiding the student’s hand toward the $\frac{1}{4}$ blocks), and full physical (e.g., gently guiding the student’s hand to grab two $\frac{1}{4}$ blocks and connect them together to make $\frac{2}{4}$ ). For all conditions during intervention, Jess solved single-digit addition with regrouping problems, Alex found equivalent fractions, and Jack solved addition of fractions with unlike denominators.

Concrete manipulatives (Base 10 or Fraction Tower)

During each concrete manipulative session, Jess was given a worksheet with five problems, a pencil, and the Base 10 blocks (orange and green units and rods). For each problem, Jess’s ability to complete the step of the task analysis (e.g., first set up the ones blocks for the first addend and then set up the ones blocks for the second addend) independently or prompts provided was recorded. During concrete manipulative sessions for Alex and Jake, each was given a worksheet of five problems, a pencil, and the fraction equivalency blocks. Alex and Jake’s ability to independently complete task analysis steps or prompts needed to use the fraction blocks to solve the problems (e.g., set up the fraction tiles for the first number) were recorded.

App-based manipulatives (Base 10 or Fraction Tiles)

During app-based manipulative sessions for the student solving single digit with regrouping addition problems, Jess was given a worksheet with five problems and a pencil, as well as an iPad with the Base 10 app open. Just as in the concrete sessions, Jess’s ability to complete task analysis steps independently or prompts provided to assist him in using the app to solve the problems were recorded. During app-based manipulative sessions for the students solving fraction problems, Jake and Jess were given a worksheet of five problems, a pencil, and an iPad with the Fraction Tiles app open for use. Students were again asked to solve the problems and steps completed independently or prompts provided for each step of the task analysis to use the app-based manipulative were recorded.

No manipulative (extended baseline)

During the extended baseline condition, participants were only provided with the worksheet problems and a pencil with no concrete or app-based manipulatives. Data collection in this condition proceeded similarly to baseline.

Best treatment

The procedures used to identify the percent of nonoverlapping data (PND) in alternating treatment designs outlined by Wolery, Gast, and Ledford (2014) were used to determine which condition was superior for each student. Corresponding data point values between conditions (i.e., app and concrete) were examined to identify which was superior based on the number of task analysis steps completed independently per minute data, and this was divided by the number of comparison sessions (five; Wolery et al., 2014). During the three best treatment sessions, students were given a pencil, a worksheet with five problems, and the predetermined best treatment manipulatives. During the best treatment phase, the same prompting procedure from intervention was used. Using similar procedures as intervention, the percent of problems solved correctly and the steps completed independently per minute were recorded.

Maintenance

Maintenance data were collected with the same procedures as baseline, 1–2 weeks after best treatment was completed. During maintenance, participants were given the worksheet with five problems to solve without use of any manipulatives or prompting.

Interobserver Agreement (IOA) and Treatment Fidelity

IOA was calculated for the percent of problems solved correctly and raw independence for at least 33% of sessions in each condition. To calculate IOA data for accuracy, a second researcher reviewed the permanent product of answers written down on the worksheets by participants. The researcher determined agreement or nonagreement based on criteria of whether or not answers were solved correctly. To calculate IOA data for independence, a second researcher observed intervention sessions and noted prompts delivered by the primary researcher. The number of agreements for prompts delivered for each step of each problem was summed and divided by the sum of agreements and disagreements. To calculate IOA for duration, the primary researcher kept time with a stopwatch and the amount of time it took participants to complete the five problems was recorded individually by both researchers based on the time indicated by the stopwatch. IOA for problems solved correctly was 100% for all participants. IOA for independence for Alex was 99.33% for intervention sessions and 100% for best treatment and maintenance sessions. For Jake, IOA for independence was 100% for all conditions. Jess’s IOA for independence was 96.5% for intervention and 100% for best treatment and maintenance sessions. Duration data IOA was 100% for all participants across all phases.

Treatment fidelity data were recorded for at least 40% of intervention sessions and 33% of best treatment and maintenance sessions. This included verification of whether participants were given the worksheet with problems and whether the correct manipulative or lack thereof was both given and used. For all phases for each student, treatment fidelity was 100%.

Social Validity

Pre- and post-social validity interviews were conducted with the participants and the parents of each participant. Students were asked questions regarding which type of manipulative they preferred and whether they preferred to solve problems with or without any manipulatives. Parents were asked questions relating to how effective the apps were in helping their children solve problems, what aspects of using app-based manipulatives they liked for their child, and whether they would suggest the use of these apps to other parents of children with ASD.

Data Analysis

In order to determine whether a functional relation between the intervention and the dependent variables existed, the data were analyzed visually. Visual analysis of change in level and trend and variability between phases and conditions were conducted. Consistency and immediacy of effect were also evaluated. Effect size of percent correct data was calculated using Tau-U, a value which represents contrasts between each intervention condition with baseline conditions per participant (Parker, Vannest, Davis, & Sauber, 2011). Tau-U scores were calculated from a web-based calculator (http://www.singlecaseresearch.org/calculators/tau-u; Vannest, Parker, & Gonen, 2011). Values less than or equal to 65% indicated a small effect, 66–92% a medium effect, and 93% and above a large effect (Parker, Vannest, & Brown, 2009).

Results

In regard to efficiency of manipulative use, fractionation of data for steps completed independently per minute indicated two participants performed more steps on their own per minute during the app-based manipulative condition compared to the concrete manipulative condition (see Figure 1), while one participant was more efficient with concrete manipulatives. Two participants completed more steps independently per minute during the best treatment condition compared to intervention, while the third completed less. During intervention (i.e., following baseline and training sessions), visual analysis shows an immediate increase in the participants’ ability to solve problems correctly (see Figure 2), and these remained high and stable throughout intervention; however, accuracy was not maintained.

Figure 1.

Task analysis steps completed independently per minute when solving mathematical problems.

Figure 2.

Accuracy of mathematical problems.

Alex

Alex’s training criteria for both conditions were three problems solved 100% independently (i.e., complete all task analysis steps without prompting), and he met criteria after five concrete training sessions and five app training sessions. During intervention, fractionation of data was demonstrated in steps completed independently per minute, with higher rates during the app condition (range 3.9–8.1, average 5.8) than during the concrete condition (range 2.1–5.6, average 4.1). PND for app to concrete was 100% (concrete to app 0%) and the app condition was his best treatment condition. During best treatment, steps completed independently per minute ranged from 6.9 to 10.7, with an average of 8.7, and he solved all problems correctly.

During baseline, Alex solved an average of 1.3 of the problems correctly, and during the last three baseline sessions, his data stabilized and he had a zero celeration trend. During intervention, Alex solved all of the problems correctly during the manipulative conditions; however, his data were variable during extended baseline sessions (range 0–4, average 1.6) and averaged 0.67 during maintenance. Compared with baseline data, Alex’s Tau-U for accuracy was 100% for both concrete and app-based manipulatives, indicating high effect sizes for both manipulatives.

Jake

Jake’s training criteria for both conditions were three problems solved 80% independently, and he met criteria for training for both types of manipulatives after three sessions, respectively. Fractionation of data was demonstrated in steps completed independently per minute following session seven, where he began to solve more steps independently per minute during the concrete condition (average 5.0; range 4.2–5.7) compared to the app condition (average 4.5; range 3.8–5.0). PND for concrete to app was 60% (app to concrete, 40%), and the concrete condition was his best treatment condition. During best treatment, steps completed independently per minute averaged 6.7 (range 5.4–7.7), and he solved all problems correctly.

During baseline, Jake did not solve any problems correctly and displayed a zero celeration trend. During intervention, Jake solved an average of 4.8 problems correctly (range 4–5) with virtual manipulatives, an average of five problems correctly with the concrete manipulatives. He only solved an average of 0.8 problems correctly during the extended baseline sessions (range 0–3) and an average of one problem correctly during maintenance. Compared with baseline data, Jake’s Tau-U for accuracy was 100% for both concrete and app-based manipulatives. Both of these indicate high effect sizes.

Jess

Jess’s training criteria for both conditions were three problems solved 80% independently. Jess met criteria on the app after six training sessions and met criteria with the concrete manipulatives after 25 training sessions. Fractionation of data was demonstrated in steps completed independently per minute between conditions, and he solved more steps independently per minute during the app condition (average 5.0; range 4.0–6.0) compared to the concrete condition (average 2.1; range 1.5–2.9). PND for app to concrete was 80% (concrete to app 20%), and the app was his best treatment condition. During best treatment, he completed all problems correctly and steps completed independently per minute averaged 3.0 (range 2–4).

During baseline sessions, Jess did not solve any problems correctly and displayed a zero celeration trend. During intervention, Jess solved all problems correctly when using either manipulatives but only solved one problem correctly during one session of extended baseline and did not solve any problems correctly during maintenance. Compared with baseline data, Jess’s Tau-U for accuracy was 100% for both concrete and app-based manipulatives, indicating high effect sizes for both manipulatives.

Social Validity

Social validity survey questions were asked to participants and parents of participants. Prior to the study, Jake and Jess’s parents indicated they believed that the app-based manipulatives could be effective for their children because of their children’s preference and capability in using technology and apps specifically. They also believed the apps would be better at keeping their children focused on working on mathematics problems and would make the problems easier for them to conceptualize enabling them to solve the problems more efficiently and would help improve math skills. Jess’s parent noted that when he used concrete manipulatives previously, he could get distracted by them and felt the app-based version may be more beneficial. Prior to the study, participants reported that math was hard and they were not sure whether the app-based manipulatives would be helpful. Following the study, Alex indicated he liked the app-based manipulative and that it was helpful; he preferred the app because he did not like searching for the concrete blocks and the blocks on the iPad were easier to find. Jake similarly indicated he preferred the app-based manipulatives and “The iPad was more fun and helped me to do the problems.” Jake indicated he preferred the app and his parent indicated the app was easier than the concrete manipulatives since these could be overwhelming to him at times.

Discussion

The purpose of this study was to explore the efficiency of app-based manipulatives compared to concrete manipulative when teaching mathematical skills to elementary students with ASD. The results indicate differential impact of treatment; specifically, efficiency was higher in two participants who completed a greater number of task analysis steps independently with the app while the third participant was more efficient with the concrete manipulatives. Both types improved students’ ability to solve problems accurately during intervention; however, accuracy during maintenance was similar to initial baseline levels.

Similar to previous research (e.g., Bassette et al., 2019; Root et al., 2017), these results indicate both app-based and concrete manipulatives improved students with ASD ability to solve mathematical problems and expanded upon the previous research by including fraction-based problems and addition problems. The results for Jess and Alex are similar to previous research which compared students with ASD ability to solve problems when using Internet-based manipulatives on a computer compared to concrete manipulatives (Bouck et al., 2014) and app-based manipulatives (i.e., Base 10 math app) to concrete manipulatives (Bassette et al., 2019), finding the virtual conditions resulted in higher rates of independence and efficiency in the majority of participants. Notably, Jake’s results on fraction addition problems suggest efficiency was more comparable between conditions (i.e., less fractionation of data). His results are similar to previous research with middle school students with intellectual disability and learning disabilities who also had similar rates of independence and task completion time when using the different types of manipulatives to solve fraction addition problems (Bouck, Shurr, et al., 2018).

In examining the task analysis steps completed independently per minute for Alex and Jess, fractionation of data was seen immediately between conditions upon beginning intervention while fractionation between conditions for Jake was not observed until approximately halfway through intervention. Since prompting is frequently used to teach students with ASD mathematics (Barnett & Cleary, 2015; Gevarter et al., 2016; King et al., 2016), it is important to consider how the prompts may impact acquisition, accuracy, and efficiency when using manipulatives. In this study, an examination of the prompts required do not reveal a consistent pattern of errors. For example, at times, all participants needed prompting during the first or second problem, while, during other sessions, they were prompted on the fourth or fifth problem. The steps students were prompted on also varied across sessions. The most intrusive prompt needed for Alex and Jake was a direct verbal prompt, while Jess required partial physical prompts during multiple concrete sessions.

During app-based manipulative sessions, Alex was prompted on Step 2—“pull out the blocks for the second fraction” when completing two problems and was prompted on Step 4—“count the number of blocks for the second fraction to equal the first” during one problem in the first intervention session. During concrete sessions, he was prompted on every step for at least one problem across sessions. Jake was also prompted on various steps during concrete sessions (e.g., find common denominator; write down equivalent fraction for first fraction). He also required prompting on these steps during app sessions in addition to several other steps (e.g., write the numerator answer and write the denominator answer). During concrete sessions, Jess needed prompting across sessions on the first four steps (e.g., move down green and orange blocks to get a total of 10) but always completed the last two steps independently. He required partial physical assistance on at least one step of at least one problem during four concrete sessions. During app sessions, the most intrusive prompt Jess required was an indirect verbal. This was needed during two sessions to help him transition between problems once he finished one (i.e., he needed to be prompted to begin the first step of the next problem and set up the ones block on the app for the first number of the problem). Jess also struggled with Step 2 (set up the ones block for the second number of the problem) during two app sessions.

Student efficiency in solving problems using manipulatives is important to identify, given that manipulatives may assist with improving students’ procedural knowledge and these skills are needed to acquire advanced mathematical skills (Van de Walle, Karp, & Bay-Williams, 2010). Students were able to solve problems accurately when using manipulatives, and all the three students demonstrated differences in efficiency when using the different types of manipulatives. These results suggest it may be beneficial to expose students with ASD to both types of manipulatives to determine preference and effectiveness when presented with different instructional options.

Implications for Practice

The results of this study contribute additional information to assist practitioners in understanding how manipulatives may be used when teaching mathematics to students with ASD. Jess’s higher level of efficiency in using the Base 10 app to solve addition problems was similar to other participants with ASD who used the Base 10 app to solve subtraction problems in previous research (Bassette et al., 2019). These combined results suggest students with ASD who are working on computational problems that require regrouping may benefit from the interface of app-based manipulatives as compared to concrete manipulatives. Similar to participants in the previous research (Bassette et al., 2019), Jess required fewer training sessions to learn to use the app and had greater efficiency in using it once taught. Teachers who are teaching students with ASD to solve basic computational problems may consider using app-based manipulatives first given they provide the students with additional visual cues (e.g., show the Arabic numerals and have the option to display the answer) and ease the response effort required for regrouping.

For example, when solving single-digit addition problems with the app, students need to group 10 ones blocks and then the option to regroup automatically was enabled. This option only becomes available in the app if students correctly selected the 10 ones blocks. With the concrete manipulatives, students need to count out 10 ones blocks and then replace these with a rod. If a student failed to count the blocks correctly, no procedure is in place to prevent the error which could lead to additional errors as well as increased frustration in students. The app may also be beneficial when working with younger students who become distracted with concrete manipulatives. Furthermore, Jess needed substantially fewer training sessions to meet criteria to use the app independently (i.e., six sessions) compared to the concrete (i.e., 25 sessions) which is similar to previous research which found students learned to use the app more quickly than the Base 10 blocks when solving subtraction problems (Bassette et al., 2019). This may be important for teachers seeking to capitalize on instructional time. Alternatively, for teachers working with students who are solving more advanced mathematical problems (i.e., fraction addition), it may be beneficial to expose students to both types of manipulatives to determine efficiency (Bouck et al., 2018, Bouck et al., 2017). In regard to training, Jake and Alex required the same number of training sessions to meet criteria for the app and blocks indicating there may be less variation of instructional time needed; however, additional research is needed.

Limitations and Future Directions

While this study extends previous research of the effectiveness of manipulatives to teach mathematics to students with ASD, there were several limitations. The first was that the mastery criteria varied between participants with Alex requiring three problems solved 100% independently while Jake and Jess’s criteria to transition from training to intervention was solving three problems 80% independently due to time constraints. Another limitation is that for Alex, the individual steps he solved independently during training were not recorded and it was only recorded if he solved the entire problem independently or not. Additionally, Jake and Alex were trained on concrete manipulatives first while, due to logistical constraints (i.e., delay in shipping of his concrete manipulatives), Jess was trained to use the app first. Future research should explore how initial instruction may impact acquisition and efficiency of manipulative use.

A third limitation is that all participants had low accuracy scores during maintenance and Jess and Jake also had low scores during the extended baseline phase during intervention. During these phases, participants were told to try their best to solve the problems; however, it appeared they rushed through problems. For example, during the first extended baseline session (i.e., Intervention Session 4), Jake expressed frustration and asked the researcher to use the blocks or the app. When told he could use them next time, he appeared relieved but subsequently reported he had guessed on all of the problems in the extended baseline session. These results suggest students were not motivated to complete the task accurately or did not have the conceptual skills needed to solve the problems abstractly (i.e., without the manipulatives). Given that students with ASD have challenges with cognitive load (Ozonoff et al., 1991) and an acknowledgment that visual strategies are considered an evidence-based practice for students with ASD (Barnett & Cleary, 2015; Gevarter et al., 2016), future research may want to consider the utility of requiring students with ASD to solve problems completely abstractly.

Relatedly, future research could also further explore how skill and motivational deficits are impacted when using manipulatives and how students with ASD’s ability to solve problems abstractly can be more systematically facilitated. For example, one area of future research might be exploration of the virtual–representational–abstract (VRA) instructional sequence (Bouck, Bassette, et al., 2017, Bouck, Park et al., 2018) or VA (no representational phase; Bouck, Park, et al., 2017); implementation of both instructional sequences resulted in improved mathematical skills for secondary students with disabilities, but effects are not explored in elementary students or students with ASD. The VA sequence may be particularly beneficial when students solve fraction problems since the challenges of drawing fractions during the representational phrase were noted as a limitation. In the current study, no prompts were provided when students solved problems abstractly which may have contributed to students’ low accuracy score during maintenance. Future research should expand upon the principles of applied behavior analysis (an evidenced-based practice for students with ASD) to better understand how specific antecedent components (e.g., virtual manipulatives, prompting during abstract phase) can be used initially, then systematically faded to teach students with ASD to solve problems abstractly. Additionally, it would also be of interest for future research to systematically explore consequence-based components to assess motivational factors (e.g., initial contingent reinforcement for accuracy followed by systematic fading; Gevarter et al., 2016). Future research should also seek to gain a better understanding of why students with ASD may have preferences toward app-based manipulatives (e.g., provide students with feedback on mathematical performance after using the app-based manipulatives and ask why/how they think it helped them or why they behaved a certain way when using the app). Due to the heterogeneous and complex learning needs of students with ASD, it is important to explore a variety of strategies to identify how to best facilitate their mathematical skills.

Conclusions

In conclusion, this study supports previous research demonstrating the effectiveness of both app-based and concrete manipulatives to improve accuracy during intervention and demonstrated differences in efficiency between manipulative type with two participants being more efficient with the app. It adds to the literature by exploring app-based manipulative use to solve different types of mathematical problems which were not previously explored in elementary participants with ASD. Furthermore, exploring efficiency may assist practitioners concerned with instructional time and provides additional important social validity of manipulatives for students with ASD (King et al., 2016). The use of manipulatives improved accuracy; however, future research is needed to more thoroughly understand the role manipulative may have on students with ASD.

Footnotes

Acknowledgment

The authors would like to thank Jessica Gundlach and Laura Paulus for their contributions to this project.

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: An internal grant was awarded to the first author from her university to support theresearch.

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